Review ArticleStatistical Methods for Establishing Personalized TreatmentRules in Oncology
Junsheng Ma Brian P Hobbs and Francesco C Stingo
Department of Biostatistics The University of Texas MD Anderson Cancer Center Unit 1411 1400 Pressler StreetHouston TX 77030 USA
Correspondence should be addressed to Francesco C Stingo fstingomdandersonorg
Received 25 November 2014 Accepted 9 February 2015
Academic Editor Aurelio Ariza
Copyright copy 2015 Junsheng Ma et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The process for using statistical inference to establish personalized treatment strategies requires specific techniques for data-analysisthat optimize the combination of competing therapies with candidate genetic features and characteristics of the patient and diseaseA wide variety of methods have been developed However heretofore the usefulness of these recent advances has not been fullyrecognized by the oncology community and the scope of their applications has not been summarized In this paper we provide anoverview of statistical methods for establishing optimal treatment rules for personalized medicine and discuss specific examples invarious medical contexts with oncology as an emphasis We also point the reader to statistical software for implementation of themethods when available
1 Introduction
Cancer is a set of diseases characterized by cellular alterationsthe complexity of which is defined atmultiple levels of cellularorganization [1 2] Personalized medicine attempts to com-bine a patientrsquos genomic and clinical characteristics to devise atreatment strategy that exploits current understanding of thebiological mechanisms of the disease [3 4] Recently the fieldhas witnessed successful development of several molecularlytargeted medicines such as Trastuzumab a drug developedto treat breast cancer patients with HER2 amplification andoverexpression [5 6] However successes have been limitedOnly 13 of cancer drugs that initiated phase I from 1993 to2004 attained finalmarket approval by theUS Food andDrugAdministration (FDA) [7] Moreover from 2003 to 2011717 of new agents failed in phase II and only 105 wereapproved by the FDA [8]The low success rate can be partiallyexplained by inadequate drug development strategies [3]and an overreliance on univariate statistical models thatfail to account for the joint effects of multiple candidategenes and environmental exposures [9] For example incolorectal cancer there have been numerous attempts todevelop treatments that target a singlemutation yet only one
an EGFR-targeted therapy for metastatic disease is currentlyused in clinical practice [10]
In oncology biomarkers are typically classified as eitherpredictive or prognostic Prognostic biomarkers are cor-relates for the extent of disease or extent to which thedisease is curable Therefore prognostic biomarkers impactthe likelihood of achieving a therapeutic response regardlessof the type of treatment By way of contrast predictivebiomarkers select patients who are likely or unlikely tobenefit from a particular class of therapies [3] Thus pre-dictive biomarkers are used to guide treatment selection forindividualized therapy based on the specific attributes of apatientrsquos disease For example BRAFV600-mutant is a widelyknown predictive biomarker which is used to guide theselection of Vemurafenib for treatment metastatic melanoma[11] Biomarkers need not derive from single genes as thoseaforementioned and yet may arise from the combination of asmall set of genes ormolecular subtypes obtained from globalgene expression profiles [6] Recently studies have shownthat the Oncotype DX recurrence score which is based on 21genes can predict awomanrsquos therapeutic response to adjuvantchemotherapy for estrogen receptor-positive tumors [12 13]Interestingly Oncotype DX was originally developed as a
Hindawi Publishing CorporationBioMed Research InternationalVolume 2015 Article ID 670691 13 pageshttpdxdoiorg1011552015670691
2 BioMed Research International
prognostic biomarker In fact prognostic gene expression sig-natures are fairly common in breast cancer [12 14]The readermay note that Oncotype DX was treated as a single biomakerand referred to as a gene expression based predictive classifier[3]
Statistically predictive associations are identified usingmodels with an interaction between a candidate biomarkerand targeted therapy [15] whereas prognostic biomarkersare identified as significant main effects [16] Thus analy-sis strategies for identifying prognostic markers are oftenunsuitable for personalized medicine [17 18] In fact thediscovery of predictive biomarkers requires specific statisticaltechniques for data-analysis that optimize the combina-tion of competing therapies with candidate genetic featuresand characteristics of the patient and disease Recentlymany statistical approaches have been developed provid-ing researchers with new tools for identifying potentialbiomarkers However the usefulness of these recent advanceshas not been fully recognized by the oncology communityand the scope of their applications has not been sum-marized
In this paper we provide an overview of statistical meth-ods for establishing optimal treatment rules for personalizedmedicine and discuss specific examples in various medicalcontexts with oncology as an emphasis We also point thereader to statistical software when available The variousapproaches enable investigators to ascertain the extent towhich one should expect a new untreated patient to respondto each candidate therapy and thereby select the treatmentthat maximizes the expected therapeutic response for thespecific patient [3 19] Section 2 discusses the limitationsof conventional approaches based on post hoc stratifiedanalysis Section 3 offers an overview of the process for thedevelopment of personalized regimes Section 4 discussesthe selection of an appropriate statistical method for differ-ent types of clinical outcomes and data sources Section 5presents technical details for deriving optimal treatmentselection rules In Section 6 we discuss approaches forevaluating model performance and assessing the extent towhich treatment selection using the derived optimal rule islikely to benefit future patients
2 Limitations of Subgroup Analysis
Cancer is an inherently heterogeneous disease Yet oftenefforts to personalize therapy rely on the application ofanalysis strategies that neglect to account for the extent of het-erogeneity intrinsic to the patient and disease and thereforeare too reductive for personalizing treatment inmany areas ofoncology [20ndash23] Subgroup analysis is often used to evaluatetreatment effects among stratified subsets of patients definedby one or a few baseline characteristics [23ndash26] For exampleThatcher et al [21] conducted a series of preplanned subgroupanalyses for refractory advanced non-small-cell lung cancerpatients treated with Gefitinib plus best supportive careagainst placebo Heterogeneous treatment effects were foundin subgroups defined by smoking status that is significantprolonged survival was observed for nonsmokers while notreatment benefit was found for smokers
Though very useful when well planned and properlyconducted the reliance on subgroup analysis for developingpersonalized treatment has been criticized [24 25] Obvi-ously a subgroup defined by a few factors is inadequatefor characterizing individualized treatment regimes thatdepends onmultivariate synthesisMoreover post hoc imple-mentation of multiple subgroup analyses considers a set ofstatistical inferences simultaneously (multiple testing) anderrors such as incorrectly rejecting the null hypothesis arelikely to occur The extent to which the resulting inferenceinflates the risk of a false positive finding can be dramatic[23] Take for example a recent study that concluded thatchemotherapy followed by tamoxifen promises substantialclinical benefit for postmenopausal women with ER negativelymph node-negative breast cancer [27] through post hocapplication subgroup analysis Subsequent studies failed toreproduce this result concluding instead that the regimersquosclinical effects were largely independent of ER status [28] butmay depend on other factors including age
3 Personalized Medicine froma Statistical Perspective
From a statistical perspective personalized medicine is aprocess involving six fundamental steps provided in Figure 1[20 29 30] Intrinsic to any statistical inference initially onemust select an appropriate method of inference based on theavailable source of training data and clinical endpoints (egsteps (1) and (2)) Step (3) is the fundamental component ofpersonalized treatment selection deriving the individualizedtreatment rule (ITR) for the chosen method of inferenceAn ITR is a decision rule that identifies the optimal treat-ment given patientdisease characteristics [31 32] Section5 is dedicated to the topic of establishing ITRs for variousstatistical models and types of clinical endpoints that arecommonly used to evaluate treatment effectiveness in onco-logy
Individualized treatment rules are functions of modelparameters (usually treatment contrasts reflecting differencesin treatment effects) which must be estimated from theassumed statistical model and training data Statistical esti-mation takes place in step 4 The topic is quite generaland it thus is not covered in detail owning to the fact thatother authors have provided several effective expositionson model building strategies in this context [29 33] Afterestimating the optimal treatment rule in step (4) the resultingestimated ITRrsquos performance and reliabilitymust be evaluatedbefore the model can be used to guide treatment selection[34] The manner in which one assesses the performance ofthe derived ITR depends on the appropriate clinical utility(ie increased response rate or prolonged survival dura-tion) Evaluation of model goodness-of-fit and appropriatesummary statistics that use the available information tomeasure the extent to which future patients would benefitfrom application of the ITR is conducted in step (5) andwill be discussed in Section 6 The ITR is applied to guidetreatment selection for a future patient based on hisherbaseline clinical and genetic characteristics as the finalstep
BioMed Research International 3
(1) Acquire the training data
∙ Randomized data from a comparative trial∙ Observational cohorts
(2) Choose a method of inference based on clinicalendpoints and data dimension
∙ Binary∙ Continuous∙ Survival
∙ Low dimensional∙ High dimensional
(3) Identify the individualized treatment rule
∙ Derive the treatment contrasts∙ Select a clinically relevant decision threshold
(4) Fit the model to training data
∙ Select the important covariates∙ Estimate the model parameters
(5) Evaluate performance
∙ Assess model goodness-of-fit and prediction∙ Measure the extent of clinical benefit
(6) Apply the treatment rule to future patients
∙ Acquire prognostic and predictive covariates∙ Select the optimal treatment
Figure 1 The process for using statistical inference to establish personalized treatment rules
4 Selecting an AppropriateMethod of Inference
The quality of a treatment rule depends on the aptness ofthe study design used to acquire the training data clinicalrelevance of the primary endpoints statistical analysis plansfor model selection and inference and quality of the dataRandomized clinical trials (RCT) remain the gold standardstudy design for treatment comparison since randomizationmitigates bias arising from treatment selection Methodsfor deriving ITRs using data from RCTs are described inSection 51 Data from well conducted observational studiesprovide useful sources of information as well given thatthe available covariates can be used to account for potentialsources of confounding due to selection bias Predominatelymethods based on propensity scores are used to adjust forconfounding [35 36] Approaches for establishing ITRs usingobservational studies are discussed in Section 52
The predominate statistical challenge pertaining to theidentification of predictive biomarkers is the high-dimen-sional nature of molecular derived candidate features Clas-sical regression models cannot be directly applied since thenumber of covariates for example genes is much larger thannumber of samples Many approaches have been proposedto analyze high-dimensional data for prognostic biomarkersSection 53 discusses several that can be applied to detectpredictive biomarkers under proper modification
In oncology several endpoints are used to compare clin-ical effectiveness However the primary therapeutic goal isto extend survivorship or delay recurrenceprogressionThustime-to-event endpoints are often considered to be the mostrepresentative of clinical effectiveness [37] The approachesaforementioned were developed for ordinal or continuousoutcomes and were thus not directly applicable for survivalanalysis Methods for establishing ITRs from time-to-eventendpoints often use Cox regression or accelerated failure timemodels [38 39] The later approach is particularly appealing
in this context since the clinical benefits of prolonged survivaltime can be easily obtained [40 41] In Section 54 we willdiscuss both models
The performance of ITRs for personalized medicineis highly dependent upon the extent to which the modelassumptions are satisfied andor the posited model is cor-rectly specified Specifically performances may suffer frommisspecification of main effects andor interactions randomerror distribution violation of linear assumptions sensitivityto outliers and other potential sources of inadequacy [42]Some advanced methodologies have been developed to over-come these issues [43] including semiparametric approachesthat circumvent prespecification of the functional form ofthe relationship between biomarker and expected clinicalresponse [32 40] In addition optimal treatment rules canbe defined without regression models using classificationapproaches where patients are assigned to the treatment thatprovides the highest expected clinical benefit Appropriateclass labels can be defined by the estimated treatment differ-ence (eg gt0 versus le0) thereby enabling the use of machinelearning and data mining techniques [42 44 45] These willbe discussed in Section 55
5 Methods for Identifying IndividualizedTreatment Rules
This section provides details of analytical approaches that areappropriate identifying ITRs using a clinical data sourceThevery nature of treatment benefit is determined by the clinicalendpoint While extending overall survival is the ultimatetherapeutic goal often the extent of reduction in tumorsize as assessed by RECIST criteria (httpwwwrecistcom)is used as a categorical surrogate for long-term responseAlternatively oncology trials often compare the extent towhich the treatment delays locoregional recurrence or dis-ease progression Therefore time-to-event and binary (as inabsencepresence of partial or complete response) are the
4 BioMed Research International
most commonly used endpoints in oncologic drug develop-ment [37 46]
Let 119884 denote the observed outcome such as survivalduration or response to the treatment and let 119860 isin 0 1
denote the treatment assignment with 0 indicating standardtreatment and 1 for a new therapy Denote the collection ofobservable data for a previously treated patient by (119884 119860X)where X = 119883
1 1198832 119883
119901 represents a vector of values for
the 119901 biomarkers under study Quantitatively the optimalITRderives from the following equation relating the observedresponse to the potential outcome attained under the alterna-tive treatment
119884 = 119860119884(1)+ (1 minus 119860)119884
(0) (1)
where119884(1) and119884(0) denote the potential outcomes that wouldbe observed if the subject had been assigned to the newtherapy or the standard treatment respectively [32 43] Let119864(119884 | 119860X) = 120583(119860X) denote the expected value of 119884 given119860 and X The optimal treatment rule follows as
where 119868(sdot) is the indicator function For instance if119868120583(1 age gt 50) minus 120583(0 age gt 50) gt 0 = 1 then the optimalrule would assign patients who are older than 50 to the newtreatment However 119864(119884 | 119860X) is actually a function ofparameters 120583(119860X120573) denoted by 120573 The model needs to beldquofittedrdquo to the training data to obtain estimates of 120573 which wedenote by Hence for a patient with observed biomarkersX = x the estimated optimal treatment rule is
119892opt(X = x )
= 119868 120583 (119860 = 1X = x ) minus 120583 (119860 = 0X = x ) (3)
The above equation pertains to steps (3) and (4) in Figure 1that is the parameter estimates from a fitted model are usedto construct the personalized treatment rule The remainderof this section instructs the readers how to identify ITRs forthe various data types
We classify the statistical methods presented in thissection into five categories methods based on multivariateand generalized linear regression for analysis of data acquiredfrom RCT (Section 51) and observational studies (Section52) methods based on penalized regression techniques forhigh-dimensional data (Section 53) methods for survivaldata (Section 54) and advanced methods based on robustestimation and machine learning techniques (Section 55)
51 Multiple Regression for Randomized Clinical Trial DataClassical generalized linear models (GLM) can be usedto develop ITRs in the presence of training data derivedfrom randomized clinical study The regression frameworkassumes that the outcome 119884 is a linear function of prognosticcovariates 119883
1 putative predictive biomarkers 119883
2 the treat-
ment indicator 119860 and treatment-by-predictive interaction1198601198832
Let Δ(X) = 119864(119884 | 119860 = 1119883) minus 119864(119884 | 119860 = 0119883) = 120583(119860 = 1
119883) minus 120583(119860 = 0119883) denote the treatment contrast The optimaltreatment rule assigns a patient to the new treatment ifΔ(X) gt 0 For binary endpoints the logistic regressionmodelfor 120583(119860X) = 119875(119884 = 1 | 119860X) is defined such that
The treatment contrast Δ(X) can be calculated using 119864(119884 |
119860 = 119886119883) = 119875(119884 = 1 | 119860 = 119886X) = 119890120596(119860X)(1 + 119890120596(119860X))for 119886 = 0 1 respectively Similarly an optimal ITR assigns apatient to the new treatment if Δ(X) gt 0 This optimal treat-ment rule can be alternatively defined as 119892opt(X) = 119868(120573
3+
12057341198832) gt 0 without the need to calculate the treatment
contrast Δ(X) [43 45]Often one might want to impose a clinically meaningful
minimal threshold Δ(X) gt 120575 on the magnitude of treatmentbenefit before assigning patients to a novel therapy [45 47]For example it may be desirable to require at least a 01increase in response rate before assigning a therapy for whichthe long-term safety profile has yet to be established The useof a threshold value can be applied to all methods Withoutloss of generality we assume 120575 = 0 unless otherwise specifiedIn addition the reader should note that the approaches forconstructing an ITR described above can be easily applied tolinear regression models for continuous outcomes
This strategy was used to develop an ITR for treatmentof depression [19] using data collected from a RCT of 154patients In this case the continuous outcome was basedon posttreatment scores from the Hamilton Rating Scale forDepression The authors constructed a personalized advant-age index using the estimated treatment contrasts Δ(X)derived from five predictive biomarkers A clinically signif-icant threshold was selected 120575 = 3 based on the NationalInstitute for Health and Care Excellence criterion Theauthors identified that 60 of patients in the sample wouldobtain a clinicallymeaningful advantage if their therapy deci-sion followed the proposed treatment rule The approachesdiscussed in this section can be easily implemented withstandard statistical software such as the 119877 (httpwwwr-projectorg) using the functions lm and glm [48]
52 Methods for Observational Data Randomization attenu-ates bias arising from treatment selection thereby providingthe highest quality data for comparing competing interven-tions However due to ethical or financial constraints RCTsare often infeasible thereby necessitating an observationalstudy Treatment selection is often based on a patientrsquosprognosis In the absence of randomization the study designfails to ensure that patients on competing arms exhibit similarclinical and prognostic characteristics thereby inducing bias
However in the event that the available covariates capturethe sources of bias a well conducted observational studycan also provide useful information for constructing ITRsFor example the two-gene ratio index (HOXB13IL17BR)
BioMed Research International 5
was first discovered as an independent prognostic biomarkerfor ER+ node-negative patients using retrospective datafrom 60 patients [49] These findings were confirmed onan independent data set comprising 852 tumors which wasacquired from a tumor bank at the Breast Center of BaylorCollege of Medicine [50] Interestingly the two-gene ratioindex (HOXB13IL17BR) was reported to predict the benefitof treatment with letrozole in one recent independent study[51]
Methods based on propensity scores are commonly usedto attenuate selection bias [35] In essence these approachesuse the available covariates to attempt to diminish the effectsof imbalances among variables that are not of interest fortreatment comparison Moreover they have been shownto be robust in the presence of multiple confounders andrare events [52] Generally after adjusting for bias usingpropensity scores the same principles for deriving ITRs fromRCTs may be applied to the observational cohort
The propensity score characterizes the probability ofassigning a given treatment 119860 from the available covariatesX [35] Using our notation the propensity score is 120587(X 120585) =119875(119860 = 1 | X 120585) which can be modeled using logisticregression
where 119901 is the number of independent variables used toconstruct the propensity score and 120585
119895represents the 119895th
regression coefficient which characterizes the 119895th covariatersquospartial effect After fitting the data to obtain estimates forthe regression coefficients the estimated probability ofreceiving new treatment can be obtained for each patient(Xi) = 120587(Xi ) by inverting the logit function The eventthat asymp 0 implies that the measured independent variablesare reasonably ldquobalancedrdquo between treatment cohorts Inpractice one often includes as many baseline covariates intothe propensity score model as permitted by the sample size
Methods that use propensity scores can be categorizedinto four categories matching stratification adjusting andinverse probability weighted estimation [36 53] Matchingand stratification aim to mimic RCTs by defining a newdataset using propensity scores such that outcomes aredirectly comparable between treatment cohorts [53] Thesetwo approaches are well suited for conventional subgroupanalysis but their application to personalized medicine hasbeen limited Regression adjustment or simply adjusting canbe used to reduce bias due to residual differences in observedbaseline covariates between treatment groups This methodincorporates the propensity scores as an independent variablein a regression model and therefore can be used in con-junction with all regression-based methods [36] Methodsinvolving inverse probability weighted estimators will bediscussed in Section 551 [43]
Of course propensity scores methods may only attenuatethe effects of the important confounding variables that havebeen acquired by the study design Casual inference in gen-eral is not robust to the presence of unmeasured confounders
that influenced treatment assignment [35 54 55] For thedevelopment of ITRs predictive and important prognosticcovariates can be incorporated in the regression model forthe clinical outcome119884 alongwith the propensity scores whileother covariatesmay be utilized only in themodel for estimat-ing the propensity scores Hence propensity score methodsmay offer the researcher a useful tool for controlling forpotential confounding due to selection bias andmaintaining amanageable number of prognostic and predictive covariates
53 Methods for High-Dimensional Biomarkers The meth-ods presented in the previous sections are appropriate foridentifying an ITR using a small set of biomarkers (low-dimensional)However recent advances inmolecular biologyin oncology have enabled researchers to acquire vast amountsof genetic and genomic characteristics on individual patientsOften the number of acquired genomic covariates will exceedthe sample size Proper analysis of these high-dimensionaldata sources poses many analytical challenges Several meth-ods have been proposed specifically for analysis of high-dimensional covariates [56] although the majority of thesemethods are well suited only for the analysis of prognosticbiomarkers In what follows we introduce variable selectionmethods that were developed to detect predictive biomarkersfrom high-dimensional sources as well as describing how toconstruct optimal ITRs from the final set of biomarkers
An appropriate regressionmodel can be defined generallyas 119864(119884 | 119860X) = ℎ
0(X) + 119860(X120573) where ℎ
0(X) is an unspe-
cified baseline mean function 120573 = (1205730 1205731 120573
119902)119879 is a
column vector of regression coefficients and X = (1X) thedesign matrix Subscript 119902 denotes the total number of bio-markers which may be larger than the sample size 119899 AnITR derives from evaluating the interactions in 119860(X120573) notthe baseline effect of the high-dimensional covariates ℎ
0(X)
[32] Technically function 119860(X120573) = 119860(1205730+ 12057311198831+ 12057321198832+
sdot sdot sdot + 120573119902119883119902) cannot be uniquely estimated using traditional
maximum likelihood-based methods when 119902 gt 119899 [57]Yet practically many of the available biomarkers may notinfluence the optimal ITR [31] Thus the process for identifyITRs from a high-dimensional source requires that we firstidentify a sparse subset of predictive biomarkers that can beutilized for constructing the ITR
Parameters for the specifiedmodel can be estimated usingthe following loss function
119871119899120601(120573 120574) =
1
119899
119899
sum119894=1
[119884119894minus 120601 (X
119894 120574) minus X120573 119860
119894minus 120587 (X
119894)]2
(7)
where 120601(X 120574) represents any arbitrary function character-izing the ldquobaselinerdquo relationship between X and Y (eg anintercept or an additive model) Here we let 120587(X
119894) = 119875(119860
119894=
1 | X119894) denote either a propensity score (for observational
data) or a randomization probability (eg 05 given 1 1randomization) for RCT data If 120587(X) is known estimationusing this model yields unbiased estimates (asymptoticallyconsistent) of the interaction effects 120573 even if the main effectsare not correctly specified providing a robustness [32]
6 BioMed Research International
Penalized estimation provides the subset of relevantpredictivemarkers that are extracted from the nonzero coeffi-cients of the corresponding treatment-biomarker interactionterms of
where 120582119899is a tuning parameter which is often selected via
cross validation and 119869 is a shrinkage penalty Different choicesof 119869 lead to different types of estimators For example thelasso penalized regression corresponds to 119869 = 1 [58] andthe adaptive lasso to 119869 = 120596
119895= 1|120573init119895| where 120573init119895 is an
initial estimate of 120573119895[59] With little modification (8) can
be solved using the LARS algorithm implemented with the119877 package of 119897119886119903119904 [32 60 61] As we have shown before atreatment rule can be defined from the parameter estimatesas 119868120573
formmay have zero estimates for some coefficients (eg 1205732=
1205735= sdot sdot sdot = 120573
119902= 0) hence an ITR can be equivalently con-
structed from the final estimated nonzero coefficients and thecorresponding covariates
Alternative penalized regression approaches includeSCAD [62] and elastic-net [63] All penalized approachesproduce sparse solutions (ie identifying a small subset ofpredictive biomarkers) however the adaptive lasso is lesseffective when 119901 gt 119899 Methods that produce nonsparsemodels such as ridge regression [57] are less preferable sinceITRs based on many biomarkers are often unstable and lessuseful in practice [31] Several packages in 119877 offer imple-mentation of penalized regression such as 119901119886119903119888119900119903 for ridgelasso and adaptive lasso and 119899119888V119903119890119892 for SCAD [64 65]
Lu et al [32] used a penalized regression approach toanalyze data from the AIDS Clinical Trials Group Protocol175 (ACTG175) [66] In this protocol 2000 patients wereequally randomized to one of four treatments zidovudine(ZDV) monotherapy ZDV + didanosine (ddI) ZDV +zalcitabine and ddI monotherapy CD4 count at 15ndash25 weekspostbaselinewas the primary outcome and 12 baseline covari-ates were included in the analysis The resulting treatmentrule favored the combined regimes over ZDV monotherapyMoreover the treatment rule determined that ZDV + ddIshould be preferred to ddI when 119868(7159 + 107 times ageminus 018 timesCD40 minus 3357 times homo) = 1 where CD40 represents baselineCD4 counts and homo represents homosexual activity Basedon this treatment rule 878 patients would have benefitedfrom treatment with ZDV + ddI
54 Survival Analysis Heretofore we have discussed meth-ods for continuous or binary outcomes yet often investigatorswant to discern the extent to which a therapeutic interventionmay alter the amount of time required before an event occursThis type of statistical inference is referred to broadly assurvival analysis One challenge for survival analysis is thatthe outcomes may be only partially observable at the time ofanalysis due to censoring or incomplete follow-up Survivalanalysis has been widely applied in cancer studies often inassociation studies aimed to identify prognostic biomarkers
[56 67] Here we discuss twowidely usedmodels for derivingITRs using time-to-event data namely Cox regression andaccelerated failure time models
The Cox regression model follows as
120582 (119905 | X 119860) = 1205820(119905) exp 120573
11198831+ 12057321198832+ 119860 (120573
4+ 12057351198832)
(9)
where 119905 is the survival time 1205820(119905) is an arbitrary baseline
hazard function and 1198831 1198832represent prognostic and pre-
dictive biomarkers respectively Each 120573 characterizes themultiplicative effect on the hazard associated with a unitincrease in the corresponding covariate Therefore Coxmodels are referred to as proportional hazards (PH) models
Several authors have provided model building strategies[29] and approaches for treatment selection [20 30 68] Fol-lowing the previously outlined strategy a naive approach forderiving an ITR uses the hazard ratio (new treatment versusthe standard) as the treatment contrast which can be calcu-lated as Δ(X) = exp(120573
4+ 12057351198832) The ITR therefore is 119868(120573
4
+12057351198832) lt 0 There are obvious limitations to this approach
First violations of the PH assumption yield substantiallymis-leading results [69]Moreover evenwhen the PH assumptionis satisfied because the Cox model does not postulate adirect relationship between the covariate (treatment) and thesurvival time the hazard ratio fails to measure the extent towhich the treatment is clinically valuable [38 70]
Accelerated failure time (AFT) models provide an alter-native semiparametric model Here we introduce its appli-cation for high-dimensional data Let 119879 and 119862 denote thesurvival and censoring times and denote the observed databy ( 120575 119860X) where = min(119879 119862) and 120575 = 119868(119879 lt 119862)Define the log survival time as 119884 = log(119879) a semiparametricregression model is given as 119864(119884 | 119860X) = ℎ
0(X) + 119860(X120573)
where ℎ0(X) is the unspecified baseline mean function
Similar to the previous section the treatment rule is 119868(1205730+
independent censoring the AFT model parameters can beestimated by minimizing the following loss function
119871119899120601(120573) =
1
119899
119899
sum119894=1
120575119894
119866(119894)[119894minus 120601 (X
119894 120574) minus X120573 119860
119894minus 120587 (X
119894)]2
(10)
where 119894= log(
119894) 120587(X
119894) = 119875(119860
119894= 1 | X
119894) is the propensity
score or randomization probability 119866(sdot) is the Kaplan-Meierestimator of the survival function of the censoring time and120601(X 120574) characterizes any arbitrary function
Thismethod can be extended to accommodatemore thantwo treatments simultaneously by specifying appropriatetreatment indicators For instance the mean function can bemodeled as 119864(119884 | 119860X) = ℎ
0(X) + 119860
(1)X120573(1) + 119860(2)X120573(2)
for two treatment drugs versus the standard care The ITRassigns the winning drug Note this work was proposed by[40] and is an extension of [32] to the survival setting Henceit shares the robustness property and can be applied to obser-vational data For implementation the sameprocedure can befollowed to obtain estimates with one addition step of calcu-lating 119866(
119894) There are several 119877 packages for Kaplan-Meier
BioMed Research International 7
estimates and Cox regression models These sources can befound at httpcranr-projectorgwebviewsSurvivalhtmlMore details pertaining to statistical methods for survivalanalysis can be found here [71] To compare treatmentrules constructed from Cox and AFT models for examplemethods for measuring the extent of clinical effectiveness foran ITR will be discussed in Section 6
We here present an example when an AFT model wasused to construct an ITR for treatment of HIV [40] Theexample derives from the AIDS Clinical Trials Group Pro-tocol 175 that was discussed in Section 53 [32 66] In thiscase the primary outcome variable was time (in days) tofirst ge50 decline in CD4 count or an AIDS-defining eventor death A total of 12 covariates and four treatments (ZDVZDV + ddI ZDV + zalcitabine and ddI) were includedThe four treatments were evaluated simultaneously Patientsreceiving the standard care of ZDV monotherapy wereconsidered as the reference group Hence three treatmentcontrasts (119868ZDV+ddI 119868ZDV+zalcitabine and 119868ddI) were combinedwith various putative predictive covariates and comparedwith ZDV monotherapy For example gender was detectedas the predictive covariate only for ddI monotherapy Theinvestigators assumed 120601(X 120574) = 120574
0 The treatment rule
recommended 1 patient for ZDV monotherapy while 7291216 and 193 patients were recommended for ZDV + ddIZDV + zalcitabine and ddI respectively
55 Advanced Methods
551 Robust Inference The performances of ITRs heretoforepresented depend heavily on whether the statistical modelswere correctly specified Recently there has been much atten-tion focused on the development of more advanced methodsand modeling strategies that are robust to various aspectsof potential misspecification We have already presented afew robust models that avoid specification of functionalparametric relationships for main effects [32 40] Here weintroduce two more advanced methods widely utilized forITRs that are robust to the type of misspecification issuescommonly encountered in practice [42 43]
Recall that the ITR for a linear model 119864(119884 | 119860 =
119886X) = 120583(119860 = 119886X120573) with two predictive markers followsas 119892(X120573) = 119868(120573
4+ 12057351198832+ 12057361198833) gt 0 where 119886 = 0 1
The treatment rule of 119892(X120573) may use only a subset of thehigh-dimensional covariates (eg 119883
2 1198833) but it always
depends on the correct specification of 119864(119884 | 119860 = 119886X)Defining a scaled version of 120573 as 120578(120573) the correspondingITR is 119892(120578X) = 119892(X120573) = 119868(119883
3gt 1205780+ 12057811198832) where
1205780= minus12057341205736and 120578
1= 12057351205736 If the model for 120583(119860X120573) is
indeed correctly specified the treatment rules of 119892(X120573) and119892(120578X) lead to the same optimal ITR Hence the treatmentrule parameterized by 120578 can be derived from a regressionmodel or may be based on some key clinical considerationswhich enable evaluation of 119892(120578X) directly without referenceto the regression model for 120583(119860X120573)
Let 119862120578
= 119860119892(120578X) + (1 minus 119860)1 minus 119892(120578X) where119862120578= 1 indicates random assignment to an intervention
that is recommended by the personalized treatment rule 119892(120578X) Let 120587(X ) denote the randomization ratio or the
estimated propensity score (as in previous section) and119898(X 120578 ) denote the potential outcome under the treatmentrule estimated from the following model 119864(119884 | 119860 =
119886X) = 120583(119860X120573) For example if the treatment rule 119892(120578X)= 1 then 119898(X 120578 ) = 119892(120578X)120583(119860 = 1X ) + 1 minus 119892(120578
X)120583(119860 = 0X ) = 120583(119860 = 1X ) Two estimators ofthe expected response to treatment the inverse probabilityweighted estimator (IPWE) and doubly robust AIPWE aregiven as follows
IPWE (120578) = 1
119899
119899
sum119894=1
119862120578sdot119894119884119894
120587119888(X119894 120578 )
=1
119899
119899
sum119894=1
119862120578sdot119894119884119894
120587 (X119894 )119860119894 1 minus 120587 (X
119894 )1minus119860119894
AIPWE (120578) = 1
119899
119899
sum119894=1
119862120578sdot119894119884119894
120587119888(X119894 120578 )
minus119862120578sdot119894119884119894 minus 120587119888 (X119894 120578 )
120587119888(X119894 120578 )
119898 (X119894 120578 )
(11)
where 120587119888(X119894 120578 ) = 120587(X )119892(120578X) + 1 minus 120587(X )1 minus
119892(120578X) The optimal treatment rule follows as 119892(X = x)where is estimated from the above models a constraintsuch as 120578 = 1 is imposed to obtain a unique solution [43] If the propensity score is correctly specified theIPWE estimator yields robust (consistent) estimates AIPWEis considered a doubly robust estimator since it produces con-sistent estimates when either propensity score or the model119864(119884 | 119860 = 119886X) is misspecified but not both [42 43] Thecompanion119877 code is publicly available at httponlinelibrarywileycomdoi101111biom12191suppinfo
552 Data Mining and Machine Learning The methodspresented in Section 551 are robust against misspecificationof regression models Yet they often require prespecificationof the parametric form for the treatment rule (eg 119868(119883
3gt
1205780+ 12057811198832)) which can be practically challenging [44]
Well established classification methods and other popularmachine learning techniques can alternatively be customizedto define treatment selection rules [44 72 73] these methodsavoid prespecification of the parametric form of the ITR AnITR can be defined following a two-step approach in thefirst step treatment contrasts are estimated from a positedmodel and in the second step classification techniques areapplied to determine the personalized treatment rules Forexample when only two treatments are considered a newvariable 119885 can be defined based on the treatment contrastthat is 119885 = 1 if Δ(X) = 120583(119860 = 1X) minus 120583(119860 = 0X) gt 0 and119885 = 0 otherwiseThe absolute value of the treatment contrast119882119894= |Δ(X)| can be used in conjunction with a classification
technique to define an appropriate ITR [44]Unlike classification problems wherein the class labels
are observed for the training data the binary ldquoresponserdquovariable 119885 which serves as the class label is not availablein practice Specifically patients who are in the class 119885 = 1
8 BioMed Research International
have 120583(119860 = 1X) gt 120583(119860 = 0X) and should thereforebe treated with the new therapy however these quantitiesneed to be estimated since patients are typically assigned toonly one of the available treatments This imparts flexibilityfor estimation of the optimal treatment regimes since anyof the previously discussed regression models and even someensemble prediction methods such as random forest [74] canbe used to construct the class labels 119885
119894and weights
119894[44]
An ITR can be estimated from the dataset 119885119894X119894 119894 using
any classification approach where 119894are subject specific
misclassification weights [44 45] This includes popularclassificationmethods such as adaptive boosting [75] supportvector machines [76] and classification and regression trees(CART) [77] At least one study has suggested that SVMoutperforms other classification methods in this contextwhereas random forest and boosting perform comparativelybetter than CART [78] However the performances ofthese classification algorithms are data dependent Definitiveconclusion pertaining to their comparative effectiveness ingeneral has yet to be determined [78] It shall be also notedthat these classification methods can be also applied to high-dimensional data [45 72]
One special case of this framework is the ldquovirtue twinsrdquoapproach [45] Specifically in the first step a random forestapproach [74] is used to obtain the treatment contrasts Thenin the second step CART is used to classify subjects tothe optimal treatment regime The approach can be easilyimplemented in 119877 using packages of randomForest [79] andrpart [80] Very recently Kang et al [42] proposed amodifiedversion of the adaptive boosting technique of Friedman et al[75] The algorithm iteratively fits a simple logistic regressionmodel (ldquoworking modelrdquo) to estimate 119875(119884 = 1 | 119860X) and ateach stage assigns higher weights to subjects whose treatmentcontrast is near zero After a prespecified stopping criterionis met an average of the treatment contrasts Δ(X) is calcu-lated for each patient using all models fitted at each iterationA subject is assigned to the new therapy if Δ(X) gt 0 The 119877code for the aforementioned boosting methods is publiclyavailable at httponlinelibrarywileycomdoi101111biom12191suppinfo
Lastly we present a breast cancer example where severalbiomarkers were combined to construct an optimal ITRThe data was collected in the Southwest Oncology Group(SOWG)-SS8814 trial [13] and analyzed with the machinelearning approach of Kang et al [42] Three hundred andsixty-seven node-positive ER-positive breast cancer patientswere selected from the randomized trial of SOWG A total of219 received tamoxifen plus adjuvant chemotherapy and 148was given tamoxifen aloneThe outcome variable was definedas breast cancer recurrence at 5 years The authors selectedthree genes which had presented treatment-biomarker inter-actions in amultivariate linear logistic regressionmodel [42]Data were analyzed with logistic models IPWE AIPWElogistic boosting a single classification tree with treatment-biomarker interactions and the proposed boosting approachwith a classification tree as the working model Each methodidentified different patient cohorts that could benefit fromtamoxifen alone these cohorts consisted of 184 183 128 86
263 and 217 patients respectively (see Table 5 in [42]) In thisanalysis the clinical benefits provided by these 6 treatmentrules were not statistically different Hence investigatorsneed to evaluate and compare ITRs in terms of the extentof expected clinical impact This is considered in the nextsection
Heretofore we have discussed various methodologies for theconstruction of ITR while their performances need to beassessed before these rules can be implemented in clinicalpractice Several aspects pertaining to the performance of aconstructed ITR need to be considered The first one is howwell the ITR fits the data and the second is how well theITR performs compared with existing treatment allocationrules The former is related to the concept of goodness-of-fitor predictive performance [34] As the true optimal treatmentgroups are hidden model fits may be evaluated by measuringthe congruity between observed treatment contrasts andpredicted ones [34 47] More details can be found in a recentpaper by Janes et al [47] Performances of ITRs can becompared via assessment of a global summary measure forexample prolonged survival time or reduced disease rate [4042] Summarymeasures are also very useful for evaluating theextent to which an ITR may benefit patients when applied inpractice Moreover it is essential that performance of an ITRis considered in comparison to business-as-usual proceduressuch as a naive rule that randomly allocates patients totreatment [81] Summary measures will be discussed inSection 61 The effectiveness of an ITR should go beyond thetraining data set used to construct a treatment rule cross-validation and bootstrapping techniques are often employedto assess the impact of ITRs on future patients [81] and willbe discussed in Section 62
61 Summary Measures ITRs may be derived from differ-ent methodologies and comparisons should be conductedwith respect to the appropriate clinically summaries A fewsummary measures for different types of outcomes havebeen proposed [19 40 42] these measures quantify thedirect clinical improvements obtained by applying an ITR incomparison with default methods for treatment allocation
Binary Outcomes Clinical effectiveness for binary clinicalresponse is represented by the difference in disease rates (ortreatment failure) induced by ITR versus a default strategythat allocates all patients to a standard treatment [42 47 82]Let 119892opt(X) = 119868120583(119860 = 1X)minus120583(119860 = 0X) lt 0 be an optimalITR This difference is formally defined as
Note 120583(119860X) needs to be estimated to construct the ITRyet parameters 120573 are omitted for simplicity Larger valuesof Θ119861119892opt(X) indicate increased clinical value for the
biomarker driven ITR A subset of patients that are recom-mended for new treatment (119860 = 1) under an ITR may havebeen randomly selected to receive it while the remainingsubset of ldquounluckyrdquo patientswould have received the standardtreatment [19] The summary measure of Θ
119861119892opt(X) char-
acterizes a weighted difference in the disease rates betweenthe standard and the new treatments in a population whereinthe constructed optimal ITR would recommend the newtreatment 119892opt(X = 1) The weight is the proportion ofpatients identified by the optimal ITR for the new treatmentand can be empirically estimated using the correspondingcounts For example 119875119892opt(X) = 1 can be estimatedusing the number of patients recommended for the newtreatment divided by the total sample size A similar summarystatistic can be derived for an alternative strategy allocatingall patients to the new treatment The summary could beapplied to the aforementioned breast cancer example [42]for example with the aim of finding a subgroup of patientswho were likely to benefit from adjuvant chemotherapywhile those unlikely to benefit would be assigned tamoxifenalone to avoid the unnecessary toxicity and inconvenience ofchemotherapy
Continuous Variables Another strategy for continuous datacompares outcomes observed for ldquoluckyrdquo subjects those whoreceived the therapy that would have been recommendedby the ITR based [81] Further one business-as-usual drugallocation procedure is randomizing treatment and standardcare at the same probability of 05 A summary statisticis to measure the mean outcome under ITR compared tothat obtained under random assignment for instance themean decrease in Hamilton Rating Scale for Depression asdiscussed in Section 51 [19] Define the summary measureas Θ119862119892opt(X) = 120583119892opt(X)X minus 120583119892rand(X)X where
119892rand(X) represents the randomization allocation procedureThe quantity of 120583119892119900119901119905(X)X represents the mean outcomeunder the constructed IRT that can be empirically estimatedfrom the ldquoluckyrdquo subjects and 120583119892rand(X)X can be esti-mated empirically from the sample means
Alternatively an ITR may be compared to an ldquooptimalrdquodrug that has showed universal benefits (a better drug onaverage) in a controlled trial The clinical benefits of anldquooptimalrdquo drug can be defined as 120583119892best(X)X = max120583(119860 =
0X) 120583(119860 = 1X) 120583(119860 = 119886X) and can be empiricallyestimated from the sample means of the new and standardtreatments respectively Then the alternative summary mea-sure is defined as Θ
Survival Data For survival data a clinically relevant measureis mean overall (or progression free) survival time As
survival time is continuous in nature the identical strategyprovided above for continuous outcomes can be employedhere However because the mean survival time may notbe well estimated from the observed data due to a highpercentage of censored observations [40] an alternativemean restricted survival duration was proposed and definedas the population average event-free durations for a restrictedtime of 119905lowast [41 83]Often 119905lowast is chosen to cover the trialrsquos follow-up period Mathematically it can be calculated by integratingthe survival function of 119878(119905) over the domain of (0 119905lowast) that is120583119892opt(X)X 119905lowast = int119905
lowast
0119878(119905)119889119905 and often estimated by the area
under the Kaplan-Meier curve up to 119905lowast [84] Thus an ITRrsquospotential to prolong survival can be calculated asΘ
119878119892opt(X)
119905lowast = 120583119892opt(X)X 119905lowast minus 120583119892rand(X)X 119905lowast
62 AssessingModel Performance The summaries heretoforediscussed evaluate an optimal ITR for a given model andestimating procedure Because these quantities are estimatedconditionally given the observed covariates they neglectto quantify the extent of marginal uncertainty for futurepatients Hence an ITR needs to be internally validated ifexternal data is not available [34] Cross-validation (CV) andbootstrap resampling techniques are commonly used for thispurpose [19 42 45 81] and expositions on both approachesare well described elsewhere [33 85 86]
We here briefly introduce a process that was proposed byKapelner et al [81] in the setting of personalized medicineTenfold CV is commonly used in practice where the wholedata is randomly partitioned into 10 roughly equal-sizedexclusive subsamples All methods under consideration areapplied to 910 of the data excluding 110 as an independenttesting data set The process is repeated 10 times for eachsubsample Considering the assignments recommended bythe optimal ITRs the summary measures can be calculatedusing results from each testing fold [45]TheCVprocess givesthe estimated summary measures and its variation can beevaluated using bootstrap procedures Specifically one drawsa sample with replacement from the entire data and calculatesthe summary measure from 10-fold CV This process willbe repeated 119861 times where 119861 is chosen for resolution ofthe resulting confidence intervals [81] Using the summarymeasures as119861 new random samples the correspondingmeanand variances can be calculated empirically Note that thesummary measures compare two treatment rules one for theoptimal ITR and another naive rule (eg randomization)
The above procedure can be applied to all the meth-ods we have discussed so far The 119877 software package119879119903119890119886119905119898119890119899119905119878119890119897119890119888119905119894119900119899 (httplabsfhcrcorgjanesindexhtml)can be used to implement these methods for evaluatingand comparing biomarkers for binary outcomes [47] Veryrecently an inferential procedure was proposed for contin-uous outcomes that is implemented in the publicly available119877 package ldquoPersonalized Treatment Evaluatorrdquo [81 87] Bothmethods consider data from RCTs with two arms for com-parative treatmentsThesemethods are in general applicableto regression model based methods but are not suitable forapproaches based on classification techniques or penalizedregression
10 BioMed Research International
Next we present two examples Recall in Section 55 thatKang et al [42] reported the estimated clinical benefits ofan ITR for breast cancer when compared to the defaultstrategy of assigning all patients to adjuvant chemotherapyThe proposed approach (based on boosting and classificationtrees) achieved the highest value of the summary measureat 0081 with 95 confidence interval (CI) (0000 0159)[42] In the second example introduced in Section 51 [19]the authors calculated the mean score of the HamiltonRating Scale for Depression for two groups of subjectsgroups were defined by randomly assigning patients to theldquooptimalrdquo and ldquononoptimalrdquo therapy as defined by the ITRThe reported difference between the two groups was minus178with a 119875 value of 009 which fails to attain a clinicalsignificant difference of 3 [19] The same data was analyzedby Kapelner et al [81] Following the discussed procedurethe authors reported the estimated values (and 95 CI)of Θ119862119892opt(X) and Θ
119862alt119892opt(X) as minus0842(minus2657 minus0441)
and minus0765(minus2362 0134) respectively The results whichfail to achieve clinical significance were based on rigorousstatistical methods and thus can be considered reliableestimates of the ITRrsquos performance
7 Discussion
As our understanding tumor heterogeneity evolves person-alized medicine will become standard medical practice inoncology Therefore it is essential that the oncology com-munity uses appropriate analytical methods for identifyingand evaluating the performance of personalized treatmentrules This paper provided an exposition of the process forusing statistical inference to establish optimal individualizedtreatment rules using data acquired from clinical study Thequality of an ITR depends on the quality of the design used toacquire the dataMoreover an ITRmust be properly validatedbefore it is integrated into clinical practice Personalizedmedicine in some areas of oncologymay be limited by the factthat biomarkers arising from a small panel of genesmay neveradequately characterize the extent of tumor heterogeneityinherent to the disease Consequently the available statisticalmethodology needs to evolve in order to optimally exploitglobal gene signatures for personalized medicine
The bulk of our review focused on statistical approachesfor treatment selection at a single time point The readershould note that another important area of research considersoptimal dynamic treatment regimes (DTRs) [88 89] whereintreatment decisions are considered sequentially over thecourse ofmultiple periods of intervention using each patientrsquosprior treatment history Zhao and Zeng provide a summaryof recent developments in this area [90]
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
JunshengMa was fully funded by the University of Texas MDAnderson Cancer Center internal funds Brian P Hobbs and
Francesco C Stingo were partially supported by the CancerCenter Support Grant (CCSG) (P30 CA016672)
References
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[2] D Hanahan and R AWeinberg ldquoHallmarks of cancer the nextgenerationrdquo Cell vol 144 no 5 pp 646ndash674 2011
[3] R Simon ldquoClinical trial designs for evaluating the medicalutility of prognostic and predictive biomarkers in oncologyrdquoPersonalized Medicine vol 7 no 1 pp 33ndash47 2010
[4] P L Bedard A R Hansen M J Ratain and L L Siu ldquoTumourheterogeneity in the clinicrdquo Nature vol 501 no 7467 pp 355ndash364 2013
[5] M D Pegram G Pauletti and D J Slamon ldquoHer-2neu as apredictive marker of response to breast cancer therapyrdquo BreastCancer Research and Treatment vol 52 no 1ndash3 pp 65ndash77 1998
[6] G J Kelloff andC C Sigman ldquoCancer biomarkers selecting theright drug for the right patientrdquoNature Reviews Drug Discoveryvol 11 no 3 pp 201ndash214 2012
[7] J ADiMasi JM Reichert L Feldman andAMalins ldquoClinicalapproval success rates for investigational cancer drugsrdquoClinicalPharmacology andTherapeutics vol 94 no 3 pp 329ndash335 2013
[8] M Hay D W Thomas J L Craighead C Economides andJ Rosenthal ldquoClinical development success rates for investiga-tional drugsrdquo Nature Biotechnology vol 32 no 1 pp 40ndash512014
[9] S S Knox ldquoFrom lsquoomicsrsquo to complex disease a systems biologyapproach to gene-environment interactions in cancerrdquo CancerCell International vol 10 article 11 2010
[10] V Deschoolmeester M Baay P Specenier F Lardon and JB Vermorken ldquoA review of the most promising biomarkersin colorectal cancer one step closer to targeted therapyrdquo TheOncologist vol 15 no 7 pp 699ndash731 2010
[11] J A Sosman K B Kim L Schuchter et al ldquoSurvival in brafV600ndashmutant advanced melanoma treated with vemurafenibrdquoThe New England Journal of Medicine vol 366 no 8 pp 707ndash714 2012
[12] S Paik S Shak G Tang et al ldquoA multigene assay to predictrecurrence of tamoxifen-treated node-negative breast cancerrdquoThe New England Journal of Medicine vol 351 no 27 pp 2817ndash2826 2004
[13] K S Albain W E Barlow S Shak et al ldquoPrognostic and pre-dictive value of the 21-gene recurrence score assay in post-menopausal women with node-positive oestrogen-receptor-positive breast cancer on chemotherapy a retrospective analysisof a randomised trialrdquoTheLancet Oncology vol 11 no 1 pp 55ndash65 2010
[14] J E Lang J S Wecsler M F Press and D Tripathy ldquoMolecularmarkers for breast cancer diagnosis prognosis and targetedtherapyrdquo Journal of Surgical Oncology vol 111 no 1 pp 81ndash902015
[15] W Werft A Benner and A Kopp-Schneider ldquoOn the identi-fication of predictive biomarkers detecting treatment-by-geneinteraction in high-dimensional datardquo Computational Statisticsand Data Analysis vol 56 no 5 pp 1275ndash1286 2012
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[17] A J Vickers MW Kattan and D J Sargent ldquoMethod for eval-uating prediction models that apply the results of randomizedtrials to individual patientsrdquo Trials vol 8 no 1 article 14 2007
[18] H Janes M S Pepe P M Bossuyt andW E Barlow ldquoMeasur-ing the performance of markers for guiding treatment deci-sionsrdquo Annals of Internal Medicine vol 154 no 4 pp 253ndash2592011
[19] R J DeRubeis Z D Cohen N R Forand J C Fournier L AGelfand and L Lorenzo-Luaces ldquoThe personalized advantageindex translating research on prediction into individualizedtreatment recommendationsAdemonstrationrdquoPLoSONE vol9 no 1 Article ID e83875 2014
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[22] A J Vickers ldquoPrediction models in cancer carerdquo CA A CancerJournal for Clinicians vol 61 no 5 pp 315ndash326 2011
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[24] S J Pocock S E Assmann L E Enos and L E Kasten ldquoSub-group analysis covariate adjustment and baseline comparisonsin clinical trial reporting current practice and problemsrdquoStatistics in Medicine vol 21 no 19 pp 2917ndash2930 2002
[25] PM Rothwell ZMehta S CHoward S A Gutnikov andC PWarlow ldquoFrom subgroups to individuals general principles andthe example of carotid endarterectomyrdquoTheLancet vol 365 no9455 pp 256ndash265 2005
[26] R Wang S W Lagakos J H Ware D J Hunter and J MDrazen ldquoStatistics in medicinemdashreporting of subgroup ana-lyses in clinical trialsrdquoTheNewEngland Journal ofMedicine vol357 no 21 pp 2108ndash2194 2007
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[30] S Gill C L Loprinzi D J Sargent et al ldquoPooled analysis offluorouracil-based adjuvant therapy for stage II and III coloncancer who benefits and by how muchrdquo Journal of ClinicalOncology vol 22 no 10 pp 1797ndash1806 2004
[31] M Qian and S A Murphy ldquoPerformance guarantees forindividualized treatment rulesrdquoThe Annals of Statistics vol 39no 2 pp 1180ndash1210 2011
[32] W Lu H H Zhang and D Zeng ldquoVariable selection for opti-mal treatment decisionrdquo StatisticalMethods inMedical Researchvol 22 no 5 pp 493ndash504 2013
[33] R Kohavi ldquoA study of cross-validation and bootstrap for accu-racy estimation and model selectionrdquo in Proceedings of the 14thInternational Joint Conference on Artificial Intelligence (IJCAIrsquo95) vol 2 pp 1137ndash1145 1995
[34] E W Steyerberg A J Vickers N R Cook et al ldquoAssessing theperformance of prediction models a framework for traditionaland novel measuresrdquo Epidemiology vol 21 no 1 pp 128ndash1382010
[35] P R Rosenbaum and D B Rubin ldquoThe central role of the pro-pensity score in observational studies for causal effectsrdquo Bio-metrika vol 70 no 1 pp 41ndash55 1983
[36] R B drsquoAgostino Jr ldquoTutorial in biostatistics propensity scoremethods for bias reduction in the comparison of a treatment toa non-randomized control grouprdquo Statistics in Medicine vol 17no 19 pp 2265ndash2281 1998
[37] R Pazdur ldquoEndpoints for assessing drug activity in clinicaltrialsrdquoThe Oncologist vol 13 supplement 2 pp 19ndash21 2008
[38] S L Spruance J E Reid M Grace and M Samore ldquoHazardratio in clinical trialsrdquo Antimicrobial Agents and Chemotherapyvol 48 no 8 pp 2787ndash2792 2004
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[40] Y Geng Flexible Statistical Learning Methods for SurvivalData Risk Prediction and Optimal Treatment Decision NorthCarolina State University 2013
[41] J Li L Zhao L Tian et alAPredictive Enrichment Procedure toIdentify Potential Responders to a NewTherapy for RandomizedComparative Controlled Clinical Studies Harvard UniversityBiostatisticsWorking Paper SeriesWorking Paper 169 HarvardUniversity 2014
[42] C Kang H Janes and Y Huang ldquoCombining biomarkers tooptimize patient treatment recommendationsrdquo Biometrics vol70 no 3 pp 695ndash720 2014
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[44] B Zhang A A Tsiatis M Davidian M Zhang and E LaberldquoEstimating optimal treatment regimes from a classificationperspectiverdquo Stat vol 1 no 1 pp 103ndash114 2012
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the HOXB13IL17BR biomarkerrdquo Journal of the National CancerInstitute vol 105 no 14 pp 1036ndash1042 2013
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[53] P C Austin ldquoAn introduction to propensity score methods forreducing the effects of confounding in observational studiesrdquoMultivariate Behavioral Research vol 46 no 3 pp 399ndash4242011
[54] G Heinze and P Juni ldquoAn overview of the objectives of andthe approaches to propensity score analysesrdquo European HeartJournal vol 32 no 14 Article ID ehr031 pp 1704ndash1708 2011
[55] L E Braitman and P R Rosenbaum ldquoRare outcomes commontreatments analytic strategies using propensity scoresrdquo Annalsof Internal Medicine vol 137 no 8 pp 693ndash695 2002
[56] D M Witten and R Tibshirani ldquoSurvival analysis withhigh-dimensional covariatesrdquo Statistical Methods in MedicalResearch vol 19 no 1 pp 29ndash51 2010
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[59] H Zou ldquoThe adaptive lasso and its oracle propertiesrdquo Journal ofthe American Statistical Association vol 101 no 476 pp 1418ndash1429 2006
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[61] T Hastie and B Efron ldquolars Least angle regression lasso andforward stagewiserdquo R package version 12 2013 httpcranr-projectorgwebpackageslarsindexhtml
[62] J Fan and R Li ldquoVariable selection via nonconcave penalizedlikelihood and its oracle propertiesrdquo Journal of the AmericanStatistical Association vol 96 no 456 pp 1348ndash1360 2001
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[67] H M Boslashvelstad S Nygard H L Stoslashrvold et al ldquoPredictingsurvival from microarray datamdasha comparative studyrdquo Bioinfor-matics vol 23 no 16 pp 2080ndash2087 2007
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[69] P Royston and M K Parmar ldquoThe use of restricted meansurvival time to estimate the treatment effect in randomizedclinical trials when the proportional hazards assumption is indoubtrdquo Statistics inMedicine vol 30 no 19 pp 2409ndash2421 2011
[70] P Royston andM K B Parmar ldquoRestrictedmean survival timean alternative to the hazard ratio for the design and analysis ofrandomized trials with a time-to-event outcomerdquo BMCMedicalResearch Methodology vol 13 no 1 article 152 2013
[71] E T Lee and J W Wang Statistical Methods for Survival DataAnalysis John Wiley amp Sons Hoboken NJ USA 2013
[72] Y Zhao D Zeng A J Rush and M R Kosorok ldquoEstimatingindividualized treatment rules using outcome weighted learn-ingrdquo Journal of the American Statistical Association vol 107 no499 pp 1106ndash1118 2012
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[74] L Breiman ldquoRandom forestsrdquoMachine Learning vol 45 no 1pp 5ndash32 2001
[75] J Friedman T Hastie and R Tibshirani ldquoAdditive logisticregression a statistical view of boostingrdquo The Annals of Statis-tics vol 28 no 2 pp 337ndash407 2000
[76] C Cortes and V Vapnik ldquoSupport-vector networksrdquo MachineLearning vol 20 no 3 pp 273ndash297 1995
[77] L Breiman J Friedman C J Stone and R A Olshen Classi-fication and Regression Trees CRC Press New York NY USA1984
[78] S Dudoit J Fridlyand and T P Speed ldquoComparison of dis-criminationmethods for the classification of tumors using geneexpression datardquo Journal of the American Statistical Associationvol 97 no 457 pp 77ndash87 2002
[79] A Liaw and MWiener ldquoClassification and regression by rand-omforestrdquoRNews vol 2 no 3 pp 18ndash22 2002 httpCRANR-projectorgdocRnews
[80] T Therneau B Atkinson and B Ripley ldquorpart RecursivePartitioning and Regression Treesrdquo R package version 41-3httpcranr-projectorgwebpackagesrpartindexhtml
[81] A Kapelner J Bleich Z D Cohen R J DeRubeis and RBerk ldquoInference for treatment regime models in personalizedmedicinerdquo httparxivorgabs14047844
[82] X Song and M S Pepe ldquoEvaluating markers for selecting apatientrsquos treatmentrdquoBiometrics vol 60 no 4 pp 874ndash883 2004
[83] T Karrison ldquoRestricted mean life with adjustment for covari-atesrdquo Journal of the American Statistical Association vol 82 no400 pp 1169ndash1176 1987
[84] C Barker ldquoThe mean median and confidence intervals ofthe kaplan-meier survival estimatemdashcomputations and appli-cationsrdquo Journal of the American Statistical Association vol 63no 1 pp 78ndash80 2009
[85] B Efron and R J Tibshirani An Introduction to the Bootstrapvol 57 CRC Press 1994
[86] S Arlot andA Celisse ldquoA survey of cross-validation proceduresfor model selectionrdquo Statistics Surveys vol 4 pp 40ndash79 2010
[87] A Kapelner and J Bleich ldquoPTE Personalized Treatment Eva-luatorrdquo 2014 R package version 10 httpCRANR-projectorgpackage=PTE
[88] S A Murphy ldquoOptimal dynamic treatment regimesrdquo Journal ofthe Royal Statistical Society Series B StatisticalMethodology vol65 no 2 pp 331ndash355 2003
BioMed Research International 13
[89] J M Robins ldquoOptimal structural nested models for optimalsequential decisionsrdquo in Proceedings of the Second Seattle Sym-posium in Biostatistics vol 179 of Lecture Notes in Statistics pp189ndash326 Springer Berlin Germany 2004
[90] Y Zhao and D Zeng ldquoRecent development on statistical meth-ods for personalized medicine discoveryrdquo Frontiers of Medicinein China vol 7 no 1 pp 102ndash110 2013
prognostic biomarker In fact prognostic gene expression sig-natures are fairly common in breast cancer [12 14]The readermay note that Oncotype DX was treated as a single biomakerand referred to as a gene expression based predictive classifier[3]
Statistically predictive associations are identified usingmodels with an interaction between a candidate biomarkerand targeted therapy [15] whereas prognostic biomarkersare identified as significant main effects [16] Thus analy-sis strategies for identifying prognostic markers are oftenunsuitable for personalized medicine [17 18] In fact thediscovery of predictive biomarkers requires specific statisticaltechniques for data-analysis that optimize the combina-tion of competing therapies with candidate genetic featuresand characteristics of the patient and disease Recentlymany statistical approaches have been developed provid-ing researchers with new tools for identifying potentialbiomarkers However the usefulness of these recent advanceshas not been fully recognized by the oncology communityand the scope of their applications has not been sum-marized
In this paper we provide an overview of statistical meth-ods for establishing optimal treatment rules for personalizedmedicine and discuss specific examples in various medicalcontexts with oncology as an emphasis We also point thereader to statistical software when available The variousapproaches enable investigators to ascertain the extent towhich one should expect a new untreated patient to respondto each candidate therapy and thereby select the treatmentthat maximizes the expected therapeutic response for thespecific patient [3 19] Section 2 discusses the limitationsof conventional approaches based on post hoc stratifiedanalysis Section 3 offers an overview of the process for thedevelopment of personalized regimes Section 4 discussesthe selection of an appropriate statistical method for differ-ent types of clinical outcomes and data sources Section 5presents technical details for deriving optimal treatmentselection rules In Section 6 we discuss approaches forevaluating model performance and assessing the extent towhich treatment selection using the derived optimal rule islikely to benefit future patients
2 Limitations of Subgroup Analysis
Cancer is an inherently heterogeneous disease Yet oftenefforts to personalize therapy rely on the application ofanalysis strategies that neglect to account for the extent of het-erogeneity intrinsic to the patient and disease and thereforeare too reductive for personalizing treatment inmany areas ofoncology [20ndash23] Subgroup analysis is often used to evaluatetreatment effects among stratified subsets of patients definedby one or a few baseline characteristics [23ndash26] For exampleThatcher et al [21] conducted a series of preplanned subgroupanalyses for refractory advanced non-small-cell lung cancerpatients treated with Gefitinib plus best supportive careagainst placebo Heterogeneous treatment effects were foundin subgroups defined by smoking status that is significantprolonged survival was observed for nonsmokers while notreatment benefit was found for smokers
Though very useful when well planned and properlyconducted the reliance on subgroup analysis for developingpersonalized treatment has been criticized [24 25] Obvi-ously a subgroup defined by a few factors is inadequatefor characterizing individualized treatment regimes thatdepends onmultivariate synthesisMoreover post hoc imple-mentation of multiple subgroup analyses considers a set ofstatistical inferences simultaneously (multiple testing) anderrors such as incorrectly rejecting the null hypothesis arelikely to occur The extent to which the resulting inferenceinflates the risk of a false positive finding can be dramatic[23] Take for example a recent study that concluded thatchemotherapy followed by tamoxifen promises substantialclinical benefit for postmenopausal women with ER negativelymph node-negative breast cancer [27] through post hocapplication subgroup analysis Subsequent studies failed toreproduce this result concluding instead that the regimersquosclinical effects were largely independent of ER status [28] butmay depend on other factors including age
3 Personalized Medicine froma Statistical Perspective
From a statistical perspective personalized medicine is aprocess involving six fundamental steps provided in Figure 1[20 29 30] Intrinsic to any statistical inference initially onemust select an appropriate method of inference based on theavailable source of training data and clinical endpoints (egsteps (1) and (2)) Step (3) is the fundamental component ofpersonalized treatment selection deriving the individualizedtreatment rule (ITR) for the chosen method of inferenceAn ITR is a decision rule that identifies the optimal treat-ment given patientdisease characteristics [31 32] Section5 is dedicated to the topic of establishing ITRs for variousstatistical models and types of clinical endpoints that arecommonly used to evaluate treatment effectiveness in onco-logy
Individualized treatment rules are functions of modelparameters (usually treatment contrasts reflecting differencesin treatment effects) which must be estimated from theassumed statistical model and training data Statistical esti-mation takes place in step 4 The topic is quite generaland it thus is not covered in detail owning to the fact thatother authors have provided several effective expositionson model building strategies in this context [29 33] Afterestimating the optimal treatment rule in step (4) the resultingestimated ITRrsquos performance and reliabilitymust be evaluatedbefore the model can be used to guide treatment selection[34] The manner in which one assesses the performance ofthe derived ITR depends on the appropriate clinical utility(ie increased response rate or prolonged survival dura-tion) Evaluation of model goodness-of-fit and appropriatesummary statistics that use the available information tomeasure the extent to which future patients would benefitfrom application of the ITR is conducted in step (5) andwill be discussed in Section 6 The ITR is applied to guidetreatment selection for a future patient based on hisherbaseline clinical and genetic characteristics as the finalstep
BioMed Research International 3
(1) Acquire the training data
∙ Randomized data from a comparative trial∙ Observational cohorts
(2) Choose a method of inference based on clinicalendpoints and data dimension
∙ Binary∙ Continuous∙ Survival
∙ Low dimensional∙ High dimensional
(3) Identify the individualized treatment rule
∙ Derive the treatment contrasts∙ Select a clinically relevant decision threshold
(4) Fit the model to training data
∙ Select the important covariates∙ Estimate the model parameters
(5) Evaluate performance
∙ Assess model goodness-of-fit and prediction∙ Measure the extent of clinical benefit
(6) Apply the treatment rule to future patients
∙ Acquire prognostic and predictive covariates∙ Select the optimal treatment
Figure 1 The process for using statistical inference to establish personalized treatment rules
4 Selecting an AppropriateMethod of Inference
The quality of a treatment rule depends on the aptness ofthe study design used to acquire the training data clinicalrelevance of the primary endpoints statistical analysis plansfor model selection and inference and quality of the dataRandomized clinical trials (RCT) remain the gold standardstudy design for treatment comparison since randomizationmitigates bias arising from treatment selection Methodsfor deriving ITRs using data from RCTs are described inSection 51 Data from well conducted observational studiesprovide useful sources of information as well given thatthe available covariates can be used to account for potentialsources of confounding due to selection bias Predominatelymethods based on propensity scores are used to adjust forconfounding [35 36] Approaches for establishing ITRs usingobservational studies are discussed in Section 52
The predominate statistical challenge pertaining to theidentification of predictive biomarkers is the high-dimen-sional nature of molecular derived candidate features Clas-sical regression models cannot be directly applied since thenumber of covariates for example genes is much larger thannumber of samples Many approaches have been proposedto analyze high-dimensional data for prognostic biomarkersSection 53 discusses several that can be applied to detectpredictive biomarkers under proper modification
In oncology several endpoints are used to compare clin-ical effectiveness However the primary therapeutic goal isto extend survivorship or delay recurrenceprogressionThustime-to-event endpoints are often considered to be the mostrepresentative of clinical effectiveness [37] The approachesaforementioned were developed for ordinal or continuousoutcomes and were thus not directly applicable for survivalanalysis Methods for establishing ITRs from time-to-eventendpoints often use Cox regression or accelerated failure timemodels [38 39] The later approach is particularly appealing
in this context since the clinical benefits of prolonged survivaltime can be easily obtained [40 41] In Section 54 we willdiscuss both models
The performance of ITRs for personalized medicineis highly dependent upon the extent to which the modelassumptions are satisfied andor the posited model is cor-rectly specified Specifically performances may suffer frommisspecification of main effects andor interactions randomerror distribution violation of linear assumptions sensitivityto outliers and other potential sources of inadequacy [42]Some advanced methodologies have been developed to over-come these issues [43] including semiparametric approachesthat circumvent prespecification of the functional form ofthe relationship between biomarker and expected clinicalresponse [32 40] In addition optimal treatment rules canbe defined without regression models using classificationapproaches where patients are assigned to the treatment thatprovides the highest expected clinical benefit Appropriateclass labels can be defined by the estimated treatment differ-ence (eg gt0 versus le0) thereby enabling the use of machinelearning and data mining techniques [42 44 45] These willbe discussed in Section 55
5 Methods for Identifying IndividualizedTreatment Rules
This section provides details of analytical approaches that areappropriate identifying ITRs using a clinical data sourceThevery nature of treatment benefit is determined by the clinicalendpoint While extending overall survival is the ultimatetherapeutic goal often the extent of reduction in tumorsize as assessed by RECIST criteria (httpwwwrecistcom)is used as a categorical surrogate for long-term responseAlternatively oncology trials often compare the extent towhich the treatment delays locoregional recurrence or dis-ease progression Therefore time-to-event and binary (as inabsencepresence of partial or complete response) are the
4 BioMed Research International
most commonly used endpoints in oncologic drug develop-ment [37 46]
Let 119884 denote the observed outcome such as survivalduration or response to the treatment and let 119860 isin 0 1
denote the treatment assignment with 0 indicating standardtreatment and 1 for a new therapy Denote the collection ofobservable data for a previously treated patient by (119884 119860X)where X = 119883
1 1198832 119883
119901 represents a vector of values for
the 119901 biomarkers under study Quantitatively the optimalITRderives from the following equation relating the observedresponse to the potential outcome attained under the alterna-tive treatment
119884 = 119860119884(1)+ (1 minus 119860)119884
(0) (1)
where119884(1) and119884(0) denote the potential outcomes that wouldbe observed if the subject had been assigned to the newtherapy or the standard treatment respectively [32 43] Let119864(119884 | 119860X) = 120583(119860X) denote the expected value of 119884 given119860 and X The optimal treatment rule follows as
where 119868(sdot) is the indicator function For instance if119868120583(1 age gt 50) minus 120583(0 age gt 50) gt 0 = 1 then the optimalrule would assign patients who are older than 50 to the newtreatment However 119864(119884 | 119860X) is actually a function ofparameters 120583(119860X120573) denoted by 120573 The model needs to beldquofittedrdquo to the training data to obtain estimates of 120573 which wedenote by Hence for a patient with observed biomarkersX = x the estimated optimal treatment rule is
119892opt(X = x )
= 119868 120583 (119860 = 1X = x ) minus 120583 (119860 = 0X = x ) (3)
The above equation pertains to steps (3) and (4) in Figure 1that is the parameter estimates from a fitted model are usedto construct the personalized treatment rule The remainderof this section instructs the readers how to identify ITRs forthe various data types
We classify the statistical methods presented in thissection into five categories methods based on multivariateand generalized linear regression for analysis of data acquiredfrom RCT (Section 51) and observational studies (Section52) methods based on penalized regression techniques forhigh-dimensional data (Section 53) methods for survivaldata (Section 54) and advanced methods based on robustestimation and machine learning techniques (Section 55)
51 Multiple Regression for Randomized Clinical Trial DataClassical generalized linear models (GLM) can be usedto develop ITRs in the presence of training data derivedfrom randomized clinical study The regression frameworkassumes that the outcome 119884 is a linear function of prognosticcovariates 119883
1 putative predictive biomarkers 119883
2 the treat-
ment indicator 119860 and treatment-by-predictive interaction1198601198832
Let Δ(X) = 119864(119884 | 119860 = 1119883) minus 119864(119884 | 119860 = 0119883) = 120583(119860 = 1
119883) minus 120583(119860 = 0119883) denote the treatment contrast The optimaltreatment rule assigns a patient to the new treatment ifΔ(X) gt 0 For binary endpoints the logistic regressionmodelfor 120583(119860X) = 119875(119884 = 1 | 119860X) is defined such that
The treatment contrast Δ(X) can be calculated using 119864(119884 |
119860 = 119886119883) = 119875(119884 = 1 | 119860 = 119886X) = 119890120596(119860X)(1 + 119890120596(119860X))for 119886 = 0 1 respectively Similarly an optimal ITR assigns apatient to the new treatment if Δ(X) gt 0 This optimal treat-ment rule can be alternatively defined as 119892opt(X) = 119868(120573
3+
12057341198832) gt 0 without the need to calculate the treatment
contrast Δ(X) [43 45]Often one might want to impose a clinically meaningful
minimal threshold Δ(X) gt 120575 on the magnitude of treatmentbenefit before assigning patients to a novel therapy [45 47]For example it may be desirable to require at least a 01increase in response rate before assigning a therapy for whichthe long-term safety profile has yet to be established The useof a threshold value can be applied to all methods Withoutloss of generality we assume 120575 = 0 unless otherwise specifiedIn addition the reader should note that the approaches forconstructing an ITR described above can be easily applied tolinear regression models for continuous outcomes
This strategy was used to develop an ITR for treatmentof depression [19] using data collected from a RCT of 154patients In this case the continuous outcome was basedon posttreatment scores from the Hamilton Rating Scale forDepression The authors constructed a personalized advant-age index using the estimated treatment contrasts Δ(X)derived from five predictive biomarkers A clinically signif-icant threshold was selected 120575 = 3 based on the NationalInstitute for Health and Care Excellence criterion Theauthors identified that 60 of patients in the sample wouldobtain a clinicallymeaningful advantage if their therapy deci-sion followed the proposed treatment rule The approachesdiscussed in this section can be easily implemented withstandard statistical software such as the 119877 (httpwwwr-projectorg) using the functions lm and glm [48]
52 Methods for Observational Data Randomization attenu-ates bias arising from treatment selection thereby providingthe highest quality data for comparing competing interven-tions However due to ethical or financial constraints RCTsare often infeasible thereby necessitating an observationalstudy Treatment selection is often based on a patientrsquosprognosis In the absence of randomization the study designfails to ensure that patients on competing arms exhibit similarclinical and prognostic characteristics thereby inducing bias
However in the event that the available covariates capturethe sources of bias a well conducted observational studycan also provide useful information for constructing ITRsFor example the two-gene ratio index (HOXB13IL17BR)
BioMed Research International 5
was first discovered as an independent prognostic biomarkerfor ER+ node-negative patients using retrospective datafrom 60 patients [49] These findings were confirmed onan independent data set comprising 852 tumors which wasacquired from a tumor bank at the Breast Center of BaylorCollege of Medicine [50] Interestingly the two-gene ratioindex (HOXB13IL17BR) was reported to predict the benefitof treatment with letrozole in one recent independent study[51]
Methods based on propensity scores are commonly usedto attenuate selection bias [35] In essence these approachesuse the available covariates to attempt to diminish the effectsof imbalances among variables that are not of interest fortreatment comparison Moreover they have been shownto be robust in the presence of multiple confounders andrare events [52] Generally after adjusting for bias usingpropensity scores the same principles for deriving ITRs fromRCTs may be applied to the observational cohort
The propensity score characterizes the probability ofassigning a given treatment 119860 from the available covariatesX [35] Using our notation the propensity score is 120587(X 120585) =119875(119860 = 1 | X 120585) which can be modeled using logisticregression
where 119901 is the number of independent variables used toconstruct the propensity score and 120585
119895represents the 119895th
regression coefficient which characterizes the 119895th covariatersquospartial effect After fitting the data to obtain estimates forthe regression coefficients the estimated probability ofreceiving new treatment can be obtained for each patient(Xi) = 120587(Xi ) by inverting the logit function The eventthat asymp 0 implies that the measured independent variablesare reasonably ldquobalancedrdquo between treatment cohorts Inpractice one often includes as many baseline covariates intothe propensity score model as permitted by the sample size
Methods that use propensity scores can be categorizedinto four categories matching stratification adjusting andinverse probability weighted estimation [36 53] Matchingand stratification aim to mimic RCTs by defining a newdataset using propensity scores such that outcomes aredirectly comparable between treatment cohorts [53] Thesetwo approaches are well suited for conventional subgroupanalysis but their application to personalized medicine hasbeen limited Regression adjustment or simply adjusting canbe used to reduce bias due to residual differences in observedbaseline covariates between treatment groups This methodincorporates the propensity scores as an independent variablein a regression model and therefore can be used in con-junction with all regression-based methods [36] Methodsinvolving inverse probability weighted estimators will bediscussed in Section 551 [43]
Of course propensity scores methods may only attenuatethe effects of the important confounding variables that havebeen acquired by the study design Casual inference in gen-eral is not robust to the presence of unmeasured confounders
that influenced treatment assignment [35 54 55] For thedevelopment of ITRs predictive and important prognosticcovariates can be incorporated in the regression model forthe clinical outcome119884 alongwith the propensity scores whileother covariatesmay be utilized only in themodel for estimat-ing the propensity scores Hence propensity score methodsmay offer the researcher a useful tool for controlling forpotential confounding due to selection bias andmaintaining amanageable number of prognostic and predictive covariates
53 Methods for High-Dimensional Biomarkers The meth-ods presented in the previous sections are appropriate foridentifying an ITR using a small set of biomarkers (low-dimensional)However recent advances inmolecular biologyin oncology have enabled researchers to acquire vast amountsof genetic and genomic characteristics on individual patientsOften the number of acquired genomic covariates will exceedthe sample size Proper analysis of these high-dimensionaldata sources poses many analytical challenges Several meth-ods have been proposed specifically for analysis of high-dimensional covariates [56] although the majority of thesemethods are well suited only for the analysis of prognosticbiomarkers In what follows we introduce variable selectionmethods that were developed to detect predictive biomarkersfrom high-dimensional sources as well as describing how toconstruct optimal ITRs from the final set of biomarkers
An appropriate regressionmodel can be defined generallyas 119864(119884 | 119860X) = ℎ
0(X) + 119860(X120573) where ℎ
0(X) is an unspe-
cified baseline mean function 120573 = (1205730 1205731 120573
119902)119879 is a
column vector of regression coefficients and X = (1X) thedesign matrix Subscript 119902 denotes the total number of bio-markers which may be larger than the sample size 119899 AnITR derives from evaluating the interactions in 119860(X120573) notthe baseline effect of the high-dimensional covariates ℎ
0(X)
[32] Technically function 119860(X120573) = 119860(1205730+ 12057311198831+ 12057321198832+
sdot sdot sdot + 120573119902119883119902) cannot be uniquely estimated using traditional
maximum likelihood-based methods when 119902 gt 119899 [57]Yet practically many of the available biomarkers may notinfluence the optimal ITR [31] Thus the process for identifyITRs from a high-dimensional source requires that we firstidentify a sparse subset of predictive biomarkers that can beutilized for constructing the ITR
Parameters for the specifiedmodel can be estimated usingthe following loss function
119871119899120601(120573 120574) =
1
119899
119899
sum119894=1
[119884119894minus 120601 (X
119894 120574) minus X120573 119860
119894minus 120587 (X
119894)]2
(7)
where 120601(X 120574) represents any arbitrary function character-izing the ldquobaselinerdquo relationship between X and Y (eg anintercept or an additive model) Here we let 120587(X
119894) = 119875(119860
119894=
1 | X119894) denote either a propensity score (for observational
data) or a randomization probability (eg 05 given 1 1randomization) for RCT data If 120587(X) is known estimationusing this model yields unbiased estimates (asymptoticallyconsistent) of the interaction effects 120573 even if the main effectsare not correctly specified providing a robustness [32]
6 BioMed Research International
Penalized estimation provides the subset of relevantpredictivemarkers that are extracted from the nonzero coeffi-cients of the corresponding treatment-biomarker interactionterms of
where 120582119899is a tuning parameter which is often selected via
cross validation and 119869 is a shrinkage penalty Different choicesof 119869 lead to different types of estimators For example thelasso penalized regression corresponds to 119869 = 1 [58] andthe adaptive lasso to 119869 = 120596
119895= 1|120573init119895| where 120573init119895 is an
initial estimate of 120573119895[59] With little modification (8) can
be solved using the LARS algorithm implemented with the119877 package of 119897119886119903119904 [32 60 61] As we have shown before atreatment rule can be defined from the parameter estimatesas 119868120573
formmay have zero estimates for some coefficients (eg 1205732=
1205735= sdot sdot sdot = 120573
119902= 0) hence an ITR can be equivalently con-
structed from the final estimated nonzero coefficients and thecorresponding covariates
Alternative penalized regression approaches includeSCAD [62] and elastic-net [63] All penalized approachesproduce sparse solutions (ie identifying a small subset ofpredictive biomarkers) however the adaptive lasso is lesseffective when 119901 gt 119899 Methods that produce nonsparsemodels such as ridge regression [57] are less preferable sinceITRs based on many biomarkers are often unstable and lessuseful in practice [31] Several packages in 119877 offer imple-mentation of penalized regression such as 119901119886119903119888119900119903 for ridgelasso and adaptive lasso and 119899119888V119903119890119892 for SCAD [64 65]
Lu et al [32] used a penalized regression approach toanalyze data from the AIDS Clinical Trials Group Protocol175 (ACTG175) [66] In this protocol 2000 patients wereequally randomized to one of four treatments zidovudine(ZDV) monotherapy ZDV + didanosine (ddI) ZDV +zalcitabine and ddI monotherapy CD4 count at 15ndash25 weekspostbaselinewas the primary outcome and 12 baseline covari-ates were included in the analysis The resulting treatmentrule favored the combined regimes over ZDV monotherapyMoreover the treatment rule determined that ZDV + ddIshould be preferred to ddI when 119868(7159 + 107 times ageminus 018 timesCD40 minus 3357 times homo) = 1 where CD40 represents baselineCD4 counts and homo represents homosexual activity Basedon this treatment rule 878 patients would have benefitedfrom treatment with ZDV + ddI
54 Survival Analysis Heretofore we have discussed meth-ods for continuous or binary outcomes yet often investigatorswant to discern the extent to which a therapeutic interventionmay alter the amount of time required before an event occursThis type of statistical inference is referred to broadly assurvival analysis One challenge for survival analysis is thatthe outcomes may be only partially observable at the time ofanalysis due to censoring or incomplete follow-up Survivalanalysis has been widely applied in cancer studies often inassociation studies aimed to identify prognostic biomarkers
[56 67] Here we discuss twowidely usedmodels for derivingITRs using time-to-event data namely Cox regression andaccelerated failure time models
The Cox regression model follows as
120582 (119905 | X 119860) = 1205820(119905) exp 120573
11198831+ 12057321198832+ 119860 (120573
4+ 12057351198832)
(9)
where 119905 is the survival time 1205820(119905) is an arbitrary baseline
hazard function and 1198831 1198832represent prognostic and pre-
dictive biomarkers respectively Each 120573 characterizes themultiplicative effect on the hazard associated with a unitincrease in the corresponding covariate Therefore Coxmodels are referred to as proportional hazards (PH) models
Several authors have provided model building strategies[29] and approaches for treatment selection [20 30 68] Fol-lowing the previously outlined strategy a naive approach forderiving an ITR uses the hazard ratio (new treatment versusthe standard) as the treatment contrast which can be calcu-lated as Δ(X) = exp(120573
4+ 12057351198832) The ITR therefore is 119868(120573
4
+12057351198832) lt 0 There are obvious limitations to this approach
First violations of the PH assumption yield substantiallymis-leading results [69]Moreover evenwhen the PH assumptionis satisfied because the Cox model does not postulate adirect relationship between the covariate (treatment) and thesurvival time the hazard ratio fails to measure the extent towhich the treatment is clinically valuable [38 70]
Accelerated failure time (AFT) models provide an alter-native semiparametric model Here we introduce its appli-cation for high-dimensional data Let 119879 and 119862 denote thesurvival and censoring times and denote the observed databy ( 120575 119860X) where = min(119879 119862) and 120575 = 119868(119879 lt 119862)Define the log survival time as 119884 = log(119879) a semiparametricregression model is given as 119864(119884 | 119860X) = ℎ
0(X) + 119860(X120573)
where ℎ0(X) is the unspecified baseline mean function
Similar to the previous section the treatment rule is 119868(1205730+
independent censoring the AFT model parameters can beestimated by minimizing the following loss function
119871119899120601(120573) =
1
119899
119899
sum119894=1
120575119894
119866(119894)[119894minus 120601 (X
119894 120574) minus X120573 119860
119894minus 120587 (X
119894)]2
(10)
where 119894= log(
119894) 120587(X
119894) = 119875(119860
119894= 1 | X
119894) is the propensity
score or randomization probability 119866(sdot) is the Kaplan-Meierestimator of the survival function of the censoring time and120601(X 120574) characterizes any arbitrary function
Thismethod can be extended to accommodatemore thantwo treatments simultaneously by specifying appropriatetreatment indicators For instance the mean function can bemodeled as 119864(119884 | 119860X) = ℎ
0(X) + 119860
(1)X120573(1) + 119860(2)X120573(2)
for two treatment drugs versus the standard care The ITRassigns the winning drug Note this work was proposed by[40] and is an extension of [32] to the survival setting Henceit shares the robustness property and can be applied to obser-vational data For implementation the sameprocedure can befollowed to obtain estimates with one addition step of calcu-lating 119866(
119894) There are several 119877 packages for Kaplan-Meier
BioMed Research International 7
estimates and Cox regression models These sources can befound at httpcranr-projectorgwebviewsSurvivalhtmlMore details pertaining to statistical methods for survivalanalysis can be found here [71] To compare treatmentrules constructed from Cox and AFT models for examplemethods for measuring the extent of clinical effectiveness foran ITR will be discussed in Section 6
We here present an example when an AFT model wasused to construct an ITR for treatment of HIV [40] Theexample derives from the AIDS Clinical Trials Group Pro-tocol 175 that was discussed in Section 53 [32 66] In thiscase the primary outcome variable was time (in days) tofirst ge50 decline in CD4 count or an AIDS-defining eventor death A total of 12 covariates and four treatments (ZDVZDV + ddI ZDV + zalcitabine and ddI) were includedThe four treatments were evaluated simultaneously Patientsreceiving the standard care of ZDV monotherapy wereconsidered as the reference group Hence three treatmentcontrasts (119868ZDV+ddI 119868ZDV+zalcitabine and 119868ddI) were combinedwith various putative predictive covariates and comparedwith ZDV monotherapy For example gender was detectedas the predictive covariate only for ddI monotherapy Theinvestigators assumed 120601(X 120574) = 120574
0 The treatment rule
recommended 1 patient for ZDV monotherapy while 7291216 and 193 patients were recommended for ZDV + ddIZDV + zalcitabine and ddI respectively
55 Advanced Methods
551 Robust Inference The performances of ITRs heretoforepresented depend heavily on whether the statistical modelswere correctly specified Recently there has been much atten-tion focused on the development of more advanced methodsand modeling strategies that are robust to various aspectsof potential misspecification We have already presented afew robust models that avoid specification of functionalparametric relationships for main effects [32 40] Here weintroduce two more advanced methods widely utilized forITRs that are robust to the type of misspecification issuescommonly encountered in practice [42 43]
Recall that the ITR for a linear model 119864(119884 | 119860 =
119886X) = 120583(119860 = 119886X120573) with two predictive markers followsas 119892(X120573) = 119868(120573
4+ 12057351198832+ 12057361198833) gt 0 where 119886 = 0 1
The treatment rule of 119892(X120573) may use only a subset of thehigh-dimensional covariates (eg 119883
2 1198833) but it always
depends on the correct specification of 119864(119884 | 119860 = 119886X)Defining a scaled version of 120573 as 120578(120573) the correspondingITR is 119892(120578X) = 119892(X120573) = 119868(119883
3gt 1205780+ 12057811198832) where
1205780= minus12057341205736and 120578
1= 12057351205736 If the model for 120583(119860X120573) is
indeed correctly specified the treatment rules of 119892(X120573) and119892(120578X) lead to the same optimal ITR Hence the treatmentrule parameterized by 120578 can be derived from a regressionmodel or may be based on some key clinical considerationswhich enable evaluation of 119892(120578X) directly without referenceto the regression model for 120583(119860X120573)
Let 119862120578
= 119860119892(120578X) + (1 minus 119860)1 minus 119892(120578X) where119862120578= 1 indicates random assignment to an intervention
that is recommended by the personalized treatment rule 119892(120578X) Let 120587(X ) denote the randomization ratio or the
estimated propensity score (as in previous section) and119898(X 120578 ) denote the potential outcome under the treatmentrule estimated from the following model 119864(119884 | 119860 =
119886X) = 120583(119860X120573) For example if the treatment rule 119892(120578X)= 1 then 119898(X 120578 ) = 119892(120578X)120583(119860 = 1X ) + 1 minus 119892(120578
X)120583(119860 = 0X ) = 120583(119860 = 1X ) Two estimators ofthe expected response to treatment the inverse probabilityweighted estimator (IPWE) and doubly robust AIPWE aregiven as follows
IPWE (120578) = 1
119899
119899
sum119894=1
119862120578sdot119894119884119894
120587119888(X119894 120578 )
=1
119899
119899
sum119894=1
119862120578sdot119894119884119894
120587 (X119894 )119860119894 1 minus 120587 (X
119894 )1minus119860119894
AIPWE (120578) = 1
119899
119899
sum119894=1
119862120578sdot119894119884119894
120587119888(X119894 120578 )
minus119862120578sdot119894119884119894 minus 120587119888 (X119894 120578 )
120587119888(X119894 120578 )
119898 (X119894 120578 )
(11)
where 120587119888(X119894 120578 ) = 120587(X )119892(120578X) + 1 minus 120587(X )1 minus
119892(120578X) The optimal treatment rule follows as 119892(X = x)where is estimated from the above models a constraintsuch as 120578 = 1 is imposed to obtain a unique solution [43] If the propensity score is correctly specified theIPWE estimator yields robust (consistent) estimates AIPWEis considered a doubly robust estimator since it produces con-sistent estimates when either propensity score or the model119864(119884 | 119860 = 119886X) is misspecified but not both [42 43] Thecompanion119877 code is publicly available at httponlinelibrarywileycomdoi101111biom12191suppinfo
552 Data Mining and Machine Learning The methodspresented in Section 551 are robust against misspecificationof regression models Yet they often require prespecificationof the parametric form for the treatment rule (eg 119868(119883
3gt
1205780+ 12057811198832)) which can be practically challenging [44]
Well established classification methods and other popularmachine learning techniques can alternatively be customizedto define treatment selection rules [44 72 73] these methodsavoid prespecification of the parametric form of the ITR AnITR can be defined following a two-step approach in thefirst step treatment contrasts are estimated from a positedmodel and in the second step classification techniques areapplied to determine the personalized treatment rules Forexample when only two treatments are considered a newvariable 119885 can be defined based on the treatment contrastthat is 119885 = 1 if Δ(X) = 120583(119860 = 1X) minus 120583(119860 = 0X) gt 0 and119885 = 0 otherwiseThe absolute value of the treatment contrast119882119894= |Δ(X)| can be used in conjunction with a classification
technique to define an appropriate ITR [44]Unlike classification problems wherein the class labels
are observed for the training data the binary ldquoresponserdquovariable 119885 which serves as the class label is not availablein practice Specifically patients who are in the class 119885 = 1
8 BioMed Research International
have 120583(119860 = 1X) gt 120583(119860 = 0X) and should thereforebe treated with the new therapy however these quantitiesneed to be estimated since patients are typically assigned toonly one of the available treatments This imparts flexibilityfor estimation of the optimal treatment regimes since anyof the previously discussed regression models and even someensemble prediction methods such as random forest [74] canbe used to construct the class labels 119885
119894and weights
119894[44]
An ITR can be estimated from the dataset 119885119894X119894 119894 using
any classification approach where 119894are subject specific
misclassification weights [44 45] This includes popularclassificationmethods such as adaptive boosting [75] supportvector machines [76] and classification and regression trees(CART) [77] At least one study has suggested that SVMoutperforms other classification methods in this contextwhereas random forest and boosting perform comparativelybetter than CART [78] However the performances ofthese classification algorithms are data dependent Definitiveconclusion pertaining to their comparative effectiveness ingeneral has yet to be determined [78] It shall be also notedthat these classification methods can be also applied to high-dimensional data [45 72]
One special case of this framework is the ldquovirtue twinsrdquoapproach [45] Specifically in the first step a random forestapproach [74] is used to obtain the treatment contrasts Thenin the second step CART is used to classify subjects tothe optimal treatment regime The approach can be easilyimplemented in 119877 using packages of randomForest [79] andrpart [80] Very recently Kang et al [42] proposed amodifiedversion of the adaptive boosting technique of Friedman et al[75] The algorithm iteratively fits a simple logistic regressionmodel (ldquoworking modelrdquo) to estimate 119875(119884 = 1 | 119860X) and ateach stage assigns higher weights to subjects whose treatmentcontrast is near zero After a prespecified stopping criterionis met an average of the treatment contrasts Δ(X) is calcu-lated for each patient using all models fitted at each iterationA subject is assigned to the new therapy if Δ(X) gt 0 The 119877code for the aforementioned boosting methods is publiclyavailable at httponlinelibrarywileycomdoi101111biom12191suppinfo
Lastly we present a breast cancer example where severalbiomarkers were combined to construct an optimal ITRThe data was collected in the Southwest Oncology Group(SOWG)-SS8814 trial [13] and analyzed with the machinelearning approach of Kang et al [42] Three hundred andsixty-seven node-positive ER-positive breast cancer patientswere selected from the randomized trial of SOWG A total of219 received tamoxifen plus adjuvant chemotherapy and 148was given tamoxifen aloneThe outcome variable was definedas breast cancer recurrence at 5 years The authors selectedthree genes which had presented treatment-biomarker inter-actions in amultivariate linear logistic regressionmodel [42]Data were analyzed with logistic models IPWE AIPWElogistic boosting a single classification tree with treatment-biomarker interactions and the proposed boosting approachwith a classification tree as the working model Each methodidentified different patient cohorts that could benefit fromtamoxifen alone these cohorts consisted of 184 183 128 86
263 and 217 patients respectively (see Table 5 in [42]) In thisanalysis the clinical benefits provided by these 6 treatmentrules were not statistically different Hence investigatorsneed to evaluate and compare ITRs in terms of the extentof expected clinical impact This is considered in the nextsection
Heretofore we have discussed various methodologies for theconstruction of ITR while their performances need to beassessed before these rules can be implemented in clinicalpractice Several aspects pertaining to the performance of aconstructed ITR need to be considered The first one is howwell the ITR fits the data and the second is how well theITR performs compared with existing treatment allocationrules The former is related to the concept of goodness-of-fitor predictive performance [34] As the true optimal treatmentgroups are hidden model fits may be evaluated by measuringthe congruity between observed treatment contrasts andpredicted ones [34 47] More details can be found in a recentpaper by Janes et al [47] Performances of ITRs can becompared via assessment of a global summary measure forexample prolonged survival time or reduced disease rate [4042] Summarymeasures are also very useful for evaluating theextent to which an ITR may benefit patients when applied inpractice Moreover it is essential that performance of an ITRis considered in comparison to business-as-usual proceduressuch as a naive rule that randomly allocates patients totreatment [81] Summary measures will be discussed inSection 61 The effectiveness of an ITR should go beyond thetraining data set used to construct a treatment rule cross-validation and bootstrapping techniques are often employedto assess the impact of ITRs on future patients [81] and willbe discussed in Section 62
61 Summary Measures ITRs may be derived from differ-ent methodologies and comparisons should be conductedwith respect to the appropriate clinically summaries A fewsummary measures for different types of outcomes havebeen proposed [19 40 42] these measures quantify thedirect clinical improvements obtained by applying an ITR incomparison with default methods for treatment allocation
Binary Outcomes Clinical effectiveness for binary clinicalresponse is represented by the difference in disease rates (ortreatment failure) induced by ITR versus a default strategythat allocates all patients to a standard treatment [42 47 82]Let 119892opt(X) = 119868120583(119860 = 1X)minus120583(119860 = 0X) lt 0 be an optimalITR This difference is formally defined as
Note 120583(119860X) needs to be estimated to construct the ITRyet parameters 120573 are omitted for simplicity Larger valuesof Θ119861119892opt(X) indicate increased clinical value for the
biomarker driven ITR A subset of patients that are recom-mended for new treatment (119860 = 1) under an ITR may havebeen randomly selected to receive it while the remainingsubset of ldquounluckyrdquo patientswould have received the standardtreatment [19] The summary measure of Θ
119861119892opt(X) char-
acterizes a weighted difference in the disease rates betweenthe standard and the new treatments in a population whereinthe constructed optimal ITR would recommend the newtreatment 119892opt(X = 1) The weight is the proportion ofpatients identified by the optimal ITR for the new treatmentand can be empirically estimated using the correspondingcounts For example 119875119892opt(X) = 1 can be estimatedusing the number of patients recommended for the newtreatment divided by the total sample size A similar summarystatistic can be derived for an alternative strategy allocatingall patients to the new treatment The summary could beapplied to the aforementioned breast cancer example [42]for example with the aim of finding a subgroup of patientswho were likely to benefit from adjuvant chemotherapywhile those unlikely to benefit would be assigned tamoxifenalone to avoid the unnecessary toxicity and inconvenience ofchemotherapy
Continuous Variables Another strategy for continuous datacompares outcomes observed for ldquoluckyrdquo subjects those whoreceived the therapy that would have been recommendedby the ITR based [81] Further one business-as-usual drugallocation procedure is randomizing treatment and standardcare at the same probability of 05 A summary statisticis to measure the mean outcome under ITR compared tothat obtained under random assignment for instance themean decrease in Hamilton Rating Scale for Depression asdiscussed in Section 51 [19] Define the summary measureas Θ119862119892opt(X) = 120583119892opt(X)X minus 120583119892rand(X)X where
119892rand(X) represents the randomization allocation procedureThe quantity of 120583119892119900119901119905(X)X represents the mean outcomeunder the constructed IRT that can be empirically estimatedfrom the ldquoluckyrdquo subjects and 120583119892rand(X)X can be esti-mated empirically from the sample means
Alternatively an ITR may be compared to an ldquooptimalrdquodrug that has showed universal benefits (a better drug onaverage) in a controlled trial The clinical benefits of anldquooptimalrdquo drug can be defined as 120583119892best(X)X = max120583(119860 =
0X) 120583(119860 = 1X) 120583(119860 = 119886X) and can be empiricallyestimated from the sample means of the new and standardtreatments respectively Then the alternative summary mea-sure is defined as Θ
Survival Data For survival data a clinically relevant measureis mean overall (or progression free) survival time As
survival time is continuous in nature the identical strategyprovided above for continuous outcomes can be employedhere However because the mean survival time may notbe well estimated from the observed data due to a highpercentage of censored observations [40] an alternativemean restricted survival duration was proposed and definedas the population average event-free durations for a restrictedtime of 119905lowast [41 83]Often 119905lowast is chosen to cover the trialrsquos follow-up period Mathematically it can be calculated by integratingthe survival function of 119878(119905) over the domain of (0 119905lowast) that is120583119892opt(X)X 119905lowast = int119905
lowast
0119878(119905)119889119905 and often estimated by the area
under the Kaplan-Meier curve up to 119905lowast [84] Thus an ITRrsquospotential to prolong survival can be calculated asΘ
119878119892opt(X)
119905lowast = 120583119892opt(X)X 119905lowast minus 120583119892rand(X)X 119905lowast
62 AssessingModel Performance The summaries heretoforediscussed evaluate an optimal ITR for a given model andestimating procedure Because these quantities are estimatedconditionally given the observed covariates they neglectto quantify the extent of marginal uncertainty for futurepatients Hence an ITR needs to be internally validated ifexternal data is not available [34] Cross-validation (CV) andbootstrap resampling techniques are commonly used for thispurpose [19 42 45 81] and expositions on both approachesare well described elsewhere [33 85 86]
We here briefly introduce a process that was proposed byKapelner et al [81] in the setting of personalized medicineTenfold CV is commonly used in practice where the wholedata is randomly partitioned into 10 roughly equal-sizedexclusive subsamples All methods under consideration areapplied to 910 of the data excluding 110 as an independenttesting data set The process is repeated 10 times for eachsubsample Considering the assignments recommended bythe optimal ITRs the summary measures can be calculatedusing results from each testing fold [45]TheCVprocess givesthe estimated summary measures and its variation can beevaluated using bootstrap procedures Specifically one drawsa sample with replacement from the entire data and calculatesthe summary measure from 10-fold CV This process willbe repeated 119861 times where 119861 is chosen for resolution ofthe resulting confidence intervals [81] Using the summarymeasures as119861 new random samples the correspondingmeanand variances can be calculated empirically Note that thesummary measures compare two treatment rules one for theoptimal ITR and another naive rule (eg randomization)
The above procedure can be applied to all the meth-ods we have discussed so far The 119877 software package119879119903119890119886119905119898119890119899119905119878119890119897119890119888119905119894119900119899 (httplabsfhcrcorgjanesindexhtml)can be used to implement these methods for evaluatingand comparing biomarkers for binary outcomes [47] Veryrecently an inferential procedure was proposed for contin-uous outcomes that is implemented in the publicly available119877 package ldquoPersonalized Treatment Evaluatorrdquo [81 87] Bothmethods consider data from RCTs with two arms for com-parative treatmentsThesemethods are in general applicableto regression model based methods but are not suitable forapproaches based on classification techniques or penalizedregression
10 BioMed Research International
Next we present two examples Recall in Section 55 thatKang et al [42] reported the estimated clinical benefits ofan ITR for breast cancer when compared to the defaultstrategy of assigning all patients to adjuvant chemotherapyThe proposed approach (based on boosting and classificationtrees) achieved the highest value of the summary measureat 0081 with 95 confidence interval (CI) (0000 0159)[42] In the second example introduced in Section 51 [19]the authors calculated the mean score of the HamiltonRating Scale for Depression for two groups of subjectsgroups were defined by randomly assigning patients to theldquooptimalrdquo and ldquononoptimalrdquo therapy as defined by the ITRThe reported difference between the two groups was minus178with a 119875 value of 009 which fails to attain a clinicalsignificant difference of 3 [19] The same data was analyzedby Kapelner et al [81] Following the discussed procedurethe authors reported the estimated values (and 95 CI)of Θ119862119892opt(X) and Θ
119862alt119892opt(X) as minus0842(minus2657 minus0441)
and minus0765(minus2362 0134) respectively The results whichfail to achieve clinical significance were based on rigorousstatistical methods and thus can be considered reliableestimates of the ITRrsquos performance
7 Discussion
As our understanding tumor heterogeneity evolves person-alized medicine will become standard medical practice inoncology Therefore it is essential that the oncology com-munity uses appropriate analytical methods for identifyingand evaluating the performance of personalized treatmentrules This paper provided an exposition of the process forusing statistical inference to establish optimal individualizedtreatment rules using data acquired from clinical study Thequality of an ITR depends on the quality of the design used toacquire the dataMoreover an ITRmust be properly validatedbefore it is integrated into clinical practice Personalizedmedicine in some areas of oncologymay be limited by the factthat biomarkers arising from a small panel of genesmay neveradequately characterize the extent of tumor heterogeneityinherent to the disease Consequently the available statisticalmethodology needs to evolve in order to optimally exploitglobal gene signatures for personalized medicine
The bulk of our review focused on statistical approachesfor treatment selection at a single time point The readershould note that another important area of research considersoptimal dynamic treatment regimes (DTRs) [88 89] whereintreatment decisions are considered sequentially over thecourse ofmultiple periods of intervention using each patientrsquosprior treatment history Zhao and Zeng provide a summaryof recent developments in this area [90]
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
JunshengMa was fully funded by the University of Texas MDAnderson Cancer Center internal funds Brian P Hobbs and
Francesco C Stingo were partially supported by the CancerCenter Support Grant (CCSG) (P30 CA016672)
References
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[2] D Hanahan and R AWeinberg ldquoHallmarks of cancer the nextgenerationrdquo Cell vol 144 no 5 pp 646ndash674 2011
[3] R Simon ldquoClinical trial designs for evaluating the medicalutility of prognostic and predictive biomarkers in oncologyrdquoPersonalized Medicine vol 7 no 1 pp 33ndash47 2010
[4] P L Bedard A R Hansen M J Ratain and L L Siu ldquoTumourheterogeneity in the clinicrdquo Nature vol 501 no 7467 pp 355ndash364 2013
[5] M D Pegram G Pauletti and D J Slamon ldquoHer-2neu as apredictive marker of response to breast cancer therapyrdquo BreastCancer Research and Treatment vol 52 no 1ndash3 pp 65ndash77 1998
[6] G J Kelloff andC C Sigman ldquoCancer biomarkers selecting theright drug for the right patientrdquoNature Reviews Drug Discoveryvol 11 no 3 pp 201ndash214 2012
[7] J ADiMasi JM Reichert L Feldman andAMalins ldquoClinicalapproval success rates for investigational cancer drugsrdquoClinicalPharmacology andTherapeutics vol 94 no 3 pp 329ndash335 2013
[8] M Hay D W Thomas J L Craighead C Economides andJ Rosenthal ldquoClinical development success rates for investiga-tional drugsrdquo Nature Biotechnology vol 32 no 1 pp 40ndash512014
[9] S S Knox ldquoFrom lsquoomicsrsquo to complex disease a systems biologyapproach to gene-environment interactions in cancerrdquo CancerCell International vol 10 article 11 2010
[10] V Deschoolmeester M Baay P Specenier F Lardon and JB Vermorken ldquoA review of the most promising biomarkersin colorectal cancer one step closer to targeted therapyrdquo TheOncologist vol 15 no 7 pp 699ndash731 2010
[11] J A Sosman K B Kim L Schuchter et al ldquoSurvival in brafV600ndashmutant advanced melanoma treated with vemurafenibrdquoThe New England Journal of Medicine vol 366 no 8 pp 707ndash714 2012
[12] S Paik S Shak G Tang et al ldquoA multigene assay to predictrecurrence of tamoxifen-treated node-negative breast cancerrdquoThe New England Journal of Medicine vol 351 no 27 pp 2817ndash2826 2004
[13] K S Albain W E Barlow S Shak et al ldquoPrognostic and pre-dictive value of the 21-gene recurrence score assay in post-menopausal women with node-positive oestrogen-receptor-positive breast cancer on chemotherapy a retrospective analysisof a randomised trialrdquoTheLancet Oncology vol 11 no 1 pp 55ndash65 2010
[14] J E Lang J S Wecsler M F Press and D Tripathy ldquoMolecularmarkers for breast cancer diagnosis prognosis and targetedtherapyrdquo Journal of Surgical Oncology vol 111 no 1 pp 81ndash902015
[15] W Werft A Benner and A Kopp-Schneider ldquoOn the identi-fication of predictive biomarkers detecting treatment-by-geneinteraction in high-dimensional datardquo Computational Statisticsand Data Analysis vol 56 no 5 pp 1275ndash1286 2012
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[17] A J Vickers MW Kattan and D J Sargent ldquoMethod for eval-uating prediction models that apply the results of randomizedtrials to individual patientsrdquo Trials vol 8 no 1 article 14 2007
[18] H Janes M S Pepe P M Bossuyt andW E Barlow ldquoMeasur-ing the performance of markers for guiding treatment deci-sionsrdquo Annals of Internal Medicine vol 154 no 4 pp 253ndash2592011
[19] R J DeRubeis Z D Cohen N R Forand J C Fournier L AGelfand and L Lorenzo-Luaces ldquoThe personalized advantageindex translating research on prediction into individualizedtreatment recommendationsAdemonstrationrdquoPLoSONE vol9 no 1 Article ID e83875 2014
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[22] A J Vickers ldquoPrediction models in cancer carerdquo CA A CancerJournal for Clinicians vol 61 no 5 pp 315ndash326 2011
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[24] S J Pocock S E Assmann L E Enos and L E Kasten ldquoSub-group analysis covariate adjustment and baseline comparisonsin clinical trial reporting current practice and problemsrdquoStatistics in Medicine vol 21 no 19 pp 2917ndash2930 2002
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[26] R Wang S W Lagakos J H Ware D J Hunter and J MDrazen ldquoStatistics in medicinemdashreporting of subgroup ana-lyses in clinical trialsrdquoTheNewEngland Journal ofMedicine vol357 no 21 pp 2108ndash2194 2007
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[31] M Qian and S A Murphy ldquoPerformance guarantees forindividualized treatment rulesrdquoThe Annals of Statistics vol 39no 2 pp 1180ndash1210 2011
[32] W Lu H H Zhang and D Zeng ldquoVariable selection for opti-mal treatment decisionrdquo StatisticalMethods inMedical Researchvol 22 no 5 pp 493ndash504 2013
[33] R Kohavi ldquoA study of cross-validation and bootstrap for accu-racy estimation and model selectionrdquo in Proceedings of the 14thInternational Joint Conference on Artificial Intelligence (IJCAIrsquo95) vol 2 pp 1137ndash1145 1995
[34] E W Steyerberg A J Vickers N R Cook et al ldquoAssessing theperformance of prediction models a framework for traditionaland novel measuresrdquo Epidemiology vol 21 no 1 pp 128ndash1382010
[35] P R Rosenbaum and D B Rubin ldquoThe central role of the pro-pensity score in observational studies for causal effectsrdquo Bio-metrika vol 70 no 1 pp 41ndash55 1983
[36] R B drsquoAgostino Jr ldquoTutorial in biostatistics propensity scoremethods for bias reduction in the comparison of a treatment toa non-randomized control grouprdquo Statistics in Medicine vol 17no 19 pp 2265ndash2281 1998
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[38] S L Spruance J E Reid M Grace and M Samore ldquoHazardratio in clinical trialsrdquo Antimicrobial Agents and Chemotherapyvol 48 no 8 pp 2787ndash2792 2004
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[40] Y Geng Flexible Statistical Learning Methods for SurvivalData Risk Prediction and Optimal Treatment Decision NorthCarolina State University 2013
[41] J Li L Zhao L Tian et alAPredictive Enrichment Procedure toIdentify Potential Responders to a NewTherapy for RandomizedComparative Controlled Clinical Studies Harvard UniversityBiostatisticsWorking Paper SeriesWorking Paper 169 HarvardUniversity 2014
[42] C Kang H Janes and Y Huang ldquoCombining biomarkers tooptimize patient treatment recommendationsrdquo Biometrics vol70 no 3 pp 695ndash720 2014
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[44] B Zhang A A Tsiatis M Davidian M Zhang and E LaberldquoEstimating optimal treatment regimes from a classificationperspectiverdquo Stat vol 1 no 1 pp 103ndash114 2012
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[47] H Janes M D Brown M Pepe and Y Huang ldquoStatisticalmethods for evaluating and comparing biomarkers for patienttreatment selectionrdquo UW Biostatistics Working Paper SeriesWorking Paper 389 2013
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the HOXB13IL17BR biomarkerrdquo Journal of the National CancerInstitute vol 105 no 14 pp 1036ndash1042 2013
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[53] P C Austin ldquoAn introduction to propensity score methods forreducing the effects of confounding in observational studiesrdquoMultivariate Behavioral Research vol 46 no 3 pp 399ndash4242011
[54] G Heinze and P Juni ldquoAn overview of the objectives of andthe approaches to propensity score analysesrdquo European HeartJournal vol 32 no 14 Article ID ehr031 pp 1704ndash1708 2011
[55] L E Braitman and P R Rosenbaum ldquoRare outcomes commontreatments analytic strategies using propensity scoresrdquo Annalsof Internal Medicine vol 137 no 8 pp 693ndash695 2002
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[59] H Zou ldquoThe adaptive lasso and its oracle propertiesrdquo Journal ofthe American Statistical Association vol 101 no 476 pp 1418ndash1429 2006
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[61] T Hastie and B Efron ldquolars Least angle regression lasso andforward stagewiserdquo R package version 12 2013 httpcranr-projectorgwebpackageslarsindexhtml
[62] J Fan and R Li ldquoVariable selection via nonconcave penalizedlikelihood and its oracle propertiesrdquo Journal of the AmericanStatistical Association vol 96 no 456 pp 1348ndash1360 2001
[63] H Zou and T Hastie ldquoRegularization and variable selection viathe elastic netrdquo Journal of the Royal Statistical Society Series BStatistical Methodology vol 67 no 2 pp 301ndash320 2005
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[67] H M Boslashvelstad S Nygard H L Stoslashrvold et al ldquoPredictingsurvival from microarray datamdasha comparative studyrdquo Bioinfor-matics vol 23 no 16 pp 2080ndash2087 2007
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[69] P Royston and M K Parmar ldquoThe use of restricted meansurvival time to estimate the treatment effect in randomizedclinical trials when the proportional hazards assumption is indoubtrdquo Statistics inMedicine vol 30 no 19 pp 2409ndash2421 2011
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[71] E T Lee and J W Wang Statistical Methods for Survival DataAnalysis John Wiley amp Sons Hoboken NJ USA 2013
[72] Y Zhao D Zeng A J Rush and M R Kosorok ldquoEstimatingindividualized treatment rules using outcome weighted learn-ingrdquo Journal of the American Statistical Association vol 107 no499 pp 1106ndash1118 2012
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[74] L Breiman ldquoRandom forestsrdquoMachine Learning vol 45 no 1pp 5ndash32 2001
[75] J Friedman T Hastie and R Tibshirani ldquoAdditive logisticregression a statistical view of boostingrdquo The Annals of Statis-tics vol 28 no 2 pp 337ndash407 2000
[76] C Cortes and V Vapnik ldquoSupport-vector networksrdquo MachineLearning vol 20 no 3 pp 273ndash297 1995
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[78] S Dudoit J Fridlyand and T P Speed ldquoComparison of dis-criminationmethods for the classification of tumors using geneexpression datardquo Journal of the American Statistical Associationvol 97 no 457 pp 77ndash87 2002
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[80] T Therneau B Atkinson and B Ripley ldquorpart RecursivePartitioning and Regression Treesrdquo R package version 41-3httpcranr-projectorgwebpackagesrpartindexhtml
[81] A Kapelner J Bleich Z D Cohen R J DeRubeis and RBerk ldquoInference for treatment regime models in personalizedmedicinerdquo httparxivorgabs14047844
[82] X Song and M S Pepe ldquoEvaluating markers for selecting apatientrsquos treatmentrdquoBiometrics vol 60 no 4 pp 874ndash883 2004
[83] T Karrison ldquoRestricted mean life with adjustment for covari-atesrdquo Journal of the American Statistical Association vol 82 no400 pp 1169ndash1176 1987
[84] C Barker ldquoThe mean median and confidence intervals ofthe kaplan-meier survival estimatemdashcomputations and appli-cationsrdquo Journal of the American Statistical Association vol 63no 1 pp 78ndash80 2009
[85] B Efron and R J Tibshirani An Introduction to the Bootstrapvol 57 CRC Press 1994
[86] S Arlot andA Celisse ldquoA survey of cross-validation proceduresfor model selectionrdquo Statistics Surveys vol 4 pp 40ndash79 2010
[87] A Kapelner and J Bleich ldquoPTE Personalized Treatment Eva-luatorrdquo 2014 R package version 10 httpCRANR-projectorgpackage=PTE
[88] S A Murphy ldquoOptimal dynamic treatment regimesrdquo Journal ofthe Royal Statistical Society Series B StatisticalMethodology vol65 no 2 pp 331ndash355 2003
BioMed Research International 13
[89] J M Robins ldquoOptimal structural nested models for optimalsequential decisionsrdquo in Proceedings of the Second Seattle Sym-posium in Biostatistics vol 179 of Lecture Notes in Statistics pp189ndash326 Springer Berlin Germany 2004
[90] Y Zhao and D Zeng ldquoRecent development on statistical meth-ods for personalized medicine discoveryrdquo Frontiers of Medicinein China vol 7 no 1 pp 102ndash110 2013
∙ Randomized data from a comparative trial∙ Observational cohorts
(2) Choose a method of inference based on clinicalendpoints and data dimension
∙ Binary∙ Continuous∙ Survival
∙ Low dimensional∙ High dimensional
(3) Identify the individualized treatment rule
∙ Derive the treatment contrasts∙ Select a clinically relevant decision threshold
(4) Fit the model to training data
∙ Select the important covariates∙ Estimate the model parameters
(5) Evaluate performance
∙ Assess model goodness-of-fit and prediction∙ Measure the extent of clinical benefit
(6) Apply the treatment rule to future patients
∙ Acquire prognostic and predictive covariates∙ Select the optimal treatment
Figure 1 The process for using statistical inference to establish personalized treatment rules
4 Selecting an AppropriateMethod of Inference
The quality of a treatment rule depends on the aptness ofthe study design used to acquire the training data clinicalrelevance of the primary endpoints statistical analysis plansfor model selection and inference and quality of the dataRandomized clinical trials (RCT) remain the gold standardstudy design for treatment comparison since randomizationmitigates bias arising from treatment selection Methodsfor deriving ITRs using data from RCTs are described inSection 51 Data from well conducted observational studiesprovide useful sources of information as well given thatthe available covariates can be used to account for potentialsources of confounding due to selection bias Predominatelymethods based on propensity scores are used to adjust forconfounding [35 36] Approaches for establishing ITRs usingobservational studies are discussed in Section 52
The predominate statistical challenge pertaining to theidentification of predictive biomarkers is the high-dimen-sional nature of molecular derived candidate features Clas-sical regression models cannot be directly applied since thenumber of covariates for example genes is much larger thannumber of samples Many approaches have been proposedto analyze high-dimensional data for prognostic biomarkersSection 53 discusses several that can be applied to detectpredictive biomarkers under proper modification
In oncology several endpoints are used to compare clin-ical effectiveness However the primary therapeutic goal isto extend survivorship or delay recurrenceprogressionThustime-to-event endpoints are often considered to be the mostrepresentative of clinical effectiveness [37] The approachesaforementioned were developed for ordinal or continuousoutcomes and were thus not directly applicable for survivalanalysis Methods for establishing ITRs from time-to-eventendpoints often use Cox regression or accelerated failure timemodels [38 39] The later approach is particularly appealing
in this context since the clinical benefits of prolonged survivaltime can be easily obtained [40 41] In Section 54 we willdiscuss both models
The performance of ITRs for personalized medicineis highly dependent upon the extent to which the modelassumptions are satisfied andor the posited model is cor-rectly specified Specifically performances may suffer frommisspecification of main effects andor interactions randomerror distribution violation of linear assumptions sensitivityto outliers and other potential sources of inadequacy [42]Some advanced methodologies have been developed to over-come these issues [43] including semiparametric approachesthat circumvent prespecification of the functional form ofthe relationship between biomarker and expected clinicalresponse [32 40] In addition optimal treatment rules canbe defined without regression models using classificationapproaches where patients are assigned to the treatment thatprovides the highest expected clinical benefit Appropriateclass labels can be defined by the estimated treatment differ-ence (eg gt0 versus le0) thereby enabling the use of machinelearning and data mining techniques [42 44 45] These willbe discussed in Section 55
5 Methods for Identifying IndividualizedTreatment Rules
This section provides details of analytical approaches that areappropriate identifying ITRs using a clinical data sourceThevery nature of treatment benefit is determined by the clinicalendpoint While extending overall survival is the ultimatetherapeutic goal often the extent of reduction in tumorsize as assessed by RECIST criteria (httpwwwrecistcom)is used as a categorical surrogate for long-term responseAlternatively oncology trials often compare the extent towhich the treatment delays locoregional recurrence or dis-ease progression Therefore time-to-event and binary (as inabsencepresence of partial or complete response) are the
4 BioMed Research International
most commonly used endpoints in oncologic drug develop-ment [37 46]
Let 119884 denote the observed outcome such as survivalduration or response to the treatment and let 119860 isin 0 1
denote the treatment assignment with 0 indicating standardtreatment and 1 for a new therapy Denote the collection ofobservable data for a previously treated patient by (119884 119860X)where X = 119883
1 1198832 119883
119901 represents a vector of values for
the 119901 biomarkers under study Quantitatively the optimalITRderives from the following equation relating the observedresponse to the potential outcome attained under the alterna-tive treatment
119884 = 119860119884(1)+ (1 minus 119860)119884
(0) (1)
where119884(1) and119884(0) denote the potential outcomes that wouldbe observed if the subject had been assigned to the newtherapy or the standard treatment respectively [32 43] Let119864(119884 | 119860X) = 120583(119860X) denote the expected value of 119884 given119860 and X The optimal treatment rule follows as
where 119868(sdot) is the indicator function For instance if119868120583(1 age gt 50) minus 120583(0 age gt 50) gt 0 = 1 then the optimalrule would assign patients who are older than 50 to the newtreatment However 119864(119884 | 119860X) is actually a function ofparameters 120583(119860X120573) denoted by 120573 The model needs to beldquofittedrdquo to the training data to obtain estimates of 120573 which wedenote by Hence for a patient with observed biomarkersX = x the estimated optimal treatment rule is
119892opt(X = x )
= 119868 120583 (119860 = 1X = x ) minus 120583 (119860 = 0X = x ) (3)
The above equation pertains to steps (3) and (4) in Figure 1that is the parameter estimates from a fitted model are usedto construct the personalized treatment rule The remainderof this section instructs the readers how to identify ITRs forthe various data types
We classify the statistical methods presented in thissection into five categories methods based on multivariateand generalized linear regression for analysis of data acquiredfrom RCT (Section 51) and observational studies (Section52) methods based on penalized regression techniques forhigh-dimensional data (Section 53) methods for survivaldata (Section 54) and advanced methods based on robustestimation and machine learning techniques (Section 55)
51 Multiple Regression for Randomized Clinical Trial DataClassical generalized linear models (GLM) can be usedto develop ITRs in the presence of training data derivedfrom randomized clinical study The regression frameworkassumes that the outcome 119884 is a linear function of prognosticcovariates 119883
1 putative predictive biomarkers 119883
2 the treat-
ment indicator 119860 and treatment-by-predictive interaction1198601198832
Let Δ(X) = 119864(119884 | 119860 = 1119883) minus 119864(119884 | 119860 = 0119883) = 120583(119860 = 1
119883) minus 120583(119860 = 0119883) denote the treatment contrast The optimaltreatment rule assigns a patient to the new treatment ifΔ(X) gt 0 For binary endpoints the logistic regressionmodelfor 120583(119860X) = 119875(119884 = 1 | 119860X) is defined such that
The treatment contrast Δ(X) can be calculated using 119864(119884 |
119860 = 119886119883) = 119875(119884 = 1 | 119860 = 119886X) = 119890120596(119860X)(1 + 119890120596(119860X))for 119886 = 0 1 respectively Similarly an optimal ITR assigns apatient to the new treatment if Δ(X) gt 0 This optimal treat-ment rule can be alternatively defined as 119892opt(X) = 119868(120573
3+
12057341198832) gt 0 without the need to calculate the treatment
contrast Δ(X) [43 45]Often one might want to impose a clinically meaningful
minimal threshold Δ(X) gt 120575 on the magnitude of treatmentbenefit before assigning patients to a novel therapy [45 47]For example it may be desirable to require at least a 01increase in response rate before assigning a therapy for whichthe long-term safety profile has yet to be established The useof a threshold value can be applied to all methods Withoutloss of generality we assume 120575 = 0 unless otherwise specifiedIn addition the reader should note that the approaches forconstructing an ITR described above can be easily applied tolinear regression models for continuous outcomes
This strategy was used to develop an ITR for treatmentof depression [19] using data collected from a RCT of 154patients In this case the continuous outcome was basedon posttreatment scores from the Hamilton Rating Scale forDepression The authors constructed a personalized advant-age index using the estimated treatment contrasts Δ(X)derived from five predictive biomarkers A clinically signif-icant threshold was selected 120575 = 3 based on the NationalInstitute for Health and Care Excellence criterion Theauthors identified that 60 of patients in the sample wouldobtain a clinicallymeaningful advantage if their therapy deci-sion followed the proposed treatment rule The approachesdiscussed in this section can be easily implemented withstandard statistical software such as the 119877 (httpwwwr-projectorg) using the functions lm and glm [48]
52 Methods for Observational Data Randomization attenu-ates bias arising from treatment selection thereby providingthe highest quality data for comparing competing interven-tions However due to ethical or financial constraints RCTsare often infeasible thereby necessitating an observationalstudy Treatment selection is often based on a patientrsquosprognosis In the absence of randomization the study designfails to ensure that patients on competing arms exhibit similarclinical and prognostic characteristics thereby inducing bias
However in the event that the available covariates capturethe sources of bias a well conducted observational studycan also provide useful information for constructing ITRsFor example the two-gene ratio index (HOXB13IL17BR)
BioMed Research International 5
was first discovered as an independent prognostic biomarkerfor ER+ node-negative patients using retrospective datafrom 60 patients [49] These findings were confirmed onan independent data set comprising 852 tumors which wasacquired from a tumor bank at the Breast Center of BaylorCollege of Medicine [50] Interestingly the two-gene ratioindex (HOXB13IL17BR) was reported to predict the benefitof treatment with letrozole in one recent independent study[51]
Methods based on propensity scores are commonly usedto attenuate selection bias [35] In essence these approachesuse the available covariates to attempt to diminish the effectsof imbalances among variables that are not of interest fortreatment comparison Moreover they have been shownto be robust in the presence of multiple confounders andrare events [52] Generally after adjusting for bias usingpropensity scores the same principles for deriving ITRs fromRCTs may be applied to the observational cohort
The propensity score characterizes the probability ofassigning a given treatment 119860 from the available covariatesX [35] Using our notation the propensity score is 120587(X 120585) =119875(119860 = 1 | X 120585) which can be modeled using logisticregression
where 119901 is the number of independent variables used toconstruct the propensity score and 120585
119895represents the 119895th
regression coefficient which characterizes the 119895th covariatersquospartial effect After fitting the data to obtain estimates forthe regression coefficients the estimated probability ofreceiving new treatment can be obtained for each patient(Xi) = 120587(Xi ) by inverting the logit function The eventthat asymp 0 implies that the measured independent variablesare reasonably ldquobalancedrdquo between treatment cohorts Inpractice one often includes as many baseline covariates intothe propensity score model as permitted by the sample size
Methods that use propensity scores can be categorizedinto four categories matching stratification adjusting andinverse probability weighted estimation [36 53] Matchingand stratification aim to mimic RCTs by defining a newdataset using propensity scores such that outcomes aredirectly comparable between treatment cohorts [53] Thesetwo approaches are well suited for conventional subgroupanalysis but their application to personalized medicine hasbeen limited Regression adjustment or simply adjusting canbe used to reduce bias due to residual differences in observedbaseline covariates between treatment groups This methodincorporates the propensity scores as an independent variablein a regression model and therefore can be used in con-junction with all regression-based methods [36] Methodsinvolving inverse probability weighted estimators will bediscussed in Section 551 [43]
Of course propensity scores methods may only attenuatethe effects of the important confounding variables that havebeen acquired by the study design Casual inference in gen-eral is not robust to the presence of unmeasured confounders
that influenced treatment assignment [35 54 55] For thedevelopment of ITRs predictive and important prognosticcovariates can be incorporated in the regression model forthe clinical outcome119884 alongwith the propensity scores whileother covariatesmay be utilized only in themodel for estimat-ing the propensity scores Hence propensity score methodsmay offer the researcher a useful tool for controlling forpotential confounding due to selection bias andmaintaining amanageable number of prognostic and predictive covariates
53 Methods for High-Dimensional Biomarkers The meth-ods presented in the previous sections are appropriate foridentifying an ITR using a small set of biomarkers (low-dimensional)However recent advances inmolecular biologyin oncology have enabled researchers to acquire vast amountsof genetic and genomic characteristics on individual patientsOften the number of acquired genomic covariates will exceedthe sample size Proper analysis of these high-dimensionaldata sources poses many analytical challenges Several meth-ods have been proposed specifically for analysis of high-dimensional covariates [56] although the majority of thesemethods are well suited only for the analysis of prognosticbiomarkers In what follows we introduce variable selectionmethods that were developed to detect predictive biomarkersfrom high-dimensional sources as well as describing how toconstruct optimal ITRs from the final set of biomarkers
An appropriate regressionmodel can be defined generallyas 119864(119884 | 119860X) = ℎ
0(X) + 119860(X120573) where ℎ
0(X) is an unspe-
cified baseline mean function 120573 = (1205730 1205731 120573
119902)119879 is a
column vector of regression coefficients and X = (1X) thedesign matrix Subscript 119902 denotes the total number of bio-markers which may be larger than the sample size 119899 AnITR derives from evaluating the interactions in 119860(X120573) notthe baseline effect of the high-dimensional covariates ℎ
0(X)
[32] Technically function 119860(X120573) = 119860(1205730+ 12057311198831+ 12057321198832+
sdot sdot sdot + 120573119902119883119902) cannot be uniquely estimated using traditional
maximum likelihood-based methods when 119902 gt 119899 [57]Yet practically many of the available biomarkers may notinfluence the optimal ITR [31] Thus the process for identifyITRs from a high-dimensional source requires that we firstidentify a sparse subset of predictive biomarkers that can beutilized for constructing the ITR
Parameters for the specifiedmodel can be estimated usingthe following loss function
119871119899120601(120573 120574) =
1
119899
119899
sum119894=1
[119884119894minus 120601 (X
119894 120574) minus X120573 119860
119894minus 120587 (X
119894)]2
(7)
where 120601(X 120574) represents any arbitrary function character-izing the ldquobaselinerdquo relationship between X and Y (eg anintercept or an additive model) Here we let 120587(X
119894) = 119875(119860
119894=
1 | X119894) denote either a propensity score (for observational
data) or a randomization probability (eg 05 given 1 1randomization) for RCT data If 120587(X) is known estimationusing this model yields unbiased estimates (asymptoticallyconsistent) of the interaction effects 120573 even if the main effectsare not correctly specified providing a robustness [32]
6 BioMed Research International
Penalized estimation provides the subset of relevantpredictivemarkers that are extracted from the nonzero coeffi-cients of the corresponding treatment-biomarker interactionterms of
where 120582119899is a tuning parameter which is often selected via
cross validation and 119869 is a shrinkage penalty Different choicesof 119869 lead to different types of estimators For example thelasso penalized regression corresponds to 119869 = 1 [58] andthe adaptive lasso to 119869 = 120596
119895= 1|120573init119895| where 120573init119895 is an
initial estimate of 120573119895[59] With little modification (8) can
be solved using the LARS algorithm implemented with the119877 package of 119897119886119903119904 [32 60 61] As we have shown before atreatment rule can be defined from the parameter estimatesas 119868120573
formmay have zero estimates for some coefficients (eg 1205732=
1205735= sdot sdot sdot = 120573
119902= 0) hence an ITR can be equivalently con-
structed from the final estimated nonzero coefficients and thecorresponding covariates
Alternative penalized regression approaches includeSCAD [62] and elastic-net [63] All penalized approachesproduce sparse solutions (ie identifying a small subset ofpredictive biomarkers) however the adaptive lasso is lesseffective when 119901 gt 119899 Methods that produce nonsparsemodels such as ridge regression [57] are less preferable sinceITRs based on many biomarkers are often unstable and lessuseful in practice [31] Several packages in 119877 offer imple-mentation of penalized regression such as 119901119886119903119888119900119903 for ridgelasso and adaptive lasso and 119899119888V119903119890119892 for SCAD [64 65]
Lu et al [32] used a penalized regression approach toanalyze data from the AIDS Clinical Trials Group Protocol175 (ACTG175) [66] In this protocol 2000 patients wereequally randomized to one of four treatments zidovudine(ZDV) monotherapy ZDV + didanosine (ddI) ZDV +zalcitabine and ddI monotherapy CD4 count at 15ndash25 weekspostbaselinewas the primary outcome and 12 baseline covari-ates were included in the analysis The resulting treatmentrule favored the combined regimes over ZDV monotherapyMoreover the treatment rule determined that ZDV + ddIshould be preferred to ddI when 119868(7159 + 107 times ageminus 018 timesCD40 minus 3357 times homo) = 1 where CD40 represents baselineCD4 counts and homo represents homosexual activity Basedon this treatment rule 878 patients would have benefitedfrom treatment with ZDV + ddI
54 Survival Analysis Heretofore we have discussed meth-ods for continuous or binary outcomes yet often investigatorswant to discern the extent to which a therapeutic interventionmay alter the amount of time required before an event occursThis type of statistical inference is referred to broadly assurvival analysis One challenge for survival analysis is thatthe outcomes may be only partially observable at the time ofanalysis due to censoring or incomplete follow-up Survivalanalysis has been widely applied in cancer studies often inassociation studies aimed to identify prognostic biomarkers
[56 67] Here we discuss twowidely usedmodels for derivingITRs using time-to-event data namely Cox regression andaccelerated failure time models
The Cox regression model follows as
120582 (119905 | X 119860) = 1205820(119905) exp 120573
11198831+ 12057321198832+ 119860 (120573
4+ 12057351198832)
(9)
where 119905 is the survival time 1205820(119905) is an arbitrary baseline
hazard function and 1198831 1198832represent prognostic and pre-
dictive biomarkers respectively Each 120573 characterizes themultiplicative effect on the hazard associated with a unitincrease in the corresponding covariate Therefore Coxmodels are referred to as proportional hazards (PH) models
Several authors have provided model building strategies[29] and approaches for treatment selection [20 30 68] Fol-lowing the previously outlined strategy a naive approach forderiving an ITR uses the hazard ratio (new treatment versusthe standard) as the treatment contrast which can be calcu-lated as Δ(X) = exp(120573
4+ 12057351198832) The ITR therefore is 119868(120573
4
+12057351198832) lt 0 There are obvious limitations to this approach
First violations of the PH assumption yield substantiallymis-leading results [69]Moreover evenwhen the PH assumptionis satisfied because the Cox model does not postulate adirect relationship between the covariate (treatment) and thesurvival time the hazard ratio fails to measure the extent towhich the treatment is clinically valuable [38 70]
Accelerated failure time (AFT) models provide an alter-native semiparametric model Here we introduce its appli-cation for high-dimensional data Let 119879 and 119862 denote thesurvival and censoring times and denote the observed databy ( 120575 119860X) where = min(119879 119862) and 120575 = 119868(119879 lt 119862)Define the log survival time as 119884 = log(119879) a semiparametricregression model is given as 119864(119884 | 119860X) = ℎ
0(X) + 119860(X120573)
where ℎ0(X) is the unspecified baseline mean function
Similar to the previous section the treatment rule is 119868(1205730+
independent censoring the AFT model parameters can beestimated by minimizing the following loss function
119871119899120601(120573) =
1
119899
119899
sum119894=1
120575119894
119866(119894)[119894minus 120601 (X
119894 120574) minus X120573 119860
119894minus 120587 (X
119894)]2
(10)
where 119894= log(
119894) 120587(X
119894) = 119875(119860
119894= 1 | X
119894) is the propensity
score or randomization probability 119866(sdot) is the Kaplan-Meierestimator of the survival function of the censoring time and120601(X 120574) characterizes any arbitrary function
Thismethod can be extended to accommodatemore thantwo treatments simultaneously by specifying appropriatetreatment indicators For instance the mean function can bemodeled as 119864(119884 | 119860X) = ℎ
0(X) + 119860
(1)X120573(1) + 119860(2)X120573(2)
for two treatment drugs versus the standard care The ITRassigns the winning drug Note this work was proposed by[40] and is an extension of [32] to the survival setting Henceit shares the robustness property and can be applied to obser-vational data For implementation the sameprocedure can befollowed to obtain estimates with one addition step of calcu-lating 119866(
119894) There are several 119877 packages for Kaplan-Meier
BioMed Research International 7
estimates and Cox regression models These sources can befound at httpcranr-projectorgwebviewsSurvivalhtmlMore details pertaining to statistical methods for survivalanalysis can be found here [71] To compare treatmentrules constructed from Cox and AFT models for examplemethods for measuring the extent of clinical effectiveness foran ITR will be discussed in Section 6
We here present an example when an AFT model wasused to construct an ITR for treatment of HIV [40] Theexample derives from the AIDS Clinical Trials Group Pro-tocol 175 that was discussed in Section 53 [32 66] In thiscase the primary outcome variable was time (in days) tofirst ge50 decline in CD4 count or an AIDS-defining eventor death A total of 12 covariates and four treatments (ZDVZDV + ddI ZDV + zalcitabine and ddI) were includedThe four treatments were evaluated simultaneously Patientsreceiving the standard care of ZDV monotherapy wereconsidered as the reference group Hence three treatmentcontrasts (119868ZDV+ddI 119868ZDV+zalcitabine and 119868ddI) were combinedwith various putative predictive covariates and comparedwith ZDV monotherapy For example gender was detectedas the predictive covariate only for ddI monotherapy Theinvestigators assumed 120601(X 120574) = 120574
0 The treatment rule
recommended 1 patient for ZDV monotherapy while 7291216 and 193 patients were recommended for ZDV + ddIZDV + zalcitabine and ddI respectively
55 Advanced Methods
551 Robust Inference The performances of ITRs heretoforepresented depend heavily on whether the statistical modelswere correctly specified Recently there has been much atten-tion focused on the development of more advanced methodsand modeling strategies that are robust to various aspectsof potential misspecification We have already presented afew robust models that avoid specification of functionalparametric relationships for main effects [32 40] Here weintroduce two more advanced methods widely utilized forITRs that are robust to the type of misspecification issuescommonly encountered in practice [42 43]
Recall that the ITR for a linear model 119864(119884 | 119860 =
119886X) = 120583(119860 = 119886X120573) with two predictive markers followsas 119892(X120573) = 119868(120573
4+ 12057351198832+ 12057361198833) gt 0 where 119886 = 0 1
The treatment rule of 119892(X120573) may use only a subset of thehigh-dimensional covariates (eg 119883
2 1198833) but it always
depends on the correct specification of 119864(119884 | 119860 = 119886X)Defining a scaled version of 120573 as 120578(120573) the correspondingITR is 119892(120578X) = 119892(X120573) = 119868(119883
3gt 1205780+ 12057811198832) where
1205780= minus12057341205736and 120578
1= 12057351205736 If the model for 120583(119860X120573) is
indeed correctly specified the treatment rules of 119892(X120573) and119892(120578X) lead to the same optimal ITR Hence the treatmentrule parameterized by 120578 can be derived from a regressionmodel or may be based on some key clinical considerationswhich enable evaluation of 119892(120578X) directly without referenceto the regression model for 120583(119860X120573)
Let 119862120578
= 119860119892(120578X) + (1 minus 119860)1 minus 119892(120578X) where119862120578= 1 indicates random assignment to an intervention
that is recommended by the personalized treatment rule 119892(120578X) Let 120587(X ) denote the randomization ratio or the
estimated propensity score (as in previous section) and119898(X 120578 ) denote the potential outcome under the treatmentrule estimated from the following model 119864(119884 | 119860 =
119886X) = 120583(119860X120573) For example if the treatment rule 119892(120578X)= 1 then 119898(X 120578 ) = 119892(120578X)120583(119860 = 1X ) + 1 minus 119892(120578
X)120583(119860 = 0X ) = 120583(119860 = 1X ) Two estimators ofthe expected response to treatment the inverse probabilityweighted estimator (IPWE) and doubly robust AIPWE aregiven as follows
IPWE (120578) = 1
119899
119899
sum119894=1
119862120578sdot119894119884119894
120587119888(X119894 120578 )
=1
119899
119899
sum119894=1
119862120578sdot119894119884119894
120587 (X119894 )119860119894 1 minus 120587 (X
119894 )1minus119860119894
AIPWE (120578) = 1
119899
119899
sum119894=1
119862120578sdot119894119884119894
120587119888(X119894 120578 )
minus119862120578sdot119894119884119894 minus 120587119888 (X119894 120578 )
120587119888(X119894 120578 )
119898 (X119894 120578 )
(11)
where 120587119888(X119894 120578 ) = 120587(X )119892(120578X) + 1 minus 120587(X )1 minus
119892(120578X) The optimal treatment rule follows as 119892(X = x)where is estimated from the above models a constraintsuch as 120578 = 1 is imposed to obtain a unique solution [43] If the propensity score is correctly specified theIPWE estimator yields robust (consistent) estimates AIPWEis considered a doubly robust estimator since it produces con-sistent estimates when either propensity score or the model119864(119884 | 119860 = 119886X) is misspecified but not both [42 43] Thecompanion119877 code is publicly available at httponlinelibrarywileycomdoi101111biom12191suppinfo
552 Data Mining and Machine Learning The methodspresented in Section 551 are robust against misspecificationof regression models Yet they often require prespecificationof the parametric form for the treatment rule (eg 119868(119883
3gt
1205780+ 12057811198832)) which can be practically challenging [44]
Well established classification methods and other popularmachine learning techniques can alternatively be customizedto define treatment selection rules [44 72 73] these methodsavoid prespecification of the parametric form of the ITR AnITR can be defined following a two-step approach in thefirst step treatment contrasts are estimated from a positedmodel and in the second step classification techniques areapplied to determine the personalized treatment rules Forexample when only two treatments are considered a newvariable 119885 can be defined based on the treatment contrastthat is 119885 = 1 if Δ(X) = 120583(119860 = 1X) minus 120583(119860 = 0X) gt 0 and119885 = 0 otherwiseThe absolute value of the treatment contrast119882119894= |Δ(X)| can be used in conjunction with a classification
technique to define an appropriate ITR [44]Unlike classification problems wherein the class labels
are observed for the training data the binary ldquoresponserdquovariable 119885 which serves as the class label is not availablein practice Specifically patients who are in the class 119885 = 1
8 BioMed Research International
have 120583(119860 = 1X) gt 120583(119860 = 0X) and should thereforebe treated with the new therapy however these quantitiesneed to be estimated since patients are typically assigned toonly one of the available treatments This imparts flexibilityfor estimation of the optimal treatment regimes since anyof the previously discussed regression models and even someensemble prediction methods such as random forest [74] canbe used to construct the class labels 119885
119894and weights
119894[44]
An ITR can be estimated from the dataset 119885119894X119894 119894 using
any classification approach where 119894are subject specific
misclassification weights [44 45] This includes popularclassificationmethods such as adaptive boosting [75] supportvector machines [76] and classification and regression trees(CART) [77] At least one study has suggested that SVMoutperforms other classification methods in this contextwhereas random forest and boosting perform comparativelybetter than CART [78] However the performances ofthese classification algorithms are data dependent Definitiveconclusion pertaining to their comparative effectiveness ingeneral has yet to be determined [78] It shall be also notedthat these classification methods can be also applied to high-dimensional data [45 72]
One special case of this framework is the ldquovirtue twinsrdquoapproach [45] Specifically in the first step a random forestapproach [74] is used to obtain the treatment contrasts Thenin the second step CART is used to classify subjects tothe optimal treatment regime The approach can be easilyimplemented in 119877 using packages of randomForest [79] andrpart [80] Very recently Kang et al [42] proposed amodifiedversion of the adaptive boosting technique of Friedman et al[75] The algorithm iteratively fits a simple logistic regressionmodel (ldquoworking modelrdquo) to estimate 119875(119884 = 1 | 119860X) and ateach stage assigns higher weights to subjects whose treatmentcontrast is near zero After a prespecified stopping criterionis met an average of the treatment contrasts Δ(X) is calcu-lated for each patient using all models fitted at each iterationA subject is assigned to the new therapy if Δ(X) gt 0 The 119877code for the aforementioned boosting methods is publiclyavailable at httponlinelibrarywileycomdoi101111biom12191suppinfo
Lastly we present a breast cancer example where severalbiomarkers were combined to construct an optimal ITRThe data was collected in the Southwest Oncology Group(SOWG)-SS8814 trial [13] and analyzed with the machinelearning approach of Kang et al [42] Three hundred andsixty-seven node-positive ER-positive breast cancer patientswere selected from the randomized trial of SOWG A total of219 received tamoxifen plus adjuvant chemotherapy and 148was given tamoxifen aloneThe outcome variable was definedas breast cancer recurrence at 5 years The authors selectedthree genes which had presented treatment-biomarker inter-actions in amultivariate linear logistic regressionmodel [42]Data were analyzed with logistic models IPWE AIPWElogistic boosting a single classification tree with treatment-biomarker interactions and the proposed boosting approachwith a classification tree as the working model Each methodidentified different patient cohorts that could benefit fromtamoxifen alone these cohorts consisted of 184 183 128 86
263 and 217 patients respectively (see Table 5 in [42]) In thisanalysis the clinical benefits provided by these 6 treatmentrules were not statistically different Hence investigatorsneed to evaluate and compare ITRs in terms of the extentof expected clinical impact This is considered in the nextsection
Heretofore we have discussed various methodologies for theconstruction of ITR while their performances need to beassessed before these rules can be implemented in clinicalpractice Several aspects pertaining to the performance of aconstructed ITR need to be considered The first one is howwell the ITR fits the data and the second is how well theITR performs compared with existing treatment allocationrules The former is related to the concept of goodness-of-fitor predictive performance [34] As the true optimal treatmentgroups are hidden model fits may be evaluated by measuringthe congruity between observed treatment contrasts andpredicted ones [34 47] More details can be found in a recentpaper by Janes et al [47] Performances of ITRs can becompared via assessment of a global summary measure forexample prolonged survival time or reduced disease rate [4042] Summarymeasures are also very useful for evaluating theextent to which an ITR may benefit patients when applied inpractice Moreover it is essential that performance of an ITRis considered in comparison to business-as-usual proceduressuch as a naive rule that randomly allocates patients totreatment [81] Summary measures will be discussed inSection 61 The effectiveness of an ITR should go beyond thetraining data set used to construct a treatment rule cross-validation and bootstrapping techniques are often employedto assess the impact of ITRs on future patients [81] and willbe discussed in Section 62
61 Summary Measures ITRs may be derived from differ-ent methodologies and comparisons should be conductedwith respect to the appropriate clinically summaries A fewsummary measures for different types of outcomes havebeen proposed [19 40 42] these measures quantify thedirect clinical improvements obtained by applying an ITR incomparison with default methods for treatment allocation
Binary Outcomes Clinical effectiveness for binary clinicalresponse is represented by the difference in disease rates (ortreatment failure) induced by ITR versus a default strategythat allocates all patients to a standard treatment [42 47 82]Let 119892opt(X) = 119868120583(119860 = 1X)minus120583(119860 = 0X) lt 0 be an optimalITR This difference is formally defined as
Note 120583(119860X) needs to be estimated to construct the ITRyet parameters 120573 are omitted for simplicity Larger valuesof Θ119861119892opt(X) indicate increased clinical value for the
biomarker driven ITR A subset of patients that are recom-mended for new treatment (119860 = 1) under an ITR may havebeen randomly selected to receive it while the remainingsubset of ldquounluckyrdquo patientswould have received the standardtreatment [19] The summary measure of Θ
119861119892opt(X) char-
acterizes a weighted difference in the disease rates betweenthe standard and the new treatments in a population whereinthe constructed optimal ITR would recommend the newtreatment 119892opt(X = 1) The weight is the proportion ofpatients identified by the optimal ITR for the new treatmentand can be empirically estimated using the correspondingcounts For example 119875119892opt(X) = 1 can be estimatedusing the number of patients recommended for the newtreatment divided by the total sample size A similar summarystatistic can be derived for an alternative strategy allocatingall patients to the new treatment The summary could beapplied to the aforementioned breast cancer example [42]for example with the aim of finding a subgroup of patientswho were likely to benefit from adjuvant chemotherapywhile those unlikely to benefit would be assigned tamoxifenalone to avoid the unnecessary toxicity and inconvenience ofchemotherapy
Continuous Variables Another strategy for continuous datacompares outcomes observed for ldquoluckyrdquo subjects those whoreceived the therapy that would have been recommendedby the ITR based [81] Further one business-as-usual drugallocation procedure is randomizing treatment and standardcare at the same probability of 05 A summary statisticis to measure the mean outcome under ITR compared tothat obtained under random assignment for instance themean decrease in Hamilton Rating Scale for Depression asdiscussed in Section 51 [19] Define the summary measureas Θ119862119892opt(X) = 120583119892opt(X)X minus 120583119892rand(X)X where
119892rand(X) represents the randomization allocation procedureThe quantity of 120583119892119900119901119905(X)X represents the mean outcomeunder the constructed IRT that can be empirically estimatedfrom the ldquoluckyrdquo subjects and 120583119892rand(X)X can be esti-mated empirically from the sample means
Alternatively an ITR may be compared to an ldquooptimalrdquodrug that has showed universal benefits (a better drug onaverage) in a controlled trial The clinical benefits of anldquooptimalrdquo drug can be defined as 120583119892best(X)X = max120583(119860 =
0X) 120583(119860 = 1X) 120583(119860 = 119886X) and can be empiricallyestimated from the sample means of the new and standardtreatments respectively Then the alternative summary mea-sure is defined as Θ
Survival Data For survival data a clinically relevant measureis mean overall (or progression free) survival time As
survival time is continuous in nature the identical strategyprovided above for continuous outcomes can be employedhere However because the mean survival time may notbe well estimated from the observed data due to a highpercentage of censored observations [40] an alternativemean restricted survival duration was proposed and definedas the population average event-free durations for a restrictedtime of 119905lowast [41 83]Often 119905lowast is chosen to cover the trialrsquos follow-up period Mathematically it can be calculated by integratingthe survival function of 119878(119905) over the domain of (0 119905lowast) that is120583119892opt(X)X 119905lowast = int119905
lowast
0119878(119905)119889119905 and often estimated by the area
under the Kaplan-Meier curve up to 119905lowast [84] Thus an ITRrsquospotential to prolong survival can be calculated asΘ
119878119892opt(X)
119905lowast = 120583119892opt(X)X 119905lowast minus 120583119892rand(X)X 119905lowast
62 AssessingModel Performance The summaries heretoforediscussed evaluate an optimal ITR for a given model andestimating procedure Because these quantities are estimatedconditionally given the observed covariates they neglectto quantify the extent of marginal uncertainty for futurepatients Hence an ITR needs to be internally validated ifexternal data is not available [34] Cross-validation (CV) andbootstrap resampling techniques are commonly used for thispurpose [19 42 45 81] and expositions on both approachesare well described elsewhere [33 85 86]
We here briefly introduce a process that was proposed byKapelner et al [81] in the setting of personalized medicineTenfold CV is commonly used in practice where the wholedata is randomly partitioned into 10 roughly equal-sizedexclusive subsamples All methods under consideration areapplied to 910 of the data excluding 110 as an independenttesting data set The process is repeated 10 times for eachsubsample Considering the assignments recommended bythe optimal ITRs the summary measures can be calculatedusing results from each testing fold [45]TheCVprocess givesthe estimated summary measures and its variation can beevaluated using bootstrap procedures Specifically one drawsa sample with replacement from the entire data and calculatesthe summary measure from 10-fold CV This process willbe repeated 119861 times where 119861 is chosen for resolution ofthe resulting confidence intervals [81] Using the summarymeasures as119861 new random samples the correspondingmeanand variances can be calculated empirically Note that thesummary measures compare two treatment rules one for theoptimal ITR and another naive rule (eg randomization)
The above procedure can be applied to all the meth-ods we have discussed so far The 119877 software package119879119903119890119886119905119898119890119899119905119878119890119897119890119888119905119894119900119899 (httplabsfhcrcorgjanesindexhtml)can be used to implement these methods for evaluatingand comparing biomarkers for binary outcomes [47] Veryrecently an inferential procedure was proposed for contin-uous outcomes that is implemented in the publicly available119877 package ldquoPersonalized Treatment Evaluatorrdquo [81 87] Bothmethods consider data from RCTs with two arms for com-parative treatmentsThesemethods are in general applicableto regression model based methods but are not suitable forapproaches based on classification techniques or penalizedregression
10 BioMed Research International
Next we present two examples Recall in Section 55 thatKang et al [42] reported the estimated clinical benefits ofan ITR for breast cancer when compared to the defaultstrategy of assigning all patients to adjuvant chemotherapyThe proposed approach (based on boosting and classificationtrees) achieved the highest value of the summary measureat 0081 with 95 confidence interval (CI) (0000 0159)[42] In the second example introduced in Section 51 [19]the authors calculated the mean score of the HamiltonRating Scale for Depression for two groups of subjectsgroups were defined by randomly assigning patients to theldquooptimalrdquo and ldquononoptimalrdquo therapy as defined by the ITRThe reported difference between the two groups was minus178with a 119875 value of 009 which fails to attain a clinicalsignificant difference of 3 [19] The same data was analyzedby Kapelner et al [81] Following the discussed procedurethe authors reported the estimated values (and 95 CI)of Θ119862119892opt(X) and Θ
119862alt119892opt(X) as minus0842(minus2657 minus0441)
and minus0765(minus2362 0134) respectively The results whichfail to achieve clinical significance were based on rigorousstatistical methods and thus can be considered reliableestimates of the ITRrsquos performance
7 Discussion
As our understanding tumor heterogeneity evolves person-alized medicine will become standard medical practice inoncology Therefore it is essential that the oncology com-munity uses appropriate analytical methods for identifyingand evaluating the performance of personalized treatmentrules This paper provided an exposition of the process forusing statistical inference to establish optimal individualizedtreatment rules using data acquired from clinical study Thequality of an ITR depends on the quality of the design used toacquire the dataMoreover an ITRmust be properly validatedbefore it is integrated into clinical practice Personalizedmedicine in some areas of oncologymay be limited by the factthat biomarkers arising from a small panel of genesmay neveradequately characterize the extent of tumor heterogeneityinherent to the disease Consequently the available statisticalmethodology needs to evolve in order to optimally exploitglobal gene signatures for personalized medicine
The bulk of our review focused on statistical approachesfor treatment selection at a single time point The readershould note that another important area of research considersoptimal dynamic treatment regimes (DTRs) [88 89] whereintreatment decisions are considered sequentially over thecourse ofmultiple periods of intervention using each patientrsquosprior treatment history Zhao and Zeng provide a summaryof recent developments in this area [90]
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
JunshengMa was fully funded by the University of Texas MDAnderson Cancer Center internal funds Brian P Hobbs and
Francesco C Stingo were partially supported by the CancerCenter Support Grant (CCSG) (P30 CA016672)
References
[1] J Reimand O Wagih and G D Bader ldquoThe mutational land-scape of phosphorylation signaling in cancerrdquo Scientific Reportsvol 3 article 2651 2013
[2] D Hanahan and R AWeinberg ldquoHallmarks of cancer the nextgenerationrdquo Cell vol 144 no 5 pp 646ndash674 2011
[3] R Simon ldquoClinical trial designs for evaluating the medicalutility of prognostic and predictive biomarkers in oncologyrdquoPersonalized Medicine vol 7 no 1 pp 33ndash47 2010
[4] P L Bedard A R Hansen M J Ratain and L L Siu ldquoTumourheterogeneity in the clinicrdquo Nature vol 501 no 7467 pp 355ndash364 2013
[5] M D Pegram G Pauletti and D J Slamon ldquoHer-2neu as apredictive marker of response to breast cancer therapyrdquo BreastCancer Research and Treatment vol 52 no 1ndash3 pp 65ndash77 1998
[6] G J Kelloff andC C Sigman ldquoCancer biomarkers selecting theright drug for the right patientrdquoNature Reviews Drug Discoveryvol 11 no 3 pp 201ndash214 2012
[7] J ADiMasi JM Reichert L Feldman andAMalins ldquoClinicalapproval success rates for investigational cancer drugsrdquoClinicalPharmacology andTherapeutics vol 94 no 3 pp 329ndash335 2013
[8] M Hay D W Thomas J L Craighead C Economides andJ Rosenthal ldquoClinical development success rates for investiga-tional drugsrdquo Nature Biotechnology vol 32 no 1 pp 40ndash512014
[9] S S Knox ldquoFrom lsquoomicsrsquo to complex disease a systems biologyapproach to gene-environment interactions in cancerrdquo CancerCell International vol 10 article 11 2010
[10] V Deschoolmeester M Baay P Specenier F Lardon and JB Vermorken ldquoA review of the most promising biomarkersin colorectal cancer one step closer to targeted therapyrdquo TheOncologist vol 15 no 7 pp 699ndash731 2010
[11] J A Sosman K B Kim L Schuchter et al ldquoSurvival in brafV600ndashmutant advanced melanoma treated with vemurafenibrdquoThe New England Journal of Medicine vol 366 no 8 pp 707ndash714 2012
[12] S Paik S Shak G Tang et al ldquoA multigene assay to predictrecurrence of tamoxifen-treated node-negative breast cancerrdquoThe New England Journal of Medicine vol 351 no 27 pp 2817ndash2826 2004
[13] K S Albain W E Barlow S Shak et al ldquoPrognostic and pre-dictive value of the 21-gene recurrence score assay in post-menopausal women with node-positive oestrogen-receptor-positive breast cancer on chemotherapy a retrospective analysisof a randomised trialrdquoTheLancet Oncology vol 11 no 1 pp 55ndash65 2010
[14] J E Lang J S Wecsler M F Press and D Tripathy ldquoMolecularmarkers for breast cancer diagnosis prognosis and targetedtherapyrdquo Journal of Surgical Oncology vol 111 no 1 pp 81ndash902015
[15] W Werft A Benner and A Kopp-Schneider ldquoOn the identi-fication of predictive biomarkers detecting treatment-by-geneinteraction in high-dimensional datardquo Computational Statisticsand Data Analysis vol 56 no 5 pp 1275ndash1286 2012
[16] M Jenkins A Flynn T Smart et al ldquoA statisticianrsquos perspectiveon biomarkers in drug developmentrdquo Pharmaceutical Statisticsvol 10 no 6 pp 494ndash507 2011
BioMed Research International 11
[17] A J Vickers MW Kattan and D J Sargent ldquoMethod for eval-uating prediction models that apply the results of randomizedtrials to individual patientsrdquo Trials vol 8 no 1 article 14 2007
[18] H Janes M S Pepe P M Bossuyt andW E Barlow ldquoMeasur-ing the performance of markers for guiding treatment deci-sionsrdquo Annals of Internal Medicine vol 154 no 4 pp 253ndash2592011
[19] R J DeRubeis Z D Cohen N R Forand J C Fournier L AGelfand and L Lorenzo-Luaces ldquoThe personalized advantageindex translating research on prediction into individualizedtreatment recommendationsAdemonstrationrdquoPLoSONE vol9 no 1 Article ID e83875 2014
[20] D P Byar and D K Corle ldquoSelecting optimal treatment inclinical trials using covariate informationrdquo Journal of ChronicDiseases vol 30 no 7 pp 445ndash459 1977
[21] N Thatcher A Chang P Parikh et al ldquoGefitinib plus bestsupportive care in previously treated patients with refractoryadvanced non-small-cell lung cancer results from a ran-domised placebo-controlled multicentre study (iressa survivalevaluation in lung cancer)rdquo The Lancet vol 366 no 9496 pp1527ndash1537 2005
[22] A J Vickers ldquoPrediction models in cancer carerdquo CA A CancerJournal for Clinicians vol 61 no 5 pp 315ndash326 2011
[23] RM Simon ldquoSubgroup analysisrdquo inWiley Encyclopedia of Clin-ical Trials John Wiley amp Sons Hoboken NJ USA 2007
[24] S J Pocock S E Assmann L E Enos and L E Kasten ldquoSub-group analysis covariate adjustment and baseline comparisonsin clinical trial reporting current practice and problemsrdquoStatistics in Medicine vol 21 no 19 pp 2917ndash2930 2002
[25] PM Rothwell ZMehta S CHoward S A Gutnikov andC PWarlow ldquoFrom subgroups to individuals general principles andthe example of carotid endarterectomyrdquoTheLancet vol 365 no9455 pp 256ndash265 2005
[26] R Wang S W Lagakos J H Ware D J Hunter and J MDrazen ldquoStatistics in medicinemdashreporting of subgroup ana-lyses in clinical trialsrdquoTheNewEngland Journal ofMedicine vol357 no 21 pp 2108ndash2194 2007
[27] International Breast Cancer Study Group ldquoEndocrine respon-siveness and tailoring adjuvant therapy for postmenopausallymph node-negative breast cancer a randomized trialrdquo Journalof the National Cancer Institute vol 94 no 14 pp 1054ndash10652002
[28] Early Breast Cancer Trialistsrsquo Collaborative Group (EBCTCG)ldquoEffects of chemotherapy and hormonal therapy for early breastcancer on recurrence and 15-year survival an overview of therandomised trialsrdquoThe Lancet vol 365 no 9472 pp 1687ndash17172005
[29] F E Harrell K L Lee and D B Mark ldquoTutorial in biostatisticsmultivariable prognostic models issues in developing modelsevaluating assumptions and adequacy and measuring andreducing errorsrdquo Statistics in Medicine vol 15 no 4 pp 361ndash387 1996
[30] S Gill C L Loprinzi D J Sargent et al ldquoPooled analysis offluorouracil-based adjuvant therapy for stage II and III coloncancer who benefits and by how muchrdquo Journal of ClinicalOncology vol 22 no 10 pp 1797ndash1806 2004
[31] M Qian and S A Murphy ldquoPerformance guarantees forindividualized treatment rulesrdquoThe Annals of Statistics vol 39no 2 pp 1180ndash1210 2011
[32] W Lu H H Zhang and D Zeng ldquoVariable selection for opti-mal treatment decisionrdquo StatisticalMethods inMedical Researchvol 22 no 5 pp 493ndash504 2013
[33] R Kohavi ldquoA study of cross-validation and bootstrap for accu-racy estimation and model selectionrdquo in Proceedings of the 14thInternational Joint Conference on Artificial Intelligence (IJCAIrsquo95) vol 2 pp 1137ndash1145 1995
[34] E W Steyerberg A J Vickers N R Cook et al ldquoAssessing theperformance of prediction models a framework for traditionaland novel measuresrdquo Epidemiology vol 21 no 1 pp 128ndash1382010
[35] P R Rosenbaum and D B Rubin ldquoThe central role of the pro-pensity score in observational studies for causal effectsrdquo Bio-metrika vol 70 no 1 pp 41ndash55 1983
[36] R B drsquoAgostino Jr ldquoTutorial in biostatistics propensity scoremethods for bias reduction in the comparison of a treatment toa non-randomized control grouprdquo Statistics in Medicine vol 17no 19 pp 2265ndash2281 1998
[37] R Pazdur ldquoEndpoints for assessing drug activity in clinicaltrialsrdquoThe Oncologist vol 13 supplement 2 pp 19ndash21 2008
[38] S L Spruance J E Reid M Grace and M Samore ldquoHazardratio in clinical trialsrdquo Antimicrobial Agents and Chemotherapyvol 48 no 8 pp 2787ndash2792 2004
[39] J D Kalbeisch and R L Prentice The Statistical Analysis ofFailure Time Data vol 360 John Wiley amp Sons 2011
[40] Y Geng Flexible Statistical Learning Methods for SurvivalData Risk Prediction and Optimal Treatment Decision NorthCarolina State University 2013
[41] J Li L Zhao L Tian et alAPredictive Enrichment Procedure toIdentify Potential Responders to a NewTherapy for RandomizedComparative Controlled Clinical Studies Harvard UniversityBiostatisticsWorking Paper SeriesWorking Paper 169 HarvardUniversity 2014
[42] C Kang H Janes and Y Huang ldquoCombining biomarkers tooptimize patient treatment recommendationsrdquo Biometrics vol70 no 3 pp 695ndash720 2014
[43] B Zhang A A Tsiatis E B Laber and M Davidian ldquoA robustmethod for estimating optimal treatment regimesrdquo Biometricsvol 68 no 4 pp 1010ndash1018 2012
[44] B Zhang A A Tsiatis M Davidian M Zhang and E LaberldquoEstimating optimal treatment regimes from a classificationperspectiverdquo Stat vol 1 no 1 pp 103ndash114 2012
[45] J C Foster J M G Taylor and S J Ruberg ldquoSubgroup identi-fication from randomized clinical trial datardquo Statistics in Medi-cine vol 30 no 24 pp 2867ndash2880 2011
[46] US Food and Drug Administration Guidance for IndustryClinical Trial Endpoints for the Approval of Cancer Drugs andBiologics US Food and Drug AdministrationWashington DCUSA 2007
[47] H Janes M D Brown M Pepe and Y Huang ldquoStatisticalmethods for evaluating and comparing biomarkers for patienttreatment selectionrdquo UW Biostatistics Working Paper SeriesWorking Paper 389 2013
[48] R Development Core Team R a language and environment forstatistical computing 2008 httpwwwR-projectorg
[49] X-JMa ZWang P D Ryan et al ldquoA two-gene expression ratiopredicts clinical outcome in breast cancer patients treated withtamoxifenrdquo Cancer Cell vol 5 no 6 pp 607ndash616 2004
[50] X-JMa S GHilsenbeckWWang et al ldquoTheHOXB13IL17BRexpression index is a prognostic factor in early-stage breastcancerrdquo Journal of Clinical Oncology vol 24 no 28 pp 4611ndash4619 2006
[51] D C Sgroi E Carney E Zarrella et al ldquoPrediction of latedisease recurrence and extended adjuvant letrozole benefit by
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the HOXB13IL17BR biomarkerrdquo Journal of the National CancerInstitute vol 105 no 14 pp 1036ndash1042 2013
[52] M S Cepeda R Boston J T Farrar and B L Strom ldquoCom-parison of logistic regression versus propensity score when thenumber of events is low and there are multiple confoundersrdquoThe American Journal of Epidemiology vol 158 no 3 pp 280ndash287 2003
[53] P C Austin ldquoAn introduction to propensity score methods forreducing the effects of confounding in observational studiesrdquoMultivariate Behavioral Research vol 46 no 3 pp 399ndash4242011
[54] G Heinze and P Juni ldquoAn overview of the objectives of andthe approaches to propensity score analysesrdquo European HeartJournal vol 32 no 14 Article ID ehr031 pp 1704ndash1708 2011
[55] L E Braitman and P R Rosenbaum ldquoRare outcomes commontreatments analytic strategies using propensity scoresrdquo Annalsof Internal Medicine vol 137 no 8 pp 693ndash695 2002
[56] D M Witten and R Tibshirani ldquoSurvival analysis withhigh-dimensional covariatesrdquo Statistical Methods in MedicalResearch vol 19 no 1 pp 29ndash51 2010
[57] A E Hoerl and R W Kennard ldquoRidge regression biasedestimation for nonorthogonal problemsrdquoTechnometrics vol 42no 1 pp 80ndash86 2000
[58] R Tibshirani ldquoRegression shrinkage and selection via the lassordquoJournal of the Royal Statistical Society Series B Methodologicalvol 58 no 1 pp 267ndash288 1996
[59] H Zou ldquoThe adaptive lasso and its oracle propertiesrdquo Journal ofthe American Statistical Association vol 101 no 476 pp 1418ndash1429 2006
[60] B Efron T Hastie I Johnstone and R Tibshirani ldquoLeast angleregressionrdquo The Annals of Statistics vol 32 no 2 pp 407ndash4992004
[61] T Hastie and B Efron ldquolars Least angle regression lasso andforward stagewiserdquo R package version 12 2013 httpcranr-projectorgwebpackageslarsindexhtml
[62] J Fan and R Li ldquoVariable selection via nonconcave penalizedlikelihood and its oracle propertiesrdquo Journal of the AmericanStatistical Association vol 96 no 456 pp 1348ndash1360 2001
[63] H Zou and T Hastie ldquoRegularization and variable selection viathe elastic netrdquo Journal of the Royal Statistical Society Series BStatistical Methodology vol 67 no 2 pp 301ndash320 2005
[64] N Kramer J Schafer and A-L Boulesteix ldquoRegularized esti-mation of large-scale gene association networks using graphicalgaussian modelsrdquo BMC Bioinformatics vol 10 no 1 article 3842009
[65] P Breheny and J Huang ldquoCoordinate descent algorithms fornonconvex penalized regression with applications to biologicalfeature selectionrdquo The Annals of Applied Statistics vol 5 no 1pp 232ndash253 2011
[66] S M Hammer D A Katzenstein M D Hughes et al ldquoA trialcomparing nucleoside monotherapy with combination therapyin HIV-infected adults with CD4 cell counts from 200 to 500per cubicmillimeterrdquoTheNew England Journal ofMedicine vol335 no 15 pp 1081ndash1090 1996
[67] H M Boslashvelstad S Nygard H L Stoslashrvold et al ldquoPredictingsurvival from microarray datamdasha comparative studyrdquo Bioinfor-matics vol 23 no 16 pp 2080ndash2087 2007
[68] V Kehl and K Ulm ldquoResponder identification in clinical trialswith censored datardquoComputational Statistics andDataAnalysisvol 50 no 5 pp 1338ndash1355 2006
[69] P Royston and M K Parmar ldquoThe use of restricted meansurvival time to estimate the treatment effect in randomizedclinical trials when the proportional hazards assumption is indoubtrdquo Statistics inMedicine vol 30 no 19 pp 2409ndash2421 2011
[70] P Royston andM K B Parmar ldquoRestrictedmean survival timean alternative to the hazard ratio for the design and analysis ofrandomized trials with a time-to-event outcomerdquo BMCMedicalResearch Methodology vol 13 no 1 article 152 2013
[71] E T Lee and J W Wang Statistical Methods for Survival DataAnalysis John Wiley amp Sons Hoboken NJ USA 2013
[72] Y Zhao D Zeng A J Rush and M R Kosorok ldquoEstimatingindividualized treatment rules using outcome weighted learn-ingrdquo Journal of the American Statistical Association vol 107 no499 pp 1106ndash1118 2012
[73] D B Rubin and M J van der Laan ldquoStatistical issues andlimitations in personalized medicine research with clinicaltrialsrdquoThe International Journal of Biostatistics vol 8 no 1 pp1ndash20 2012
[74] L Breiman ldquoRandom forestsrdquoMachine Learning vol 45 no 1pp 5ndash32 2001
[75] J Friedman T Hastie and R Tibshirani ldquoAdditive logisticregression a statistical view of boostingrdquo The Annals of Statis-tics vol 28 no 2 pp 337ndash407 2000
[76] C Cortes and V Vapnik ldquoSupport-vector networksrdquo MachineLearning vol 20 no 3 pp 273ndash297 1995
[77] L Breiman J Friedman C J Stone and R A Olshen Classi-fication and Regression Trees CRC Press New York NY USA1984
[78] S Dudoit J Fridlyand and T P Speed ldquoComparison of dis-criminationmethods for the classification of tumors using geneexpression datardquo Journal of the American Statistical Associationvol 97 no 457 pp 77ndash87 2002
[79] A Liaw and MWiener ldquoClassification and regression by rand-omforestrdquoRNews vol 2 no 3 pp 18ndash22 2002 httpCRANR-projectorgdocRnews
[80] T Therneau B Atkinson and B Ripley ldquorpart RecursivePartitioning and Regression Treesrdquo R package version 41-3httpcranr-projectorgwebpackagesrpartindexhtml
[81] A Kapelner J Bleich Z D Cohen R J DeRubeis and RBerk ldquoInference for treatment regime models in personalizedmedicinerdquo httparxivorgabs14047844
[82] X Song and M S Pepe ldquoEvaluating markers for selecting apatientrsquos treatmentrdquoBiometrics vol 60 no 4 pp 874ndash883 2004
[83] T Karrison ldquoRestricted mean life with adjustment for covari-atesrdquo Journal of the American Statistical Association vol 82 no400 pp 1169ndash1176 1987
[84] C Barker ldquoThe mean median and confidence intervals ofthe kaplan-meier survival estimatemdashcomputations and appli-cationsrdquo Journal of the American Statistical Association vol 63no 1 pp 78ndash80 2009
[85] B Efron and R J Tibshirani An Introduction to the Bootstrapvol 57 CRC Press 1994
[86] S Arlot andA Celisse ldquoA survey of cross-validation proceduresfor model selectionrdquo Statistics Surveys vol 4 pp 40ndash79 2010
[87] A Kapelner and J Bleich ldquoPTE Personalized Treatment Eva-luatorrdquo 2014 R package version 10 httpCRANR-projectorgpackage=PTE
[88] S A Murphy ldquoOptimal dynamic treatment regimesrdquo Journal ofthe Royal Statistical Society Series B StatisticalMethodology vol65 no 2 pp 331ndash355 2003
BioMed Research International 13
[89] J M Robins ldquoOptimal structural nested models for optimalsequential decisionsrdquo in Proceedings of the Second Seattle Sym-posium in Biostatistics vol 179 of Lecture Notes in Statistics pp189ndash326 Springer Berlin Germany 2004
[90] Y Zhao and D Zeng ldquoRecent development on statistical meth-ods for personalized medicine discoveryrdquo Frontiers of Medicinein China vol 7 no 1 pp 102ndash110 2013
most commonly used endpoints in oncologic drug develop-ment [37 46]
Let 119884 denote the observed outcome such as survivalduration or response to the treatment and let 119860 isin 0 1
denote the treatment assignment with 0 indicating standardtreatment and 1 for a new therapy Denote the collection ofobservable data for a previously treated patient by (119884 119860X)where X = 119883
1 1198832 119883
119901 represents a vector of values for
the 119901 biomarkers under study Quantitatively the optimalITRderives from the following equation relating the observedresponse to the potential outcome attained under the alterna-tive treatment
119884 = 119860119884(1)+ (1 minus 119860)119884
(0) (1)
where119884(1) and119884(0) denote the potential outcomes that wouldbe observed if the subject had been assigned to the newtherapy or the standard treatment respectively [32 43] Let119864(119884 | 119860X) = 120583(119860X) denote the expected value of 119884 given119860 and X The optimal treatment rule follows as
where 119868(sdot) is the indicator function For instance if119868120583(1 age gt 50) minus 120583(0 age gt 50) gt 0 = 1 then the optimalrule would assign patients who are older than 50 to the newtreatment However 119864(119884 | 119860X) is actually a function ofparameters 120583(119860X120573) denoted by 120573 The model needs to beldquofittedrdquo to the training data to obtain estimates of 120573 which wedenote by Hence for a patient with observed biomarkersX = x the estimated optimal treatment rule is
119892opt(X = x )
= 119868 120583 (119860 = 1X = x ) minus 120583 (119860 = 0X = x ) (3)
The above equation pertains to steps (3) and (4) in Figure 1that is the parameter estimates from a fitted model are usedto construct the personalized treatment rule The remainderof this section instructs the readers how to identify ITRs forthe various data types
We classify the statistical methods presented in thissection into five categories methods based on multivariateand generalized linear regression for analysis of data acquiredfrom RCT (Section 51) and observational studies (Section52) methods based on penalized regression techniques forhigh-dimensional data (Section 53) methods for survivaldata (Section 54) and advanced methods based on robustestimation and machine learning techniques (Section 55)
51 Multiple Regression for Randomized Clinical Trial DataClassical generalized linear models (GLM) can be usedto develop ITRs in the presence of training data derivedfrom randomized clinical study The regression frameworkassumes that the outcome 119884 is a linear function of prognosticcovariates 119883
1 putative predictive biomarkers 119883
2 the treat-
ment indicator 119860 and treatment-by-predictive interaction1198601198832
Let Δ(X) = 119864(119884 | 119860 = 1119883) minus 119864(119884 | 119860 = 0119883) = 120583(119860 = 1
119883) minus 120583(119860 = 0119883) denote the treatment contrast The optimaltreatment rule assigns a patient to the new treatment ifΔ(X) gt 0 For binary endpoints the logistic regressionmodelfor 120583(119860X) = 119875(119884 = 1 | 119860X) is defined such that
The treatment contrast Δ(X) can be calculated using 119864(119884 |
119860 = 119886119883) = 119875(119884 = 1 | 119860 = 119886X) = 119890120596(119860X)(1 + 119890120596(119860X))for 119886 = 0 1 respectively Similarly an optimal ITR assigns apatient to the new treatment if Δ(X) gt 0 This optimal treat-ment rule can be alternatively defined as 119892opt(X) = 119868(120573
3+
12057341198832) gt 0 without the need to calculate the treatment
contrast Δ(X) [43 45]Often one might want to impose a clinically meaningful
minimal threshold Δ(X) gt 120575 on the magnitude of treatmentbenefit before assigning patients to a novel therapy [45 47]For example it may be desirable to require at least a 01increase in response rate before assigning a therapy for whichthe long-term safety profile has yet to be established The useof a threshold value can be applied to all methods Withoutloss of generality we assume 120575 = 0 unless otherwise specifiedIn addition the reader should note that the approaches forconstructing an ITR described above can be easily applied tolinear regression models for continuous outcomes
This strategy was used to develop an ITR for treatmentof depression [19] using data collected from a RCT of 154patients In this case the continuous outcome was basedon posttreatment scores from the Hamilton Rating Scale forDepression The authors constructed a personalized advant-age index using the estimated treatment contrasts Δ(X)derived from five predictive biomarkers A clinically signif-icant threshold was selected 120575 = 3 based on the NationalInstitute for Health and Care Excellence criterion Theauthors identified that 60 of patients in the sample wouldobtain a clinicallymeaningful advantage if their therapy deci-sion followed the proposed treatment rule The approachesdiscussed in this section can be easily implemented withstandard statistical software such as the 119877 (httpwwwr-projectorg) using the functions lm and glm [48]
52 Methods for Observational Data Randomization attenu-ates bias arising from treatment selection thereby providingthe highest quality data for comparing competing interven-tions However due to ethical or financial constraints RCTsare often infeasible thereby necessitating an observationalstudy Treatment selection is often based on a patientrsquosprognosis In the absence of randomization the study designfails to ensure that patients on competing arms exhibit similarclinical and prognostic characteristics thereby inducing bias
However in the event that the available covariates capturethe sources of bias a well conducted observational studycan also provide useful information for constructing ITRsFor example the two-gene ratio index (HOXB13IL17BR)
BioMed Research International 5
was first discovered as an independent prognostic biomarkerfor ER+ node-negative patients using retrospective datafrom 60 patients [49] These findings were confirmed onan independent data set comprising 852 tumors which wasacquired from a tumor bank at the Breast Center of BaylorCollege of Medicine [50] Interestingly the two-gene ratioindex (HOXB13IL17BR) was reported to predict the benefitof treatment with letrozole in one recent independent study[51]
Methods based on propensity scores are commonly usedto attenuate selection bias [35] In essence these approachesuse the available covariates to attempt to diminish the effectsof imbalances among variables that are not of interest fortreatment comparison Moreover they have been shownto be robust in the presence of multiple confounders andrare events [52] Generally after adjusting for bias usingpropensity scores the same principles for deriving ITRs fromRCTs may be applied to the observational cohort
The propensity score characterizes the probability ofassigning a given treatment 119860 from the available covariatesX [35] Using our notation the propensity score is 120587(X 120585) =119875(119860 = 1 | X 120585) which can be modeled using logisticregression
where 119901 is the number of independent variables used toconstruct the propensity score and 120585
119895represents the 119895th
regression coefficient which characterizes the 119895th covariatersquospartial effect After fitting the data to obtain estimates forthe regression coefficients the estimated probability ofreceiving new treatment can be obtained for each patient(Xi) = 120587(Xi ) by inverting the logit function The eventthat asymp 0 implies that the measured independent variablesare reasonably ldquobalancedrdquo between treatment cohorts Inpractice one often includes as many baseline covariates intothe propensity score model as permitted by the sample size
Methods that use propensity scores can be categorizedinto four categories matching stratification adjusting andinverse probability weighted estimation [36 53] Matchingand stratification aim to mimic RCTs by defining a newdataset using propensity scores such that outcomes aredirectly comparable between treatment cohorts [53] Thesetwo approaches are well suited for conventional subgroupanalysis but their application to personalized medicine hasbeen limited Regression adjustment or simply adjusting canbe used to reduce bias due to residual differences in observedbaseline covariates between treatment groups This methodincorporates the propensity scores as an independent variablein a regression model and therefore can be used in con-junction with all regression-based methods [36] Methodsinvolving inverse probability weighted estimators will bediscussed in Section 551 [43]
Of course propensity scores methods may only attenuatethe effects of the important confounding variables that havebeen acquired by the study design Casual inference in gen-eral is not robust to the presence of unmeasured confounders
that influenced treatment assignment [35 54 55] For thedevelopment of ITRs predictive and important prognosticcovariates can be incorporated in the regression model forthe clinical outcome119884 alongwith the propensity scores whileother covariatesmay be utilized only in themodel for estimat-ing the propensity scores Hence propensity score methodsmay offer the researcher a useful tool for controlling forpotential confounding due to selection bias andmaintaining amanageable number of prognostic and predictive covariates
53 Methods for High-Dimensional Biomarkers The meth-ods presented in the previous sections are appropriate foridentifying an ITR using a small set of biomarkers (low-dimensional)However recent advances inmolecular biologyin oncology have enabled researchers to acquire vast amountsof genetic and genomic characteristics on individual patientsOften the number of acquired genomic covariates will exceedthe sample size Proper analysis of these high-dimensionaldata sources poses many analytical challenges Several meth-ods have been proposed specifically for analysis of high-dimensional covariates [56] although the majority of thesemethods are well suited only for the analysis of prognosticbiomarkers In what follows we introduce variable selectionmethods that were developed to detect predictive biomarkersfrom high-dimensional sources as well as describing how toconstruct optimal ITRs from the final set of biomarkers
An appropriate regressionmodel can be defined generallyas 119864(119884 | 119860X) = ℎ
0(X) + 119860(X120573) where ℎ
0(X) is an unspe-
cified baseline mean function 120573 = (1205730 1205731 120573
119902)119879 is a
column vector of regression coefficients and X = (1X) thedesign matrix Subscript 119902 denotes the total number of bio-markers which may be larger than the sample size 119899 AnITR derives from evaluating the interactions in 119860(X120573) notthe baseline effect of the high-dimensional covariates ℎ
0(X)
[32] Technically function 119860(X120573) = 119860(1205730+ 12057311198831+ 12057321198832+
sdot sdot sdot + 120573119902119883119902) cannot be uniquely estimated using traditional
maximum likelihood-based methods when 119902 gt 119899 [57]Yet practically many of the available biomarkers may notinfluence the optimal ITR [31] Thus the process for identifyITRs from a high-dimensional source requires that we firstidentify a sparse subset of predictive biomarkers that can beutilized for constructing the ITR
Parameters for the specifiedmodel can be estimated usingthe following loss function
119871119899120601(120573 120574) =
1
119899
119899
sum119894=1
[119884119894minus 120601 (X
119894 120574) minus X120573 119860
119894minus 120587 (X
119894)]2
(7)
where 120601(X 120574) represents any arbitrary function character-izing the ldquobaselinerdquo relationship between X and Y (eg anintercept or an additive model) Here we let 120587(X
119894) = 119875(119860
119894=
1 | X119894) denote either a propensity score (for observational
data) or a randomization probability (eg 05 given 1 1randomization) for RCT data If 120587(X) is known estimationusing this model yields unbiased estimates (asymptoticallyconsistent) of the interaction effects 120573 even if the main effectsare not correctly specified providing a robustness [32]
6 BioMed Research International
Penalized estimation provides the subset of relevantpredictivemarkers that are extracted from the nonzero coeffi-cients of the corresponding treatment-biomarker interactionterms of
where 120582119899is a tuning parameter which is often selected via
cross validation and 119869 is a shrinkage penalty Different choicesof 119869 lead to different types of estimators For example thelasso penalized regression corresponds to 119869 = 1 [58] andthe adaptive lasso to 119869 = 120596
119895= 1|120573init119895| where 120573init119895 is an
initial estimate of 120573119895[59] With little modification (8) can
be solved using the LARS algorithm implemented with the119877 package of 119897119886119903119904 [32 60 61] As we have shown before atreatment rule can be defined from the parameter estimatesas 119868120573
formmay have zero estimates for some coefficients (eg 1205732=
1205735= sdot sdot sdot = 120573
119902= 0) hence an ITR can be equivalently con-
structed from the final estimated nonzero coefficients and thecorresponding covariates
Alternative penalized regression approaches includeSCAD [62] and elastic-net [63] All penalized approachesproduce sparse solutions (ie identifying a small subset ofpredictive biomarkers) however the adaptive lasso is lesseffective when 119901 gt 119899 Methods that produce nonsparsemodels such as ridge regression [57] are less preferable sinceITRs based on many biomarkers are often unstable and lessuseful in practice [31] Several packages in 119877 offer imple-mentation of penalized regression such as 119901119886119903119888119900119903 for ridgelasso and adaptive lasso and 119899119888V119903119890119892 for SCAD [64 65]
Lu et al [32] used a penalized regression approach toanalyze data from the AIDS Clinical Trials Group Protocol175 (ACTG175) [66] In this protocol 2000 patients wereequally randomized to one of four treatments zidovudine(ZDV) monotherapy ZDV + didanosine (ddI) ZDV +zalcitabine and ddI monotherapy CD4 count at 15ndash25 weekspostbaselinewas the primary outcome and 12 baseline covari-ates were included in the analysis The resulting treatmentrule favored the combined regimes over ZDV monotherapyMoreover the treatment rule determined that ZDV + ddIshould be preferred to ddI when 119868(7159 + 107 times ageminus 018 timesCD40 minus 3357 times homo) = 1 where CD40 represents baselineCD4 counts and homo represents homosexual activity Basedon this treatment rule 878 patients would have benefitedfrom treatment with ZDV + ddI
54 Survival Analysis Heretofore we have discussed meth-ods for continuous or binary outcomes yet often investigatorswant to discern the extent to which a therapeutic interventionmay alter the amount of time required before an event occursThis type of statistical inference is referred to broadly assurvival analysis One challenge for survival analysis is thatthe outcomes may be only partially observable at the time ofanalysis due to censoring or incomplete follow-up Survivalanalysis has been widely applied in cancer studies often inassociation studies aimed to identify prognostic biomarkers
[56 67] Here we discuss twowidely usedmodels for derivingITRs using time-to-event data namely Cox regression andaccelerated failure time models
The Cox regression model follows as
120582 (119905 | X 119860) = 1205820(119905) exp 120573
11198831+ 12057321198832+ 119860 (120573
4+ 12057351198832)
(9)
where 119905 is the survival time 1205820(119905) is an arbitrary baseline
hazard function and 1198831 1198832represent prognostic and pre-
dictive biomarkers respectively Each 120573 characterizes themultiplicative effect on the hazard associated with a unitincrease in the corresponding covariate Therefore Coxmodels are referred to as proportional hazards (PH) models
Several authors have provided model building strategies[29] and approaches for treatment selection [20 30 68] Fol-lowing the previously outlined strategy a naive approach forderiving an ITR uses the hazard ratio (new treatment versusthe standard) as the treatment contrast which can be calcu-lated as Δ(X) = exp(120573
4+ 12057351198832) The ITR therefore is 119868(120573
4
+12057351198832) lt 0 There are obvious limitations to this approach
First violations of the PH assumption yield substantiallymis-leading results [69]Moreover evenwhen the PH assumptionis satisfied because the Cox model does not postulate adirect relationship between the covariate (treatment) and thesurvival time the hazard ratio fails to measure the extent towhich the treatment is clinically valuable [38 70]
Accelerated failure time (AFT) models provide an alter-native semiparametric model Here we introduce its appli-cation for high-dimensional data Let 119879 and 119862 denote thesurvival and censoring times and denote the observed databy ( 120575 119860X) where = min(119879 119862) and 120575 = 119868(119879 lt 119862)Define the log survival time as 119884 = log(119879) a semiparametricregression model is given as 119864(119884 | 119860X) = ℎ
0(X) + 119860(X120573)
where ℎ0(X) is the unspecified baseline mean function
Similar to the previous section the treatment rule is 119868(1205730+
independent censoring the AFT model parameters can beestimated by minimizing the following loss function
119871119899120601(120573) =
1
119899
119899
sum119894=1
120575119894
119866(119894)[119894minus 120601 (X
119894 120574) minus X120573 119860
119894minus 120587 (X
119894)]2
(10)
where 119894= log(
119894) 120587(X
119894) = 119875(119860
119894= 1 | X
119894) is the propensity
score or randomization probability 119866(sdot) is the Kaplan-Meierestimator of the survival function of the censoring time and120601(X 120574) characterizes any arbitrary function
Thismethod can be extended to accommodatemore thantwo treatments simultaneously by specifying appropriatetreatment indicators For instance the mean function can bemodeled as 119864(119884 | 119860X) = ℎ
0(X) + 119860
(1)X120573(1) + 119860(2)X120573(2)
for two treatment drugs versus the standard care The ITRassigns the winning drug Note this work was proposed by[40] and is an extension of [32] to the survival setting Henceit shares the robustness property and can be applied to obser-vational data For implementation the sameprocedure can befollowed to obtain estimates with one addition step of calcu-lating 119866(
119894) There are several 119877 packages for Kaplan-Meier
BioMed Research International 7
estimates and Cox regression models These sources can befound at httpcranr-projectorgwebviewsSurvivalhtmlMore details pertaining to statistical methods for survivalanalysis can be found here [71] To compare treatmentrules constructed from Cox and AFT models for examplemethods for measuring the extent of clinical effectiveness foran ITR will be discussed in Section 6
We here present an example when an AFT model wasused to construct an ITR for treatment of HIV [40] Theexample derives from the AIDS Clinical Trials Group Pro-tocol 175 that was discussed in Section 53 [32 66] In thiscase the primary outcome variable was time (in days) tofirst ge50 decline in CD4 count or an AIDS-defining eventor death A total of 12 covariates and four treatments (ZDVZDV + ddI ZDV + zalcitabine and ddI) were includedThe four treatments were evaluated simultaneously Patientsreceiving the standard care of ZDV monotherapy wereconsidered as the reference group Hence three treatmentcontrasts (119868ZDV+ddI 119868ZDV+zalcitabine and 119868ddI) were combinedwith various putative predictive covariates and comparedwith ZDV monotherapy For example gender was detectedas the predictive covariate only for ddI monotherapy Theinvestigators assumed 120601(X 120574) = 120574
0 The treatment rule
recommended 1 patient for ZDV monotherapy while 7291216 and 193 patients were recommended for ZDV + ddIZDV + zalcitabine and ddI respectively
55 Advanced Methods
551 Robust Inference The performances of ITRs heretoforepresented depend heavily on whether the statistical modelswere correctly specified Recently there has been much atten-tion focused on the development of more advanced methodsand modeling strategies that are robust to various aspectsof potential misspecification We have already presented afew robust models that avoid specification of functionalparametric relationships for main effects [32 40] Here weintroduce two more advanced methods widely utilized forITRs that are robust to the type of misspecification issuescommonly encountered in practice [42 43]
Recall that the ITR for a linear model 119864(119884 | 119860 =
119886X) = 120583(119860 = 119886X120573) with two predictive markers followsas 119892(X120573) = 119868(120573
4+ 12057351198832+ 12057361198833) gt 0 where 119886 = 0 1
The treatment rule of 119892(X120573) may use only a subset of thehigh-dimensional covariates (eg 119883
2 1198833) but it always
depends on the correct specification of 119864(119884 | 119860 = 119886X)Defining a scaled version of 120573 as 120578(120573) the correspondingITR is 119892(120578X) = 119892(X120573) = 119868(119883
3gt 1205780+ 12057811198832) where
1205780= minus12057341205736and 120578
1= 12057351205736 If the model for 120583(119860X120573) is
indeed correctly specified the treatment rules of 119892(X120573) and119892(120578X) lead to the same optimal ITR Hence the treatmentrule parameterized by 120578 can be derived from a regressionmodel or may be based on some key clinical considerationswhich enable evaluation of 119892(120578X) directly without referenceto the regression model for 120583(119860X120573)
Let 119862120578
= 119860119892(120578X) + (1 minus 119860)1 minus 119892(120578X) where119862120578= 1 indicates random assignment to an intervention
that is recommended by the personalized treatment rule 119892(120578X) Let 120587(X ) denote the randomization ratio or the
estimated propensity score (as in previous section) and119898(X 120578 ) denote the potential outcome under the treatmentrule estimated from the following model 119864(119884 | 119860 =
119886X) = 120583(119860X120573) For example if the treatment rule 119892(120578X)= 1 then 119898(X 120578 ) = 119892(120578X)120583(119860 = 1X ) + 1 minus 119892(120578
X)120583(119860 = 0X ) = 120583(119860 = 1X ) Two estimators ofthe expected response to treatment the inverse probabilityweighted estimator (IPWE) and doubly robust AIPWE aregiven as follows
IPWE (120578) = 1
119899
119899
sum119894=1
119862120578sdot119894119884119894
120587119888(X119894 120578 )
=1
119899
119899
sum119894=1
119862120578sdot119894119884119894
120587 (X119894 )119860119894 1 minus 120587 (X
119894 )1minus119860119894
AIPWE (120578) = 1
119899
119899
sum119894=1
119862120578sdot119894119884119894
120587119888(X119894 120578 )
minus119862120578sdot119894119884119894 minus 120587119888 (X119894 120578 )
120587119888(X119894 120578 )
119898 (X119894 120578 )
(11)
where 120587119888(X119894 120578 ) = 120587(X )119892(120578X) + 1 minus 120587(X )1 minus
119892(120578X) The optimal treatment rule follows as 119892(X = x)where is estimated from the above models a constraintsuch as 120578 = 1 is imposed to obtain a unique solution [43] If the propensity score is correctly specified theIPWE estimator yields robust (consistent) estimates AIPWEis considered a doubly robust estimator since it produces con-sistent estimates when either propensity score or the model119864(119884 | 119860 = 119886X) is misspecified but not both [42 43] Thecompanion119877 code is publicly available at httponlinelibrarywileycomdoi101111biom12191suppinfo
552 Data Mining and Machine Learning The methodspresented in Section 551 are robust against misspecificationof regression models Yet they often require prespecificationof the parametric form for the treatment rule (eg 119868(119883
3gt
1205780+ 12057811198832)) which can be practically challenging [44]
Well established classification methods and other popularmachine learning techniques can alternatively be customizedto define treatment selection rules [44 72 73] these methodsavoid prespecification of the parametric form of the ITR AnITR can be defined following a two-step approach in thefirst step treatment contrasts are estimated from a positedmodel and in the second step classification techniques areapplied to determine the personalized treatment rules Forexample when only two treatments are considered a newvariable 119885 can be defined based on the treatment contrastthat is 119885 = 1 if Δ(X) = 120583(119860 = 1X) minus 120583(119860 = 0X) gt 0 and119885 = 0 otherwiseThe absolute value of the treatment contrast119882119894= |Δ(X)| can be used in conjunction with a classification
technique to define an appropriate ITR [44]Unlike classification problems wherein the class labels
are observed for the training data the binary ldquoresponserdquovariable 119885 which serves as the class label is not availablein practice Specifically patients who are in the class 119885 = 1
8 BioMed Research International
have 120583(119860 = 1X) gt 120583(119860 = 0X) and should thereforebe treated with the new therapy however these quantitiesneed to be estimated since patients are typically assigned toonly one of the available treatments This imparts flexibilityfor estimation of the optimal treatment regimes since anyof the previously discussed regression models and even someensemble prediction methods such as random forest [74] canbe used to construct the class labels 119885
119894and weights
119894[44]
An ITR can be estimated from the dataset 119885119894X119894 119894 using
any classification approach where 119894are subject specific
misclassification weights [44 45] This includes popularclassificationmethods such as adaptive boosting [75] supportvector machines [76] and classification and regression trees(CART) [77] At least one study has suggested that SVMoutperforms other classification methods in this contextwhereas random forest and boosting perform comparativelybetter than CART [78] However the performances ofthese classification algorithms are data dependent Definitiveconclusion pertaining to their comparative effectiveness ingeneral has yet to be determined [78] It shall be also notedthat these classification methods can be also applied to high-dimensional data [45 72]
One special case of this framework is the ldquovirtue twinsrdquoapproach [45] Specifically in the first step a random forestapproach [74] is used to obtain the treatment contrasts Thenin the second step CART is used to classify subjects tothe optimal treatment regime The approach can be easilyimplemented in 119877 using packages of randomForest [79] andrpart [80] Very recently Kang et al [42] proposed amodifiedversion of the adaptive boosting technique of Friedman et al[75] The algorithm iteratively fits a simple logistic regressionmodel (ldquoworking modelrdquo) to estimate 119875(119884 = 1 | 119860X) and ateach stage assigns higher weights to subjects whose treatmentcontrast is near zero After a prespecified stopping criterionis met an average of the treatment contrasts Δ(X) is calcu-lated for each patient using all models fitted at each iterationA subject is assigned to the new therapy if Δ(X) gt 0 The 119877code for the aforementioned boosting methods is publiclyavailable at httponlinelibrarywileycomdoi101111biom12191suppinfo
Lastly we present a breast cancer example where severalbiomarkers were combined to construct an optimal ITRThe data was collected in the Southwest Oncology Group(SOWG)-SS8814 trial [13] and analyzed with the machinelearning approach of Kang et al [42] Three hundred andsixty-seven node-positive ER-positive breast cancer patientswere selected from the randomized trial of SOWG A total of219 received tamoxifen plus adjuvant chemotherapy and 148was given tamoxifen aloneThe outcome variable was definedas breast cancer recurrence at 5 years The authors selectedthree genes which had presented treatment-biomarker inter-actions in amultivariate linear logistic regressionmodel [42]Data were analyzed with logistic models IPWE AIPWElogistic boosting a single classification tree with treatment-biomarker interactions and the proposed boosting approachwith a classification tree as the working model Each methodidentified different patient cohorts that could benefit fromtamoxifen alone these cohorts consisted of 184 183 128 86
263 and 217 patients respectively (see Table 5 in [42]) In thisanalysis the clinical benefits provided by these 6 treatmentrules were not statistically different Hence investigatorsneed to evaluate and compare ITRs in terms of the extentof expected clinical impact This is considered in the nextsection
Heretofore we have discussed various methodologies for theconstruction of ITR while their performances need to beassessed before these rules can be implemented in clinicalpractice Several aspects pertaining to the performance of aconstructed ITR need to be considered The first one is howwell the ITR fits the data and the second is how well theITR performs compared with existing treatment allocationrules The former is related to the concept of goodness-of-fitor predictive performance [34] As the true optimal treatmentgroups are hidden model fits may be evaluated by measuringthe congruity between observed treatment contrasts andpredicted ones [34 47] More details can be found in a recentpaper by Janes et al [47] Performances of ITRs can becompared via assessment of a global summary measure forexample prolonged survival time or reduced disease rate [4042] Summarymeasures are also very useful for evaluating theextent to which an ITR may benefit patients when applied inpractice Moreover it is essential that performance of an ITRis considered in comparison to business-as-usual proceduressuch as a naive rule that randomly allocates patients totreatment [81] Summary measures will be discussed inSection 61 The effectiveness of an ITR should go beyond thetraining data set used to construct a treatment rule cross-validation and bootstrapping techniques are often employedto assess the impact of ITRs on future patients [81] and willbe discussed in Section 62
61 Summary Measures ITRs may be derived from differ-ent methodologies and comparisons should be conductedwith respect to the appropriate clinically summaries A fewsummary measures for different types of outcomes havebeen proposed [19 40 42] these measures quantify thedirect clinical improvements obtained by applying an ITR incomparison with default methods for treatment allocation
Binary Outcomes Clinical effectiveness for binary clinicalresponse is represented by the difference in disease rates (ortreatment failure) induced by ITR versus a default strategythat allocates all patients to a standard treatment [42 47 82]Let 119892opt(X) = 119868120583(119860 = 1X)minus120583(119860 = 0X) lt 0 be an optimalITR This difference is formally defined as
Note 120583(119860X) needs to be estimated to construct the ITRyet parameters 120573 are omitted for simplicity Larger valuesof Θ119861119892opt(X) indicate increased clinical value for the
biomarker driven ITR A subset of patients that are recom-mended for new treatment (119860 = 1) under an ITR may havebeen randomly selected to receive it while the remainingsubset of ldquounluckyrdquo patientswould have received the standardtreatment [19] The summary measure of Θ
119861119892opt(X) char-
acterizes a weighted difference in the disease rates betweenthe standard and the new treatments in a population whereinthe constructed optimal ITR would recommend the newtreatment 119892opt(X = 1) The weight is the proportion ofpatients identified by the optimal ITR for the new treatmentand can be empirically estimated using the correspondingcounts For example 119875119892opt(X) = 1 can be estimatedusing the number of patients recommended for the newtreatment divided by the total sample size A similar summarystatistic can be derived for an alternative strategy allocatingall patients to the new treatment The summary could beapplied to the aforementioned breast cancer example [42]for example with the aim of finding a subgroup of patientswho were likely to benefit from adjuvant chemotherapywhile those unlikely to benefit would be assigned tamoxifenalone to avoid the unnecessary toxicity and inconvenience ofchemotherapy
Continuous Variables Another strategy for continuous datacompares outcomes observed for ldquoluckyrdquo subjects those whoreceived the therapy that would have been recommendedby the ITR based [81] Further one business-as-usual drugallocation procedure is randomizing treatment and standardcare at the same probability of 05 A summary statisticis to measure the mean outcome under ITR compared tothat obtained under random assignment for instance themean decrease in Hamilton Rating Scale for Depression asdiscussed in Section 51 [19] Define the summary measureas Θ119862119892opt(X) = 120583119892opt(X)X minus 120583119892rand(X)X where
119892rand(X) represents the randomization allocation procedureThe quantity of 120583119892119900119901119905(X)X represents the mean outcomeunder the constructed IRT that can be empirically estimatedfrom the ldquoluckyrdquo subjects and 120583119892rand(X)X can be esti-mated empirically from the sample means
Alternatively an ITR may be compared to an ldquooptimalrdquodrug that has showed universal benefits (a better drug onaverage) in a controlled trial The clinical benefits of anldquooptimalrdquo drug can be defined as 120583119892best(X)X = max120583(119860 =
0X) 120583(119860 = 1X) 120583(119860 = 119886X) and can be empiricallyestimated from the sample means of the new and standardtreatments respectively Then the alternative summary mea-sure is defined as Θ
Survival Data For survival data a clinically relevant measureis mean overall (or progression free) survival time As
survival time is continuous in nature the identical strategyprovided above for continuous outcomes can be employedhere However because the mean survival time may notbe well estimated from the observed data due to a highpercentage of censored observations [40] an alternativemean restricted survival duration was proposed and definedas the population average event-free durations for a restrictedtime of 119905lowast [41 83]Often 119905lowast is chosen to cover the trialrsquos follow-up period Mathematically it can be calculated by integratingthe survival function of 119878(119905) over the domain of (0 119905lowast) that is120583119892opt(X)X 119905lowast = int119905
lowast
0119878(119905)119889119905 and often estimated by the area
under the Kaplan-Meier curve up to 119905lowast [84] Thus an ITRrsquospotential to prolong survival can be calculated asΘ
119878119892opt(X)
119905lowast = 120583119892opt(X)X 119905lowast minus 120583119892rand(X)X 119905lowast
62 AssessingModel Performance The summaries heretoforediscussed evaluate an optimal ITR for a given model andestimating procedure Because these quantities are estimatedconditionally given the observed covariates they neglectto quantify the extent of marginal uncertainty for futurepatients Hence an ITR needs to be internally validated ifexternal data is not available [34] Cross-validation (CV) andbootstrap resampling techniques are commonly used for thispurpose [19 42 45 81] and expositions on both approachesare well described elsewhere [33 85 86]
We here briefly introduce a process that was proposed byKapelner et al [81] in the setting of personalized medicineTenfold CV is commonly used in practice where the wholedata is randomly partitioned into 10 roughly equal-sizedexclusive subsamples All methods under consideration areapplied to 910 of the data excluding 110 as an independenttesting data set The process is repeated 10 times for eachsubsample Considering the assignments recommended bythe optimal ITRs the summary measures can be calculatedusing results from each testing fold [45]TheCVprocess givesthe estimated summary measures and its variation can beevaluated using bootstrap procedures Specifically one drawsa sample with replacement from the entire data and calculatesthe summary measure from 10-fold CV This process willbe repeated 119861 times where 119861 is chosen for resolution ofthe resulting confidence intervals [81] Using the summarymeasures as119861 new random samples the correspondingmeanand variances can be calculated empirically Note that thesummary measures compare two treatment rules one for theoptimal ITR and another naive rule (eg randomization)
The above procedure can be applied to all the meth-ods we have discussed so far The 119877 software package119879119903119890119886119905119898119890119899119905119878119890119897119890119888119905119894119900119899 (httplabsfhcrcorgjanesindexhtml)can be used to implement these methods for evaluatingand comparing biomarkers for binary outcomes [47] Veryrecently an inferential procedure was proposed for contin-uous outcomes that is implemented in the publicly available119877 package ldquoPersonalized Treatment Evaluatorrdquo [81 87] Bothmethods consider data from RCTs with two arms for com-parative treatmentsThesemethods are in general applicableto regression model based methods but are not suitable forapproaches based on classification techniques or penalizedregression
10 BioMed Research International
Next we present two examples Recall in Section 55 thatKang et al [42] reported the estimated clinical benefits ofan ITR for breast cancer when compared to the defaultstrategy of assigning all patients to adjuvant chemotherapyThe proposed approach (based on boosting and classificationtrees) achieved the highest value of the summary measureat 0081 with 95 confidence interval (CI) (0000 0159)[42] In the second example introduced in Section 51 [19]the authors calculated the mean score of the HamiltonRating Scale for Depression for two groups of subjectsgroups were defined by randomly assigning patients to theldquooptimalrdquo and ldquononoptimalrdquo therapy as defined by the ITRThe reported difference between the two groups was minus178with a 119875 value of 009 which fails to attain a clinicalsignificant difference of 3 [19] The same data was analyzedby Kapelner et al [81] Following the discussed procedurethe authors reported the estimated values (and 95 CI)of Θ119862119892opt(X) and Θ
119862alt119892opt(X) as minus0842(minus2657 minus0441)
and minus0765(minus2362 0134) respectively The results whichfail to achieve clinical significance were based on rigorousstatistical methods and thus can be considered reliableestimates of the ITRrsquos performance
7 Discussion
As our understanding tumor heterogeneity evolves person-alized medicine will become standard medical practice inoncology Therefore it is essential that the oncology com-munity uses appropriate analytical methods for identifyingand evaluating the performance of personalized treatmentrules This paper provided an exposition of the process forusing statistical inference to establish optimal individualizedtreatment rules using data acquired from clinical study Thequality of an ITR depends on the quality of the design used toacquire the dataMoreover an ITRmust be properly validatedbefore it is integrated into clinical practice Personalizedmedicine in some areas of oncologymay be limited by the factthat biomarkers arising from a small panel of genesmay neveradequately characterize the extent of tumor heterogeneityinherent to the disease Consequently the available statisticalmethodology needs to evolve in order to optimally exploitglobal gene signatures for personalized medicine
The bulk of our review focused on statistical approachesfor treatment selection at a single time point The readershould note that another important area of research considersoptimal dynamic treatment regimes (DTRs) [88 89] whereintreatment decisions are considered sequentially over thecourse ofmultiple periods of intervention using each patientrsquosprior treatment history Zhao and Zeng provide a summaryof recent developments in this area [90]
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
JunshengMa was fully funded by the University of Texas MDAnderson Cancer Center internal funds Brian P Hobbs and
Francesco C Stingo were partially supported by the CancerCenter Support Grant (CCSG) (P30 CA016672)
References
[1] J Reimand O Wagih and G D Bader ldquoThe mutational land-scape of phosphorylation signaling in cancerrdquo Scientific Reportsvol 3 article 2651 2013
[2] D Hanahan and R AWeinberg ldquoHallmarks of cancer the nextgenerationrdquo Cell vol 144 no 5 pp 646ndash674 2011
[3] R Simon ldquoClinical trial designs for evaluating the medicalutility of prognostic and predictive biomarkers in oncologyrdquoPersonalized Medicine vol 7 no 1 pp 33ndash47 2010
[4] P L Bedard A R Hansen M J Ratain and L L Siu ldquoTumourheterogeneity in the clinicrdquo Nature vol 501 no 7467 pp 355ndash364 2013
[5] M D Pegram G Pauletti and D J Slamon ldquoHer-2neu as apredictive marker of response to breast cancer therapyrdquo BreastCancer Research and Treatment vol 52 no 1ndash3 pp 65ndash77 1998
[6] G J Kelloff andC C Sigman ldquoCancer biomarkers selecting theright drug for the right patientrdquoNature Reviews Drug Discoveryvol 11 no 3 pp 201ndash214 2012
[7] J ADiMasi JM Reichert L Feldman andAMalins ldquoClinicalapproval success rates for investigational cancer drugsrdquoClinicalPharmacology andTherapeutics vol 94 no 3 pp 329ndash335 2013
[8] M Hay D W Thomas J L Craighead C Economides andJ Rosenthal ldquoClinical development success rates for investiga-tional drugsrdquo Nature Biotechnology vol 32 no 1 pp 40ndash512014
[9] S S Knox ldquoFrom lsquoomicsrsquo to complex disease a systems biologyapproach to gene-environment interactions in cancerrdquo CancerCell International vol 10 article 11 2010
[10] V Deschoolmeester M Baay P Specenier F Lardon and JB Vermorken ldquoA review of the most promising biomarkersin colorectal cancer one step closer to targeted therapyrdquo TheOncologist vol 15 no 7 pp 699ndash731 2010
[11] J A Sosman K B Kim L Schuchter et al ldquoSurvival in brafV600ndashmutant advanced melanoma treated with vemurafenibrdquoThe New England Journal of Medicine vol 366 no 8 pp 707ndash714 2012
[12] S Paik S Shak G Tang et al ldquoA multigene assay to predictrecurrence of tamoxifen-treated node-negative breast cancerrdquoThe New England Journal of Medicine vol 351 no 27 pp 2817ndash2826 2004
[13] K S Albain W E Barlow S Shak et al ldquoPrognostic and pre-dictive value of the 21-gene recurrence score assay in post-menopausal women with node-positive oestrogen-receptor-positive breast cancer on chemotherapy a retrospective analysisof a randomised trialrdquoTheLancet Oncology vol 11 no 1 pp 55ndash65 2010
[14] J E Lang J S Wecsler M F Press and D Tripathy ldquoMolecularmarkers for breast cancer diagnosis prognosis and targetedtherapyrdquo Journal of Surgical Oncology vol 111 no 1 pp 81ndash902015
[15] W Werft A Benner and A Kopp-Schneider ldquoOn the identi-fication of predictive biomarkers detecting treatment-by-geneinteraction in high-dimensional datardquo Computational Statisticsand Data Analysis vol 56 no 5 pp 1275ndash1286 2012
[16] M Jenkins A Flynn T Smart et al ldquoA statisticianrsquos perspectiveon biomarkers in drug developmentrdquo Pharmaceutical Statisticsvol 10 no 6 pp 494ndash507 2011
BioMed Research International 11
[17] A J Vickers MW Kattan and D J Sargent ldquoMethod for eval-uating prediction models that apply the results of randomizedtrials to individual patientsrdquo Trials vol 8 no 1 article 14 2007
[18] H Janes M S Pepe P M Bossuyt andW E Barlow ldquoMeasur-ing the performance of markers for guiding treatment deci-sionsrdquo Annals of Internal Medicine vol 154 no 4 pp 253ndash2592011
[19] R J DeRubeis Z D Cohen N R Forand J C Fournier L AGelfand and L Lorenzo-Luaces ldquoThe personalized advantageindex translating research on prediction into individualizedtreatment recommendationsAdemonstrationrdquoPLoSONE vol9 no 1 Article ID e83875 2014
[20] D P Byar and D K Corle ldquoSelecting optimal treatment inclinical trials using covariate informationrdquo Journal of ChronicDiseases vol 30 no 7 pp 445ndash459 1977
[21] N Thatcher A Chang P Parikh et al ldquoGefitinib plus bestsupportive care in previously treated patients with refractoryadvanced non-small-cell lung cancer results from a ran-domised placebo-controlled multicentre study (iressa survivalevaluation in lung cancer)rdquo The Lancet vol 366 no 9496 pp1527ndash1537 2005
[22] A J Vickers ldquoPrediction models in cancer carerdquo CA A CancerJournal for Clinicians vol 61 no 5 pp 315ndash326 2011
[23] RM Simon ldquoSubgroup analysisrdquo inWiley Encyclopedia of Clin-ical Trials John Wiley amp Sons Hoboken NJ USA 2007
[24] S J Pocock S E Assmann L E Enos and L E Kasten ldquoSub-group analysis covariate adjustment and baseline comparisonsin clinical trial reporting current practice and problemsrdquoStatistics in Medicine vol 21 no 19 pp 2917ndash2930 2002
[25] PM Rothwell ZMehta S CHoward S A Gutnikov andC PWarlow ldquoFrom subgroups to individuals general principles andthe example of carotid endarterectomyrdquoTheLancet vol 365 no9455 pp 256ndash265 2005
[26] R Wang S W Lagakos J H Ware D J Hunter and J MDrazen ldquoStatistics in medicinemdashreporting of subgroup ana-lyses in clinical trialsrdquoTheNewEngland Journal ofMedicine vol357 no 21 pp 2108ndash2194 2007
[27] International Breast Cancer Study Group ldquoEndocrine respon-siveness and tailoring adjuvant therapy for postmenopausallymph node-negative breast cancer a randomized trialrdquo Journalof the National Cancer Institute vol 94 no 14 pp 1054ndash10652002
[28] Early Breast Cancer Trialistsrsquo Collaborative Group (EBCTCG)ldquoEffects of chemotherapy and hormonal therapy for early breastcancer on recurrence and 15-year survival an overview of therandomised trialsrdquoThe Lancet vol 365 no 9472 pp 1687ndash17172005
[29] F E Harrell K L Lee and D B Mark ldquoTutorial in biostatisticsmultivariable prognostic models issues in developing modelsevaluating assumptions and adequacy and measuring andreducing errorsrdquo Statistics in Medicine vol 15 no 4 pp 361ndash387 1996
[30] S Gill C L Loprinzi D J Sargent et al ldquoPooled analysis offluorouracil-based adjuvant therapy for stage II and III coloncancer who benefits and by how muchrdquo Journal of ClinicalOncology vol 22 no 10 pp 1797ndash1806 2004
[31] M Qian and S A Murphy ldquoPerformance guarantees forindividualized treatment rulesrdquoThe Annals of Statistics vol 39no 2 pp 1180ndash1210 2011
[32] W Lu H H Zhang and D Zeng ldquoVariable selection for opti-mal treatment decisionrdquo StatisticalMethods inMedical Researchvol 22 no 5 pp 493ndash504 2013
[33] R Kohavi ldquoA study of cross-validation and bootstrap for accu-racy estimation and model selectionrdquo in Proceedings of the 14thInternational Joint Conference on Artificial Intelligence (IJCAIrsquo95) vol 2 pp 1137ndash1145 1995
[34] E W Steyerberg A J Vickers N R Cook et al ldquoAssessing theperformance of prediction models a framework for traditionaland novel measuresrdquo Epidemiology vol 21 no 1 pp 128ndash1382010
[35] P R Rosenbaum and D B Rubin ldquoThe central role of the pro-pensity score in observational studies for causal effectsrdquo Bio-metrika vol 70 no 1 pp 41ndash55 1983
[36] R B drsquoAgostino Jr ldquoTutorial in biostatistics propensity scoremethods for bias reduction in the comparison of a treatment toa non-randomized control grouprdquo Statistics in Medicine vol 17no 19 pp 2265ndash2281 1998
[37] R Pazdur ldquoEndpoints for assessing drug activity in clinicaltrialsrdquoThe Oncologist vol 13 supplement 2 pp 19ndash21 2008
[38] S L Spruance J E Reid M Grace and M Samore ldquoHazardratio in clinical trialsrdquo Antimicrobial Agents and Chemotherapyvol 48 no 8 pp 2787ndash2792 2004
[39] J D Kalbeisch and R L Prentice The Statistical Analysis ofFailure Time Data vol 360 John Wiley amp Sons 2011
[40] Y Geng Flexible Statistical Learning Methods for SurvivalData Risk Prediction and Optimal Treatment Decision NorthCarolina State University 2013
[41] J Li L Zhao L Tian et alAPredictive Enrichment Procedure toIdentify Potential Responders to a NewTherapy for RandomizedComparative Controlled Clinical Studies Harvard UniversityBiostatisticsWorking Paper SeriesWorking Paper 169 HarvardUniversity 2014
[42] C Kang H Janes and Y Huang ldquoCombining biomarkers tooptimize patient treatment recommendationsrdquo Biometrics vol70 no 3 pp 695ndash720 2014
[43] B Zhang A A Tsiatis E B Laber and M Davidian ldquoA robustmethod for estimating optimal treatment regimesrdquo Biometricsvol 68 no 4 pp 1010ndash1018 2012
[44] B Zhang A A Tsiatis M Davidian M Zhang and E LaberldquoEstimating optimal treatment regimes from a classificationperspectiverdquo Stat vol 1 no 1 pp 103ndash114 2012
[45] J C Foster J M G Taylor and S J Ruberg ldquoSubgroup identi-fication from randomized clinical trial datardquo Statistics in Medi-cine vol 30 no 24 pp 2867ndash2880 2011
[46] US Food and Drug Administration Guidance for IndustryClinical Trial Endpoints for the Approval of Cancer Drugs andBiologics US Food and Drug AdministrationWashington DCUSA 2007
[47] H Janes M D Brown M Pepe and Y Huang ldquoStatisticalmethods for evaluating and comparing biomarkers for patienttreatment selectionrdquo UW Biostatistics Working Paper SeriesWorking Paper 389 2013
[48] R Development Core Team R a language and environment forstatistical computing 2008 httpwwwR-projectorg
[49] X-JMa ZWang P D Ryan et al ldquoA two-gene expression ratiopredicts clinical outcome in breast cancer patients treated withtamoxifenrdquo Cancer Cell vol 5 no 6 pp 607ndash616 2004
[50] X-JMa S GHilsenbeckWWang et al ldquoTheHOXB13IL17BRexpression index is a prognostic factor in early-stage breastcancerrdquo Journal of Clinical Oncology vol 24 no 28 pp 4611ndash4619 2006
[51] D C Sgroi E Carney E Zarrella et al ldquoPrediction of latedisease recurrence and extended adjuvant letrozole benefit by
12 BioMed Research International
the HOXB13IL17BR biomarkerrdquo Journal of the National CancerInstitute vol 105 no 14 pp 1036ndash1042 2013
[52] M S Cepeda R Boston J T Farrar and B L Strom ldquoCom-parison of logistic regression versus propensity score when thenumber of events is low and there are multiple confoundersrdquoThe American Journal of Epidemiology vol 158 no 3 pp 280ndash287 2003
[53] P C Austin ldquoAn introduction to propensity score methods forreducing the effects of confounding in observational studiesrdquoMultivariate Behavioral Research vol 46 no 3 pp 399ndash4242011
[54] G Heinze and P Juni ldquoAn overview of the objectives of andthe approaches to propensity score analysesrdquo European HeartJournal vol 32 no 14 Article ID ehr031 pp 1704ndash1708 2011
[55] L E Braitman and P R Rosenbaum ldquoRare outcomes commontreatments analytic strategies using propensity scoresrdquo Annalsof Internal Medicine vol 137 no 8 pp 693ndash695 2002
[56] D M Witten and R Tibshirani ldquoSurvival analysis withhigh-dimensional covariatesrdquo Statistical Methods in MedicalResearch vol 19 no 1 pp 29ndash51 2010
[57] A E Hoerl and R W Kennard ldquoRidge regression biasedestimation for nonorthogonal problemsrdquoTechnometrics vol 42no 1 pp 80ndash86 2000
[58] R Tibshirani ldquoRegression shrinkage and selection via the lassordquoJournal of the Royal Statistical Society Series B Methodologicalvol 58 no 1 pp 267ndash288 1996
[59] H Zou ldquoThe adaptive lasso and its oracle propertiesrdquo Journal ofthe American Statistical Association vol 101 no 476 pp 1418ndash1429 2006
[60] B Efron T Hastie I Johnstone and R Tibshirani ldquoLeast angleregressionrdquo The Annals of Statistics vol 32 no 2 pp 407ndash4992004
[61] T Hastie and B Efron ldquolars Least angle regression lasso andforward stagewiserdquo R package version 12 2013 httpcranr-projectorgwebpackageslarsindexhtml
[62] J Fan and R Li ldquoVariable selection via nonconcave penalizedlikelihood and its oracle propertiesrdquo Journal of the AmericanStatistical Association vol 96 no 456 pp 1348ndash1360 2001
[63] H Zou and T Hastie ldquoRegularization and variable selection viathe elastic netrdquo Journal of the Royal Statistical Society Series BStatistical Methodology vol 67 no 2 pp 301ndash320 2005
[64] N Kramer J Schafer and A-L Boulesteix ldquoRegularized esti-mation of large-scale gene association networks using graphicalgaussian modelsrdquo BMC Bioinformatics vol 10 no 1 article 3842009
[65] P Breheny and J Huang ldquoCoordinate descent algorithms fornonconvex penalized regression with applications to biologicalfeature selectionrdquo The Annals of Applied Statistics vol 5 no 1pp 232ndash253 2011
[66] S M Hammer D A Katzenstein M D Hughes et al ldquoA trialcomparing nucleoside monotherapy with combination therapyin HIV-infected adults with CD4 cell counts from 200 to 500per cubicmillimeterrdquoTheNew England Journal ofMedicine vol335 no 15 pp 1081ndash1090 1996
[67] H M Boslashvelstad S Nygard H L Stoslashrvold et al ldquoPredictingsurvival from microarray datamdasha comparative studyrdquo Bioinfor-matics vol 23 no 16 pp 2080ndash2087 2007
[68] V Kehl and K Ulm ldquoResponder identification in clinical trialswith censored datardquoComputational Statistics andDataAnalysisvol 50 no 5 pp 1338ndash1355 2006
[69] P Royston and M K Parmar ldquoThe use of restricted meansurvival time to estimate the treatment effect in randomizedclinical trials when the proportional hazards assumption is indoubtrdquo Statistics inMedicine vol 30 no 19 pp 2409ndash2421 2011
[70] P Royston andM K B Parmar ldquoRestrictedmean survival timean alternative to the hazard ratio for the design and analysis ofrandomized trials with a time-to-event outcomerdquo BMCMedicalResearch Methodology vol 13 no 1 article 152 2013
[71] E T Lee and J W Wang Statistical Methods for Survival DataAnalysis John Wiley amp Sons Hoboken NJ USA 2013
[72] Y Zhao D Zeng A J Rush and M R Kosorok ldquoEstimatingindividualized treatment rules using outcome weighted learn-ingrdquo Journal of the American Statistical Association vol 107 no499 pp 1106ndash1118 2012
[73] D B Rubin and M J van der Laan ldquoStatistical issues andlimitations in personalized medicine research with clinicaltrialsrdquoThe International Journal of Biostatistics vol 8 no 1 pp1ndash20 2012
[74] L Breiman ldquoRandom forestsrdquoMachine Learning vol 45 no 1pp 5ndash32 2001
[75] J Friedman T Hastie and R Tibshirani ldquoAdditive logisticregression a statistical view of boostingrdquo The Annals of Statis-tics vol 28 no 2 pp 337ndash407 2000
[76] C Cortes and V Vapnik ldquoSupport-vector networksrdquo MachineLearning vol 20 no 3 pp 273ndash297 1995
[77] L Breiman J Friedman C J Stone and R A Olshen Classi-fication and Regression Trees CRC Press New York NY USA1984
[78] S Dudoit J Fridlyand and T P Speed ldquoComparison of dis-criminationmethods for the classification of tumors using geneexpression datardquo Journal of the American Statistical Associationvol 97 no 457 pp 77ndash87 2002
[79] A Liaw and MWiener ldquoClassification and regression by rand-omforestrdquoRNews vol 2 no 3 pp 18ndash22 2002 httpCRANR-projectorgdocRnews
[80] T Therneau B Atkinson and B Ripley ldquorpart RecursivePartitioning and Regression Treesrdquo R package version 41-3httpcranr-projectorgwebpackagesrpartindexhtml
[81] A Kapelner J Bleich Z D Cohen R J DeRubeis and RBerk ldquoInference for treatment regime models in personalizedmedicinerdquo httparxivorgabs14047844
[82] X Song and M S Pepe ldquoEvaluating markers for selecting apatientrsquos treatmentrdquoBiometrics vol 60 no 4 pp 874ndash883 2004
[83] T Karrison ldquoRestricted mean life with adjustment for covari-atesrdquo Journal of the American Statistical Association vol 82 no400 pp 1169ndash1176 1987
[84] C Barker ldquoThe mean median and confidence intervals ofthe kaplan-meier survival estimatemdashcomputations and appli-cationsrdquo Journal of the American Statistical Association vol 63no 1 pp 78ndash80 2009
[85] B Efron and R J Tibshirani An Introduction to the Bootstrapvol 57 CRC Press 1994
[86] S Arlot andA Celisse ldquoA survey of cross-validation proceduresfor model selectionrdquo Statistics Surveys vol 4 pp 40ndash79 2010
[87] A Kapelner and J Bleich ldquoPTE Personalized Treatment Eva-luatorrdquo 2014 R package version 10 httpCRANR-projectorgpackage=PTE
[88] S A Murphy ldquoOptimal dynamic treatment regimesrdquo Journal ofthe Royal Statistical Society Series B StatisticalMethodology vol65 no 2 pp 331ndash355 2003
BioMed Research International 13
[89] J M Robins ldquoOptimal structural nested models for optimalsequential decisionsrdquo in Proceedings of the Second Seattle Sym-posium in Biostatistics vol 179 of Lecture Notes in Statistics pp189ndash326 Springer Berlin Germany 2004
[90] Y Zhao and D Zeng ldquoRecent development on statistical meth-ods for personalized medicine discoveryrdquo Frontiers of Medicinein China vol 7 no 1 pp 102ndash110 2013
was first discovered as an independent prognostic biomarkerfor ER+ node-negative patients using retrospective datafrom 60 patients [49] These findings were confirmed onan independent data set comprising 852 tumors which wasacquired from a tumor bank at the Breast Center of BaylorCollege of Medicine [50] Interestingly the two-gene ratioindex (HOXB13IL17BR) was reported to predict the benefitof treatment with letrozole in one recent independent study[51]
Methods based on propensity scores are commonly usedto attenuate selection bias [35] In essence these approachesuse the available covariates to attempt to diminish the effectsof imbalances among variables that are not of interest fortreatment comparison Moreover they have been shownto be robust in the presence of multiple confounders andrare events [52] Generally after adjusting for bias usingpropensity scores the same principles for deriving ITRs fromRCTs may be applied to the observational cohort
The propensity score characterizes the probability ofassigning a given treatment 119860 from the available covariatesX [35] Using our notation the propensity score is 120587(X 120585) =119875(119860 = 1 | X 120585) which can be modeled using logisticregression
where 119901 is the number of independent variables used toconstruct the propensity score and 120585
119895represents the 119895th
regression coefficient which characterizes the 119895th covariatersquospartial effect After fitting the data to obtain estimates forthe regression coefficients the estimated probability ofreceiving new treatment can be obtained for each patient(Xi) = 120587(Xi ) by inverting the logit function The eventthat asymp 0 implies that the measured independent variablesare reasonably ldquobalancedrdquo between treatment cohorts Inpractice one often includes as many baseline covariates intothe propensity score model as permitted by the sample size
Methods that use propensity scores can be categorizedinto four categories matching stratification adjusting andinverse probability weighted estimation [36 53] Matchingand stratification aim to mimic RCTs by defining a newdataset using propensity scores such that outcomes aredirectly comparable between treatment cohorts [53] Thesetwo approaches are well suited for conventional subgroupanalysis but their application to personalized medicine hasbeen limited Regression adjustment or simply adjusting canbe used to reduce bias due to residual differences in observedbaseline covariates between treatment groups This methodincorporates the propensity scores as an independent variablein a regression model and therefore can be used in con-junction with all regression-based methods [36] Methodsinvolving inverse probability weighted estimators will bediscussed in Section 551 [43]
Of course propensity scores methods may only attenuatethe effects of the important confounding variables that havebeen acquired by the study design Casual inference in gen-eral is not robust to the presence of unmeasured confounders
that influenced treatment assignment [35 54 55] For thedevelopment of ITRs predictive and important prognosticcovariates can be incorporated in the regression model forthe clinical outcome119884 alongwith the propensity scores whileother covariatesmay be utilized only in themodel for estimat-ing the propensity scores Hence propensity score methodsmay offer the researcher a useful tool for controlling forpotential confounding due to selection bias andmaintaining amanageable number of prognostic and predictive covariates
53 Methods for High-Dimensional Biomarkers The meth-ods presented in the previous sections are appropriate foridentifying an ITR using a small set of biomarkers (low-dimensional)However recent advances inmolecular biologyin oncology have enabled researchers to acquire vast amountsof genetic and genomic characteristics on individual patientsOften the number of acquired genomic covariates will exceedthe sample size Proper analysis of these high-dimensionaldata sources poses many analytical challenges Several meth-ods have been proposed specifically for analysis of high-dimensional covariates [56] although the majority of thesemethods are well suited only for the analysis of prognosticbiomarkers In what follows we introduce variable selectionmethods that were developed to detect predictive biomarkersfrom high-dimensional sources as well as describing how toconstruct optimal ITRs from the final set of biomarkers
An appropriate regressionmodel can be defined generallyas 119864(119884 | 119860X) = ℎ
0(X) + 119860(X120573) where ℎ
0(X) is an unspe-
cified baseline mean function 120573 = (1205730 1205731 120573
119902)119879 is a
column vector of regression coefficients and X = (1X) thedesign matrix Subscript 119902 denotes the total number of bio-markers which may be larger than the sample size 119899 AnITR derives from evaluating the interactions in 119860(X120573) notthe baseline effect of the high-dimensional covariates ℎ
0(X)
[32] Technically function 119860(X120573) = 119860(1205730+ 12057311198831+ 12057321198832+
sdot sdot sdot + 120573119902119883119902) cannot be uniquely estimated using traditional
maximum likelihood-based methods when 119902 gt 119899 [57]Yet practically many of the available biomarkers may notinfluence the optimal ITR [31] Thus the process for identifyITRs from a high-dimensional source requires that we firstidentify a sparse subset of predictive biomarkers that can beutilized for constructing the ITR
Parameters for the specifiedmodel can be estimated usingthe following loss function
119871119899120601(120573 120574) =
1
119899
119899
sum119894=1
[119884119894minus 120601 (X
119894 120574) minus X120573 119860
119894minus 120587 (X
119894)]2
(7)
where 120601(X 120574) represents any arbitrary function character-izing the ldquobaselinerdquo relationship between X and Y (eg anintercept or an additive model) Here we let 120587(X
119894) = 119875(119860
119894=
1 | X119894) denote either a propensity score (for observational
data) or a randomization probability (eg 05 given 1 1randomization) for RCT data If 120587(X) is known estimationusing this model yields unbiased estimates (asymptoticallyconsistent) of the interaction effects 120573 even if the main effectsare not correctly specified providing a robustness [32]
6 BioMed Research International
Penalized estimation provides the subset of relevantpredictivemarkers that are extracted from the nonzero coeffi-cients of the corresponding treatment-biomarker interactionterms of
where 120582119899is a tuning parameter which is often selected via
cross validation and 119869 is a shrinkage penalty Different choicesof 119869 lead to different types of estimators For example thelasso penalized regression corresponds to 119869 = 1 [58] andthe adaptive lasso to 119869 = 120596
119895= 1|120573init119895| where 120573init119895 is an
initial estimate of 120573119895[59] With little modification (8) can
be solved using the LARS algorithm implemented with the119877 package of 119897119886119903119904 [32 60 61] As we have shown before atreatment rule can be defined from the parameter estimatesas 119868120573
formmay have zero estimates for some coefficients (eg 1205732=
1205735= sdot sdot sdot = 120573
119902= 0) hence an ITR can be equivalently con-
structed from the final estimated nonzero coefficients and thecorresponding covariates
Alternative penalized regression approaches includeSCAD [62] and elastic-net [63] All penalized approachesproduce sparse solutions (ie identifying a small subset ofpredictive biomarkers) however the adaptive lasso is lesseffective when 119901 gt 119899 Methods that produce nonsparsemodels such as ridge regression [57] are less preferable sinceITRs based on many biomarkers are often unstable and lessuseful in practice [31] Several packages in 119877 offer imple-mentation of penalized regression such as 119901119886119903119888119900119903 for ridgelasso and adaptive lasso and 119899119888V119903119890119892 for SCAD [64 65]
Lu et al [32] used a penalized regression approach toanalyze data from the AIDS Clinical Trials Group Protocol175 (ACTG175) [66] In this protocol 2000 patients wereequally randomized to one of four treatments zidovudine(ZDV) monotherapy ZDV + didanosine (ddI) ZDV +zalcitabine and ddI monotherapy CD4 count at 15ndash25 weekspostbaselinewas the primary outcome and 12 baseline covari-ates were included in the analysis The resulting treatmentrule favored the combined regimes over ZDV monotherapyMoreover the treatment rule determined that ZDV + ddIshould be preferred to ddI when 119868(7159 + 107 times ageminus 018 timesCD40 minus 3357 times homo) = 1 where CD40 represents baselineCD4 counts and homo represents homosexual activity Basedon this treatment rule 878 patients would have benefitedfrom treatment with ZDV + ddI
54 Survival Analysis Heretofore we have discussed meth-ods for continuous or binary outcomes yet often investigatorswant to discern the extent to which a therapeutic interventionmay alter the amount of time required before an event occursThis type of statistical inference is referred to broadly assurvival analysis One challenge for survival analysis is thatthe outcomes may be only partially observable at the time ofanalysis due to censoring or incomplete follow-up Survivalanalysis has been widely applied in cancer studies often inassociation studies aimed to identify prognostic biomarkers
[56 67] Here we discuss twowidely usedmodels for derivingITRs using time-to-event data namely Cox regression andaccelerated failure time models
The Cox regression model follows as
120582 (119905 | X 119860) = 1205820(119905) exp 120573
11198831+ 12057321198832+ 119860 (120573
4+ 12057351198832)
(9)
where 119905 is the survival time 1205820(119905) is an arbitrary baseline
hazard function and 1198831 1198832represent prognostic and pre-
dictive biomarkers respectively Each 120573 characterizes themultiplicative effect on the hazard associated with a unitincrease in the corresponding covariate Therefore Coxmodels are referred to as proportional hazards (PH) models
Several authors have provided model building strategies[29] and approaches for treatment selection [20 30 68] Fol-lowing the previously outlined strategy a naive approach forderiving an ITR uses the hazard ratio (new treatment versusthe standard) as the treatment contrast which can be calcu-lated as Δ(X) = exp(120573
4+ 12057351198832) The ITR therefore is 119868(120573
4
+12057351198832) lt 0 There are obvious limitations to this approach
First violations of the PH assumption yield substantiallymis-leading results [69]Moreover evenwhen the PH assumptionis satisfied because the Cox model does not postulate adirect relationship between the covariate (treatment) and thesurvival time the hazard ratio fails to measure the extent towhich the treatment is clinically valuable [38 70]
Accelerated failure time (AFT) models provide an alter-native semiparametric model Here we introduce its appli-cation for high-dimensional data Let 119879 and 119862 denote thesurvival and censoring times and denote the observed databy ( 120575 119860X) where = min(119879 119862) and 120575 = 119868(119879 lt 119862)Define the log survival time as 119884 = log(119879) a semiparametricregression model is given as 119864(119884 | 119860X) = ℎ
0(X) + 119860(X120573)
where ℎ0(X) is the unspecified baseline mean function
Similar to the previous section the treatment rule is 119868(1205730+
independent censoring the AFT model parameters can beestimated by minimizing the following loss function
119871119899120601(120573) =
1
119899
119899
sum119894=1
120575119894
119866(119894)[119894minus 120601 (X
119894 120574) minus X120573 119860
119894minus 120587 (X
119894)]2
(10)
where 119894= log(
119894) 120587(X
119894) = 119875(119860
119894= 1 | X
119894) is the propensity
score or randomization probability 119866(sdot) is the Kaplan-Meierestimator of the survival function of the censoring time and120601(X 120574) characterizes any arbitrary function
Thismethod can be extended to accommodatemore thantwo treatments simultaneously by specifying appropriatetreatment indicators For instance the mean function can bemodeled as 119864(119884 | 119860X) = ℎ
0(X) + 119860
(1)X120573(1) + 119860(2)X120573(2)
for two treatment drugs versus the standard care The ITRassigns the winning drug Note this work was proposed by[40] and is an extension of [32] to the survival setting Henceit shares the robustness property and can be applied to obser-vational data For implementation the sameprocedure can befollowed to obtain estimates with one addition step of calcu-lating 119866(
119894) There are several 119877 packages for Kaplan-Meier
BioMed Research International 7
estimates and Cox regression models These sources can befound at httpcranr-projectorgwebviewsSurvivalhtmlMore details pertaining to statistical methods for survivalanalysis can be found here [71] To compare treatmentrules constructed from Cox and AFT models for examplemethods for measuring the extent of clinical effectiveness foran ITR will be discussed in Section 6
We here present an example when an AFT model wasused to construct an ITR for treatment of HIV [40] Theexample derives from the AIDS Clinical Trials Group Pro-tocol 175 that was discussed in Section 53 [32 66] In thiscase the primary outcome variable was time (in days) tofirst ge50 decline in CD4 count or an AIDS-defining eventor death A total of 12 covariates and four treatments (ZDVZDV + ddI ZDV + zalcitabine and ddI) were includedThe four treatments were evaluated simultaneously Patientsreceiving the standard care of ZDV monotherapy wereconsidered as the reference group Hence three treatmentcontrasts (119868ZDV+ddI 119868ZDV+zalcitabine and 119868ddI) were combinedwith various putative predictive covariates and comparedwith ZDV monotherapy For example gender was detectedas the predictive covariate only for ddI monotherapy Theinvestigators assumed 120601(X 120574) = 120574
0 The treatment rule
recommended 1 patient for ZDV monotherapy while 7291216 and 193 patients were recommended for ZDV + ddIZDV + zalcitabine and ddI respectively
55 Advanced Methods
551 Robust Inference The performances of ITRs heretoforepresented depend heavily on whether the statistical modelswere correctly specified Recently there has been much atten-tion focused on the development of more advanced methodsand modeling strategies that are robust to various aspectsof potential misspecification We have already presented afew robust models that avoid specification of functionalparametric relationships for main effects [32 40] Here weintroduce two more advanced methods widely utilized forITRs that are robust to the type of misspecification issuescommonly encountered in practice [42 43]
Recall that the ITR for a linear model 119864(119884 | 119860 =
119886X) = 120583(119860 = 119886X120573) with two predictive markers followsas 119892(X120573) = 119868(120573
4+ 12057351198832+ 12057361198833) gt 0 where 119886 = 0 1
The treatment rule of 119892(X120573) may use only a subset of thehigh-dimensional covariates (eg 119883
2 1198833) but it always
depends on the correct specification of 119864(119884 | 119860 = 119886X)Defining a scaled version of 120573 as 120578(120573) the correspondingITR is 119892(120578X) = 119892(X120573) = 119868(119883
3gt 1205780+ 12057811198832) where
1205780= minus12057341205736and 120578
1= 12057351205736 If the model for 120583(119860X120573) is
indeed correctly specified the treatment rules of 119892(X120573) and119892(120578X) lead to the same optimal ITR Hence the treatmentrule parameterized by 120578 can be derived from a regressionmodel or may be based on some key clinical considerationswhich enable evaluation of 119892(120578X) directly without referenceto the regression model for 120583(119860X120573)
Let 119862120578
= 119860119892(120578X) + (1 minus 119860)1 minus 119892(120578X) where119862120578= 1 indicates random assignment to an intervention
that is recommended by the personalized treatment rule 119892(120578X) Let 120587(X ) denote the randomization ratio or the
estimated propensity score (as in previous section) and119898(X 120578 ) denote the potential outcome under the treatmentrule estimated from the following model 119864(119884 | 119860 =
119886X) = 120583(119860X120573) For example if the treatment rule 119892(120578X)= 1 then 119898(X 120578 ) = 119892(120578X)120583(119860 = 1X ) + 1 minus 119892(120578
X)120583(119860 = 0X ) = 120583(119860 = 1X ) Two estimators ofthe expected response to treatment the inverse probabilityweighted estimator (IPWE) and doubly robust AIPWE aregiven as follows
IPWE (120578) = 1
119899
119899
sum119894=1
119862120578sdot119894119884119894
120587119888(X119894 120578 )
=1
119899
119899
sum119894=1
119862120578sdot119894119884119894
120587 (X119894 )119860119894 1 minus 120587 (X
119894 )1minus119860119894
AIPWE (120578) = 1
119899
119899
sum119894=1
119862120578sdot119894119884119894
120587119888(X119894 120578 )
minus119862120578sdot119894119884119894 minus 120587119888 (X119894 120578 )
120587119888(X119894 120578 )
119898 (X119894 120578 )
(11)
where 120587119888(X119894 120578 ) = 120587(X )119892(120578X) + 1 minus 120587(X )1 minus
119892(120578X) The optimal treatment rule follows as 119892(X = x)where is estimated from the above models a constraintsuch as 120578 = 1 is imposed to obtain a unique solution [43] If the propensity score is correctly specified theIPWE estimator yields robust (consistent) estimates AIPWEis considered a doubly robust estimator since it produces con-sistent estimates when either propensity score or the model119864(119884 | 119860 = 119886X) is misspecified but not both [42 43] Thecompanion119877 code is publicly available at httponlinelibrarywileycomdoi101111biom12191suppinfo
552 Data Mining and Machine Learning The methodspresented in Section 551 are robust against misspecificationof regression models Yet they often require prespecificationof the parametric form for the treatment rule (eg 119868(119883
3gt
1205780+ 12057811198832)) which can be practically challenging [44]
Well established classification methods and other popularmachine learning techniques can alternatively be customizedto define treatment selection rules [44 72 73] these methodsavoid prespecification of the parametric form of the ITR AnITR can be defined following a two-step approach in thefirst step treatment contrasts are estimated from a positedmodel and in the second step classification techniques areapplied to determine the personalized treatment rules Forexample when only two treatments are considered a newvariable 119885 can be defined based on the treatment contrastthat is 119885 = 1 if Δ(X) = 120583(119860 = 1X) minus 120583(119860 = 0X) gt 0 and119885 = 0 otherwiseThe absolute value of the treatment contrast119882119894= |Δ(X)| can be used in conjunction with a classification
technique to define an appropriate ITR [44]Unlike classification problems wherein the class labels
are observed for the training data the binary ldquoresponserdquovariable 119885 which serves as the class label is not availablein practice Specifically patients who are in the class 119885 = 1
8 BioMed Research International
have 120583(119860 = 1X) gt 120583(119860 = 0X) and should thereforebe treated with the new therapy however these quantitiesneed to be estimated since patients are typically assigned toonly one of the available treatments This imparts flexibilityfor estimation of the optimal treatment regimes since anyof the previously discussed regression models and even someensemble prediction methods such as random forest [74] canbe used to construct the class labels 119885
119894and weights
119894[44]
An ITR can be estimated from the dataset 119885119894X119894 119894 using
any classification approach where 119894are subject specific
misclassification weights [44 45] This includes popularclassificationmethods such as adaptive boosting [75] supportvector machines [76] and classification and regression trees(CART) [77] At least one study has suggested that SVMoutperforms other classification methods in this contextwhereas random forest and boosting perform comparativelybetter than CART [78] However the performances ofthese classification algorithms are data dependent Definitiveconclusion pertaining to their comparative effectiveness ingeneral has yet to be determined [78] It shall be also notedthat these classification methods can be also applied to high-dimensional data [45 72]
One special case of this framework is the ldquovirtue twinsrdquoapproach [45] Specifically in the first step a random forestapproach [74] is used to obtain the treatment contrasts Thenin the second step CART is used to classify subjects tothe optimal treatment regime The approach can be easilyimplemented in 119877 using packages of randomForest [79] andrpart [80] Very recently Kang et al [42] proposed amodifiedversion of the adaptive boosting technique of Friedman et al[75] The algorithm iteratively fits a simple logistic regressionmodel (ldquoworking modelrdquo) to estimate 119875(119884 = 1 | 119860X) and ateach stage assigns higher weights to subjects whose treatmentcontrast is near zero After a prespecified stopping criterionis met an average of the treatment contrasts Δ(X) is calcu-lated for each patient using all models fitted at each iterationA subject is assigned to the new therapy if Δ(X) gt 0 The 119877code for the aforementioned boosting methods is publiclyavailable at httponlinelibrarywileycomdoi101111biom12191suppinfo
Lastly we present a breast cancer example where severalbiomarkers were combined to construct an optimal ITRThe data was collected in the Southwest Oncology Group(SOWG)-SS8814 trial [13] and analyzed with the machinelearning approach of Kang et al [42] Three hundred andsixty-seven node-positive ER-positive breast cancer patientswere selected from the randomized trial of SOWG A total of219 received tamoxifen plus adjuvant chemotherapy and 148was given tamoxifen aloneThe outcome variable was definedas breast cancer recurrence at 5 years The authors selectedthree genes which had presented treatment-biomarker inter-actions in amultivariate linear logistic regressionmodel [42]Data were analyzed with logistic models IPWE AIPWElogistic boosting a single classification tree with treatment-biomarker interactions and the proposed boosting approachwith a classification tree as the working model Each methodidentified different patient cohorts that could benefit fromtamoxifen alone these cohorts consisted of 184 183 128 86
263 and 217 patients respectively (see Table 5 in [42]) In thisanalysis the clinical benefits provided by these 6 treatmentrules were not statistically different Hence investigatorsneed to evaluate and compare ITRs in terms of the extentof expected clinical impact This is considered in the nextsection
Heretofore we have discussed various methodologies for theconstruction of ITR while their performances need to beassessed before these rules can be implemented in clinicalpractice Several aspects pertaining to the performance of aconstructed ITR need to be considered The first one is howwell the ITR fits the data and the second is how well theITR performs compared with existing treatment allocationrules The former is related to the concept of goodness-of-fitor predictive performance [34] As the true optimal treatmentgroups are hidden model fits may be evaluated by measuringthe congruity between observed treatment contrasts andpredicted ones [34 47] More details can be found in a recentpaper by Janes et al [47] Performances of ITRs can becompared via assessment of a global summary measure forexample prolonged survival time or reduced disease rate [4042] Summarymeasures are also very useful for evaluating theextent to which an ITR may benefit patients when applied inpractice Moreover it is essential that performance of an ITRis considered in comparison to business-as-usual proceduressuch as a naive rule that randomly allocates patients totreatment [81] Summary measures will be discussed inSection 61 The effectiveness of an ITR should go beyond thetraining data set used to construct a treatment rule cross-validation and bootstrapping techniques are often employedto assess the impact of ITRs on future patients [81] and willbe discussed in Section 62
61 Summary Measures ITRs may be derived from differ-ent methodologies and comparisons should be conductedwith respect to the appropriate clinically summaries A fewsummary measures for different types of outcomes havebeen proposed [19 40 42] these measures quantify thedirect clinical improvements obtained by applying an ITR incomparison with default methods for treatment allocation
Binary Outcomes Clinical effectiveness for binary clinicalresponse is represented by the difference in disease rates (ortreatment failure) induced by ITR versus a default strategythat allocates all patients to a standard treatment [42 47 82]Let 119892opt(X) = 119868120583(119860 = 1X)minus120583(119860 = 0X) lt 0 be an optimalITR This difference is formally defined as
Note 120583(119860X) needs to be estimated to construct the ITRyet parameters 120573 are omitted for simplicity Larger valuesof Θ119861119892opt(X) indicate increased clinical value for the
biomarker driven ITR A subset of patients that are recom-mended for new treatment (119860 = 1) under an ITR may havebeen randomly selected to receive it while the remainingsubset of ldquounluckyrdquo patientswould have received the standardtreatment [19] The summary measure of Θ
119861119892opt(X) char-
acterizes a weighted difference in the disease rates betweenthe standard and the new treatments in a population whereinthe constructed optimal ITR would recommend the newtreatment 119892opt(X = 1) The weight is the proportion ofpatients identified by the optimal ITR for the new treatmentand can be empirically estimated using the correspondingcounts For example 119875119892opt(X) = 1 can be estimatedusing the number of patients recommended for the newtreatment divided by the total sample size A similar summarystatistic can be derived for an alternative strategy allocatingall patients to the new treatment The summary could beapplied to the aforementioned breast cancer example [42]for example with the aim of finding a subgroup of patientswho were likely to benefit from adjuvant chemotherapywhile those unlikely to benefit would be assigned tamoxifenalone to avoid the unnecessary toxicity and inconvenience ofchemotherapy
Continuous Variables Another strategy for continuous datacompares outcomes observed for ldquoluckyrdquo subjects those whoreceived the therapy that would have been recommendedby the ITR based [81] Further one business-as-usual drugallocation procedure is randomizing treatment and standardcare at the same probability of 05 A summary statisticis to measure the mean outcome under ITR compared tothat obtained under random assignment for instance themean decrease in Hamilton Rating Scale for Depression asdiscussed in Section 51 [19] Define the summary measureas Θ119862119892opt(X) = 120583119892opt(X)X minus 120583119892rand(X)X where
119892rand(X) represents the randomization allocation procedureThe quantity of 120583119892119900119901119905(X)X represents the mean outcomeunder the constructed IRT that can be empirically estimatedfrom the ldquoluckyrdquo subjects and 120583119892rand(X)X can be esti-mated empirically from the sample means
Alternatively an ITR may be compared to an ldquooptimalrdquodrug that has showed universal benefits (a better drug onaverage) in a controlled trial The clinical benefits of anldquooptimalrdquo drug can be defined as 120583119892best(X)X = max120583(119860 =
0X) 120583(119860 = 1X) 120583(119860 = 119886X) and can be empiricallyestimated from the sample means of the new and standardtreatments respectively Then the alternative summary mea-sure is defined as Θ
Survival Data For survival data a clinically relevant measureis mean overall (or progression free) survival time As
survival time is continuous in nature the identical strategyprovided above for continuous outcomes can be employedhere However because the mean survival time may notbe well estimated from the observed data due to a highpercentage of censored observations [40] an alternativemean restricted survival duration was proposed and definedas the population average event-free durations for a restrictedtime of 119905lowast [41 83]Often 119905lowast is chosen to cover the trialrsquos follow-up period Mathematically it can be calculated by integratingthe survival function of 119878(119905) over the domain of (0 119905lowast) that is120583119892opt(X)X 119905lowast = int119905
lowast
0119878(119905)119889119905 and often estimated by the area
under the Kaplan-Meier curve up to 119905lowast [84] Thus an ITRrsquospotential to prolong survival can be calculated asΘ
119878119892opt(X)
119905lowast = 120583119892opt(X)X 119905lowast minus 120583119892rand(X)X 119905lowast
62 AssessingModel Performance The summaries heretoforediscussed evaluate an optimal ITR for a given model andestimating procedure Because these quantities are estimatedconditionally given the observed covariates they neglectto quantify the extent of marginal uncertainty for futurepatients Hence an ITR needs to be internally validated ifexternal data is not available [34] Cross-validation (CV) andbootstrap resampling techniques are commonly used for thispurpose [19 42 45 81] and expositions on both approachesare well described elsewhere [33 85 86]
We here briefly introduce a process that was proposed byKapelner et al [81] in the setting of personalized medicineTenfold CV is commonly used in practice where the wholedata is randomly partitioned into 10 roughly equal-sizedexclusive subsamples All methods under consideration areapplied to 910 of the data excluding 110 as an independenttesting data set The process is repeated 10 times for eachsubsample Considering the assignments recommended bythe optimal ITRs the summary measures can be calculatedusing results from each testing fold [45]TheCVprocess givesthe estimated summary measures and its variation can beevaluated using bootstrap procedures Specifically one drawsa sample with replacement from the entire data and calculatesthe summary measure from 10-fold CV This process willbe repeated 119861 times where 119861 is chosen for resolution ofthe resulting confidence intervals [81] Using the summarymeasures as119861 new random samples the correspondingmeanand variances can be calculated empirically Note that thesummary measures compare two treatment rules one for theoptimal ITR and another naive rule (eg randomization)
The above procedure can be applied to all the meth-ods we have discussed so far The 119877 software package119879119903119890119886119905119898119890119899119905119878119890119897119890119888119905119894119900119899 (httplabsfhcrcorgjanesindexhtml)can be used to implement these methods for evaluatingand comparing biomarkers for binary outcomes [47] Veryrecently an inferential procedure was proposed for contin-uous outcomes that is implemented in the publicly available119877 package ldquoPersonalized Treatment Evaluatorrdquo [81 87] Bothmethods consider data from RCTs with two arms for com-parative treatmentsThesemethods are in general applicableto regression model based methods but are not suitable forapproaches based on classification techniques or penalizedregression
10 BioMed Research International
Next we present two examples Recall in Section 55 thatKang et al [42] reported the estimated clinical benefits ofan ITR for breast cancer when compared to the defaultstrategy of assigning all patients to adjuvant chemotherapyThe proposed approach (based on boosting and classificationtrees) achieved the highest value of the summary measureat 0081 with 95 confidence interval (CI) (0000 0159)[42] In the second example introduced in Section 51 [19]the authors calculated the mean score of the HamiltonRating Scale for Depression for two groups of subjectsgroups were defined by randomly assigning patients to theldquooptimalrdquo and ldquononoptimalrdquo therapy as defined by the ITRThe reported difference between the two groups was minus178with a 119875 value of 009 which fails to attain a clinicalsignificant difference of 3 [19] The same data was analyzedby Kapelner et al [81] Following the discussed procedurethe authors reported the estimated values (and 95 CI)of Θ119862119892opt(X) and Θ
119862alt119892opt(X) as minus0842(minus2657 minus0441)
and minus0765(minus2362 0134) respectively The results whichfail to achieve clinical significance were based on rigorousstatistical methods and thus can be considered reliableestimates of the ITRrsquos performance
7 Discussion
As our understanding tumor heterogeneity evolves person-alized medicine will become standard medical practice inoncology Therefore it is essential that the oncology com-munity uses appropriate analytical methods for identifyingand evaluating the performance of personalized treatmentrules This paper provided an exposition of the process forusing statistical inference to establish optimal individualizedtreatment rules using data acquired from clinical study Thequality of an ITR depends on the quality of the design used toacquire the dataMoreover an ITRmust be properly validatedbefore it is integrated into clinical practice Personalizedmedicine in some areas of oncologymay be limited by the factthat biomarkers arising from a small panel of genesmay neveradequately characterize the extent of tumor heterogeneityinherent to the disease Consequently the available statisticalmethodology needs to evolve in order to optimally exploitglobal gene signatures for personalized medicine
The bulk of our review focused on statistical approachesfor treatment selection at a single time point The readershould note that another important area of research considersoptimal dynamic treatment regimes (DTRs) [88 89] whereintreatment decisions are considered sequentially over thecourse ofmultiple periods of intervention using each patientrsquosprior treatment history Zhao and Zeng provide a summaryof recent developments in this area [90]
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
JunshengMa was fully funded by the University of Texas MDAnderson Cancer Center internal funds Brian P Hobbs and
Francesco C Stingo were partially supported by the CancerCenter Support Grant (CCSG) (P30 CA016672)
References
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[2] D Hanahan and R AWeinberg ldquoHallmarks of cancer the nextgenerationrdquo Cell vol 144 no 5 pp 646ndash674 2011
[3] R Simon ldquoClinical trial designs for evaluating the medicalutility of prognostic and predictive biomarkers in oncologyrdquoPersonalized Medicine vol 7 no 1 pp 33ndash47 2010
[4] P L Bedard A R Hansen M J Ratain and L L Siu ldquoTumourheterogeneity in the clinicrdquo Nature vol 501 no 7467 pp 355ndash364 2013
[5] M D Pegram G Pauletti and D J Slamon ldquoHer-2neu as apredictive marker of response to breast cancer therapyrdquo BreastCancer Research and Treatment vol 52 no 1ndash3 pp 65ndash77 1998
[6] G J Kelloff andC C Sigman ldquoCancer biomarkers selecting theright drug for the right patientrdquoNature Reviews Drug Discoveryvol 11 no 3 pp 201ndash214 2012
[7] J ADiMasi JM Reichert L Feldman andAMalins ldquoClinicalapproval success rates for investigational cancer drugsrdquoClinicalPharmacology andTherapeutics vol 94 no 3 pp 329ndash335 2013
[8] M Hay D W Thomas J L Craighead C Economides andJ Rosenthal ldquoClinical development success rates for investiga-tional drugsrdquo Nature Biotechnology vol 32 no 1 pp 40ndash512014
[9] S S Knox ldquoFrom lsquoomicsrsquo to complex disease a systems biologyapproach to gene-environment interactions in cancerrdquo CancerCell International vol 10 article 11 2010
[10] V Deschoolmeester M Baay P Specenier F Lardon and JB Vermorken ldquoA review of the most promising biomarkersin colorectal cancer one step closer to targeted therapyrdquo TheOncologist vol 15 no 7 pp 699ndash731 2010
[11] J A Sosman K B Kim L Schuchter et al ldquoSurvival in brafV600ndashmutant advanced melanoma treated with vemurafenibrdquoThe New England Journal of Medicine vol 366 no 8 pp 707ndash714 2012
[12] S Paik S Shak G Tang et al ldquoA multigene assay to predictrecurrence of tamoxifen-treated node-negative breast cancerrdquoThe New England Journal of Medicine vol 351 no 27 pp 2817ndash2826 2004
[13] K S Albain W E Barlow S Shak et al ldquoPrognostic and pre-dictive value of the 21-gene recurrence score assay in post-menopausal women with node-positive oestrogen-receptor-positive breast cancer on chemotherapy a retrospective analysisof a randomised trialrdquoTheLancet Oncology vol 11 no 1 pp 55ndash65 2010
[14] J E Lang J S Wecsler M F Press and D Tripathy ldquoMolecularmarkers for breast cancer diagnosis prognosis and targetedtherapyrdquo Journal of Surgical Oncology vol 111 no 1 pp 81ndash902015
[15] W Werft A Benner and A Kopp-Schneider ldquoOn the identi-fication of predictive biomarkers detecting treatment-by-geneinteraction in high-dimensional datardquo Computational Statisticsand Data Analysis vol 56 no 5 pp 1275ndash1286 2012
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[17] A J Vickers MW Kattan and D J Sargent ldquoMethod for eval-uating prediction models that apply the results of randomizedtrials to individual patientsrdquo Trials vol 8 no 1 article 14 2007
[18] H Janes M S Pepe P M Bossuyt andW E Barlow ldquoMeasur-ing the performance of markers for guiding treatment deci-sionsrdquo Annals of Internal Medicine vol 154 no 4 pp 253ndash2592011
[19] R J DeRubeis Z D Cohen N R Forand J C Fournier L AGelfand and L Lorenzo-Luaces ldquoThe personalized advantageindex translating research on prediction into individualizedtreatment recommendationsAdemonstrationrdquoPLoSONE vol9 no 1 Article ID e83875 2014
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[21] N Thatcher A Chang P Parikh et al ldquoGefitinib plus bestsupportive care in previously treated patients with refractoryadvanced non-small-cell lung cancer results from a ran-domised placebo-controlled multicentre study (iressa survivalevaluation in lung cancer)rdquo The Lancet vol 366 no 9496 pp1527ndash1537 2005
[22] A J Vickers ldquoPrediction models in cancer carerdquo CA A CancerJournal for Clinicians vol 61 no 5 pp 315ndash326 2011
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[24] S J Pocock S E Assmann L E Enos and L E Kasten ldquoSub-group analysis covariate adjustment and baseline comparisonsin clinical trial reporting current practice and problemsrdquoStatistics in Medicine vol 21 no 19 pp 2917ndash2930 2002
[25] PM Rothwell ZMehta S CHoward S A Gutnikov andC PWarlow ldquoFrom subgroups to individuals general principles andthe example of carotid endarterectomyrdquoTheLancet vol 365 no9455 pp 256ndash265 2005
[26] R Wang S W Lagakos J H Ware D J Hunter and J MDrazen ldquoStatistics in medicinemdashreporting of subgroup ana-lyses in clinical trialsrdquoTheNewEngland Journal ofMedicine vol357 no 21 pp 2108ndash2194 2007
[27] International Breast Cancer Study Group ldquoEndocrine respon-siveness and tailoring adjuvant therapy for postmenopausallymph node-negative breast cancer a randomized trialrdquo Journalof the National Cancer Institute vol 94 no 14 pp 1054ndash10652002
[28] Early Breast Cancer Trialistsrsquo Collaborative Group (EBCTCG)ldquoEffects of chemotherapy and hormonal therapy for early breastcancer on recurrence and 15-year survival an overview of therandomised trialsrdquoThe Lancet vol 365 no 9472 pp 1687ndash17172005
[29] F E Harrell K L Lee and D B Mark ldquoTutorial in biostatisticsmultivariable prognostic models issues in developing modelsevaluating assumptions and adequacy and measuring andreducing errorsrdquo Statistics in Medicine vol 15 no 4 pp 361ndash387 1996
[30] S Gill C L Loprinzi D J Sargent et al ldquoPooled analysis offluorouracil-based adjuvant therapy for stage II and III coloncancer who benefits and by how muchrdquo Journal of ClinicalOncology vol 22 no 10 pp 1797ndash1806 2004
[31] M Qian and S A Murphy ldquoPerformance guarantees forindividualized treatment rulesrdquoThe Annals of Statistics vol 39no 2 pp 1180ndash1210 2011
[32] W Lu H H Zhang and D Zeng ldquoVariable selection for opti-mal treatment decisionrdquo StatisticalMethods inMedical Researchvol 22 no 5 pp 493ndash504 2013
[33] R Kohavi ldquoA study of cross-validation and bootstrap for accu-racy estimation and model selectionrdquo in Proceedings of the 14thInternational Joint Conference on Artificial Intelligence (IJCAIrsquo95) vol 2 pp 1137ndash1145 1995
[34] E W Steyerberg A J Vickers N R Cook et al ldquoAssessing theperformance of prediction models a framework for traditionaland novel measuresrdquo Epidemiology vol 21 no 1 pp 128ndash1382010
[35] P R Rosenbaum and D B Rubin ldquoThe central role of the pro-pensity score in observational studies for causal effectsrdquo Bio-metrika vol 70 no 1 pp 41ndash55 1983
[36] R B drsquoAgostino Jr ldquoTutorial in biostatistics propensity scoremethods for bias reduction in the comparison of a treatment toa non-randomized control grouprdquo Statistics in Medicine vol 17no 19 pp 2265ndash2281 1998
[37] R Pazdur ldquoEndpoints for assessing drug activity in clinicaltrialsrdquoThe Oncologist vol 13 supplement 2 pp 19ndash21 2008
[38] S L Spruance J E Reid M Grace and M Samore ldquoHazardratio in clinical trialsrdquo Antimicrobial Agents and Chemotherapyvol 48 no 8 pp 2787ndash2792 2004
[39] J D Kalbeisch and R L Prentice The Statistical Analysis ofFailure Time Data vol 360 John Wiley amp Sons 2011
[40] Y Geng Flexible Statistical Learning Methods for SurvivalData Risk Prediction and Optimal Treatment Decision NorthCarolina State University 2013
[41] J Li L Zhao L Tian et alAPredictive Enrichment Procedure toIdentify Potential Responders to a NewTherapy for RandomizedComparative Controlled Clinical Studies Harvard UniversityBiostatisticsWorking Paper SeriesWorking Paper 169 HarvardUniversity 2014
[42] C Kang H Janes and Y Huang ldquoCombining biomarkers tooptimize patient treatment recommendationsrdquo Biometrics vol70 no 3 pp 695ndash720 2014
[43] B Zhang A A Tsiatis E B Laber and M Davidian ldquoA robustmethod for estimating optimal treatment regimesrdquo Biometricsvol 68 no 4 pp 1010ndash1018 2012
[44] B Zhang A A Tsiatis M Davidian M Zhang and E LaberldquoEstimating optimal treatment regimes from a classificationperspectiverdquo Stat vol 1 no 1 pp 103ndash114 2012
[45] J C Foster J M G Taylor and S J Ruberg ldquoSubgroup identi-fication from randomized clinical trial datardquo Statistics in Medi-cine vol 30 no 24 pp 2867ndash2880 2011
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[47] H Janes M D Brown M Pepe and Y Huang ldquoStatisticalmethods for evaluating and comparing biomarkers for patienttreatment selectionrdquo UW Biostatistics Working Paper SeriesWorking Paper 389 2013
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[49] X-JMa ZWang P D Ryan et al ldquoA two-gene expression ratiopredicts clinical outcome in breast cancer patients treated withtamoxifenrdquo Cancer Cell vol 5 no 6 pp 607ndash616 2004
[50] X-JMa S GHilsenbeckWWang et al ldquoTheHOXB13IL17BRexpression index is a prognostic factor in early-stage breastcancerrdquo Journal of Clinical Oncology vol 24 no 28 pp 4611ndash4619 2006
[51] D C Sgroi E Carney E Zarrella et al ldquoPrediction of latedisease recurrence and extended adjuvant letrozole benefit by
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the HOXB13IL17BR biomarkerrdquo Journal of the National CancerInstitute vol 105 no 14 pp 1036ndash1042 2013
[52] M S Cepeda R Boston J T Farrar and B L Strom ldquoCom-parison of logistic regression versus propensity score when thenumber of events is low and there are multiple confoundersrdquoThe American Journal of Epidemiology vol 158 no 3 pp 280ndash287 2003
[53] P C Austin ldquoAn introduction to propensity score methods forreducing the effects of confounding in observational studiesrdquoMultivariate Behavioral Research vol 46 no 3 pp 399ndash4242011
[54] G Heinze and P Juni ldquoAn overview of the objectives of andthe approaches to propensity score analysesrdquo European HeartJournal vol 32 no 14 Article ID ehr031 pp 1704ndash1708 2011
[55] L E Braitman and P R Rosenbaum ldquoRare outcomes commontreatments analytic strategies using propensity scoresrdquo Annalsof Internal Medicine vol 137 no 8 pp 693ndash695 2002
[56] D M Witten and R Tibshirani ldquoSurvival analysis withhigh-dimensional covariatesrdquo Statistical Methods in MedicalResearch vol 19 no 1 pp 29ndash51 2010
[57] A E Hoerl and R W Kennard ldquoRidge regression biasedestimation for nonorthogonal problemsrdquoTechnometrics vol 42no 1 pp 80ndash86 2000
[58] R Tibshirani ldquoRegression shrinkage and selection via the lassordquoJournal of the Royal Statistical Society Series B Methodologicalvol 58 no 1 pp 267ndash288 1996
[59] H Zou ldquoThe adaptive lasso and its oracle propertiesrdquo Journal ofthe American Statistical Association vol 101 no 476 pp 1418ndash1429 2006
[60] B Efron T Hastie I Johnstone and R Tibshirani ldquoLeast angleregressionrdquo The Annals of Statistics vol 32 no 2 pp 407ndash4992004
[61] T Hastie and B Efron ldquolars Least angle regression lasso andforward stagewiserdquo R package version 12 2013 httpcranr-projectorgwebpackageslarsindexhtml
[62] J Fan and R Li ldquoVariable selection via nonconcave penalizedlikelihood and its oracle propertiesrdquo Journal of the AmericanStatistical Association vol 96 no 456 pp 1348ndash1360 2001
[63] H Zou and T Hastie ldquoRegularization and variable selection viathe elastic netrdquo Journal of the Royal Statistical Society Series BStatistical Methodology vol 67 no 2 pp 301ndash320 2005
[64] N Kramer J Schafer and A-L Boulesteix ldquoRegularized esti-mation of large-scale gene association networks using graphicalgaussian modelsrdquo BMC Bioinformatics vol 10 no 1 article 3842009
[65] P Breheny and J Huang ldquoCoordinate descent algorithms fornonconvex penalized regression with applications to biologicalfeature selectionrdquo The Annals of Applied Statistics vol 5 no 1pp 232ndash253 2011
[66] S M Hammer D A Katzenstein M D Hughes et al ldquoA trialcomparing nucleoside monotherapy with combination therapyin HIV-infected adults with CD4 cell counts from 200 to 500per cubicmillimeterrdquoTheNew England Journal ofMedicine vol335 no 15 pp 1081ndash1090 1996
[67] H M Boslashvelstad S Nygard H L Stoslashrvold et al ldquoPredictingsurvival from microarray datamdasha comparative studyrdquo Bioinfor-matics vol 23 no 16 pp 2080ndash2087 2007
[68] V Kehl and K Ulm ldquoResponder identification in clinical trialswith censored datardquoComputational Statistics andDataAnalysisvol 50 no 5 pp 1338ndash1355 2006
[69] P Royston and M K Parmar ldquoThe use of restricted meansurvival time to estimate the treatment effect in randomizedclinical trials when the proportional hazards assumption is indoubtrdquo Statistics inMedicine vol 30 no 19 pp 2409ndash2421 2011
[70] P Royston andM K B Parmar ldquoRestrictedmean survival timean alternative to the hazard ratio for the design and analysis ofrandomized trials with a time-to-event outcomerdquo BMCMedicalResearch Methodology vol 13 no 1 article 152 2013
[71] E T Lee and J W Wang Statistical Methods for Survival DataAnalysis John Wiley amp Sons Hoboken NJ USA 2013
[72] Y Zhao D Zeng A J Rush and M R Kosorok ldquoEstimatingindividualized treatment rules using outcome weighted learn-ingrdquo Journal of the American Statistical Association vol 107 no499 pp 1106ndash1118 2012
[73] D B Rubin and M J van der Laan ldquoStatistical issues andlimitations in personalized medicine research with clinicaltrialsrdquoThe International Journal of Biostatistics vol 8 no 1 pp1ndash20 2012
[74] L Breiman ldquoRandom forestsrdquoMachine Learning vol 45 no 1pp 5ndash32 2001
[75] J Friedman T Hastie and R Tibshirani ldquoAdditive logisticregression a statistical view of boostingrdquo The Annals of Statis-tics vol 28 no 2 pp 337ndash407 2000
[76] C Cortes and V Vapnik ldquoSupport-vector networksrdquo MachineLearning vol 20 no 3 pp 273ndash297 1995
[77] L Breiman J Friedman C J Stone and R A Olshen Classi-fication and Regression Trees CRC Press New York NY USA1984
[78] S Dudoit J Fridlyand and T P Speed ldquoComparison of dis-criminationmethods for the classification of tumors using geneexpression datardquo Journal of the American Statistical Associationvol 97 no 457 pp 77ndash87 2002
[79] A Liaw and MWiener ldquoClassification and regression by rand-omforestrdquoRNews vol 2 no 3 pp 18ndash22 2002 httpCRANR-projectorgdocRnews
[80] T Therneau B Atkinson and B Ripley ldquorpart RecursivePartitioning and Regression Treesrdquo R package version 41-3httpcranr-projectorgwebpackagesrpartindexhtml
[81] A Kapelner J Bleich Z D Cohen R J DeRubeis and RBerk ldquoInference for treatment regime models in personalizedmedicinerdquo httparxivorgabs14047844
[82] X Song and M S Pepe ldquoEvaluating markers for selecting apatientrsquos treatmentrdquoBiometrics vol 60 no 4 pp 874ndash883 2004
[83] T Karrison ldquoRestricted mean life with adjustment for covari-atesrdquo Journal of the American Statistical Association vol 82 no400 pp 1169ndash1176 1987
[84] C Barker ldquoThe mean median and confidence intervals ofthe kaplan-meier survival estimatemdashcomputations and appli-cationsrdquo Journal of the American Statistical Association vol 63no 1 pp 78ndash80 2009
[85] B Efron and R J Tibshirani An Introduction to the Bootstrapvol 57 CRC Press 1994
[86] S Arlot andA Celisse ldquoA survey of cross-validation proceduresfor model selectionrdquo Statistics Surveys vol 4 pp 40ndash79 2010
[87] A Kapelner and J Bleich ldquoPTE Personalized Treatment Eva-luatorrdquo 2014 R package version 10 httpCRANR-projectorgpackage=PTE
[88] S A Murphy ldquoOptimal dynamic treatment regimesrdquo Journal ofthe Royal Statistical Society Series B StatisticalMethodology vol65 no 2 pp 331ndash355 2003
BioMed Research International 13
[89] J M Robins ldquoOptimal structural nested models for optimalsequential decisionsrdquo in Proceedings of the Second Seattle Sym-posium in Biostatistics vol 179 of Lecture Notes in Statistics pp189ndash326 Springer Berlin Germany 2004
[90] Y Zhao and D Zeng ldquoRecent development on statistical meth-ods for personalized medicine discoveryrdquo Frontiers of Medicinein China vol 7 no 1 pp 102ndash110 2013
Penalized estimation provides the subset of relevantpredictivemarkers that are extracted from the nonzero coeffi-cients of the corresponding treatment-biomarker interactionterms of
where 120582119899is a tuning parameter which is often selected via
cross validation and 119869 is a shrinkage penalty Different choicesof 119869 lead to different types of estimators For example thelasso penalized regression corresponds to 119869 = 1 [58] andthe adaptive lasso to 119869 = 120596
119895= 1|120573init119895| where 120573init119895 is an
initial estimate of 120573119895[59] With little modification (8) can
be solved using the LARS algorithm implemented with the119877 package of 119897119886119903119904 [32 60 61] As we have shown before atreatment rule can be defined from the parameter estimatesas 119868120573
formmay have zero estimates for some coefficients (eg 1205732=
1205735= sdot sdot sdot = 120573
119902= 0) hence an ITR can be equivalently con-
structed from the final estimated nonzero coefficients and thecorresponding covariates
Alternative penalized regression approaches includeSCAD [62] and elastic-net [63] All penalized approachesproduce sparse solutions (ie identifying a small subset ofpredictive biomarkers) however the adaptive lasso is lesseffective when 119901 gt 119899 Methods that produce nonsparsemodels such as ridge regression [57] are less preferable sinceITRs based on many biomarkers are often unstable and lessuseful in practice [31] Several packages in 119877 offer imple-mentation of penalized regression such as 119901119886119903119888119900119903 for ridgelasso and adaptive lasso and 119899119888V119903119890119892 for SCAD [64 65]
Lu et al [32] used a penalized regression approach toanalyze data from the AIDS Clinical Trials Group Protocol175 (ACTG175) [66] In this protocol 2000 patients wereequally randomized to one of four treatments zidovudine(ZDV) monotherapy ZDV + didanosine (ddI) ZDV +zalcitabine and ddI monotherapy CD4 count at 15ndash25 weekspostbaselinewas the primary outcome and 12 baseline covari-ates were included in the analysis The resulting treatmentrule favored the combined regimes over ZDV monotherapyMoreover the treatment rule determined that ZDV + ddIshould be preferred to ddI when 119868(7159 + 107 times ageminus 018 timesCD40 minus 3357 times homo) = 1 where CD40 represents baselineCD4 counts and homo represents homosexual activity Basedon this treatment rule 878 patients would have benefitedfrom treatment with ZDV + ddI
54 Survival Analysis Heretofore we have discussed meth-ods for continuous or binary outcomes yet often investigatorswant to discern the extent to which a therapeutic interventionmay alter the amount of time required before an event occursThis type of statistical inference is referred to broadly assurvival analysis One challenge for survival analysis is thatthe outcomes may be only partially observable at the time ofanalysis due to censoring or incomplete follow-up Survivalanalysis has been widely applied in cancer studies often inassociation studies aimed to identify prognostic biomarkers
[56 67] Here we discuss twowidely usedmodels for derivingITRs using time-to-event data namely Cox regression andaccelerated failure time models
The Cox regression model follows as
120582 (119905 | X 119860) = 1205820(119905) exp 120573
11198831+ 12057321198832+ 119860 (120573
4+ 12057351198832)
(9)
where 119905 is the survival time 1205820(119905) is an arbitrary baseline
hazard function and 1198831 1198832represent prognostic and pre-
dictive biomarkers respectively Each 120573 characterizes themultiplicative effect on the hazard associated with a unitincrease in the corresponding covariate Therefore Coxmodels are referred to as proportional hazards (PH) models
Several authors have provided model building strategies[29] and approaches for treatment selection [20 30 68] Fol-lowing the previously outlined strategy a naive approach forderiving an ITR uses the hazard ratio (new treatment versusthe standard) as the treatment contrast which can be calcu-lated as Δ(X) = exp(120573
4+ 12057351198832) The ITR therefore is 119868(120573
4
+12057351198832) lt 0 There are obvious limitations to this approach
First violations of the PH assumption yield substantiallymis-leading results [69]Moreover evenwhen the PH assumptionis satisfied because the Cox model does not postulate adirect relationship between the covariate (treatment) and thesurvival time the hazard ratio fails to measure the extent towhich the treatment is clinically valuable [38 70]
Accelerated failure time (AFT) models provide an alter-native semiparametric model Here we introduce its appli-cation for high-dimensional data Let 119879 and 119862 denote thesurvival and censoring times and denote the observed databy ( 120575 119860X) where = min(119879 119862) and 120575 = 119868(119879 lt 119862)Define the log survival time as 119884 = log(119879) a semiparametricregression model is given as 119864(119884 | 119860X) = ℎ
0(X) + 119860(X120573)
where ℎ0(X) is the unspecified baseline mean function
Similar to the previous section the treatment rule is 119868(1205730+
independent censoring the AFT model parameters can beestimated by minimizing the following loss function
119871119899120601(120573) =
1
119899
119899
sum119894=1
120575119894
119866(119894)[119894minus 120601 (X
119894 120574) minus X120573 119860
119894minus 120587 (X
119894)]2
(10)
where 119894= log(
119894) 120587(X
119894) = 119875(119860
119894= 1 | X
119894) is the propensity
score or randomization probability 119866(sdot) is the Kaplan-Meierestimator of the survival function of the censoring time and120601(X 120574) characterizes any arbitrary function
Thismethod can be extended to accommodatemore thantwo treatments simultaneously by specifying appropriatetreatment indicators For instance the mean function can bemodeled as 119864(119884 | 119860X) = ℎ
0(X) + 119860
(1)X120573(1) + 119860(2)X120573(2)
for two treatment drugs versus the standard care The ITRassigns the winning drug Note this work was proposed by[40] and is an extension of [32] to the survival setting Henceit shares the robustness property and can be applied to obser-vational data For implementation the sameprocedure can befollowed to obtain estimates with one addition step of calcu-lating 119866(
119894) There are several 119877 packages for Kaplan-Meier
BioMed Research International 7
estimates and Cox regression models These sources can befound at httpcranr-projectorgwebviewsSurvivalhtmlMore details pertaining to statistical methods for survivalanalysis can be found here [71] To compare treatmentrules constructed from Cox and AFT models for examplemethods for measuring the extent of clinical effectiveness foran ITR will be discussed in Section 6
We here present an example when an AFT model wasused to construct an ITR for treatment of HIV [40] Theexample derives from the AIDS Clinical Trials Group Pro-tocol 175 that was discussed in Section 53 [32 66] In thiscase the primary outcome variable was time (in days) tofirst ge50 decline in CD4 count or an AIDS-defining eventor death A total of 12 covariates and four treatments (ZDVZDV + ddI ZDV + zalcitabine and ddI) were includedThe four treatments were evaluated simultaneously Patientsreceiving the standard care of ZDV monotherapy wereconsidered as the reference group Hence three treatmentcontrasts (119868ZDV+ddI 119868ZDV+zalcitabine and 119868ddI) were combinedwith various putative predictive covariates and comparedwith ZDV monotherapy For example gender was detectedas the predictive covariate only for ddI monotherapy Theinvestigators assumed 120601(X 120574) = 120574
0 The treatment rule
recommended 1 patient for ZDV monotherapy while 7291216 and 193 patients were recommended for ZDV + ddIZDV + zalcitabine and ddI respectively
55 Advanced Methods
551 Robust Inference The performances of ITRs heretoforepresented depend heavily on whether the statistical modelswere correctly specified Recently there has been much atten-tion focused on the development of more advanced methodsand modeling strategies that are robust to various aspectsof potential misspecification We have already presented afew robust models that avoid specification of functionalparametric relationships for main effects [32 40] Here weintroduce two more advanced methods widely utilized forITRs that are robust to the type of misspecification issuescommonly encountered in practice [42 43]
Recall that the ITR for a linear model 119864(119884 | 119860 =
119886X) = 120583(119860 = 119886X120573) with two predictive markers followsas 119892(X120573) = 119868(120573
4+ 12057351198832+ 12057361198833) gt 0 where 119886 = 0 1
The treatment rule of 119892(X120573) may use only a subset of thehigh-dimensional covariates (eg 119883
2 1198833) but it always
depends on the correct specification of 119864(119884 | 119860 = 119886X)Defining a scaled version of 120573 as 120578(120573) the correspondingITR is 119892(120578X) = 119892(X120573) = 119868(119883
3gt 1205780+ 12057811198832) where
1205780= minus12057341205736and 120578
1= 12057351205736 If the model for 120583(119860X120573) is
indeed correctly specified the treatment rules of 119892(X120573) and119892(120578X) lead to the same optimal ITR Hence the treatmentrule parameterized by 120578 can be derived from a regressionmodel or may be based on some key clinical considerationswhich enable evaluation of 119892(120578X) directly without referenceto the regression model for 120583(119860X120573)
Let 119862120578
= 119860119892(120578X) + (1 minus 119860)1 minus 119892(120578X) where119862120578= 1 indicates random assignment to an intervention
that is recommended by the personalized treatment rule 119892(120578X) Let 120587(X ) denote the randomization ratio or the
estimated propensity score (as in previous section) and119898(X 120578 ) denote the potential outcome under the treatmentrule estimated from the following model 119864(119884 | 119860 =
119886X) = 120583(119860X120573) For example if the treatment rule 119892(120578X)= 1 then 119898(X 120578 ) = 119892(120578X)120583(119860 = 1X ) + 1 minus 119892(120578
X)120583(119860 = 0X ) = 120583(119860 = 1X ) Two estimators ofthe expected response to treatment the inverse probabilityweighted estimator (IPWE) and doubly robust AIPWE aregiven as follows
IPWE (120578) = 1
119899
119899
sum119894=1
119862120578sdot119894119884119894
120587119888(X119894 120578 )
=1
119899
119899
sum119894=1
119862120578sdot119894119884119894
120587 (X119894 )119860119894 1 minus 120587 (X
119894 )1minus119860119894
AIPWE (120578) = 1
119899
119899
sum119894=1
119862120578sdot119894119884119894
120587119888(X119894 120578 )
minus119862120578sdot119894119884119894 minus 120587119888 (X119894 120578 )
120587119888(X119894 120578 )
119898 (X119894 120578 )
(11)
where 120587119888(X119894 120578 ) = 120587(X )119892(120578X) + 1 minus 120587(X )1 minus
119892(120578X) The optimal treatment rule follows as 119892(X = x)where is estimated from the above models a constraintsuch as 120578 = 1 is imposed to obtain a unique solution [43] If the propensity score is correctly specified theIPWE estimator yields robust (consistent) estimates AIPWEis considered a doubly robust estimator since it produces con-sistent estimates when either propensity score or the model119864(119884 | 119860 = 119886X) is misspecified but not both [42 43] Thecompanion119877 code is publicly available at httponlinelibrarywileycomdoi101111biom12191suppinfo
552 Data Mining and Machine Learning The methodspresented in Section 551 are robust against misspecificationof regression models Yet they often require prespecificationof the parametric form for the treatment rule (eg 119868(119883
3gt
1205780+ 12057811198832)) which can be practically challenging [44]
Well established classification methods and other popularmachine learning techniques can alternatively be customizedto define treatment selection rules [44 72 73] these methodsavoid prespecification of the parametric form of the ITR AnITR can be defined following a two-step approach in thefirst step treatment contrasts are estimated from a positedmodel and in the second step classification techniques areapplied to determine the personalized treatment rules Forexample when only two treatments are considered a newvariable 119885 can be defined based on the treatment contrastthat is 119885 = 1 if Δ(X) = 120583(119860 = 1X) minus 120583(119860 = 0X) gt 0 and119885 = 0 otherwiseThe absolute value of the treatment contrast119882119894= |Δ(X)| can be used in conjunction with a classification
technique to define an appropriate ITR [44]Unlike classification problems wherein the class labels
are observed for the training data the binary ldquoresponserdquovariable 119885 which serves as the class label is not availablein practice Specifically patients who are in the class 119885 = 1
8 BioMed Research International
have 120583(119860 = 1X) gt 120583(119860 = 0X) and should thereforebe treated with the new therapy however these quantitiesneed to be estimated since patients are typically assigned toonly one of the available treatments This imparts flexibilityfor estimation of the optimal treatment regimes since anyof the previously discussed regression models and even someensemble prediction methods such as random forest [74] canbe used to construct the class labels 119885
119894and weights
119894[44]
An ITR can be estimated from the dataset 119885119894X119894 119894 using
any classification approach where 119894are subject specific
misclassification weights [44 45] This includes popularclassificationmethods such as adaptive boosting [75] supportvector machines [76] and classification and regression trees(CART) [77] At least one study has suggested that SVMoutperforms other classification methods in this contextwhereas random forest and boosting perform comparativelybetter than CART [78] However the performances ofthese classification algorithms are data dependent Definitiveconclusion pertaining to their comparative effectiveness ingeneral has yet to be determined [78] It shall be also notedthat these classification methods can be also applied to high-dimensional data [45 72]
One special case of this framework is the ldquovirtue twinsrdquoapproach [45] Specifically in the first step a random forestapproach [74] is used to obtain the treatment contrasts Thenin the second step CART is used to classify subjects tothe optimal treatment regime The approach can be easilyimplemented in 119877 using packages of randomForest [79] andrpart [80] Very recently Kang et al [42] proposed amodifiedversion of the adaptive boosting technique of Friedman et al[75] The algorithm iteratively fits a simple logistic regressionmodel (ldquoworking modelrdquo) to estimate 119875(119884 = 1 | 119860X) and ateach stage assigns higher weights to subjects whose treatmentcontrast is near zero After a prespecified stopping criterionis met an average of the treatment contrasts Δ(X) is calcu-lated for each patient using all models fitted at each iterationA subject is assigned to the new therapy if Δ(X) gt 0 The 119877code for the aforementioned boosting methods is publiclyavailable at httponlinelibrarywileycomdoi101111biom12191suppinfo
Lastly we present a breast cancer example where severalbiomarkers were combined to construct an optimal ITRThe data was collected in the Southwest Oncology Group(SOWG)-SS8814 trial [13] and analyzed with the machinelearning approach of Kang et al [42] Three hundred andsixty-seven node-positive ER-positive breast cancer patientswere selected from the randomized trial of SOWG A total of219 received tamoxifen plus adjuvant chemotherapy and 148was given tamoxifen aloneThe outcome variable was definedas breast cancer recurrence at 5 years The authors selectedthree genes which had presented treatment-biomarker inter-actions in amultivariate linear logistic regressionmodel [42]Data were analyzed with logistic models IPWE AIPWElogistic boosting a single classification tree with treatment-biomarker interactions and the proposed boosting approachwith a classification tree as the working model Each methodidentified different patient cohorts that could benefit fromtamoxifen alone these cohorts consisted of 184 183 128 86
263 and 217 patients respectively (see Table 5 in [42]) In thisanalysis the clinical benefits provided by these 6 treatmentrules were not statistically different Hence investigatorsneed to evaluate and compare ITRs in terms of the extentof expected clinical impact This is considered in the nextsection
Heretofore we have discussed various methodologies for theconstruction of ITR while their performances need to beassessed before these rules can be implemented in clinicalpractice Several aspects pertaining to the performance of aconstructed ITR need to be considered The first one is howwell the ITR fits the data and the second is how well theITR performs compared with existing treatment allocationrules The former is related to the concept of goodness-of-fitor predictive performance [34] As the true optimal treatmentgroups are hidden model fits may be evaluated by measuringthe congruity between observed treatment contrasts andpredicted ones [34 47] More details can be found in a recentpaper by Janes et al [47] Performances of ITRs can becompared via assessment of a global summary measure forexample prolonged survival time or reduced disease rate [4042] Summarymeasures are also very useful for evaluating theextent to which an ITR may benefit patients when applied inpractice Moreover it is essential that performance of an ITRis considered in comparison to business-as-usual proceduressuch as a naive rule that randomly allocates patients totreatment [81] Summary measures will be discussed inSection 61 The effectiveness of an ITR should go beyond thetraining data set used to construct a treatment rule cross-validation and bootstrapping techniques are often employedto assess the impact of ITRs on future patients [81] and willbe discussed in Section 62
61 Summary Measures ITRs may be derived from differ-ent methodologies and comparisons should be conductedwith respect to the appropriate clinically summaries A fewsummary measures for different types of outcomes havebeen proposed [19 40 42] these measures quantify thedirect clinical improvements obtained by applying an ITR incomparison with default methods for treatment allocation
Binary Outcomes Clinical effectiveness for binary clinicalresponse is represented by the difference in disease rates (ortreatment failure) induced by ITR versus a default strategythat allocates all patients to a standard treatment [42 47 82]Let 119892opt(X) = 119868120583(119860 = 1X)minus120583(119860 = 0X) lt 0 be an optimalITR This difference is formally defined as
Note 120583(119860X) needs to be estimated to construct the ITRyet parameters 120573 are omitted for simplicity Larger valuesof Θ119861119892opt(X) indicate increased clinical value for the
biomarker driven ITR A subset of patients that are recom-mended for new treatment (119860 = 1) under an ITR may havebeen randomly selected to receive it while the remainingsubset of ldquounluckyrdquo patientswould have received the standardtreatment [19] The summary measure of Θ
119861119892opt(X) char-
acterizes a weighted difference in the disease rates betweenthe standard and the new treatments in a population whereinthe constructed optimal ITR would recommend the newtreatment 119892opt(X = 1) The weight is the proportion ofpatients identified by the optimal ITR for the new treatmentand can be empirically estimated using the correspondingcounts For example 119875119892opt(X) = 1 can be estimatedusing the number of patients recommended for the newtreatment divided by the total sample size A similar summarystatistic can be derived for an alternative strategy allocatingall patients to the new treatment The summary could beapplied to the aforementioned breast cancer example [42]for example with the aim of finding a subgroup of patientswho were likely to benefit from adjuvant chemotherapywhile those unlikely to benefit would be assigned tamoxifenalone to avoid the unnecessary toxicity and inconvenience ofchemotherapy
Continuous Variables Another strategy for continuous datacompares outcomes observed for ldquoluckyrdquo subjects those whoreceived the therapy that would have been recommendedby the ITR based [81] Further one business-as-usual drugallocation procedure is randomizing treatment and standardcare at the same probability of 05 A summary statisticis to measure the mean outcome under ITR compared tothat obtained under random assignment for instance themean decrease in Hamilton Rating Scale for Depression asdiscussed in Section 51 [19] Define the summary measureas Θ119862119892opt(X) = 120583119892opt(X)X minus 120583119892rand(X)X where
119892rand(X) represents the randomization allocation procedureThe quantity of 120583119892119900119901119905(X)X represents the mean outcomeunder the constructed IRT that can be empirically estimatedfrom the ldquoluckyrdquo subjects and 120583119892rand(X)X can be esti-mated empirically from the sample means
Alternatively an ITR may be compared to an ldquooptimalrdquodrug that has showed universal benefits (a better drug onaverage) in a controlled trial The clinical benefits of anldquooptimalrdquo drug can be defined as 120583119892best(X)X = max120583(119860 =
0X) 120583(119860 = 1X) 120583(119860 = 119886X) and can be empiricallyestimated from the sample means of the new and standardtreatments respectively Then the alternative summary mea-sure is defined as Θ
Survival Data For survival data a clinically relevant measureis mean overall (or progression free) survival time As
survival time is continuous in nature the identical strategyprovided above for continuous outcomes can be employedhere However because the mean survival time may notbe well estimated from the observed data due to a highpercentage of censored observations [40] an alternativemean restricted survival duration was proposed and definedas the population average event-free durations for a restrictedtime of 119905lowast [41 83]Often 119905lowast is chosen to cover the trialrsquos follow-up period Mathematically it can be calculated by integratingthe survival function of 119878(119905) over the domain of (0 119905lowast) that is120583119892opt(X)X 119905lowast = int119905
lowast
0119878(119905)119889119905 and often estimated by the area
under the Kaplan-Meier curve up to 119905lowast [84] Thus an ITRrsquospotential to prolong survival can be calculated asΘ
119878119892opt(X)
119905lowast = 120583119892opt(X)X 119905lowast minus 120583119892rand(X)X 119905lowast
62 AssessingModel Performance The summaries heretoforediscussed evaluate an optimal ITR for a given model andestimating procedure Because these quantities are estimatedconditionally given the observed covariates they neglectto quantify the extent of marginal uncertainty for futurepatients Hence an ITR needs to be internally validated ifexternal data is not available [34] Cross-validation (CV) andbootstrap resampling techniques are commonly used for thispurpose [19 42 45 81] and expositions on both approachesare well described elsewhere [33 85 86]
We here briefly introduce a process that was proposed byKapelner et al [81] in the setting of personalized medicineTenfold CV is commonly used in practice where the wholedata is randomly partitioned into 10 roughly equal-sizedexclusive subsamples All methods under consideration areapplied to 910 of the data excluding 110 as an independenttesting data set The process is repeated 10 times for eachsubsample Considering the assignments recommended bythe optimal ITRs the summary measures can be calculatedusing results from each testing fold [45]TheCVprocess givesthe estimated summary measures and its variation can beevaluated using bootstrap procedures Specifically one drawsa sample with replacement from the entire data and calculatesthe summary measure from 10-fold CV This process willbe repeated 119861 times where 119861 is chosen for resolution ofthe resulting confidence intervals [81] Using the summarymeasures as119861 new random samples the correspondingmeanand variances can be calculated empirically Note that thesummary measures compare two treatment rules one for theoptimal ITR and another naive rule (eg randomization)
The above procedure can be applied to all the meth-ods we have discussed so far The 119877 software package119879119903119890119886119905119898119890119899119905119878119890119897119890119888119905119894119900119899 (httplabsfhcrcorgjanesindexhtml)can be used to implement these methods for evaluatingand comparing biomarkers for binary outcomes [47] Veryrecently an inferential procedure was proposed for contin-uous outcomes that is implemented in the publicly available119877 package ldquoPersonalized Treatment Evaluatorrdquo [81 87] Bothmethods consider data from RCTs with two arms for com-parative treatmentsThesemethods are in general applicableto regression model based methods but are not suitable forapproaches based on classification techniques or penalizedregression
10 BioMed Research International
Next we present two examples Recall in Section 55 thatKang et al [42] reported the estimated clinical benefits ofan ITR for breast cancer when compared to the defaultstrategy of assigning all patients to adjuvant chemotherapyThe proposed approach (based on boosting and classificationtrees) achieved the highest value of the summary measureat 0081 with 95 confidence interval (CI) (0000 0159)[42] In the second example introduced in Section 51 [19]the authors calculated the mean score of the HamiltonRating Scale for Depression for two groups of subjectsgroups were defined by randomly assigning patients to theldquooptimalrdquo and ldquononoptimalrdquo therapy as defined by the ITRThe reported difference between the two groups was minus178with a 119875 value of 009 which fails to attain a clinicalsignificant difference of 3 [19] The same data was analyzedby Kapelner et al [81] Following the discussed procedurethe authors reported the estimated values (and 95 CI)of Θ119862119892opt(X) and Θ
119862alt119892opt(X) as minus0842(minus2657 minus0441)
and minus0765(minus2362 0134) respectively The results whichfail to achieve clinical significance were based on rigorousstatistical methods and thus can be considered reliableestimates of the ITRrsquos performance
7 Discussion
As our understanding tumor heterogeneity evolves person-alized medicine will become standard medical practice inoncology Therefore it is essential that the oncology com-munity uses appropriate analytical methods for identifyingand evaluating the performance of personalized treatmentrules This paper provided an exposition of the process forusing statistical inference to establish optimal individualizedtreatment rules using data acquired from clinical study Thequality of an ITR depends on the quality of the design used toacquire the dataMoreover an ITRmust be properly validatedbefore it is integrated into clinical practice Personalizedmedicine in some areas of oncologymay be limited by the factthat biomarkers arising from a small panel of genesmay neveradequately characterize the extent of tumor heterogeneityinherent to the disease Consequently the available statisticalmethodology needs to evolve in order to optimally exploitglobal gene signatures for personalized medicine
The bulk of our review focused on statistical approachesfor treatment selection at a single time point The readershould note that another important area of research considersoptimal dynamic treatment regimes (DTRs) [88 89] whereintreatment decisions are considered sequentially over thecourse ofmultiple periods of intervention using each patientrsquosprior treatment history Zhao and Zeng provide a summaryof recent developments in this area [90]
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
JunshengMa was fully funded by the University of Texas MDAnderson Cancer Center internal funds Brian P Hobbs and
Francesco C Stingo were partially supported by the CancerCenter Support Grant (CCSG) (P30 CA016672)
References
[1] J Reimand O Wagih and G D Bader ldquoThe mutational land-scape of phosphorylation signaling in cancerrdquo Scientific Reportsvol 3 article 2651 2013
[2] D Hanahan and R AWeinberg ldquoHallmarks of cancer the nextgenerationrdquo Cell vol 144 no 5 pp 646ndash674 2011
[3] R Simon ldquoClinical trial designs for evaluating the medicalutility of prognostic and predictive biomarkers in oncologyrdquoPersonalized Medicine vol 7 no 1 pp 33ndash47 2010
[4] P L Bedard A R Hansen M J Ratain and L L Siu ldquoTumourheterogeneity in the clinicrdquo Nature vol 501 no 7467 pp 355ndash364 2013
[5] M D Pegram G Pauletti and D J Slamon ldquoHer-2neu as apredictive marker of response to breast cancer therapyrdquo BreastCancer Research and Treatment vol 52 no 1ndash3 pp 65ndash77 1998
[6] G J Kelloff andC C Sigman ldquoCancer biomarkers selecting theright drug for the right patientrdquoNature Reviews Drug Discoveryvol 11 no 3 pp 201ndash214 2012
[7] J ADiMasi JM Reichert L Feldman andAMalins ldquoClinicalapproval success rates for investigational cancer drugsrdquoClinicalPharmacology andTherapeutics vol 94 no 3 pp 329ndash335 2013
[8] M Hay D W Thomas J L Craighead C Economides andJ Rosenthal ldquoClinical development success rates for investiga-tional drugsrdquo Nature Biotechnology vol 32 no 1 pp 40ndash512014
[9] S S Knox ldquoFrom lsquoomicsrsquo to complex disease a systems biologyapproach to gene-environment interactions in cancerrdquo CancerCell International vol 10 article 11 2010
[10] V Deschoolmeester M Baay P Specenier F Lardon and JB Vermorken ldquoA review of the most promising biomarkersin colorectal cancer one step closer to targeted therapyrdquo TheOncologist vol 15 no 7 pp 699ndash731 2010
[11] J A Sosman K B Kim L Schuchter et al ldquoSurvival in brafV600ndashmutant advanced melanoma treated with vemurafenibrdquoThe New England Journal of Medicine vol 366 no 8 pp 707ndash714 2012
[12] S Paik S Shak G Tang et al ldquoA multigene assay to predictrecurrence of tamoxifen-treated node-negative breast cancerrdquoThe New England Journal of Medicine vol 351 no 27 pp 2817ndash2826 2004
[13] K S Albain W E Barlow S Shak et al ldquoPrognostic and pre-dictive value of the 21-gene recurrence score assay in post-menopausal women with node-positive oestrogen-receptor-positive breast cancer on chemotherapy a retrospective analysisof a randomised trialrdquoTheLancet Oncology vol 11 no 1 pp 55ndash65 2010
[14] J E Lang J S Wecsler M F Press and D Tripathy ldquoMolecularmarkers for breast cancer diagnosis prognosis and targetedtherapyrdquo Journal of Surgical Oncology vol 111 no 1 pp 81ndash902015
[15] W Werft A Benner and A Kopp-Schneider ldquoOn the identi-fication of predictive biomarkers detecting treatment-by-geneinteraction in high-dimensional datardquo Computational Statisticsand Data Analysis vol 56 no 5 pp 1275ndash1286 2012
[16] M Jenkins A Flynn T Smart et al ldquoA statisticianrsquos perspectiveon biomarkers in drug developmentrdquo Pharmaceutical Statisticsvol 10 no 6 pp 494ndash507 2011
BioMed Research International 11
[17] A J Vickers MW Kattan and D J Sargent ldquoMethod for eval-uating prediction models that apply the results of randomizedtrials to individual patientsrdquo Trials vol 8 no 1 article 14 2007
[18] H Janes M S Pepe P M Bossuyt andW E Barlow ldquoMeasur-ing the performance of markers for guiding treatment deci-sionsrdquo Annals of Internal Medicine vol 154 no 4 pp 253ndash2592011
[19] R J DeRubeis Z D Cohen N R Forand J C Fournier L AGelfand and L Lorenzo-Luaces ldquoThe personalized advantageindex translating research on prediction into individualizedtreatment recommendationsAdemonstrationrdquoPLoSONE vol9 no 1 Article ID e83875 2014
[20] D P Byar and D K Corle ldquoSelecting optimal treatment inclinical trials using covariate informationrdquo Journal of ChronicDiseases vol 30 no 7 pp 445ndash459 1977
[21] N Thatcher A Chang P Parikh et al ldquoGefitinib plus bestsupportive care in previously treated patients with refractoryadvanced non-small-cell lung cancer results from a ran-domised placebo-controlled multicentre study (iressa survivalevaluation in lung cancer)rdquo The Lancet vol 366 no 9496 pp1527ndash1537 2005
[22] A J Vickers ldquoPrediction models in cancer carerdquo CA A CancerJournal for Clinicians vol 61 no 5 pp 315ndash326 2011
[23] RM Simon ldquoSubgroup analysisrdquo inWiley Encyclopedia of Clin-ical Trials John Wiley amp Sons Hoboken NJ USA 2007
[24] S J Pocock S E Assmann L E Enos and L E Kasten ldquoSub-group analysis covariate adjustment and baseline comparisonsin clinical trial reporting current practice and problemsrdquoStatistics in Medicine vol 21 no 19 pp 2917ndash2930 2002
[25] PM Rothwell ZMehta S CHoward S A Gutnikov andC PWarlow ldquoFrom subgroups to individuals general principles andthe example of carotid endarterectomyrdquoTheLancet vol 365 no9455 pp 256ndash265 2005
[26] R Wang S W Lagakos J H Ware D J Hunter and J MDrazen ldquoStatistics in medicinemdashreporting of subgroup ana-lyses in clinical trialsrdquoTheNewEngland Journal ofMedicine vol357 no 21 pp 2108ndash2194 2007
[27] International Breast Cancer Study Group ldquoEndocrine respon-siveness and tailoring adjuvant therapy for postmenopausallymph node-negative breast cancer a randomized trialrdquo Journalof the National Cancer Institute vol 94 no 14 pp 1054ndash10652002
[28] Early Breast Cancer Trialistsrsquo Collaborative Group (EBCTCG)ldquoEffects of chemotherapy and hormonal therapy for early breastcancer on recurrence and 15-year survival an overview of therandomised trialsrdquoThe Lancet vol 365 no 9472 pp 1687ndash17172005
[29] F E Harrell K L Lee and D B Mark ldquoTutorial in biostatisticsmultivariable prognostic models issues in developing modelsevaluating assumptions and adequacy and measuring andreducing errorsrdquo Statistics in Medicine vol 15 no 4 pp 361ndash387 1996
[30] S Gill C L Loprinzi D J Sargent et al ldquoPooled analysis offluorouracil-based adjuvant therapy for stage II and III coloncancer who benefits and by how muchrdquo Journal of ClinicalOncology vol 22 no 10 pp 1797ndash1806 2004
[31] M Qian and S A Murphy ldquoPerformance guarantees forindividualized treatment rulesrdquoThe Annals of Statistics vol 39no 2 pp 1180ndash1210 2011
[32] W Lu H H Zhang and D Zeng ldquoVariable selection for opti-mal treatment decisionrdquo StatisticalMethods inMedical Researchvol 22 no 5 pp 493ndash504 2013
[33] R Kohavi ldquoA study of cross-validation and bootstrap for accu-racy estimation and model selectionrdquo in Proceedings of the 14thInternational Joint Conference on Artificial Intelligence (IJCAIrsquo95) vol 2 pp 1137ndash1145 1995
[34] E W Steyerberg A J Vickers N R Cook et al ldquoAssessing theperformance of prediction models a framework for traditionaland novel measuresrdquo Epidemiology vol 21 no 1 pp 128ndash1382010
[35] P R Rosenbaum and D B Rubin ldquoThe central role of the pro-pensity score in observational studies for causal effectsrdquo Bio-metrika vol 70 no 1 pp 41ndash55 1983
[36] R B drsquoAgostino Jr ldquoTutorial in biostatistics propensity scoremethods for bias reduction in the comparison of a treatment toa non-randomized control grouprdquo Statistics in Medicine vol 17no 19 pp 2265ndash2281 1998
[37] R Pazdur ldquoEndpoints for assessing drug activity in clinicaltrialsrdquoThe Oncologist vol 13 supplement 2 pp 19ndash21 2008
[38] S L Spruance J E Reid M Grace and M Samore ldquoHazardratio in clinical trialsrdquo Antimicrobial Agents and Chemotherapyvol 48 no 8 pp 2787ndash2792 2004
[39] J D Kalbeisch and R L Prentice The Statistical Analysis ofFailure Time Data vol 360 John Wiley amp Sons 2011
[40] Y Geng Flexible Statistical Learning Methods for SurvivalData Risk Prediction and Optimal Treatment Decision NorthCarolina State University 2013
[41] J Li L Zhao L Tian et alAPredictive Enrichment Procedure toIdentify Potential Responders to a NewTherapy for RandomizedComparative Controlled Clinical Studies Harvard UniversityBiostatisticsWorking Paper SeriesWorking Paper 169 HarvardUniversity 2014
[42] C Kang H Janes and Y Huang ldquoCombining biomarkers tooptimize patient treatment recommendationsrdquo Biometrics vol70 no 3 pp 695ndash720 2014
[43] B Zhang A A Tsiatis E B Laber and M Davidian ldquoA robustmethod for estimating optimal treatment regimesrdquo Biometricsvol 68 no 4 pp 1010ndash1018 2012
[44] B Zhang A A Tsiatis M Davidian M Zhang and E LaberldquoEstimating optimal treatment regimes from a classificationperspectiverdquo Stat vol 1 no 1 pp 103ndash114 2012
[45] J C Foster J M G Taylor and S J Ruberg ldquoSubgroup identi-fication from randomized clinical trial datardquo Statistics in Medi-cine vol 30 no 24 pp 2867ndash2880 2011
[46] US Food and Drug Administration Guidance for IndustryClinical Trial Endpoints for the Approval of Cancer Drugs andBiologics US Food and Drug AdministrationWashington DCUSA 2007
[47] H Janes M D Brown M Pepe and Y Huang ldquoStatisticalmethods for evaluating and comparing biomarkers for patienttreatment selectionrdquo UW Biostatistics Working Paper SeriesWorking Paper 389 2013
[48] R Development Core Team R a language and environment forstatistical computing 2008 httpwwwR-projectorg
[49] X-JMa ZWang P D Ryan et al ldquoA two-gene expression ratiopredicts clinical outcome in breast cancer patients treated withtamoxifenrdquo Cancer Cell vol 5 no 6 pp 607ndash616 2004
[50] X-JMa S GHilsenbeckWWang et al ldquoTheHOXB13IL17BRexpression index is a prognostic factor in early-stage breastcancerrdquo Journal of Clinical Oncology vol 24 no 28 pp 4611ndash4619 2006
[51] D C Sgroi E Carney E Zarrella et al ldquoPrediction of latedisease recurrence and extended adjuvant letrozole benefit by
12 BioMed Research International
the HOXB13IL17BR biomarkerrdquo Journal of the National CancerInstitute vol 105 no 14 pp 1036ndash1042 2013
[52] M S Cepeda R Boston J T Farrar and B L Strom ldquoCom-parison of logistic regression versus propensity score when thenumber of events is low and there are multiple confoundersrdquoThe American Journal of Epidemiology vol 158 no 3 pp 280ndash287 2003
[53] P C Austin ldquoAn introduction to propensity score methods forreducing the effects of confounding in observational studiesrdquoMultivariate Behavioral Research vol 46 no 3 pp 399ndash4242011
[54] G Heinze and P Juni ldquoAn overview of the objectives of andthe approaches to propensity score analysesrdquo European HeartJournal vol 32 no 14 Article ID ehr031 pp 1704ndash1708 2011
[55] L E Braitman and P R Rosenbaum ldquoRare outcomes commontreatments analytic strategies using propensity scoresrdquo Annalsof Internal Medicine vol 137 no 8 pp 693ndash695 2002
[56] D M Witten and R Tibshirani ldquoSurvival analysis withhigh-dimensional covariatesrdquo Statistical Methods in MedicalResearch vol 19 no 1 pp 29ndash51 2010
[57] A E Hoerl and R W Kennard ldquoRidge regression biasedestimation for nonorthogonal problemsrdquoTechnometrics vol 42no 1 pp 80ndash86 2000
[58] R Tibshirani ldquoRegression shrinkage and selection via the lassordquoJournal of the Royal Statistical Society Series B Methodologicalvol 58 no 1 pp 267ndash288 1996
[59] H Zou ldquoThe adaptive lasso and its oracle propertiesrdquo Journal ofthe American Statistical Association vol 101 no 476 pp 1418ndash1429 2006
[60] B Efron T Hastie I Johnstone and R Tibshirani ldquoLeast angleregressionrdquo The Annals of Statistics vol 32 no 2 pp 407ndash4992004
[61] T Hastie and B Efron ldquolars Least angle regression lasso andforward stagewiserdquo R package version 12 2013 httpcranr-projectorgwebpackageslarsindexhtml
[62] J Fan and R Li ldquoVariable selection via nonconcave penalizedlikelihood and its oracle propertiesrdquo Journal of the AmericanStatistical Association vol 96 no 456 pp 1348ndash1360 2001
[63] H Zou and T Hastie ldquoRegularization and variable selection viathe elastic netrdquo Journal of the Royal Statistical Society Series BStatistical Methodology vol 67 no 2 pp 301ndash320 2005
[64] N Kramer J Schafer and A-L Boulesteix ldquoRegularized esti-mation of large-scale gene association networks using graphicalgaussian modelsrdquo BMC Bioinformatics vol 10 no 1 article 3842009
[65] P Breheny and J Huang ldquoCoordinate descent algorithms fornonconvex penalized regression with applications to biologicalfeature selectionrdquo The Annals of Applied Statistics vol 5 no 1pp 232ndash253 2011
[66] S M Hammer D A Katzenstein M D Hughes et al ldquoA trialcomparing nucleoside monotherapy with combination therapyin HIV-infected adults with CD4 cell counts from 200 to 500per cubicmillimeterrdquoTheNew England Journal ofMedicine vol335 no 15 pp 1081ndash1090 1996
[67] H M Boslashvelstad S Nygard H L Stoslashrvold et al ldquoPredictingsurvival from microarray datamdasha comparative studyrdquo Bioinfor-matics vol 23 no 16 pp 2080ndash2087 2007
[68] V Kehl and K Ulm ldquoResponder identification in clinical trialswith censored datardquoComputational Statistics andDataAnalysisvol 50 no 5 pp 1338ndash1355 2006
[69] P Royston and M K Parmar ldquoThe use of restricted meansurvival time to estimate the treatment effect in randomizedclinical trials when the proportional hazards assumption is indoubtrdquo Statistics inMedicine vol 30 no 19 pp 2409ndash2421 2011
[70] P Royston andM K B Parmar ldquoRestrictedmean survival timean alternative to the hazard ratio for the design and analysis ofrandomized trials with a time-to-event outcomerdquo BMCMedicalResearch Methodology vol 13 no 1 article 152 2013
[71] E T Lee and J W Wang Statistical Methods for Survival DataAnalysis John Wiley amp Sons Hoboken NJ USA 2013
[72] Y Zhao D Zeng A J Rush and M R Kosorok ldquoEstimatingindividualized treatment rules using outcome weighted learn-ingrdquo Journal of the American Statistical Association vol 107 no499 pp 1106ndash1118 2012
[73] D B Rubin and M J van der Laan ldquoStatistical issues andlimitations in personalized medicine research with clinicaltrialsrdquoThe International Journal of Biostatistics vol 8 no 1 pp1ndash20 2012
[74] L Breiman ldquoRandom forestsrdquoMachine Learning vol 45 no 1pp 5ndash32 2001
[75] J Friedman T Hastie and R Tibshirani ldquoAdditive logisticregression a statistical view of boostingrdquo The Annals of Statis-tics vol 28 no 2 pp 337ndash407 2000
[76] C Cortes and V Vapnik ldquoSupport-vector networksrdquo MachineLearning vol 20 no 3 pp 273ndash297 1995
[77] L Breiman J Friedman C J Stone and R A Olshen Classi-fication and Regression Trees CRC Press New York NY USA1984
[78] S Dudoit J Fridlyand and T P Speed ldquoComparison of dis-criminationmethods for the classification of tumors using geneexpression datardquo Journal of the American Statistical Associationvol 97 no 457 pp 77ndash87 2002
[79] A Liaw and MWiener ldquoClassification and regression by rand-omforestrdquoRNews vol 2 no 3 pp 18ndash22 2002 httpCRANR-projectorgdocRnews
[80] T Therneau B Atkinson and B Ripley ldquorpart RecursivePartitioning and Regression Treesrdquo R package version 41-3httpcranr-projectorgwebpackagesrpartindexhtml
[81] A Kapelner J Bleich Z D Cohen R J DeRubeis and RBerk ldquoInference for treatment regime models in personalizedmedicinerdquo httparxivorgabs14047844
[82] X Song and M S Pepe ldquoEvaluating markers for selecting apatientrsquos treatmentrdquoBiometrics vol 60 no 4 pp 874ndash883 2004
[83] T Karrison ldquoRestricted mean life with adjustment for covari-atesrdquo Journal of the American Statistical Association vol 82 no400 pp 1169ndash1176 1987
[84] C Barker ldquoThe mean median and confidence intervals ofthe kaplan-meier survival estimatemdashcomputations and appli-cationsrdquo Journal of the American Statistical Association vol 63no 1 pp 78ndash80 2009
[85] B Efron and R J Tibshirani An Introduction to the Bootstrapvol 57 CRC Press 1994
[86] S Arlot andA Celisse ldquoA survey of cross-validation proceduresfor model selectionrdquo Statistics Surveys vol 4 pp 40ndash79 2010
[87] A Kapelner and J Bleich ldquoPTE Personalized Treatment Eva-luatorrdquo 2014 R package version 10 httpCRANR-projectorgpackage=PTE
[88] S A Murphy ldquoOptimal dynamic treatment regimesrdquo Journal ofthe Royal Statistical Society Series B StatisticalMethodology vol65 no 2 pp 331ndash355 2003
BioMed Research International 13
[89] J M Robins ldquoOptimal structural nested models for optimalsequential decisionsrdquo in Proceedings of the Second Seattle Sym-posium in Biostatistics vol 179 of Lecture Notes in Statistics pp189ndash326 Springer Berlin Germany 2004
[90] Y Zhao and D Zeng ldquoRecent development on statistical meth-ods for personalized medicine discoveryrdquo Frontiers of Medicinein China vol 7 no 1 pp 102ndash110 2013
estimates and Cox regression models These sources can befound at httpcranr-projectorgwebviewsSurvivalhtmlMore details pertaining to statistical methods for survivalanalysis can be found here [71] To compare treatmentrules constructed from Cox and AFT models for examplemethods for measuring the extent of clinical effectiveness foran ITR will be discussed in Section 6
We here present an example when an AFT model wasused to construct an ITR for treatment of HIV [40] Theexample derives from the AIDS Clinical Trials Group Pro-tocol 175 that was discussed in Section 53 [32 66] In thiscase the primary outcome variable was time (in days) tofirst ge50 decline in CD4 count or an AIDS-defining eventor death A total of 12 covariates and four treatments (ZDVZDV + ddI ZDV + zalcitabine and ddI) were includedThe four treatments were evaluated simultaneously Patientsreceiving the standard care of ZDV monotherapy wereconsidered as the reference group Hence three treatmentcontrasts (119868ZDV+ddI 119868ZDV+zalcitabine and 119868ddI) were combinedwith various putative predictive covariates and comparedwith ZDV monotherapy For example gender was detectedas the predictive covariate only for ddI monotherapy Theinvestigators assumed 120601(X 120574) = 120574
0 The treatment rule
recommended 1 patient for ZDV monotherapy while 7291216 and 193 patients were recommended for ZDV + ddIZDV + zalcitabine and ddI respectively
55 Advanced Methods
551 Robust Inference The performances of ITRs heretoforepresented depend heavily on whether the statistical modelswere correctly specified Recently there has been much atten-tion focused on the development of more advanced methodsand modeling strategies that are robust to various aspectsof potential misspecification We have already presented afew robust models that avoid specification of functionalparametric relationships for main effects [32 40] Here weintroduce two more advanced methods widely utilized forITRs that are robust to the type of misspecification issuescommonly encountered in practice [42 43]
Recall that the ITR for a linear model 119864(119884 | 119860 =
119886X) = 120583(119860 = 119886X120573) with two predictive markers followsas 119892(X120573) = 119868(120573
4+ 12057351198832+ 12057361198833) gt 0 where 119886 = 0 1
The treatment rule of 119892(X120573) may use only a subset of thehigh-dimensional covariates (eg 119883
2 1198833) but it always
depends on the correct specification of 119864(119884 | 119860 = 119886X)Defining a scaled version of 120573 as 120578(120573) the correspondingITR is 119892(120578X) = 119892(X120573) = 119868(119883
3gt 1205780+ 12057811198832) where
1205780= minus12057341205736and 120578
1= 12057351205736 If the model for 120583(119860X120573) is
indeed correctly specified the treatment rules of 119892(X120573) and119892(120578X) lead to the same optimal ITR Hence the treatmentrule parameterized by 120578 can be derived from a regressionmodel or may be based on some key clinical considerationswhich enable evaluation of 119892(120578X) directly without referenceto the regression model for 120583(119860X120573)
Let 119862120578
= 119860119892(120578X) + (1 minus 119860)1 minus 119892(120578X) where119862120578= 1 indicates random assignment to an intervention
that is recommended by the personalized treatment rule 119892(120578X) Let 120587(X ) denote the randomization ratio or the
estimated propensity score (as in previous section) and119898(X 120578 ) denote the potential outcome under the treatmentrule estimated from the following model 119864(119884 | 119860 =
119886X) = 120583(119860X120573) For example if the treatment rule 119892(120578X)= 1 then 119898(X 120578 ) = 119892(120578X)120583(119860 = 1X ) + 1 minus 119892(120578
X)120583(119860 = 0X ) = 120583(119860 = 1X ) Two estimators ofthe expected response to treatment the inverse probabilityweighted estimator (IPWE) and doubly robust AIPWE aregiven as follows
IPWE (120578) = 1
119899
119899
sum119894=1
119862120578sdot119894119884119894
120587119888(X119894 120578 )
=1
119899
119899
sum119894=1
119862120578sdot119894119884119894
120587 (X119894 )119860119894 1 minus 120587 (X
119894 )1minus119860119894
AIPWE (120578) = 1
119899
119899
sum119894=1
119862120578sdot119894119884119894
120587119888(X119894 120578 )
minus119862120578sdot119894119884119894 minus 120587119888 (X119894 120578 )
120587119888(X119894 120578 )
119898 (X119894 120578 )
(11)
where 120587119888(X119894 120578 ) = 120587(X )119892(120578X) + 1 minus 120587(X )1 minus
119892(120578X) The optimal treatment rule follows as 119892(X = x)where is estimated from the above models a constraintsuch as 120578 = 1 is imposed to obtain a unique solution [43] If the propensity score is correctly specified theIPWE estimator yields robust (consistent) estimates AIPWEis considered a doubly robust estimator since it produces con-sistent estimates when either propensity score or the model119864(119884 | 119860 = 119886X) is misspecified but not both [42 43] Thecompanion119877 code is publicly available at httponlinelibrarywileycomdoi101111biom12191suppinfo
552 Data Mining and Machine Learning The methodspresented in Section 551 are robust against misspecificationof regression models Yet they often require prespecificationof the parametric form for the treatment rule (eg 119868(119883
3gt
1205780+ 12057811198832)) which can be practically challenging [44]
Well established classification methods and other popularmachine learning techniques can alternatively be customizedto define treatment selection rules [44 72 73] these methodsavoid prespecification of the parametric form of the ITR AnITR can be defined following a two-step approach in thefirst step treatment contrasts are estimated from a positedmodel and in the second step classification techniques areapplied to determine the personalized treatment rules Forexample when only two treatments are considered a newvariable 119885 can be defined based on the treatment contrastthat is 119885 = 1 if Δ(X) = 120583(119860 = 1X) minus 120583(119860 = 0X) gt 0 and119885 = 0 otherwiseThe absolute value of the treatment contrast119882119894= |Δ(X)| can be used in conjunction with a classification
technique to define an appropriate ITR [44]Unlike classification problems wherein the class labels
are observed for the training data the binary ldquoresponserdquovariable 119885 which serves as the class label is not availablein practice Specifically patients who are in the class 119885 = 1
8 BioMed Research International
have 120583(119860 = 1X) gt 120583(119860 = 0X) and should thereforebe treated with the new therapy however these quantitiesneed to be estimated since patients are typically assigned toonly one of the available treatments This imparts flexibilityfor estimation of the optimal treatment regimes since anyof the previously discussed regression models and even someensemble prediction methods such as random forest [74] canbe used to construct the class labels 119885
119894and weights
119894[44]
An ITR can be estimated from the dataset 119885119894X119894 119894 using
any classification approach where 119894are subject specific
misclassification weights [44 45] This includes popularclassificationmethods such as adaptive boosting [75] supportvector machines [76] and classification and regression trees(CART) [77] At least one study has suggested that SVMoutperforms other classification methods in this contextwhereas random forest and boosting perform comparativelybetter than CART [78] However the performances ofthese classification algorithms are data dependent Definitiveconclusion pertaining to their comparative effectiveness ingeneral has yet to be determined [78] It shall be also notedthat these classification methods can be also applied to high-dimensional data [45 72]
One special case of this framework is the ldquovirtue twinsrdquoapproach [45] Specifically in the first step a random forestapproach [74] is used to obtain the treatment contrasts Thenin the second step CART is used to classify subjects tothe optimal treatment regime The approach can be easilyimplemented in 119877 using packages of randomForest [79] andrpart [80] Very recently Kang et al [42] proposed amodifiedversion of the adaptive boosting technique of Friedman et al[75] The algorithm iteratively fits a simple logistic regressionmodel (ldquoworking modelrdquo) to estimate 119875(119884 = 1 | 119860X) and ateach stage assigns higher weights to subjects whose treatmentcontrast is near zero After a prespecified stopping criterionis met an average of the treatment contrasts Δ(X) is calcu-lated for each patient using all models fitted at each iterationA subject is assigned to the new therapy if Δ(X) gt 0 The 119877code for the aforementioned boosting methods is publiclyavailable at httponlinelibrarywileycomdoi101111biom12191suppinfo
Lastly we present a breast cancer example where severalbiomarkers were combined to construct an optimal ITRThe data was collected in the Southwest Oncology Group(SOWG)-SS8814 trial [13] and analyzed with the machinelearning approach of Kang et al [42] Three hundred andsixty-seven node-positive ER-positive breast cancer patientswere selected from the randomized trial of SOWG A total of219 received tamoxifen plus adjuvant chemotherapy and 148was given tamoxifen aloneThe outcome variable was definedas breast cancer recurrence at 5 years The authors selectedthree genes which had presented treatment-biomarker inter-actions in amultivariate linear logistic regressionmodel [42]Data were analyzed with logistic models IPWE AIPWElogistic boosting a single classification tree with treatment-biomarker interactions and the proposed boosting approachwith a classification tree as the working model Each methodidentified different patient cohorts that could benefit fromtamoxifen alone these cohorts consisted of 184 183 128 86
263 and 217 patients respectively (see Table 5 in [42]) In thisanalysis the clinical benefits provided by these 6 treatmentrules were not statistically different Hence investigatorsneed to evaluate and compare ITRs in terms of the extentof expected clinical impact This is considered in the nextsection
Heretofore we have discussed various methodologies for theconstruction of ITR while their performances need to beassessed before these rules can be implemented in clinicalpractice Several aspects pertaining to the performance of aconstructed ITR need to be considered The first one is howwell the ITR fits the data and the second is how well theITR performs compared with existing treatment allocationrules The former is related to the concept of goodness-of-fitor predictive performance [34] As the true optimal treatmentgroups are hidden model fits may be evaluated by measuringthe congruity between observed treatment contrasts andpredicted ones [34 47] More details can be found in a recentpaper by Janes et al [47] Performances of ITRs can becompared via assessment of a global summary measure forexample prolonged survival time or reduced disease rate [4042] Summarymeasures are also very useful for evaluating theextent to which an ITR may benefit patients when applied inpractice Moreover it is essential that performance of an ITRis considered in comparison to business-as-usual proceduressuch as a naive rule that randomly allocates patients totreatment [81] Summary measures will be discussed inSection 61 The effectiveness of an ITR should go beyond thetraining data set used to construct a treatment rule cross-validation and bootstrapping techniques are often employedto assess the impact of ITRs on future patients [81] and willbe discussed in Section 62
61 Summary Measures ITRs may be derived from differ-ent methodologies and comparisons should be conductedwith respect to the appropriate clinically summaries A fewsummary measures for different types of outcomes havebeen proposed [19 40 42] these measures quantify thedirect clinical improvements obtained by applying an ITR incomparison with default methods for treatment allocation
Binary Outcomes Clinical effectiveness for binary clinicalresponse is represented by the difference in disease rates (ortreatment failure) induced by ITR versus a default strategythat allocates all patients to a standard treatment [42 47 82]Let 119892opt(X) = 119868120583(119860 = 1X)minus120583(119860 = 0X) lt 0 be an optimalITR This difference is formally defined as
Note 120583(119860X) needs to be estimated to construct the ITRyet parameters 120573 are omitted for simplicity Larger valuesof Θ119861119892opt(X) indicate increased clinical value for the
biomarker driven ITR A subset of patients that are recom-mended for new treatment (119860 = 1) under an ITR may havebeen randomly selected to receive it while the remainingsubset of ldquounluckyrdquo patientswould have received the standardtreatment [19] The summary measure of Θ
119861119892opt(X) char-
acterizes a weighted difference in the disease rates betweenthe standard and the new treatments in a population whereinthe constructed optimal ITR would recommend the newtreatment 119892opt(X = 1) The weight is the proportion ofpatients identified by the optimal ITR for the new treatmentand can be empirically estimated using the correspondingcounts For example 119875119892opt(X) = 1 can be estimatedusing the number of patients recommended for the newtreatment divided by the total sample size A similar summarystatistic can be derived for an alternative strategy allocatingall patients to the new treatment The summary could beapplied to the aforementioned breast cancer example [42]for example with the aim of finding a subgroup of patientswho were likely to benefit from adjuvant chemotherapywhile those unlikely to benefit would be assigned tamoxifenalone to avoid the unnecessary toxicity and inconvenience ofchemotherapy
Continuous Variables Another strategy for continuous datacompares outcomes observed for ldquoluckyrdquo subjects those whoreceived the therapy that would have been recommendedby the ITR based [81] Further one business-as-usual drugallocation procedure is randomizing treatment and standardcare at the same probability of 05 A summary statisticis to measure the mean outcome under ITR compared tothat obtained under random assignment for instance themean decrease in Hamilton Rating Scale for Depression asdiscussed in Section 51 [19] Define the summary measureas Θ119862119892opt(X) = 120583119892opt(X)X minus 120583119892rand(X)X where
119892rand(X) represents the randomization allocation procedureThe quantity of 120583119892119900119901119905(X)X represents the mean outcomeunder the constructed IRT that can be empirically estimatedfrom the ldquoluckyrdquo subjects and 120583119892rand(X)X can be esti-mated empirically from the sample means
Alternatively an ITR may be compared to an ldquooptimalrdquodrug that has showed universal benefits (a better drug onaverage) in a controlled trial The clinical benefits of anldquooptimalrdquo drug can be defined as 120583119892best(X)X = max120583(119860 =
0X) 120583(119860 = 1X) 120583(119860 = 119886X) and can be empiricallyestimated from the sample means of the new and standardtreatments respectively Then the alternative summary mea-sure is defined as Θ
Survival Data For survival data a clinically relevant measureis mean overall (or progression free) survival time As
survival time is continuous in nature the identical strategyprovided above for continuous outcomes can be employedhere However because the mean survival time may notbe well estimated from the observed data due to a highpercentage of censored observations [40] an alternativemean restricted survival duration was proposed and definedas the population average event-free durations for a restrictedtime of 119905lowast [41 83]Often 119905lowast is chosen to cover the trialrsquos follow-up period Mathematically it can be calculated by integratingthe survival function of 119878(119905) over the domain of (0 119905lowast) that is120583119892opt(X)X 119905lowast = int119905
lowast
0119878(119905)119889119905 and often estimated by the area
under the Kaplan-Meier curve up to 119905lowast [84] Thus an ITRrsquospotential to prolong survival can be calculated asΘ
119878119892opt(X)
119905lowast = 120583119892opt(X)X 119905lowast minus 120583119892rand(X)X 119905lowast
62 AssessingModel Performance The summaries heretoforediscussed evaluate an optimal ITR for a given model andestimating procedure Because these quantities are estimatedconditionally given the observed covariates they neglectto quantify the extent of marginal uncertainty for futurepatients Hence an ITR needs to be internally validated ifexternal data is not available [34] Cross-validation (CV) andbootstrap resampling techniques are commonly used for thispurpose [19 42 45 81] and expositions on both approachesare well described elsewhere [33 85 86]
We here briefly introduce a process that was proposed byKapelner et al [81] in the setting of personalized medicineTenfold CV is commonly used in practice where the wholedata is randomly partitioned into 10 roughly equal-sizedexclusive subsamples All methods under consideration areapplied to 910 of the data excluding 110 as an independenttesting data set The process is repeated 10 times for eachsubsample Considering the assignments recommended bythe optimal ITRs the summary measures can be calculatedusing results from each testing fold [45]TheCVprocess givesthe estimated summary measures and its variation can beevaluated using bootstrap procedures Specifically one drawsa sample with replacement from the entire data and calculatesthe summary measure from 10-fold CV This process willbe repeated 119861 times where 119861 is chosen for resolution ofthe resulting confidence intervals [81] Using the summarymeasures as119861 new random samples the correspondingmeanand variances can be calculated empirically Note that thesummary measures compare two treatment rules one for theoptimal ITR and another naive rule (eg randomization)
The above procedure can be applied to all the meth-ods we have discussed so far The 119877 software package119879119903119890119886119905119898119890119899119905119878119890119897119890119888119905119894119900119899 (httplabsfhcrcorgjanesindexhtml)can be used to implement these methods for evaluatingand comparing biomarkers for binary outcomes [47] Veryrecently an inferential procedure was proposed for contin-uous outcomes that is implemented in the publicly available119877 package ldquoPersonalized Treatment Evaluatorrdquo [81 87] Bothmethods consider data from RCTs with two arms for com-parative treatmentsThesemethods are in general applicableto regression model based methods but are not suitable forapproaches based on classification techniques or penalizedregression
10 BioMed Research International
Next we present two examples Recall in Section 55 thatKang et al [42] reported the estimated clinical benefits ofan ITR for breast cancer when compared to the defaultstrategy of assigning all patients to adjuvant chemotherapyThe proposed approach (based on boosting and classificationtrees) achieved the highest value of the summary measureat 0081 with 95 confidence interval (CI) (0000 0159)[42] In the second example introduced in Section 51 [19]the authors calculated the mean score of the HamiltonRating Scale for Depression for two groups of subjectsgroups were defined by randomly assigning patients to theldquooptimalrdquo and ldquononoptimalrdquo therapy as defined by the ITRThe reported difference between the two groups was minus178with a 119875 value of 009 which fails to attain a clinicalsignificant difference of 3 [19] The same data was analyzedby Kapelner et al [81] Following the discussed procedurethe authors reported the estimated values (and 95 CI)of Θ119862119892opt(X) and Θ
119862alt119892opt(X) as minus0842(minus2657 minus0441)
and minus0765(minus2362 0134) respectively The results whichfail to achieve clinical significance were based on rigorousstatistical methods and thus can be considered reliableestimates of the ITRrsquos performance
7 Discussion
As our understanding tumor heterogeneity evolves person-alized medicine will become standard medical practice inoncology Therefore it is essential that the oncology com-munity uses appropriate analytical methods for identifyingand evaluating the performance of personalized treatmentrules This paper provided an exposition of the process forusing statistical inference to establish optimal individualizedtreatment rules using data acquired from clinical study Thequality of an ITR depends on the quality of the design used toacquire the dataMoreover an ITRmust be properly validatedbefore it is integrated into clinical practice Personalizedmedicine in some areas of oncologymay be limited by the factthat biomarkers arising from a small panel of genesmay neveradequately characterize the extent of tumor heterogeneityinherent to the disease Consequently the available statisticalmethodology needs to evolve in order to optimally exploitglobal gene signatures for personalized medicine
The bulk of our review focused on statistical approachesfor treatment selection at a single time point The readershould note that another important area of research considersoptimal dynamic treatment regimes (DTRs) [88 89] whereintreatment decisions are considered sequentially over thecourse ofmultiple periods of intervention using each patientrsquosprior treatment history Zhao and Zeng provide a summaryof recent developments in this area [90]
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
JunshengMa was fully funded by the University of Texas MDAnderson Cancer Center internal funds Brian P Hobbs and
Francesco C Stingo were partially supported by the CancerCenter Support Grant (CCSG) (P30 CA016672)
References
[1] J Reimand O Wagih and G D Bader ldquoThe mutational land-scape of phosphorylation signaling in cancerrdquo Scientific Reportsvol 3 article 2651 2013
[2] D Hanahan and R AWeinberg ldquoHallmarks of cancer the nextgenerationrdquo Cell vol 144 no 5 pp 646ndash674 2011
[3] R Simon ldquoClinical trial designs for evaluating the medicalutility of prognostic and predictive biomarkers in oncologyrdquoPersonalized Medicine vol 7 no 1 pp 33ndash47 2010
[4] P L Bedard A R Hansen M J Ratain and L L Siu ldquoTumourheterogeneity in the clinicrdquo Nature vol 501 no 7467 pp 355ndash364 2013
[5] M D Pegram G Pauletti and D J Slamon ldquoHer-2neu as apredictive marker of response to breast cancer therapyrdquo BreastCancer Research and Treatment vol 52 no 1ndash3 pp 65ndash77 1998
[6] G J Kelloff andC C Sigman ldquoCancer biomarkers selecting theright drug for the right patientrdquoNature Reviews Drug Discoveryvol 11 no 3 pp 201ndash214 2012
[7] J ADiMasi JM Reichert L Feldman andAMalins ldquoClinicalapproval success rates for investigational cancer drugsrdquoClinicalPharmacology andTherapeutics vol 94 no 3 pp 329ndash335 2013
[8] M Hay D W Thomas J L Craighead C Economides andJ Rosenthal ldquoClinical development success rates for investiga-tional drugsrdquo Nature Biotechnology vol 32 no 1 pp 40ndash512014
[9] S S Knox ldquoFrom lsquoomicsrsquo to complex disease a systems biologyapproach to gene-environment interactions in cancerrdquo CancerCell International vol 10 article 11 2010
[10] V Deschoolmeester M Baay P Specenier F Lardon and JB Vermorken ldquoA review of the most promising biomarkersin colorectal cancer one step closer to targeted therapyrdquo TheOncologist vol 15 no 7 pp 699ndash731 2010
[11] J A Sosman K B Kim L Schuchter et al ldquoSurvival in brafV600ndashmutant advanced melanoma treated with vemurafenibrdquoThe New England Journal of Medicine vol 366 no 8 pp 707ndash714 2012
[12] S Paik S Shak G Tang et al ldquoA multigene assay to predictrecurrence of tamoxifen-treated node-negative breast cancerrdquoThe New England Journal of Medicine vol 351 no 27 pp 2817ndash2826 2004
[13] K S Albain W E Barlow S Shak et al ldquoPrognostic and pre-dictive value of the 21-gene recurrence score assay in post-menopausal women with node-positive oestrogen-receptor-positive breast cancer on chemotherapy a retrospective analysisof a randomised trialrdquoTheLancet Oncology vol 11 no 1 pp 55ndash65 2010
[14] J E Lang J S Wecsler M F Press and D Tripathy ldquoMolecularmarkers for breast cancer diagnosis prognosis and targetedtherapyrdquo Journal of Surgical Oncology vol 111 no 1 pp 81ndash902015
[15] W Werft A Benner and A Kopp-Schneider ldquoOn the identi-fication of predictive biomarkers detecting treatment-by-geneinteraction in high-dimensional datardquo Computational Statisticsand Data Analysis vol 56 no 5 pp 1275ndash1286 2012
[16] M Jenkins A Flynn T Smart et al ldquoA statisticianrsquos perspectiveon biomarkers in drug developmentrdquo Pharmaceutical Statisticsvol 10 no 6 pp 494ndash507 2011
BioMed Research International 11
[17] A J Vickers MW Kattan and D J Sargent ldquoMethod for eval-uating prediction models that apply the results of randomizedtrials to individual patientsrdquo Trials vol 8 no 1 article 14 2007
[18] H Janes M S Pepe P M Bossuyt andW E Barlow ldquoMeasur-ing the performance of markers for guiding treatment deci-sionsrdquo Annals of Internal Medicine vol 154 no 4 pp 253ndash2592011
[19] R J DeRubeis Z D Cohen N R Forand J C Fournier L AGelfand and L Lorenzo-Luaces ldquoThe personalized advantageindex translating research on prediction into individualizedtreatment recommendationsAdemonstrationrdquoPLoSONE vol9 no 1 Article ID e83875 2014
[20] D P Byar and D K Corle ldquoSelecting optimal treatment inclinical trials using covariate informationrdquo Journal of ChronicDiseases vol 30 no 7 pp 445ndash459 1977
[21] N Thatcher A Chang P Parikh et al ldquoGefitinib plus bestsupportive care in previously treated patients with refractoryadvanced non-small-cell lung cancer results from a ran-domised placebo-controlled multicentre study (iressa survivalevaluation in lung cancer)rdquo The Lancet vol 366 no 9496 pp1527ndash1537 2005
[22] A J Vickers ldquoPrediction models in cancer carerdquo CA A CancerJournal for Clinicians vol 61 no 5 pp 315ndash326 2011
[23] RM Simon ldquoSubgroup analysisrdquo inWiley Encyclopedia of Clin-ical Trials John Wiley amp Sons Hoboken NJ USA 2007
[24] S J Pocock S E Assmann L E Enos and L E Kasten ldquoSub-group analysis covariate adjustment and baseline comparisonsin clinical trial reporting current practice and problemsrdquoStatistics in Medicine vol 21 no 19 pp 2917ndash2930 2002
[25] PM Rothwell ZMehta S CHoward S A Gutnikov andC PWarlow ldquoFrom subgroups to individuals general principles andthe example of carotid endarterectomyrdquoTheLancet vol 365 no9455 pp 256ndash265 2005
[26] R Wang S W Lagakos J H Ware D J Hunter and J MDrazen ldquoStatistics in medicinemdashreporting of subgroup ana-lyses in clinical trialsrdquoTheNewEngland Journal ofMedicine vol357 no 21 pp 2108ndash2194 2007
[27] International Breast Cancer Study Group ldquoEndocrine respon-siveness and tailoring adjuvant therapy for postmenopausallymph node-negative breast cancer a randomized trialrdquo Journalof the National Cancer Institute vol 94 no 14 pp 1054ndash10652002
[28] Early Breast Cancer Trialistsrsquo Collaborative Group (EBCTCG)ldquoEffects of chemotherapy and hormonal therapy for early breastcancer on recurrence and 15-year survival an overview of therandomised trialsrdquoThe Lancet vol 365 no 9472 pp 1687ndash17172005
[29] F E Harrell K L Lee and D B Mark ldquoTutorial in biostatisticsmultivariable prognostic models issues in developing modelsevaluating assumptions and adequacy and measuring andreducing errorsrdquo Statistics in Medicine vol 15 no 4 pp 361ndash387 1996
[30] S Gill C L Loprinzi D J Sargent et al ldquoPooled analysis offluorouracil-based adjuvant therapy for stage II and III coloncancer who benefits and by how muchrdquo Journal of ClinicalOncology vol 22 no 10 pp 1797ndash1806 2004
[31] M Qian and S A Murphy ldquoPerformance guarantees forindividualized treatment rulesrdquoThe Annals of Statistics vol 39no 2 pp 1180ndash1210 2011
[32] W Lu H H Zhang and D Zeng ldquoVariable selection for opti-mal treatment decisionrdquo StatisticalMethods inMedical Researchvol 22 no 5 pp 493ndash504 2013
[33] R Kohavi ldquoA study of cross-validation and bootstrap for accu-racy estimation and model selectionrdquo in Proceedings of the 14thInternational Joint Conference on Artificial Intelligence (IJCAIrsquo95) vol 2 pp 1137ndash1145 1995
[34] E W Steyerberg A J Vickers N R Cook et al ldquoAssessing theperformance of prediction models a framework for traditionaland novel measuresrdquo Epidemiology vol 21 no 1 pp 128ndash1382010
[35] P R Rosenbaum and D B Rubin ldquoThe central role of the pro-pensity score in observational studies for causal effectsrdquo Bio-metrika vol 70 no 1 pp 41ndash55 1983
[36] R B drsquoAgostino Jr ldquoTutorial in biostatistics propensity scoremethods for bias reduction in the comparison of a treatment toa non-randomized control grouprdquo Statistics in Medicine vol 17no 19 pp 2265ndash2281 1998
[37] R Pazdur ldquoEndpoints for assessing drug activity in clinicaltrialsrdquoThe Oncologist vol 13 supplement 2 pp 19ndash21 2008
[38] S L Spruance J E Reid M Grace and M Samore ldquoHazardratio in clinical trialsrdquo Antimicrobial Agents and Chemotherapyvol 48 no 8 pp 2787ndash2792 2004
[39] J D Kalbeisch and R L Prentice The Statistical Analysis ofFailure Time Data vol 360 John Wiley amp Sons 2011
[40] Y Geng Flexible Statistical Learning Methods for SurvivalData Risk Prediction and Optimal Treatment Decision NorthCarolina State University 2013
[41] J Li L Zhao L Tian et alAPredictive Enrichment Procedure toIdentify Potential Responders to a NewTherapy for RandomizedComparative Controlled Clinical Studies Harvard UniversityBiostatisticsWorking Paper SeriesWorking Paper 169 HarvardUniversity 2014
[42] C Kang H Janes and Y Huang ldquoCombining biomarkers tooptimize patient treatment recommendationsrdquo Biometrics vol70 no 3 pp 695ndash720 2014
[43] B Zhang A A Tsiatis E B Laber and M Davidian ldquoA robustmethod for estimating optimal treatment regimesrdquo Biometricsvol 68 no 4 pp 1010ndash1018 2012
[44] B Zhang A A Tsiatis M Davidian M Zhang and E LaberldquoEstimating optimal treatment regimes from a classificationperspectiverdquo Stat vol 1 no 1 pp 103ndash114 2012
[45] J C Foster J M G Taylor and S J Ruberg ldquoSubgroup identi-fication from randomized clinical trial datardquo Statistics in Medi-cine vol 30 no 24 pp 2867ndash2880 2011
[46] US Food and Drug Administration Guidance for IndustryClinical Trial Endpoints for the Approval of Cancer Drugs andBiologics US Food and Drug AdministrationWashington DCUSA 2007
[47] H Janes M D Brown M Pepe and Y Huang ldquoStatisticalmethods for evaluating and comparing biomarkers for patienttreatment selectionrdquo UW Biostatistics Working Paper SeriesWorking Paper 389 2013
[48] R Development Core Team R a language and environment forstatistical computing 2008 httpwwwR-projectorg
[49] X-JMa ZWang P D Ryan et al ldquoA two-gene expression ratiopredicts clinical outcome in breast cancer patients treated withtamoxifenrdquo Cancer Cell vol 5 no 6 pp 607ndash616 2004
[50] X-JMa S GHilsenbeckWWang et al ldquoTheHOXB13IL17BRexpression index is a prognostic factor in early-stage breastcancerrdquo Journal of Clinical Oncology vol 24 no 28 pp 4611ndash4619 2006
[51] D C Sgroi E Carney E Zarrella et al ldquoPrediction of latedisease recurrence and extended adjuvant letrozole benefit by
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the HOXB13IL17BR biomarkerrdquo Journal of the National CancerInstitute vol 105 no 14 pp 1036ndash1042 2013
[52] M S Cepeda R Boston J T Farrar and B L Strom ldquoCom-parison of logistic regression versus propensity score when thenumber of events is low and there are multiple confoundersrdquoThe American Journal of Epidemiology vol 158 no 3 pp 280ndash287 2003
[53] P C Austin ldquoAn introduction to propensity score methods forreducing the effects of confounding in observational studiesrdquoMultivariate Behavioral Research vol 46 no 3 pp 399ndash4242011
[54] G Heinze and P Juni ldquoAn overview of the objectives of andthe approaches to propensity score analysesrdquo European HeartJournal vol 32 no 14 Article ID ehr031 pp 1704ndash1708 2011
[55] L E Braitman and P R Rosenbaum ldquoRare outcomes commontreatments analytic strategies using propensity scoresrdquo Annalsof Internal Medicine vol 137 no 8 pp 693ndash695 2002
[56] D M Witten and R Tibshirani ldquoSurvival analysis withhigh-dimensional covariatesrdquo Statistical Methods in MedicalResearch vol 19 no 1 pp 29ndash51 2010
[57] A E Hoerl and R W Kennard ldquoRidge regression biasedestimation for nonorthogonal problemsrdquoTechnometrics vol 42no 1 pp 80ndash86 2000
[58] R Tibshirani ldquoRegression shrinkage and selection via the lassordquoJournal of the Royal Statistical Society Series B Methodologicalvol 58 no 1 pp 267ndash288 1996
[59] H Zou ldquoThe adaptive lasso and its oracle propertiesrdquo Journal ofthe American Statistical Association vol 101 no 476 pp 1418ndash1429 2006
[60] B Efron T Hastie I Johnstone and R Tibshirani ldquoLeast angleregressionrdquo The Annals of Statistics vol 32 no 2 pp 407ndash4992004
[61] T Hastie and B Efron ldquolars Least angle regression lasso andforward stagewiserdquo R package version 12 2013 httpcranr-projectorgwebpackageslarsindexhtml
[62] J Fan and R Li ldquoVariable selection via nonconcave penalizedlikelihood and its oracle propertiesrdquo Journal of the AmericanStatistical Association vol 96 no 456 pp 1348ndash1360 2001
[63] H Zou and T Hastie ldquoRegularization and variable selection viathe elastic netrdquo Journal of the Royal Statistical Society Series BStatistical Methodology vol 67 no 2 pp 301ndash320 2005
[64] N Kramer J Schafer and A-L Boulesteix ldquoRegularized esti-mation of large-scale gene association networks using graphicalgaussian modelsrdquo BMC Bioinformatics vol 10 no 1 article 3842009
[65] P Breheny and J Huang ldquoCoordinate descent algorithms fornonconvex penalized regression with applications to biologicalfeature selectionrdquo The Annals of Applied Statistics vol 5 no 1pp 232ndash253 2011
[66] S M Hammer D A Katzenstein M D Hughes et al ldquoA trialcomparing nucleoside monotherapy with combination therapyin HIV-infected adults with CD4 cell counts from 200 to 500per cubicmillimeterrdquoTheNew England Journal ofMedicine vol335 no 15 pp 1081ndash1090 1996
[67] H M Boslashvelstad S Nygard H L Stoslashrvold et al ldquoPredictingsurvival from microarray datamdasha comparative studyrdquo Bioinfor-matics vol 23 no 16 pp 2080ndash2087 2007
[68] V Kehl and K Ulm ldquoResponder identification in clinical trialswith censored datardquoComputational Statistics andDataAnalysisvol 50 no 5 pp 1338ndash1355 2006
[69] P Royston and M K Parmar ldquoThe use of restricted meansurvival time to estimate the treatment effect in randomizedclinical trials when the proportional hazards assumption is indoubtrdquo Statistics inMedicine vol 30 no 19 pp 2409ndash2421 2011
[70] P Royston andM K B Parmar ldquoRestrictedmean survival timean alternative to the hazard ratio for the design and analysis ofrandomized trials with a time-to-event outcomerdquo BMCMedicalResearch Methodology vol 13 no 1 article 152 2013
[71] E T Lee and J W Wang Statistical Methods for Survival DataAnalysis John Wiley amp Sons Hoboken NJ USA 2013
[72] Y Zhao D Zeng A J Rush and M R Kosorok ldquoEstimatingindividualized treatment rules using outcome weighted learn-ingrdquo Journal of the American Statistical Association vol 107 no499 pp 1106ndash1118 2012
[73] D B Rubin and M J van der Laan ldquoStatistical issues andlimitations in personalized medicine research with clinicaltrialsrdquoThe International Journal of Biostatistics vol 8 no 1 pp1ndash20 2012
[74] L Breiman ldquoRandom forestsrdquoMachine Learning vol 45 no 1pp 5ndash32 2001
[75] J Friedman T Hastie and R Tibshirani ldquoAdditive logisticregression a statistical view of boostingrdquo The Annals of Statis-tics vol 28 no 2 pp 337ndash407 2000
[76] C Cortes and V Vapnik ldquoSupport-vector networksrdquo MachineLearning vol 20 no 3 pp 273ndash297 1995
[77] L Breiman J Friedman C J Stone and R A Olshen Classi-fication and Regression Trees CRC Press New York NY USA1984
[78] S Dudoit J Fridlyand and T P Speed ldquoComparison of dis-criminationmethods for the classification of tumors using geneexpression datardquo Journal of the American Statistical Associationvol 97 no 457 pp 77ndash87 2002
[79] A Liaw and MWiener ldquoClassification and regression by rand-omforestrdquoRNews vol 2 no 3 pp 18ndash22 2002 httpCRANR-projectorgdocRnews
[80] T Therneau B Atkinson and B Ripley ldquorpart RecursivePartitioning and Regression Treesrdquo R package version 41-3httpcranr-projectorgwebpackagesrpartindexhtml
[81] A Kapelner J Bleich Z D Cohen R J DeRubeis and RBerk ldquoInference for treatment regime models in personalizedmedicinerdquo httparxivorgabs14047844
[82] X Song and M S Pepe ldquoEvaluating markers for selecting apatientrsquos treatmentrdquoBiometrics vol 60 no 4 pp 874ndash883 2004
[83] T Karrison ldquoRestricted mean life with adjustment for covari-atesrdquo Journal of the American Statistical Association vol 82 no400 pp 1169ndash1176 1987
[84] C Barker ldquoThe mean median and confidence intervals ofthe kaplan-meier survival estimatemdashcomputations and appli-cationsrdquo Journal of the American Statistical Association vol 63no 1 pp 78ndash80 2009
[85] B Efron and R J Tibshirani An Introduction to the Bootstrapvol 57 CRC Press 1994
[86] S Arlot andA Celisse ldquoA survey of cross-validation proceduresfor model selectionrdquo Statistics Surveys vol 4 pp 40ndash79 2010
[87] A Kapelner and J Bleich ldquoPTE Personalized Treatment Eva-luatorrdquo 2014 R package version 10 httpCRANR-projectorgpackage=PTE
[88] S A Murphy ldquoOptimal dynamic treatment regimesrdquo Journal ofthe Royal Statistical Society Series B StatisticalMethodology vol65 no 2 pp 331ndash355 2003
BioMed Research International 13
[89] J M Robins ldquoOptimal structural nested models for optimalsequential decisionsrdquo in Proceedings of the Second Seattle Sym-posium in Biostatistics vol 179 of Lecture Notes in Statistics pp189ndash326 Springer Berlin Germany 2004
[90] Y Zhao and D Zeng ldquoRecent development on statistical meth-ods for personalized medicine discoveryrdquo Frontiers of Medicinein China vol 7 no 1 pp 102ndash110 2013
have 120583(119860 = 1X) gt 120583(119860 = 0X) and should thereforebe treated with the new therapy however these quantitiesneed to be estimated since patients are typically assigned toonly one of the available treatments This imparts flexibilityfor estimation of the optimal treatment regimes since anyof the previously discussed regression models and even someensemble prediction methods such as random forest [74] canbe used to construct the class labels 119885
119894and weights
119894[44]
An ITR can be estimated from the dataset 119885119894X119894 119894 using
any classification approach where 119894are subject specific
misclassification weights [44 45] This includes popularclassificationmethods such as adaptive boosting [75] supportvector machines [76] and classification and regression trees(CART) [77] At least one study has suggested that SVMoutperforms other classification methods in this contextwhereas random forest and boosting perform comparativelybetter than CART [78] However the performances ofthese classification algorithms are data dependent Definitiveconclusion pertaining to their comparative effectiveness ingeneral has yet to be determined [78] It shall be also notedthat these classification methods can be also applied to high-dimensional data [45 72]
One special case of this framework is the ldquovirtue twinsrdquoapproach [45] Specifically in the first step a random forestapproach [74] is used to obtain the treatment contrasts Thenin the second step CART is used to classify subjects tothe optimal treatment regime The approach can be easilyimplemented in 119877 using packages of randomForest [79] andrpart [80] Very recently Kang et al [42] proposed amodifiedversion of the adaptive boosting technique of Friedman et al[75] The algorithm iteratively fits a simple logistic regressionmodel (ldquoworking modelrdquo) to estimate 119875(119884 = 1 | 119860X) and ateach stage assigns higher weights to subjects whose treatmentcontrast is near zero After a prespecified stopping criterionis met an average of the treatment contrasts Δ(X) is calcu-lated for each patient using all models fitted at each iterationA subject is assigned to the new therapy if Δ(X) gt 0 The 119877code for the aforementioned boosting methods is publiclyavailable at httponlinelibrarywileycomdoi101111biom12191suppinfo
Lastly we present a breast cancer example where severalbiomarkers were combined to construct an optimal ITRThe data was collected in the Southwest Oncology Group(SOWG)-SS8814 trial [13] and analyzed with the machinelearning approach of Kang et al [42] Three hundred andsixty-seven node-positive ER-positive breast cancer patientswere selected from the randomized trial of SOWG A total of219 received tamoxifen plus adjuvant chemotherapy and 148was given tamoxifen aloneThe outcome variable was definedas breast cancer recurrence at 5 years The authors selectedthree genes which had presented treatment-biomarker inter-actions in amultivariate linear logistic regressionmodel [42]Data were analyzed with logistic models IPWE AIPWElogistic boosting a single classification tree with treatment-biomarker interactions and the proposed boosting approachwith a classification tree as the working model Each methodidentified different patient cohorts that could benefit fromtamoxifen alone these cohorts consisted of 184 183 128 86
263 and 217 patients respectively (see Table 5 in [42]) In thisanalysis the clinical benefits provided by these 6 treatmentrules were not statistically different Hence investigatorsneed to evaluate and compare ITRs in terms of the extentof expected clinical impact This is considered in the nextsection
Heretofore we have discussed various methodologies for theconstruction of ITR while their performances need to beassessed before these rules can be implemented in clinicalpractice Several aspects pertaining to the performance of aconstructed ITR need to be considered The first one is howwell the ITR fits the data and the second is how well theITR performs compared with existing treatment allocationrules The former is related to the concept of goodness-of-fitor predictive performance [34] As the true optimal treatmentgroups are hidden model fits may be evaluated by measuringthe congruity between observed treatment contrasts andpredicted ones [34 47] More details can be found in a recentpaper by Janes et al [47] Performances of ITRs can becompared via assessment of a global summary measure forexample prolonged survival time or reduced disease rate [4042] Summarymeasures are also very useful for evaluating theextent to which an ITR may benefit patients when applied inpractice Moreover it is essential that performance of an ITRis considered in comparison to business-as-usual proceduressuch as a naive rule that randomly allocates patients totreatment [81] Summary measures will be discussed inSection 61 The effectiveness of an ITR should go beyond thetraining data set used to construct a treatment rule cross-validation and bootstrapping techniques are often employedto assess the impact of ITRs on future patients [81] and willbe discussed in Section 62
61 Summary Measures ITRs may be derived from differ-ent methodologies and comparisons should be conductedwith respect to the appropriate clinically summaries A fewsummary measures for different types of outcomes havebeen proposed [19 40 42] these measures quantify thedirect clinical improvements obtained by applying an ITR incomparison with default methods for treatment allocation
Binary Outcomes Clinical effectiveness for binary clinicalresponse is represented by the difference in disease rates (ortreatment failure) induced by ITR versus a default strategythat allocates all patients to a standard treatment [42 47 82]Let 119892opt(X) = 119868120583(119860 = 1X)minus120583(119860 = 0X) lt 0 be an optimalITR This difference is formally defined as
Note 120583(119860X) needs to be estimated to construct the ITRyet parameters 120573 are omitted for simplicity Larger valuesof Θ119861119892opt(X) indicate increased clinical value for the
biomarker driven ITR A subset of patients that are recom-mended for new treatment (119860 = 1) under an ITR may havebeen randomly selected to receive it while the remainingsubset of ldquounluckyrdquo patientswould have received the standardtreatment [19] The summary measure of Θ
119861119892opt(X) char-
acterizes a weighted difference in the disease rates betweenthe standard and the new treatments in a population whereinthe constructed optimal ITR would recommend the newtreatment 119892opt(X = 1) The weight is the proportion ofpatients identified by the optimal ITR for the new treatmentand can be empirically estimated using the correspondingcounts For example 119875119892opt(X) = 1 can be estimatedusing the number of patients recommended for the newtreatment divided by the total sample size A similar summarystatistic can be derived for an alternative strategy allocatingall patients to the new treatment The summary could beapplied to the aforementioned breast cancer example [42]for example with the aim of finding a subgroup of patientswho were likely to benefit from adjuvant chemotherapywhile those unlikely to benefit would be assigned tamoxifenalone to avoid the unnecessary toxicity and inconvenience ofchemotherapy
Continuous Variables Another strategy for continuous datacompares outcomes observed for ldquoluckyrdquo subjects those whoreceived the therapy that would have been recommendedby the ITR based [81] Further one business-as-usual drugallocation procedure is randomizing treatment and standardcare at the same probability of 05 A summary statisticis to measure the mean outcome under ITR compared tothat obtained under random assignment for instance themean decrease in Hamilton Rating Scale for Depression asdiscussed in Section 51 [19] Define the summary measureas Θ119862119892opt(X) = 120583119892opt(X)X minus 120583119892rand(X)X where
119892rand(X) represents the randomization allocation procedureThe quantity of 120583119892119900119901119905(X)X represents the mean outcomeunder the constructed IRT that can be empirically estimatedfrom the ldquoluckyrdquo subjects and 120583119892rand(X)X can be esti-mated empirically from the sample means
Alternatively an ITR may be compared to an ldquooptimalrdquodrug that has showed universal benefits (a better drug onaverage) in a controlled trial The clinical benefits of anldquooptimalrdquo drug can be defined as 120583119892best(X)X = max120583(119860 =
0X) 120583(119860 = 1X) 120583(119860 = 119886X) and can be empiricallyestimated from the sample means of the new and standardtreatments respectively Then the alternative summary mea-sure is defined as Θ
Survival Data For survival data a clinically relevant measureis mean overall (or progression free) survival time As
survival time is continuous in nature the identical strategyprovided above for continuous outcomes can be employedhere However because the mean survival time may notbe well estimated from the observed data due to a highpercentage of censored observations [40] an alternativemean restricted survival duration was proposed and definedas the population average event-free durations for a restrictedtime of 119905lowast [41 83]Often 119905lowast is chosen to cover the trialrsquos follow-up period Mathematically it can be calculated by integratingthe survival function of 119878(119905) over the domain of (0 119905lowast) that is120583119892opt(X)X 119905lowast = int119905
lowast
0119878(119905)119889119905 and often estimated by the area
under the Kaplan-Meier curve up to 119905lowast [84] Thus an ITRrsquospotential to prolong survival can be calculated asΘ
119878119892opt(X)
119905lowast = 120583119892opt(X)X 119905lowast minus 120583119892rand(X)X 119905lowast
62 AssessingModel Performance The summaries heretoforediscussed evaluate an optimal ITR for a given model andestimating procedure Because these quantities are estimatedconditionally given the observed covariates they neglectto quantify the extent of marginal uncertainty for futurepatients Hence an ITR needs to be internally validated ifexternal data is not available [34] Cross-validation (CV) andbootstrap resampling techniques are commonly used for thispurpose [19 42 45 81] and expositions on both approachesare well described elsewhere [33 85 86]
We here briefly introduce a process that was proposed byKapelner et al [81] in the setting of personalized medicineTenfold CV is commonly used in practice where the wholedata is randomly partitioned into 10 roughly equal-sizedexclusive subsamples All methods under consideration areapplied to 910 of the data excluding 110 as an independenttesting data set The process is repeated 10 times for eachsubsample Considering the assignments recommended bythe optimal ITRs the summary measures can be calculatedusing results from each testing fold [45]TheCVprocess givesthe estimated summary measures and its variation can beevaluated using bootstrap procedures Specifically one drawsa sample with replacement from the entire data and calculatesthe summary measure from 10-fold CV This process willbe repeated 119861 times where 119861 is chosen for resolution ofthe resulting confidence intervals [81] Using the summarymeasures as119861 new random samples the correspondingmeanand variances can be calculated empirically Note that thesummary measures compare two treatment rules one for theoptimal ITR and another naive rule (eg randomization)
The above procedure can be applied to all the meth-ods we have discussed so far The 119877 software package119879119903119890119886119905119898119890119899119905119878119890119897119890119888119905119894119900119899 (httplabsfhcrcorgjanesindexhtml)can be used to implement these methods for evaluatingand comparing biomarkers for binary outcomes [47] Veryrecently an inferential procedure was proposed for contin-uous outcomes that is implemented in the publicly available119877 package ldquoPersonalized Treatment Evaluatorrdquo [81 87] Bothmethods consider data from RCTs with two arms for com-parative treatmentsThesemethods are in general applicableto regression model based methods but are not suitable forapproaches based on classification techniques or penalizedregression
10 BioMed Research International
Next we present two examples Recall in Section 55 thatKang et al [42] reported the estimated clinical benefits ofan ITR for breast cancer when compared to the defaultstrategy of assigning all patients to adjuvant chemotherapyThe proposed approach (based on boosting and classificationtrees) achieved the highest value of the summary measureat 0081 with 95 confidence interval (CI) (0000 0159)[42] In the second example introduced in Section 51 [19]the authors calculated the mean score of the HamiltonRating Scale for Depression for two groups of subjectsgroups were defined by randomly assigning patients to theldquooptimalrdquo and ldquononoptimalrdquo therapy as defined by the ITRThe reported difference between the two groups was minus178with a 119875 value of 009 which fails to attain a clinicalsignificant difference of 3 [19] The same data was analyzedby Kapelner et al [81] Following the discussed procedurethe authors reported the estimated values (and 95 CI)of Θ119862119892opt(X) and Θ
119862alt119892opt(X) as minus0842(minus2657 minus0441)
and minus0765(minus2362 0134) respectively The results whichfail to achieve clinical significance were based on rigorousstatistical methods and thus can be considered reliableestimates of the ITRrsquos performance
7 Discussion
As our understanding tumor heterogeneity evolves person-alized medicine will become standard medical practice inoncology Therefore it is essential that the oncology com-munity uses appropriate analytical methods for identifyingand evaluating the performance of personalized treatmentrules This paper provided an exposition of the process forusing statistical inference to establish optimal individualizedtreatment rules using data acquired from clinical study Thequality of an ITR depends on the quality of the design used toacquire the dataMoreover an ITRmust be properly validatedbefore it is integrated into clinical practice Personalizedmedicine in some areas of oncologymay be limited by the factthat biomarkers arising from a small panel of genesmay neveradequately characterize the extent of tumor heterogeneityinherent to the disease Consequently the available statisticalmethodology needs to evolve in order to optimally exploitglobal gene signatures for personalized medicine
The bulk of our review focused on statistical approachesfor treatment selection at a single time point The readershould note that another important area of research considersoptimal dynamic treatment regimes (DTRs) [88 89] whereintreatment decisions are considered sequentially over thecourse ofmultiple periods of intervention using each patientrsquosprior treatment history Zhao and Zeng provide a summaryof recent developments in this area [90]
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
JunshengMa was fully funded by the University of Texas MDAnderson Cancer Center internal funds Brian P Hobbs and
Francesco C Stingo were partially supported by the CancerCenter Support Grant (CCSG) (P30 CA016672)
References
[1] J Reimand O Wagih and G D Bader ldquoThe mutational land-scape of phosphorylation signaling in cancerrdquo Scientific Reportsvol 3 article 2651 2013
[2] D Hanahan and R AWeinberg ldquoHallmarks of cancer the nextgenerationrdquo Cell vol 144 no 5 pp 646ndash674 2011
[3] R Simon ldquoClinical trial designs for evaluating the medicalutility of prognostic and predictive biomarkers in oncologyrdquoPersonalized Medicine vol 7 no 1 pp 33ndash47 2010
[4] P L Bedard A R Hansen M J Ratain and L L Siu ldquoTumourheterogeneity in the clinicrdquo Nature vol 501 no 7467 pp 355ndash364 2013
[5] M D Pegram G Pauletti and D J Slamon ldquoHer-2neu as apredictive marker of response to breast cancer therapyrdquo BreastCancer Research and Treatment vol 52 no 1ndash3 pp 65ndash77 1998
[6] G J Kelloff andC C Sigman ldquoCancer biomarkers selecting theright drug for the right patientrdquoNature Reviews Drug Discoveryvol 11 no 3 pp 201ndash214 2012
[7] J ADiMasi JM Reichert L Feldman andAMalins ldquoClinicalapproval success rates for investigational cancer drugsrdquoClinicalPharmacology andTherapeutics vol 94 no 3 pp 329ndash335 2013
[8] M Hay D W Thomas J L Craighead C Economides andJ Rosenthal ldquoClinical development success rates for investiga-tional drugsrdquo Nature Biotechnology vol 32 no 1 pp 40ndash512014
[9] S S Knox ldquoFrom lsquoomicsrsquo to complex disease a systems biologyapproach to gene-environment interactions in cancerrdquo CancerCell International vol 10 article 11 2010
[10] V Deschoolmeester M Baay P Specenier F Lardon and JB Vermorken ldquoA review of the most promising biomarkersin colorectal cancer one step closer to targeted therapyrdquo TheOncologist vol 15 no 7 pp 699ndash731 2010
[11] J A Sosman K B Kim L Schuchter et al ldquoSurvival in brafV600ndashmutant advanced melanoma treated with vemurafenibrdquoThe New England Journal of Medicine vol 366 no 8 pp 707ndash714 2012
[12] S Paik S Shak G Tang et al ldquoA multigene assay to predictrecurrence of tamoxifen-treated node-negative breast cancerrdquoThe New England Journal of Medicine vol 351 no 27 pp 2817ndash2826 2004
[13] K S Albain W E Barlow S Shak et al ldquoPrognostic and pre-dictive value of the 21-gene recurrence score assay in post-menopausal women with node-positive oestrogen-receptor-positive breast cancer on chemotherapy a retrospective analysisof a randomised trialrdquoTheLancet Oncology vol 11 no 1 pp 55ndash65 2010
[14] J E Lang J S Wecsler M F Press and D Tripathy ldquoMolecularmarkers for breast cancer diagnosis prognosis and targetedtherapyrdquo Journal of Surgical Oncology vol 111 no 1 pp 81ndash902015
[15] W Werft A Benner and A Kopp-Schneider ldquoOn the identi-fication of predictive biomarkers detecting treatment-by-geneinteraction in high-dimensional datardquo Computational Statisticsand Data Analysis vol 56 no 5 pp 1275ndash1286 2012
[16] M Jenkins A Flynn T Smart et al ldquoA statisticianrsquos perspectiveon biomarkers in drug developmentrdquo Pharmaceutical Statisticsvol 10 no 6 pp 494ndash507 2011
BioMed Research International 11
[17] A J Vickers MW Kattan and D J Sargent ldquoMethod for eval-uating prediction models that apply the results of randomizedtrials to individual patientsrdquo Trials vol 8 no 1 article 14 2007
[18] H Janes M S Pepe P M Bossuyt andW E Barlow ldquoMeasur-ing the performance of markers for guiding treatment deci-sionsrdquo Annals of Internal Medicine vol 154 no 4 pp 253ndash2592011
[19] R J DeRubeis Z D Cohen N R Forand J C Fournier L AGelfand and L Lorenzo-Luaces ldquoThe personalized advantageindex translating research on prediction into individualizedtreatment recommendationsAdemonstrationrdquoPLoSONE vol9 no 1 Article ID e83875 2014
[20] D P Byar and D K Corle ldquoSelecting optimal treatment inclinical trials using covariate informationrdquo Journal of ChronicDiseases vol 30 no 7 pp 445ndash459 1977
[21] N Thatcher A Chang P Parikh et al ldquoGefitinib plus bestsupportive care in previously treated patients with refractoryadvanced non-small-cell lung cancer results from a ran-domised placebo-controlled multicentre study (iressa survivalevaluation in lung cancer)rdquo The Lancet vol 366 no 9496 pp1527ndash1537 2005
[22] A J Vickers ldquoPrediction models in cancer carerdquo CA A CancerJournal for Clinicians vol 61 no 5 pp 315ndash326 2011
[23] RM Simon ldquoSubgroup analysisrdquo inWiley Encyclopedia of Clin-ical Trials John Wiley amp Sons Hoboken NJ USA 2007
[24] S J Pocock S E Assmann L E Enos and L E Kasten ldquoSub-group analysis covariate adjustment and baseline comparisonsin clinical trial reporting current practice and problemsrdquoStatistics in Medicine vol 21 no 19 pp 2917ndash2930 2002
[25] PM Rothwell ZMehta S CHoward S A Gutnikov andC PWarlow ldquoFrom subgroups to individuals general principles andthe example of carotid endarterectomyrdquoTheLancet vol 365 no9455 pp 256ndash265 2005
[26] R Wang S W Lagakos J H Ware D J Hunter and J MDrazen ldquoStatistics in medicinemdashreporting of subgroup ana-lyses in clinical trialsrdquoTheNewEngland Journal ofMedicine vol357 no 21 pp 2108ndash2194 2007
[27] International Breast Cancer Study Group ldquoEndocrine respon-siveness and tailoring adjuvant therapy for postmenopausallymph node-negative breast cancer a randomized trialrdquo Journalof the National Cancer Institute vol 94 no 14 pp 1054ndash10652002
[28] Early Breast Cancer Trialistsrsquo Collaborative Group (EBCTCG)ldquoEffects of chemotherapy and hormonal therapy for early breastcancer on recurrence and 15-year survival an overview of therandomised trialsrdquoThe Lancet vol 365 no 9472 pp 1687ndash17172005
[29] F E Harrell K L Lee and D B Mark ldquoTutorial in biostatisticsmultivariable prognostic models issues in developing modelsevaluating assumptions and adequacy and measuring andreducing errorsrdquo Statistics in Medicine vol 15 no 4 pp 361ndash387 1996
[30] S Gill C L Loprinzi D J Sargent et al ldquoPooled analysis offluorouracil-based adjuvant therapy for stage II and III coloncancer who benefits and by how muchrdquo Journal of ClinicalOncology vol 22 no 10 pp 1797ndash1806 2004
[31] M Qian and S A Murphy ldquoPerformance guarantees forindividualized treatment rulesrdquoThe Annals of Statistics vol 39no 2 pp 1180ndash1210 2011
[32] W Lu H H Zhang and D Zeng ldquoVariable selection for opti-mal treatment decisionrdquo StatisticalMethods inMedical Researchvol 22 no 5 pp 493ndash504 2013
[33] R Kohavi ldquoA study of cross-validation and bootstrap for accu-racy estimation and model selectionrdquo in Proceedings of the 14thInternational Joint Conference on Artificial Intelligence (IJCAIrsquo95) vol 2 pp 1137ndash1145 1995
[34] E W Steyerberg A J Vickers N R Cook et al ldquoAssessing theperformance of prediction models a framework for traditionaland novel measuresrdquo Epidemiology vol 21 no 1 pp 128ndash1382010
[35] P R Rosenbaum and D B Rubin ldquoThe central role of the pro-pensity score in observational studies for causal effectsrdquo Bio-metrika vol 70 no 1 pp 41ndash55 1983
[36] R B drsquoAgostino Jr ldquoTutorial in biostatistics propensity scoremethods for bias reduction in the comparison of a treatment toa non-randomized control grouprdquo Statistics in Medicine vol 17no 19 pp 2265ndash2281 1998
[37] R Pazdur ldquoEndpoints for assessing drug activity in clinicaltrialsrdquoThe Oncologist vol 13 supplement 2 pp 19ndash21 2008
[38] S L Spruance J E Reid M Grace and M Samore ldquoHazardratio in clinical trialsrdquo Antimicrobial Agents and Chemotherapyvol 48 no 8 pp 2787ndash2792 2004
[39] J D Kalbeisch and R L Prentice The Statistical Analysis ofFailure Time Data vol 360 John Wiley amp Sons 2011
[40] Y Geng Flexible Statistical Learning Methods for SurvivalData Risk Prediction and Optimal Treatment Decision NorthCarolina State University 2013
[41] J Li L Zhao L Tian et alAPredictive Enrichment Procedure toIdentify Potential Responders to a NewTherapy for RandomizedComparative Controlled Clinical Studies Harvard UniversityBiostatisticsWorking Paper SeriesWorking Paper 169 HarvardUniversity 2014
[42] C Kang H Janes and Y Huang ldquoCombining biomarkers tooptimize patient treatment recommendationsrdquo Biometrics vol70 no 3 pp 695ndash720 2014
[43] B Zhang A A Tsiatis E B Laber and M Davidian ldquoA robustmethod for estimating optimal treatment regimesrdquo Biometricsvol 68 no 4 pp 1010ndash1018 2012
[44] B Zhang A A Tsiatis M Davidian M Zhang and E LaberldquoEstimating optimal treatment regimes from a classificationperspectiverdquo Stat vol 1 no 1 pp 103ndash114 2012
[45] J C Foster J M G Taylor and S J Ruberg ldquoSubgroup identi-fication from randomized clinical trial datardquo Statistics in Medi-cine vol 30 no 24 pp 2867ndash2880 2011
[46] US Food and Drug Administration Guidance for IndustryClinical Trial Endpoints for the Approval of Cancer Drugs andBiologics US Food and Drug AdministrationWashington DCUSA 2007
[47] H Janes M D Brown M Pepe and Y Huang ldquoStatisticalmethods for evaluating and comparing biomarkers for patienttreatment selectionrdquo UW Biostatistics Working Paper SeriesWorking Paper 389 2013
[48] R Development Core Team R a language and environment forstatistical computing 2008 httpwwwR-projectorg
[49] X-JMa ZWang P D Ryan et al ldquoA two-gene expression ratiopredicts clinical outcome in breast cancer patients treated withtamoxifenrdquo Cancer Cell vol 5 no 6 pp 607ndash616 2004
[50] X-JMa S GHilsenbeckWWang et al ldquoTheHOXB13IL17BRexpression index is a prognostic factor in early-stage breastcancerrdquo Journal of Clinical Oncology vol 24 no 28 pp 4611ndash4619 2006
[51] D C Sgroi E Carney E Zarrella et al ldquoPrediction of latedisease recurrence and extended adjuvant letrozole benefit by
12 BioMed Research International
the HOXB13IL17BR biomarkerrdquo Journal of the National CancerInstitute vol 105 no 14 pp 1036ndash1042 2013
[52] M S Cepeda R Boston J T Farrar and B L Strom ldquoCom-parison of logistic regression versus propensity score when thenumber of events is low and there are multiple confoundersrdquoThe American Journal of Epidemiology vol 158 no 3 pp 280ndash287 2003
[53] P C Austin ldquoAn introduction to propensity score methods forreducing the effects of confounding in observational studiesrdquoMultivariate Behavioral Research vol 46 no 3 pp 399ndash4242011
[54] G Heinze and P Juni ldquoAn overview of the objectives of andthe approaches to propensity score analysesrdquo European HeartJournal vol 32 no 14 Article ID ehr031 pp 1704ndash1708 2011
[55] L E Braitman and P R Rosenbaum ldquoRare outcomes commontreatments analytic strategies using propensity scoresrdquo Annalsof Internal Medicine vol 137 no 8 pp 693ndash695 2002
[56] D M Witten and R Tibshirani ldquoSurvival analysis withhigh-dimensional covariatesrdquo Statistical Methods in MedicalResearch vol 19 no 1 pp 29ndash51 2010
[57] A E Hoerl and R W Kennard ldquoRidge regression biasedestimation for nonorthogonal problemsrdquoTechnometrics vol 42no 1 pp 80ndash86 2000
[58] R Tibshirani ldquoRegression shrinkage and selection via the lassordquoJournal of the Royal Statistical Society Series B Methodologicalvol 58 no 1 pp 267ndash288 1996
[59] H Zou ldquoThe adaptive lasso and its oracle propertiesrdquo Journal ofthe American Statistical Association vol 101 no 476 pp 1418ndash1429 2006
[60] B Efron T Hastie I Johnstone and R Tibshirani ldquoLeast angleregressionrdquo The Annals of Statistics vol 32 no 2 pp 407ndash4992004
[61] T Hastie and B Efron ldquolars Least angle regression lasso andforward stagewiserdquo R package version 12 2013 httpcranr-projectorgwebpackageslarsindexhtml
[62] J Fan and R Li ldquoVariable selection via nonconcave penalizedlikelihood and its oracle propertiesrdquo Journal of the AmericanStatistical Association vol 96 no 456 pp 1348ndash1360 2001
[63] H Zou and T Hastie ldquoRegularization and variable selection viathe elastic netrdquo Journal of the Royal Statistical Society Series BStatistical Methodology vol 67 no 2 pp 301ndash320 2005
[64] N Kramer J Schafer and A-L Boulesteix ldquoRegularized esti-mation of large-scale gene association networks using graphicalgaussian modelsrdquo BMC Bioinformatics vol 10 no 1 article 3842009
[65] P Breheny and J Huang ldquoCoordinate descent algorithms fornonconvex penalized regression with applications to biologicalfeature selectionrdquo The Annals of Applied Statistics vol 5 no 1pp 232ndash253 2011
[66] S M Hammer D A Katzenstein M D Hughes et al ldquoA trialcomparing nucleoside monotherapy with combination therapyin HIV-infected adults with CD4 cell counts from 200 to 500per cubicmillimeterrdquoTheNew England Journal ofMedicine vol335 no 15 pp 1081ndash1090 1996
[67] H M Boslashvelstad S Nygard H L Stoslashrvold et al ldquoPredictingsurvival from microarray datamdasha comparative studyrdquo Bioinfor-matics vol 23 no 16 pp 2080ndash2087 2007
[68] V Kehl and K Ulm ldquoResponder identification in clinical trialswith censored datardquoComputational Statistics andDataAnalysisvol 50 no 5 pp 1338ndash1355 2006
[69] P Royston and M K Parmar ldquoThe use of restricted meansurvival time to estimate the treatment effect in randomizedclinical trials when the proportional hazards assumption is indoubtrdquo Statistics inMedicine vol 30 no 19 pp 2409ndash2421 2011
[70] P Royston andM K B Parmar ldquoRestrictedmean survival timean alternative to the hazard ratio for the design and analysis ofrandomized trials with a time-to-event outcomerdquo BMCMedicalResearch Methodology vol 13 no 1 article 152 2013
[71] E T Lee and J W Wang Statistical Methods for Survival DataAnalysis John Wiley amp Sons Hoboken NJ USA 2013
[72] Y Zhao D Zeng A J Rush and M R Kosorok ldquoEstimatingindividualized treatment rules using outcome weighted learn-ingrdquo Journal of the American Statistical Association vol 107 no499 pp 1106ndash1118 2012
[73] D B Rubin and M J van der Laan ldquoStatistical issues andlimitations in personalized medicine research with clinicaltrialsrdquoThe International Journal of Biostatistics vol 8 no 1 pp1ndash20 2012
[74] L Breiman ldquoRandom forestsrdquoMachine Learning vol 45 no 1pp 5ndash32 2001
[75] J Friedman T Hastie and R Tibshirani ldquoAdditive logisticregression a statistical view of boostingrdquo The Annals of Statis-tics vol 28 no 2 pp 337ndash407 2000
[76] C Cortes and V Vapnik ldquoSupport-vector networksrdquo MachineLearning vol 20 no 3 pp 273ndash297 1995
[77] L Breiman J Friedman C J Stone and R A Olshen Classi-fication and Regression Trees CRC Press New York NY USA1984
[78] S Dudoit J Fridlyand and T P Speed ldquoComparison of dis-criminationmethods for the classification of tumors using geneexpression datardquo Journal of the American Statistical Associationvol 97 no 457 pp 77ndash87 2002
[79] A Liaw and MWiener ldquoClassification and regression by rand-omforestrdquoRNews vol 2 no 3 pp 18ndash22 2002 httpCRANR-projectorgdocRnews
[80] T Therneau B Atkinson and B Ripley ldquorpart RecursivePartitioning and Regression Treesrdquo R package version 41-3httpcranr-projectorgwebpackagesrpartindexhtml
[81] A Kapelner J Bleich Z D Cohen R J DeRubeis and RBerk ldquoInference for treatment regime models in personalizedmedicinerdquo httparxivorgabs14047844
[82] X Song and M S Pepe ldquoEvaluating markers for selecting apatientrsquos treatmentrdquoBiometrics vol 60 no 4 pp 874ndash883 2004
[83] T Karrison ldquoRestricted mean life with adjustment for covari-atesrdquo Journal of the American Statistical Association vol 82 no400 pp 1169ndash1176 1987
[84] C Barker ldquoThe mean median and confidence intervals ofthe kaplan-meier survival estimatemdashcomputations and appli-cationsrdquo Journal of the American Statistical Association vol 63no 1 pp 78ndash80 2009
[85] B Efron and R J Tibshirani An Introduction to the Bootstrapvol 57 CRC Press 1994
[86] S Arlot andA Celisse ldquoA survey of cross-validation proceduresfor model selectionrdquo Statistics Surveys vol 4 pp 40ndash79 2010
[87] A Kapelner and J Bleich ldquoPTE Personalized Treatment Eva-luatorrdquo 2014 R package version 10 httpCRANR-projectorgpackage=PTE
[88] S A Murphy ldquoOptimal dynamic treatment regimesrdquo Journal ofthe Royal Statistical Society Series B StatisticalMethodology vol65 no 2 pp 331ndash355 2003
BioMed Research International 13
[89] J M Robins ldquoOptimal structural nested models for optimalsequential decisionsrdquo in Proceedings of the Second Seattle Sym-posium in Biostatistics vol 179 of Lecture Notes in Statistics pp189ndash326 Springer Berlin Germany 2004
[90] Y Zhao and D Zeng ldquoRecent development on statistical meth-ods for personalized medicine discoveryrdquo Frontiers of Medicinein China vol 7 no 1 pp 102ndash110 2013
Note 120583(119860X) needs to be estimated to construct the ITRyet parameters 120573 are omitted for simplicity Larger valuesof Θ119861119892opt(X) indicate increased clinical value for the
biomarker driven ITR A subset of patients that are recom-mended for new treatment (119860 = 1) under an ITR may havebeen randomly selected to receive it while the remainingsubset of ldquounluckyrdquo patientswould have received the standardtreatment [19] The summary measure of Θ
119861119892opt(X) char-
acterizes a weighted difference in the disease rates betweenthe standard and the new treatments in a population whereinthe constructed optimal ITR would recommend the newtreatment 119892opt(X = 1) The weight is the proportion ofpatients identified by the optimal ITR for the new treatmentand can be empirically estimated using the correspondingcounts For example 119875119892opt(X) = 1 can be estimatedusing the number of patients recommended for the newtreatment divided by the total sample size A similar summarystatistic can be derived for an alternative strategy allocatingall patients to the new treatment The summary could beapplied to the aforementioned breast cancer example [42]for example with the aim of finding a subgroup of patientswho were likely to benefit from adjuvant chemotherapywhile those unlikely to benefit would be assigned tamoxifenalone to avoid the unnecessary toxicity and inconvenience ofchemotherapy
Continuous Variables Another strategy for continuous datacompares outcomes observed for ldquoluckyrdquo subjects those whoreceived the therapy that would have been recommendedby the ITR based [81] Further one business-as-usual drugallocation procedure is randomizing treatment and standardcare at the same probability of 05 A summary statisticis to measure the mean outcome under ITR compared tothat obtained under random assignment for instance themean decrease in Hamilton Rating Scale for Depression asdiscussed in Section 51 [19] Define the summary measureas Θ119862119892opt(X) = 120583119892opt(X)X minus 120583119892rand(X)X where
119892rand(X) represents the randomization allocation procedureThe quantity of 120583119892119900119901119905(X)X represents the mean outcomeunder the constructed IRT that can be empirically estimatedfrom the ldquoluckyrdquo subjects and 120583119892rand(X)X can be esti-mated empirically from the sample means
Alternatively an ITR may be compared to an ldquooptimalrdquodrug that has showed universal benefits (a better drug onaverage) in a controlled trial The clinical benefits of anldquooptimalrdquo drug can be defined as 120583119892best(X)X = max120583(119860 =
0X) 120583(119860 = 1X) 120583(119860 = 119886X) and can be empiricallyestimated from the sample means of the new and standardtreatments respectively Then the alternative summary mea-sure is defined as Θ
Survival Data For survival data a clinically relevant measureis mean overall (or progression free) survival time As
survival time is continuous in nature the identical strategyprovided above for continuous outcomes can be employedhere However because the mean survival time may notbe well estimated from the observed data due to a highpercentage of censored observations [40] an alternativemean restricted survival duration was proposed and definedas the population average event-free durations for a restrictedtime of 119905lowast [41 83]Often 119905lowast is chosen to cover the trialrsquos follow-up period Mathematically it can be calculated by integratingthe survival function of 119878(119905) over the domain of (0 119905lowast) that is120583119892opt(X)X 119905lowast = int119905
lowast
0119878(119905)119889119905 and often estimated by the area
under the Kaplan-Meier curve up to 119905lowast [84] Thus an ITRrsquospotential to prolong survival can be calculated asΘ
119878119892opt(X)
119905lowast = 120583119892opt(X)X 119905lowast minus 120583119892rand(X)X 119905lowast
62 AssessingModel Performance The summaries heretoforediscussed evaluate an optimal ITR for a given model andestimating procedure Because these quantities are estimatedconditionally given the observed covariates they neglectto quantify the extent of marginal uncertainty for futurepatients Hence an ITR needs to be internally validated ifexternal data is not available [34] Cross-validation (CV) andbootstrap resampling techniques are commonly used for thispurpose [19 42 45 81] and expositions on both approachesare well described elsewhere [33 85 86]
We here briefly introduce a process that was proposed byKapelner et al [81] in the setting of personalized medicineTenfold CV is commonly used in practice where the wholedata is randomly partitioned into 10 roughly equal-sizedexclusive subsamples All methods under consideration areapplied to 910 of the data excluding 110 as an independenttesting data set The process is repeated 10 times for eachsubsample Considering the assignments recommended bythe optimal ITRs the summary measures can be calculatedusing results from each testing fold [45]TheCVprocess givesthe estimated summary measures and its variation can beevaluated using bootstrap procedures Specifically one drawsa sample with replacement from the entire data and calculatesthe summary measure from 10-fold CV This process willbe repeated 119861 times where 119861 is chosen for resolution ofthe resulting confidence intervals [81] Using the summarymeasures as119861 new random samples the correspondingmeanand variances can be calculated empirically Note that thesummary measures compare two treatment rules one for theoptimal ITR and another naive rule (eg randomization)
The above procedure can be applied to all the meth-ods we have discussed so far The 119877 software package119879119903119890119886119905119898119890119899119905119878119890119897119890119888119905119894119900119899 (httplabsfhcrcorgjanesindexhtml)can be used to implement these methods for evaluatingand comparing biomarkers for binary outcomes [47] Veryrecently an inferential procedure was proposed for contin-uous outcomes that is implemented in the publicly available119877 package ldquoPersonalized Treatment Evaluatorrdquo [81 87] Bothmethods consider data from RCTs with two arms for com-parative treatmentsThesemethods are in general applicableto regression model based methods but are not suitable forapproaches based on classification techniques or penalizedregression
10 BioMed Research International
Next we present two examples Recall in Section 55 thatKang et al [42] reported the estimated clinical benefits ofan ITR for breast cancer when compared to the defaultstrategy of assigning all patients to adjuvant chemotherapyThe proposed approach (based on boosting and classificationtrees) achieved the highest value of the summary measureat 0081 with 95 confidence interval (CI) (0000 0159)[42] In the second example introduced in Section 51 [19]the authors calculated the mean score of the HamiltonRating Scale for Depression for two groups of subjectsgroups were defined by randomly assigning patients to theldquooptimalrdquo and ldquononoptimalrdquo therapy as defined by the ITRThe reported difference between the two groups was minus178with a 119875 value of 009 which fails to attain a clinicalsignificant difference of 3 [19] The same data was analyzedby Kapelner et al [81] Following the discussed procedurethe authors reported the estimated values (and 95 CI)of Θ119862119892opt(X) and Θ
119862alt119892opt(X) as minus0842(minus2657 minus0441)
and minus0765(minus2362 0134) respectively The results whichfail to achieve clinical significance were based on rigorousstatistical methods and thus can be considered reliableestimates of the ITRrsquos performance
7 Discussion
As our understanding tumor heterogeneity evolves person-alized medicine will become standard medical practice inoncology Therefore it is essential that the oncology com-munity uses appropriate analytical methods for identifyingand evaluating the performance of personalized treatmentrules This paper provided an exposition of the process forusing statistical inference to establish optimal individualizedtreatment rules using data acquired from clinical study Thequality of an ITR depends on the quality of the design used toacquire the dataMoreover an ITRmust be properly validatedbefore it is integrated into clinical practice Personalizedmedicine in some areas of oncologymay be limited by the factthat biomarkers arising from a small panel of genesmay neveradequately characterize the extent of tumor heterogeneityinherent to the disease Consequently the available statisticalmethodology needs to evolve in order to optimally exploitglobal gene signatures for personalized medicine
The bulk of our review focused on statistical approachesfor treatment selection at a single time point The readershould note that another important area of research considersoptimal dynamic treatment regimes (DTRs) [88 89] whereintreatment decisions are considered sequentially over thecourse ofmultiple periods of intervention using each patientrsquosprior treatment history Zhao and Zeng provide a summaryof recent developments in this area [90]
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
JunshengMa was fully funded by the University of Texas MDAnderson Cancer Center internal funds Brian P Hobbs and
Francesco C Stingo were partially supported by the CancerCenter Support Grant (CCSG) (P30 CA016672)
References
[1] J Reimand O Wagih and G D Bader ldquoThe mutational land-scape of phosphorylation signaling in cancerrdquo Scientific Reportsvol 3 article 2651 2013
[2] D Hanahan and R AWeinberg ldquoHallmarks of cancer the nextgenerationrdquo Cell vol 144 no 5 pp 646ndash674 2011
[3] R Simon ldquoClinical trial designs for evaluating the medicalutility of prognostic and predictive biomarkers in oncologyrdquoPersonalized Medicine vol 7 no 1 pp 33ndash47 2010
[4] P L Bedard A R Hansen M J Ratain and L L Siu ldquoTumourheterogeneity in the clinicrdquo Nature vol 501 no 7467 pp 355ndash364 2013
[5] M D Pegram G Pauletti and D J Slamon ldquoHer-2neu as apredictive marker of response to breast cancer therapyrdquo BreastCancer Research and Treatment vol 52 no 1ndash3 pp 65ndash77 1998
[6] G J Kelloff andC C Sigman ldquoCancer biomarkers selecting theright drug for the right patientrdquoNature Reviews Drug Discoveryvol 11 no 3 pp 201ndash214 2012
[7] J ADiMasi JM Reichert L Feldman andAMalins ldquoClinicalapproval success rates for investigational cancer drugsrdquoClinicalPharmacology andTherapeutics vol 94 no 3 pp 329ndash335 2013
[8] M Hay D W Thomas J L Craighead C Economides andJ Rosenthal ldquoClinical development success rates for investiga-tional drugsrdquo Nature Biotechnology vol 32 no 1 pp 40ndash512014
[9] S S Knox ldquoFrom lsquoomicsrsquo to complex disease a systems biologyapproach to gene-environment interactions in cancerrdquo CancerCell International vol 10 article 11 2010
[10] V Deschoolmeester M Baay P Specenier F Lardon and JB Vermorken ldquoA review of the most promising biomarkersin colorectal cancer one step closer to targeted therapyrdquo TheOncologist vol 15 no 7 pp 699ndash731 2010
[11] J A Sosman K B Kim L Schuchter et al ldquoSurvival in brafV600ndashmutant advanced melanoma treated with vemurafenibrdquoThe New England Journal of Medicine vol 366 no 8 pp 707ndash714 2012
[12] S Paik S Shak G Tang et al ldquoA multigene assay to predictrecurrence of tamoxifen-treated node-negative breast cancerrdquoThe New England Journal of Medicine vol 351 no 27 pp 2817ndash2826 2004
[13] K S Albain W E Barlow S Shak et al ldquoPrognostic and pre-dictive value of the 21-gene recurrence score assay in post-menopausal women with node-positive oestrogen-receptor-positive breast cancer on chemotherapy a retrospective analysisof a randomised trialrdquoTheLancet Oncology vol 11 no 1 pp 55ndash65 2010
[14] J E Lang J S Wecsler M F Press and D Tripathy ldquoMolecularmarkers for breast cancer diagnosis prognosis and targetedtherapyrdquo Journal of Surgical Oncology vol 111 no 1 pp 81ndash902015
[15] W Werft A Benner and A Kopp-Schneider ldquoOn the identi-fication of predictive biomarkers detecting treatment-by-geneinteraction in high-dimensional datardquo Computational Statisticsand Data Analysis vol 56 no 5 pp 1275ndash1286 2012
[16] M Jenkins A Flynn T Smart et al ldquoA statisticianrsquos perspectiveon biomarkers in drug developmentrdquo Pharmaceutical Statisticsvol 10 no 6 pp 494ndash507 2011
BioMed Research International 11
[17] A J Vickers MW Kattan and D J Sargent ldquoMethod for eval-uating prediction models that apply the results of randomizedtrials to individual patientsrdquo Trials vol 8 no 1 article 14 2007
[18] H Janes M S Pepe P M Bossuyt andW E Barlow ldquoMeasur-ing the performance of markers for guiding treatment deci-sionsrdquo Annals of Internal Medicine vol 154 no 4 pp 253ndash2592011
[19] R J DeRubeis Z D Cohen N R Forand J C Fournier L AGelfand and L Lorenzo-Luaces ldquoThe personalized advantageindex translating research on prediction into individualizedtreatment recommendationsAdemonstrationrdquoPLoSONE vol9 no 1 Article ID e83875 2014
[20] D P Byar and D K Corle ldquoSelecting optimal treatment inclinical trials using covariate informationrdquo Journal of ChronicDiseases vol 30 no 7 pp 445ndash459 1977
[21] N Thatcher A Chang P Parikh et al ldquoGefitinib plus bestsupportive care in previously treated patients with refractoryadvanced non-small-cell lung cancer results from a ran-domised placebo-controlled multicentre study (iressa survivalevaluation in lung cancer)rdquo The Lancet vol 366 no 9496 pp1527ndash1537 2005
[22] A J Vickers ldquoPrediction models in cancer carerdquo CA A CancerJournal for Clinicians vol 61 no 5 pp 315ndash326 2011
[23] RM Simon ldquoSubgroup analysisrdquo inWiley Encyclopedia of Clin-ical Trials John Wiley amp Sons Hoboken NJ USA 2007
[24] S J Pocock S E Assmann L E Enos and L E Kasten ldquoSub-group analysis covariate adjustment and baseline comparisonsin clinical trial reporting current practice and problemsrdquoStatistics in Medicine vol 21 no 19 pp 2917ndash2930 2002
[25] PM Rothwell ZMehta S CHoward S A Gutnikov andC PWarlow ldquoFrom subgroups to individuals general principles andthe example of carotid endarterectomyrdquoTheLancet vol 365 no9455 pp 256ndash265 2005
[26] R Wang S W Lagakos J H Ware D J Hunter and J MDrazen ldquoStatistics in medicinemdashreporting of subgroup ana-lyses in clinical trialsrdquoTheNewEngland Journal ofMedicine vol357 no 21 pp 2108ndash2194 2007
[27] International Breast Cancer Study Group ldquoEndocrine respon-siveness and tailoring adjuvant therapy for postmenopausallymph node-negative breast cancer a randomized trialrdquo Journalof the National Cancer Institute vol 94 no 14 pp 1054ndash10652002
[28] Early Breast Cancer Trialistsrsquo Collaborative Group (EBCTCG)ldquoEffects of chemotherapy and hormonal therapy for early breastcancer on recurrence and 15-year survival an overview of therandomised trialsrdquoThe Lancet vol 365 no 9472 pp 1687ndash17172005
[29] F E Harrell K L Lee and D B Mark ldquoTutorial in biostatisticsmultivariable prognostic models issues in developing modelsevaluating assumptions and adequacy and measuring andreducing errorsrdquo Statistics in Medicine vol 15 no 4 pp 361ndash387 1996
[30] S Gill C L Loprinzi D J Sargent et al ldquoPooled analysis offluorouracil-based adjuvant therapy for stage II and III coloncancer who benefits and by how muchrdquo Journal of ClinicalOncology vol 22 no 10 pp 1797ndash1806 2004
[31] M Qian and S A Murphy ldquoPerformance guarantees forindividualized treatment rulesrdquoThe Annals of Statistics vol 39no 2 pp 1180ndash1210 2011
[32] W Lu H H Zhang and D Zeng ldquoVariable selection for opti-mal treatment decisionrdquo StatisticalMethods inMedical Researchvol 22 no 5 pp 493ndash504 2013
[33] R Kohavi ldquoA study of cross-validation and bootstrap for accu-racy estimation and model selectionrdquo in Proceedings of the 14thInternational Joint Conference on Artificial Intelligence (IJCAIrsquo95) vol 2 pp 1137ndash1145 1995
[34] E W Steyerberg A J Vickers N R Cook et al ldquoAssessing theperformance of prediction models a framework for traditionaland novel measuresrdquo Epidemiology vol 21 no 1 pp 128ndash1382010
[35] P R Rosenbaum and D B Rubin ldquoThe central role of the pro-pensity score in observational studies for causal effectsrdquo Bio-metrika vol 70 no 1 pp 41ndash55 1983
[36] R B drsquoAgostino Jr ldquoTutorial in biostatistics propensity scoremethods for bias reduction in the comparison of a treatment toa non-randomized control grouprdquo Statistics in Medicine vol 17no 19 pp 2265ndash2281 1998
[37] R Pazdur ldquoEndpoints for assessing drug activity in clinicaltrialsrdquoThe Oncologist vol 13 supplement 2 pp 19ndash21 2008
[38] S L Spruance J E Reid M Grace and M Samore ldquoHazardratio in clinical trialsrdquo Antimicrobial Agents and Chemotherapyvol 48 no 8 pp 2787ndash2792 2004
[39] J D Kalbeisch and R L Prentice The Statistical Analysis ofFailure Time Data vol 360 John Wiley amp Sons 2011
[40] Y Geng Flexible Statistical Learning Methods for SurvivalData Risk Prediction and Optimal Treatment Decision NorthCarolina State University 2013
[41] J Li L Zhao L Tian et alAPredictive Enrichment Procedure toIdentify Potential Responders to a NewTherapy for RandomizedComparative Controlled Clinical Studies Harvard UniversityBiostatisticsWorking Paper SeriesWorking Paper 169 HarvardUniversity 2014
[42] C Kang H Janes and Y Huang ldquoCombining biomarkers tooptimize patient treatment recommendationsrdquo Biometrics vol70 no 3 pp 695ndash720 2014
[43] B Zhang A A Tsiatis E B Laber and M Davidian ldquoA robustmethod for estimating optimal treatment regimesrdquo Biometricsvol 68 no 4 pp 1010ndash1018 2012
[44] B Zhang A A Tsiatis M Davidian M Zhang and E LaberldquoEstimating optimal treatment regimes from a classificationperspectiverdquo Stat vol 1 no 1 pp 103ndash114 2012
[45] J C Foster J M G Taylor and S J Ruberg ldquoSubgroup identi-fication from randomized clinical trial datardquo Statistics in Medi-cine vol 30 no 24 pp 2867ndash2880 2011
[46] US Food and Drug Administration Guidance for IndustryClinical Trial Endpoints for the Approval of Cancer Drugs andBiologics US Food and Drug AdministrationWashington DCUSA 2007
[47] H Janes M D Brown M Pepe and Y Huang ldquoStatisticalmethods for evaluating and comparing biomarkers for patienttreatment selectionrdquo UW Biostatistics Working Paper SeriesWorking Paper 389 2013
[48] R Development Core Team R a language and environment forstatistical computing 2008 httpwwwR-projectorg
[49] X-JMa ZWang P D Ryan et al ldquoA two-gene expression ratiopredicts clinical outcome in breast cancer patients treated withtamoxifenrdquo Cancer Cell vol 5 no 6 pp 607ndash616 2004
[50] X-JMa S GHilsenbeckWWang et al ldquoTheHOXB13IL17BRexpression index is a prognostic factor in early-stage breastcancerrdquo Journal of Clinical Oncology vol 24 no 28 pp 4611ndash4619 2006
[51] D C Sgroi E Carney E Zarrella et al ldquoPrediction of latedisease recurrence and extended adjuvant letrozole benefit by
12 BioMed Research International
the HOXB13IL17BR biomarkerrdquo Journal of the National CancerInstitute vol 105 no 14 pp 1036ndash1042 2013
[52] M S Cepeda R Boston J T Farrar and B L Strom ldquoCom-parison of logistic regression versus propensity score when thenumber of events is low and there are multiple confoundersrdquoThe American Journal of Epidemiology vol 158 no 3 pp 280ndash287 2003
[53] P C Austin ldquoAn introduction to propensity score methods forreducing the effects of confounding in observational studiesrdquoMultivariate Behavioral Research vol 46 no 3 pp 399ndash4242011
[54] G Heinze and P Juni ldquoAn overview of the objectives of andthe approaches to propensity score analysesrdquo European HeartJournal vol 32 no 14 Article ID ehr031 pp 1704ndash1708 2011
[55] L E Braitman and P R Rosenbaum ldquoRare outcomes commontreatments analytic strategies using propensity scoresrdquo Annalsof Internal Medicine vol 137 no 8 pp 693ndash695 2002
[56] D M Witten and R Tibshirani ldquoSurvival analysis withhigh-dimensional covariatesrdquo Statistical Methods in MedicalResearch vol 19 no 1 pp 29ndash51 2010
[57] A E Hoerl and R W Kennard ldquoRidge regression biasedestimation for nonorthogonal problemsrdquoTechnometrics vol 42no 1 pp 80ndash86 2000
[58] R Tibshirani ldquoRegression shrinkage and selection via the lassordquoJournal of the Royal Statistical Society Series B Methodologicalvol 58 no 1 pp 267ndash288 1996
[59] H Zou ldquoThe adaptive lasso and its oracle propertiesrdquo Journal ofthe American Statistical Association vol 101 no 476 pp 1418ndash1429 2006
[60] B Efron T Hastie I Johnstone and R Tibshirani ldquoLeast angleregressionrdquo The Annals of Statistics vol 32 no 2 pp 407ndash4992004
[61] T Hastie and B Efron ldquolars Least angle regression lasso andforward stagewiserdquo R package version 12 2013 httpcranr-projectorgwebpackageslarsindexhtml
[62] J Fan and R Li ldquoVariable selection via nonconcave penalizedlikelihood and its oracle propertiesrdquo Journal of the AmericanStatistical Association vol 96 no 456 pp 1348ndash1360 2001
[63] H Zou and T Hastie ldquoRegularization and variable selection viathe elastic netrdquo Journal of the Royal Statistical Society Series BStatistical Methodology vol 67 no 2 pp 301ndash320 2005
[64] N Kramer J Schafer and A-L Boulesteix ldquoRegularized esti-mation of large-scale gene association networks using graphicalgaussian modelsrdquo BMC Bioinformatics vol 10 no 1 article 3842009
[65] P Breheny and J Huang ldquoCoordinate descent algorithms fornonconvex penalized regression with applications to biologicalfeature selectionrdquo The Annals of Applied Statistics vol 5 no 1pp 232ndash253 2011
[66] S M Hammer D A Katzenstein M D Hughes et al ldquoA trialcomparing nucleoside monotherapy with combination therapyin HIV-infected adults with CD4 cell counts from 200 to 500per cubicmillimeterrdquoTheNew England Journal ofMedicine vol335 no 15 pp 1081ndash1090 1996
[67] H M Boslashvelstad S Nygard H L Stoslashrvold et al ldquoPredictingsurvival from microarray datamdasha comparative studyrdquo Bioinfor-matics vol 23 no 16 pp 2080ndash2087 2007
[68] V Kehl and K Ulm ldquoResponder identification in clinical trialswith censored datardquoComputational Statistics andDataAnalysisvol 50 no 5 pp 1338ndash1355 2006
[69] P Royston and M K Parmar ldquoThe use of restricted meansurvival time to estimate the treatment effect in randomizedclinical trials when the proportional hazards assumption is indoubtrdquo Statistics inMedicine vol 30 no 19 pp 2409ndash2421 2011
[70] P Royston andM K B Parmar ldquoRestrictedmean survival timean alternative to the hazard ratio for the design and analysis ofrandomized trials with a time-to-event outcomerdquo BMCMedicalResearch Methodology vol 13 no 1 article 152 2013
[71] E T Lee and J W Wang Statistical Methods for Survival DataAnalysis John Wiley amp Sons Hoboken NJ USA 2013
[72] Y Zhao D Zeng A J Rush and M R Kosorok ldquoEstimatingindividualized treatment rules using outcome weighted learn-ingrdquo Journal of the American Statistical Association vol 107 no499 pp 1106ndash1118 2012
[73] D B Rubin and M J van der Laan ldquoStatistical issues andlimitations in personalized medicine research with clinicaltrialsrdquoThe International Journal of Biostatistics vol 8 no 1 pp1ndash20 2012
[74] L Breiman ldquoRandom forestsrdquoMachine Learning vol 45 no 1pp 5ndash32 2001
[75] J Friedman T Hastie and R Tibshirani ldquoAdditive logisticregression a statistical view of boostingrdquo The Annals of Statis-tics vol 28 no 2 pp 337ndash407 2000
[76] C Cortes and V Vapnik ldquoSupport-vector networksrdquo MachineLearning vol 20 no 3 pp 273ndash297 1995
[77] L Breiman J Friedman C J Stone and R A Olshen Classi-fication and Regression Trees CRC Press New York NY USA1984
[78] S Dudoit J Fridlyand and T P Speed ldquoComparison of dis-criminationmethods for the classification of tumors using geneexpression datardquo Journal of the American Statistical Associationvol 97 no 457 pp 77ndash87 2002
[79] A Liaw and MWiener ldquoClassification and regression by rand-omforestrdquoRNews vol 2 no 3 pp 18ndash22 2002 httpCRANR-projectorgdocRnews
[80] T Therneau B Atkinson and B Ripley ldquorpart RecursivePartitioning and Regression Treesrdquo R package version 41-3httpcranr-projectorgwebpackagesrpartindexhtml
[81] A Kapelner J Bleich Z D Cohen R J DeRubeis and RBerk ldquoInference for treatment regime models in personalizedmedicinerdquo httparxivorgabs14047844
[82] X Song and M S Pepe ldquoEvaluating markers for selecting apatientrsquos treatmentrdquoBiometrics vol 60 no 4 pp 874ndash883 2004
[83] T Karrison ldquoRestricted mean life with adjustment for covari-atesrdquo Journal of the American Statistical Association vol 82 no400 pp 1169ndash1176 1987
[84] C Barker ldquoThe mean median and confidence intervals ofthe kaplan-meier survival estimatemdashcomputations and appli-cationsrdquo Journal of the American Statistical Association vol 63no 1 pp 78ndash80 2009
[85] B Efron and R J Tibshirani An Introduction to the Bootstrapvol 57 CRC Press 1994
[86] S Arlot andA Celisse ldquoA survey of cross-validation proceduresfor model selectionrdquo Statistics Surveys vol 4 pp 40ndash79 2010
[87] A Kapelner and J Bleich ldquoPTE Personalized Treatment Eva-luatorrdquo 2014 R package version 10 httpCRANR-projectorgpackage=PTE
[88] S A Murphy ldquoOptimal dynamic treatment regimesrdquo Journal ofthe Royal Statistical Society Series B StatisticalMethodology vol65 no 2 pp 331ndash355 2003
BioMed Research International 13
[89] J M Robins ldquoOptimal structural nested models for optimalsequential decisionsrdquo in Proceedings of the Second Seattle Sym-posium in Biostatistics vol 179 of Lecture Notes in Statistics pp189ndash326 Springer Berlin Germany 2004
[90] Y Zhao and D Zeng ldquoRecent development on statistical meth-ods for personalized medicine discoveryrdquo Frontiers of Medicinein China vol 7 no 1 pp 102ndash110 2013
Next we present two examples Recall in Section 55 thatKang et al [42] reported the estimated clinical benefits ofan ITR for breast cancer when compared to the defaultstrategy of assigning all patients to adjuvant chemotherapyThe proposed approach (based on boosting and classificationtrees) achieved the highest value of the summary measureat 0081 with 95 confidence interval (CI) (0000 0159)[42] In the second example introduced in Section 51 [19]the authors calculated the mean score of the HamiltonRating Scale for Depression for two groups of subjectsgroups were defined by randomly assigning patients to theldquooptimalrdquo and ldquononoptimalrdquo therapy as defined by the ITRThe reported difference between the two groups was minus178with a 119875 value of 009 which fails to attain a clinicalsignificant difference of 3 [19] The same data was analyzedby Kapelner et al [81] Following the discussed procedurethe authors reported the estimated values (and 95 CI)of Θ119862119892opt(X) and Θ
119862alt119892opt(X) as minus0842(minus2657 minus0441)
and minus0765(minus2362 0134) respectively The results whichfail to achieve clinical significance were based on rigorousstatistical methods and thus can be considered reliableestimates of the ITRrsquos performance
7 Discussion
As our understanding tumor heterogeneity evolves person-alized medicine will become standard medical practice inoncology Therefore it is essential that the oncology com-munity uses appropriate analytical methods for identifyingand evaluating the performance of personalized treatmentrules This paper provided an exposition of the process forusing statistical inference to establish optimal individualizedtreatment rules using data acquired from clinical study Thequality of an ITR depends on the quality of the design used toacquire the dataMoreover an ITRmust be properly validatedbefore it is integrated into clinical practice Personalizedmedicine in some areas of oncologymay be limited by the factthat biomarkers arising from a small panel of genesmay neveradequately characterize the extent of tumor heterogeneityinherent to the disease Consequently the available statisticalmethodology needs to evolve in order to optimally exploitglobal gene signatures for personalized medicine
The bulk of our review focused on statistical approachesfor treatment selection at a single time point The readershould note that another important area of research considersoptimal dynamic treatment regimes (DTRs) [88 89] whereintreatment decisions are considered sequentially over thecourse ofmultiple periods of intervention using each patientrsquosprior treatment history Zhao and Zeng provide a summaryof recent developments in this area [90]
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
JunshengMa was fully funded by the University of Texas MDAnderson Cancer Center internal funds Brian P Hobbs and
Francesco C Stingo were partially supported by the CancerCenter Support Grant (CCSG) (P30 CA016672)
References
[1] J Reimand O Wagih and G D Bader ldquoThe mutational land-scape of phosphorylation signaling in cancerrdquo Scientific Reportsvol 3 article 2651 2013
[2] D Hanahan and R AWeinberg ldquoHallmarks of cancer the nextgenerationrdquo Cell vol 144 no 5 pp 646ndash674 2011
[3] R Simon ldquoClinical trial designs for evaluating the medicalutility of prognostic and predictive biomarkers in oncologyrdquoPersonalized Medicine vol 7 no 1 pp 33ndash47 2010
[4] P L Bedard A R Hansen M J Ratain and L L Siu ldquoTumourheterogeneity in the clinicrdquo Nature vol 501 no 7467 pp 355ndash364 2013
[5] M D Pegram G Pauletti and D J Slamon ldquoHer-2neu as apredictive marker of response to breast cancer therapyrdquo BreastCancer Research and Treatment vol 52 no 1ndash3 pp 65ndash77 1998
[6] G J Kelloff andC C Sigman ldquoCancer biomarkers selecting theright drug for the right patientrdquoNature Reviews Drug Discoveryvol 11 no 3 pp 201ndash214 2012
[7] J ADiMasi JM Reichert L Feldman andAMalins ldquoClinicalapproval success rates for investigational cancer drugsrdquoClinicalPharmacology andTherapeutics vol 94 no 3 pp 329ndash335 2013
[8] M Hay D W Thomas J L Craighead C Economides andJ Rosenthal ldquoClinical development success rates for investiga-tional drugsrdquo Nature Biotechnology vol 32 no 1 pp 40ndash512014
[9] S S Knox ldquoFrom lsquoomicsrsquo to complex disease a systems biologyapproach to gene-environment interactions in cancerrdquo CancerCell International vol 10 article 11 2010
[10] V Deschoolmeester M Baay P Specenier F Lardon and JB Vermorken ldquoA review of the most promising biomarkersin colorectal cancer one step closer to targeted therapyrdquo TheOncologist vol 15 no 7 pp 699ndash731 2010
[11] J A Sosman K B Kim L Schuchter et al ldquoSurvival in brafV600ndashmutant advanced melanoma treated with vemurafenibrdquoThe New England Journal of Medicine vol 366 no 8 pp 707ndash714 2012
[12] S Paik S Shak G Tang et al ldquoA multigene assay to predictrecurrence of tamoxifen-treated node-negative breast cancerrdquoThe New England Journal of Medicine vol 351 no 27 pp 2817ndash2826 2004
[13] K S Albain W E Barlow S Shak et al ldquoPrognostic and pre-dictive value of the 21-gene recurrence score assay in post-menopausal women with node-positive oestrogen-receptor-positive breast cancer on chemotherapy a retrospective analysisof a randomised trialrdquoTheLancet Oncology vol 11 no 1 pp 55ndash65 2010
[14] J E Lang J S Wecsler M F Press and D Tripathy ldquoMolecularmarkers for breast cancer diagnosis prognosis and targetedtherapyrdquo Journal of Surgical Oncology vol 111 no 1 pp 81ndash902015
[15] W Werft A Benner and A Kopp-Schneider ldquoOn the identi-fication of predictive biomarkers detecting treatment-by-geneinteraction in high-dimensional datardquo Computational Statisticsand Data Analysis vol 56 no 5 pp 1275ndash1286 2012
[16] M Jenkins A Flynn T Smart et al ldquoA statisticianrsquos perspectiveon biomarkers in drug developmentrdquo Pharmaceutical Statisticsvol 10 no 6 pp 494ndash507 2011
BioMed Research International 11
[17] A J Vickers MW Kattan and D J Sargent ldquoMethod for eval-uating prediction models that apply the results of randomizedtrials to individual patientsrdquo Trials vol 8 no 1 article 14 2007
[18] H Janes M S Pepe P M Bossuyt andW E Barlow ldquoMeasur-ing the performance of markers for guiding treatment deci-sionsrdquo Annals of Internal Medicine vol 154 no 4 pp 253ndash2592011
[19] R J DeRubeis Z D Cohen N R Forand J C Fournier L AGelfand and L Lorenzo-Luaces ldquoThe personalized advantageindex translating research on prediction into individualizedtreatment recommendationsAdemonstrationrdquoPLoSONE vol9 no 1 Article ID e83875 2014
[20] D P Byar and D K Corle ldquoSelecting optimal treatment inclinical trials using covariate informationrdquo Journal of ChronicDiseases vol 30 no 7 pp 445ndash459 1977
[21] N Thatcher A Chang P Parikh et al ldquoGefitinib plus bestsupportive care in previously treated patients with refractoryadvanced non-small-cell lung cancer results from a ran-domised placebo-controlled multicentre study (iressa survivalevaluation in lung cancer)rdquo The Lancet vol 366 no 9496 pp1527ndash1537 2005
[22] A J Vickers ldquoPrediction models in cancer carerdquo CA A CancerJournal for Clinicians vol 61 no 5 pp 315ndash326 2011
[23] RM Simon ldquoSubgroup analysisrdquo inWiley Encyclopedia of Clin-ical Trials John Wiley amp Sons Hoboken NJ USA 2007
[24] S J Pocock S E Assmann L E Enos and L E Kasten ldquoSub-group analysis covariate adjustment and baseline comparisonsin clinical trial reporting current practice and problemsrdquoStatistics in Medicine vol 21 no 19 pp 2917ndash2930 2002
[25] PM Rothwell ZMehta S CHoward S A Gutnikov andC PWarlow ldquoFrom subgroups to individuals general principles andthe example of carotid endarterectomyrdquoTheLancet vol 365 no9455 pp 256ndash265 2005
[26] R Wang S W Lagakos J H Ware D J Hunter and J MDrazen ldquoStatistics in medicinemdashreporting of subgroup ana-lyses in clinical trialsrdquoTheNewEngland Journal ofMedicine vol357 no 21 pp 2108ndash2194 2007
[27] International Breast Cancer Study Group ldquoEndocrine respon-siveness and tailoring adjuvant therapy for postmenopausallymph node-negative breast cancer a randomized trialrdquo Journalof the National Cancer Institute vol 94 no 14 pp 1054ndash10652002
[28] Early Breast Cancer Trialistsrsquo Collaborative Group (EBCTCG)ldquoEffects of chemotherapy and hormonal therapy for early breastcancer on recurrence and 15-year survival an overview of therandomised trialsrdquoThe Lancet vol 365 no 9472 pp 1687ndash17172005
[29] F E Harrell K L Lee and D B Mark ldquoTutorial in biostatisticsmultivariable prognostic models issues in developing modelsevaluating assumptions and adequacy and measuring andreducing errorsrdquo Statistics in Medicine vol 15 no 4 pp 361ndash387 1996
[30] S Gill C L Loprinzi D J Sargent et al ldquoPooled analysis offluorouracil-based adjuvant therapy for stage II and III coloncancer who benefits and by how muchrdquo Journal of ClinicalOncology vol 22 no 10 pp 1797ndash1806 2004
[31] M Qian and S A Murphy ldquoPerformance guarantees forindividualized treatment rulesrdquoThe Annals of Statistics vol 39no 2 pp 1180ndash1210 2011
[32] W Lu H H Zhang and D Zeng ldquoVariable selection for opti-mal treatment decisionrdquo StatisticalMethods inMedical Researchvol 22 no 5 pp 493ndash504 2013
[33] R Kohavi ldquoA study of cross-validation and bootstrap for accu-racy estimation and model selectionrdquo in Proceedings of the 14thInternational Joint Conference on Artificial Intelligence (IJCAIrsquo95) vol 2 pp 1137ndash1145 1995
[34] E W Steyerberg A J Vickers N R Cook et al ldquoAssessing theperformance of prediction models a framework for traditionaland novel measuresrdquo Epidemiology vol 21 no 1 pp 128ndash1382010
[35] P R Rosenbaum and D B Rubin ldquoThe central role of the pro-pensity score in observational studies for causal effectsrdquo Bio-metrika vol 70 no 1 pp 41ndash55 1983
[36] R B drsquoAgostino Jr ldquoTutorial in biostatistics propensity scoremethods for bias reduction in the comparison of a treatment toa non-randomized control grouprdquo Statistics in Medicine vol 17no 19 pp 2265ndash2281 1998
[37] R Pazdur ldquoEndpoints for assessing drug activity in clinicaltrialsrdquoThe Oncologist vol 13 supplement 2 pp 19ndash21 2008
[38] S L Spruance J E Reid M Grace and M Samore ldquoHazardratio in clinical trialsrdquo Antimicrobial Agents and Chemotherapyvol 48 no 8 pp 2787ndash2792 2004
[39] J D Kalbeisch and R L Prentice The Statistical Analysis ofFailure Time Data vol 360 John Wiley amp Sons 2011
[40] Y Geng Flexible Statistical Learning Methods for SurvivalData Risk Prediction and Optimal Treatment Decision NorthCarolina State University 2013
[41] J Li L Zhao L Tian et alAPredictive Enrichment Procedure toIdentify Potential Responders to a NewTherapy for RandomizedComparative Controlled Clinical Studies Harvard UniversityBiostatisticsWorking Paper SeriesWorking Paper 169 HarvardUniversity 2014
[42] C Kang H Janes and Y Huang ldquoCombining biomarkers tooptimize patient treatment recommendationsrdquo Biometrics vol70 no 3 pp 695ndash720 2014
[43] B Zhang A A Tsiatis E B Laber and M Davidian ldquoA robustmethod for estimating optimal treatment regimesrdquo Biometricsvol 68 no 4 pp 1010ndash1018 2012
[44] B Zhang A A Tsiatis M Davidian M Zhang and E LaberldquoEstimating optimal treatment regimes from a classificationperspectiverdquo Stat vol 1 no 1 pp 103ndash114 2012
[45] J C Foster J M G Taylor and S J Ruberg ldquoSubgroup identi-fication from randomized clinical trial datardquo Statistics in Medi-cine vol 30 no 24 pp 2867ndash2880 2011
[46] US Food and Drug Administration Guidance for IndustryClinical Trial Endpoints for the Approval of Cancer Drugs andBiologics US Food and Drug AdministrationWashington DCUSA 2007
[47] H Janes M D Brown M Pepe and Y Huang ldquoStatisticalmethods for evaluating and comparing biomarkers for patienttreatment selectionrdquo UW Biostatistics Working Paper SeriesWorking Paper 389 2013
[48] R Development Core Team R a language and environment forstatistical computing 2008 httpwwwR-projectorg
[49] X-JMa ZWang P D Ryan et al ldquoA two-gene expression ratiopredicts clinical outcome in breast cancer patients treated withtamoxifenrdquo Cancer Cell vol 5 no 6 pp 607ndash616 2004
[50] X-JMa S GHilsenbeckWWang et al ldquoTheHOXB13IL17BRexpression index is a prognostic factor in early-stage breastcancerrdquo Journal of Clinical Oncology vol 24 no 28 pp 4611ndash4619 2006
[51] D C Sgroi E Carney E Zarrella et al ldquoPrediction of latedisease recurrence and extended adjuvant letrozole benefit by
12 BioMed Research International
the HOXB13IL17BR biomarkerrdquo Journal of the National CancerInstitute vol 105 no 14 pp 1036ndash1042 2013
[52] M S Cepeda R Boston J T Farrar and B L Strom ldquoCom-parison of logistic regression versus propensity score when thenumber of events is low and there are multiple confoundersrdquoThe American Journal of Epidemiology vol 158 no 3 pp 280ndash287 2003
[53] P C Austin ldquoAn introduction to propensity score methods forreducing the effects of confounding in observational studiesrdquoMultivariate Behavioral Research vol 46 no 3 pp 399ndash4242011
[54] G Heinze and P Juni ldquoAn overview of the objectives of andthe approaches to propensity score analysesrdquo European HeartJournal vol 32 no 14 Article ID ehr031 pp 1704ndash1708 2011
[55] L E Braitman and P R Rosenbaum ldquoRare outcomes commontreatments analytic strategies using propensity scoresrdquo Annalsof Internal Medicine vol 137 no 8 pp 693ndash695 2002
[56] D M Witten and R Tibshirani ldquoSurvival analysis withhigh-dimensional covariatesrdquo Statistical Methods in MedicalResearch vol 19 no 1 pp 29ndash51 2010
[57] A E Hoerl and R W Kennard ldquoRidge regression biasedestimation for nonorthogonal problemsrdquoTechnometrics vol 42no 1 pp 80ndash86 2000
[58] R Tibshirani ldquoRegression shrinkage and selection via the lassordquoJournal of the Royal Statistical Society Series B Methodologicalvol 58 no 1 pp 267ndash288 1996
[59] H Zou ldquoThe adaptive lasso and its oracle propertiesrdquo Journal ofthe American Statistical Association vol 101 no 476 pp 1418ndash1429 2006
[60] B Efron T Hastie I Johnstone and R Tibshirani ldquoLeast angleregressionrdquo The Annals of Statistics vol 32 no 2 pp 407ndash4992004
[61] T Hastie and B Efron ldquolars Least angle regression lasso andforward stagewiserdquo R package version 12 2013 httpcranr-projectorgwebpackageslarsindexhtml
[62] J Fan and R Li ldquoVariable selection via nonconcave penalizedlikelihood and its oracle propertiesrdquo Journal of the AmericanStatistical Association vol 96 no 456 pp 1348ndash1360 2001
[63] H Zou and T Hastie ldquoRegularization and variable selection viathe elastic netrdquo Journal of the Royal Statistical Society Series BStatistical Methodology vol 67 no 2 pp 301ndash320 2005
[64] N Kramer J Schafer and A-L Boulesteix ldquoRegularized esti-mation of large-scale gene association networks using graphicalgaussian modelsrdquo BMC Bioinformatics vol 10 no 1 article 3842009
[65] P Breheny and J Huang ldquoCoordinate descent algorithms fornonconvex penalized regression with applications to biologicalfeature selectionrdquo The Annals of Applied Statistics vol 5 no 1pp 232ndash253 2011
[66] S M Hammer D A Katzenstein M D Hughes et al ldquoA trialcomparing nucleoside monotherapy with combination therapyin HIV-infected adults with CD4 cell counts from 200 to 500per cubicmillimeterrdquoTheNew England Journal ofMedicine vol335 no 15 pp 1081ndash1090 1996
[67] H M Boslashvelstad S Nygard H L Stoslashrvold et al ldquoPredictingsurvival from microarray datamdasha comparative studyrdquo Bioinfor-matics vol 23 no 16 pp 2080ndash2087 2007
[68] V Kehl and K Ulm ldquoResponder identification in clinical trialswith censored datardquoComputational Statistics andDataAnalysisvol 50 no 5 pp 1338ndash1355 2006
[69] P Royston and M K Parmar ldquoThe use of restricted meansurvival time to estimate the treatment effect in randomizedclinical trials when the proportional hazards assumption is indoubtrdquo Statistics inMedicine vol 30 no 19 pp 2409ndash2421 2011
[70] P Royston andM K B Parmar ldquoRestrictedmean survival timean alternative to the hazard ratio for the design and analysis ofrandomized trials with a time-to-event outcomerdquo BMCMedicalResearch Methodology vol 13 no 1 article 152 2013
[71] E T Lee and J W Wang Statistical Methods for Survival DataAnalysis John Wiley amp Sons Hoboken NJ USA 2013
[72] Y Zhao D Zeng A J Rush and M R Kosorok ldquoEstimatingindividualized treatment rules using outcome weighted learn-ingrdquo Journal of the American Statistical Association vol 107 no499 pp 1106ndash1118 2012
[73] D B Rubin and M J van der Laan ldquoStatistical issues andlimitations in personalized medicine research with clinicaltrialsrdquoThe International Journal of Biostatistics vol 8 no 1 pp1ndash20 2012
[74] L Breiman ldquoRandom forestsrdquoMachine Learning vol 45 no 1pp 5ndash32 2001
[75] J Friedman T Hastie and R Tibshirani ldquoAdditive logisticregression a statistical view of boostingrdquo The Annals of Statis-tics vol 28 no 2 pp 337ndash407 2000
[76] C Cortes and V Vapnik ldquoSupport-vector networksrdquo MachineLearning vol 20 no 3 pp 273ndash297 1995
[77] L Breiman J Friedman C J Stone and R A Olshen Classi-fication and Regression Trees CRC Press New York NY USA1984
[78] S Dudoit J Fridlyand and T P Speed ldquoComparison of dis-criminationmethods for the classification of tumors using geneexpression datardquo Journal of the American Statistical Associationvol 97 no 457 pp 77ndash87 2002
[79] A Liaw and MWiener ldquoClassification and regression by rand-omforestrdquoRNews vol 2 no 3 pp 18ndash22 2002 httpCRANR-projectorgdocRnews
[80] T Therneau B Atkinson and B Ripley ldquorpart RecursivePartitioning and Regression Treesrdquo R package version 41-3httpcranr-projectorgwebpackagesrpartindexhtml
[81] A Kapelner J Bleich Z D Cohen R J DeRubeis and RBerk ldquoInference for treatment regime models in personalizedmedicinerdquo httparxivorgabs14047844
[82] X Song and M S Pepe ldquoEvaluating markers for selecting apatientrsquos treatmentrdquoBiometrics vol 60 no 4 pp 874ndash883 2004
[83] T Karrison ldquoRestricted mean life with adjustment for covari-atesrdquo Journal of the American Statistical Association vol 82 no400 pp 1169ndash1176 1987
[84] C Barker ldquoThe mean median and confidence intervals ofthe kaplan-meier survival estimatemdashcomputations and appli-cationsrdquo Journal of the American Statistical Association vol 63no 1 pp 78ndash80 2009
[85] B Efron and R J Tibshirani An Introduction to the Bootstrapvol 57 CRC Press 1994
[86] S Arlot andA Celisse ldquoA survey of cross-validation proceduresfor model selectionrdquo Statistics Surveys vol 4 pp 40ndash79 2010
[87] A Kapelner and J Bleich ldquoPTE Personalized Treatment Eva-luatorrdquo 2014 R package version 10 httpCRANR-projectorgpackage=PTE
[88] S A Murphy ldquoOptimal dynamic treatment regimesrdquo Journal ofthe Royal Statistical Society Series B StatisticalMethodology vol65 no 2 pp 331ndash355 2003
BioMed Research International 13
[89] J M Robins ldquoOptimal structural nested models for optimalsequential decisionsrdquo in Proceedings of the Second Seattle Sym-posium in Biostatistics vol 179 of Lecture Notes in Statistics pp189ndash326 Springer Berlin Germany 2004
[90] Y Zhao and D Zeng ldquoRecent development on statistical meth-ods for personalized medicine discoveryrdquo Frontiers of Medicinein China vol 7 no 1 pp 102ndash110 2013
[17] A J Vickers MW Kattan and D J Sargent ldquoMethod for eval-uating prediction models that apply the results of randomizedtrials to individual patientsrdquo Trials vol 8 no 1 article 14 2007
[18] H Janes M S Pepe P M Bossuyt andW E Barlow ldquoMeasur-ing the performance of markers for guiding treatment deci-sionsrdquo Annals of Internal Medicine vol 154 no 4 pp 253ndash2592011
[19] R J DeRubeis Z D Cohen N R Forand J C Fournier L AGelfand and L Lorenzo-Luaces ldquoThe personalized advantageindex translating research on prediction into individualizedtreatment recommendationsAdemonstrationrdquoPLoSONE vol9 no 1 Article ID e83875 2014
[20] D P Byar and D K Corle ldquoSelecting optimal treatment inclinical trials using covariate informationrdquo Journal of ChronicDiseases vol 30 no 7 pp 445ndash459 1977
[21] N Thatcher A Chang P Parikh et al ldquoGefitinib plus bestsupportive care in previously treated patients with refractoryadvanced non-small-cell lung cancer results from a ran-domised placebo-controlled multicentre study (iressa survivalevaluation in lung cancer)rdquo The Lancet vol 366 no 9496 pp1527ndash1537 2005
[22] A J Vickers ldquoPrediction models in cancer carerdquo CA A CancerJournal for Clinicians vol 61 no 5 pp 315ndash326 2011
[23] RM Simon ldquoSubgroup analysisrdquo inWiley Encyclopedia of Clin-ical Trials John Wiley amp Sons Hoboken NJ USA 2007
[24] S J Pocock S E Assmann L E Enos and L E Kasten ldquoSub-group analysis covariate adjustment and baseline comparisonsin clinical trial reporting current practice and problemsrdquoStatistics in Medicine vol 21 no 19 pp 2917ndash2930 2002
[25] PM Rothwell ZMehta S CHoward S A Gutnikov andC PWarlow ldquoFrom subgroups to individuals general principles andthe example of carotid endarterectomyrdquoTheLancet vol 365 no9455 pp 256ndash265 2005
[26] R Wang S W Lagakos J H Ware D J Hunter and J MDrazen ldquoStatistics in medicinemdashreporting of subgroup ana-lyses in clinical trialsrdquoTheNewEngland Journal ofMedicine vol357 no 21 pp 2108ndash2194 2007
[27] International Breast Cancer Study Group ldquoEndocrine respon-siveness and tailoring adjuvant therapy for postmenopausallymph node-negative breast cancer a randomized trialrdquo Journalof the National Cancer Institute vol 94 no 14 pp 1054ndash10652002
[28] Early Breast Cancer Trialistsrsquo Collaborative Group (EBCTCG)ldquoEffects of chemotherapy and hormonal therapy for early breastcancer on recurrence and 15-year survival an overview of therandomised trialsrdquoThe Lancet vol 365 no 9472 pp 1687ndash17172005
[29] F E Harrell K L Lee and D B Mark ldquoTutorial in biostatisticsmultivariable prognostic models issues in developing modelsevaluating assumptions and adequacy and measuring andreducing errorsrdquo Statistics in Medicine vol 15 no 4 pp 361ndash387 1996
[30] S Gill C L Loprinzi D J Sargent et al ldquoPooled analysis offluorouracil-based adjuvant therapy for stage II and III coloncancer who benefits and by how muchrdquo Journal of ClinicalOncology vol 22 no 10 pp 1797ndash1806 2004
[31] M Qian and S A Murphy ldquoPerformance guarantees forindividualized treatment rulesrdquoThe Annals of Statistics vol 39no 2 pp 1180ndash1210 2011
[32] W Lu H H Zhang and D Zeng ldquoVariable selection for opti-mal treatment decisionrdquo StatisticalMethods inMedical Researchvol 22 no 5 pp 493ndash504 2013
[33] R Kohavi ldquoA study of cross-validation and bootstrap for accu-racy estimation and model selectionrdquo in Proceedings of the 14thInternational Joint Conference on Artificial Intelligence (IJCAIrsquo95) vol 2 pp 1137ndash1145 1995
[34] E W Steyerberg A J Vickers N R Cook et al ldquoAssessing theperformance of prediction models a framework for traditionaland novel measuresrdquo Epidemiology vol 21 no 1 pp 128ndash1382010
[35] P R Rosenbaum and D B Rubin ldquoThe central role of the pro-pensity score in observational studies for causal effectsrdquo Bio-metrika vol 70 no 1 pp 41ndash55 1983
[36] R B drsquoAgostino Jr ldquoTutorial in biostatistics propensity scoremethods for bias reduction in the comparison of a treatment toa non-randomized control grouprdquo Statistics in Medicine vol 17no 19 pp 2265ndash2281 1998
[37] R Pazdur ldquoEndpoints for assessing drug activity in clinicaltrialsrdquoThe Oncologist vol 13 supplement 2 pp 19ndash21 2008
[38] S L Spruance J E Reid M Grace and M Samore ldquoHazardratio in clinical trialsrdquo Antimicrobial Agents and Chemotherapyvol 48 no 8 pp 2787ndash2792 2004
[39] J D Kalbeisch and R L Prentice The Statistical Analysis ofFailure Time Data vol 360 John Wiley amp Sons 2011
[40] Y Geng Flexible Statistical Learning Methods for SurvivalData Risk Prediction and Optimal Treatment Decision NorthCarolina State University 2013
[41] J Li L Zhao L Tian et alAPredictive Enrichment Procedure toIdentify Potential Responders to a NewTherapy for RandomizedComparative Controlled Clinical Studies Harvard UniversityBiostatisticsWorking Paper SeriesWorking Paper 169 HarvardUniversity 2014
[42] C Kang H Janes and Y Huang ldquoCombining biomarkers tooptimize patient treatment recommendationsrdquo Biometrics vol70 no 3 pp 695ndash720 2014
[43] B Zhang A A Tsiatis E B Laber and M Davidian ldquoA robustmethod for estimating optimal treatment regimesrdquo Biometricsvol 68 no 4 pp 1010ndash1018 2012
[44] B Zhang A A Tsiatis M Davidian M Zhang and E LaberldquoEstimating optimal treatment regimes from a classificationperspectiverdquo Stat vol 1 no 1 pp 103ndash114 2012
[45] J C Foster J M G Taylor and S J Ruberg ldquoSubgroup identi-fication from randomized clinical trial datardquo Statistics in Medi-cine vol 30 no 24 pp 2867ndash2880 2011
[46] US Food and Drug Administration Guidance for IndustryClinical Trial Endpoints for the Approval of Cancer Drugs andBiologics US Food and Drug AdministrationWashington DCUSA 2007
[47] H Janes M D Brown M Pepe and Y Huang ldquoStatisticalmethods for evaluating and comparing biomarkers for patienttreatment selectionrdquo UW Biostatistics Working Paper SeriesWorking Paper 389 2013
[48] R Development Core Team R a language and environment forstatistical computing 2008 httpwwwR-projectorg
[49] X-JMa ZWang P D Ryan et al ldquoA two-gene expression ratiopredicts clinical outcome in breast cancer patients treated withtamoxifenrdquo Cancer Cell vol 5 no 6 pp 607ndash616 2004
[50] X-JMa S GHilsenbeckWWang et al ldquoTheHOXB13IL17BRexpression index is a prognostic factor in early-stage breastcancerrdquo Journal of Clinical Oncology vol 24 no 28 pp 4611ndash4619 2006
[51] D C Sgroi E Carney E Zarrella et al ldquoPrediction of latedisease recurrence and extended adjuvant letrozole benefit by
12 BioMed Research International
the HOXB13IL17BR biomarkerrdquo Journal of the National CancerInstitute vol 105 no 14 pp 1036ndash1042 2013
[52] M S Cepeda R Boston J T Farrar and B L Strom ldquoCom-parison of logistic regression versus propensity score when thenumber of events is low and there are multiple confoundersrdquoThe American Journal of Epidemiology vol 158 no 3 pp 280ndash287 2003
[53] P C Austin ldquoAn introduction to propensity score methods forreducing the effects of confounding in observational studiesrdquoMultivariate Behavioral Research vol 46 no 3 pp 399ndash4242011
[54] G Heinze and P Juni ldquoAn overview of the objectives of andthe approaches to propensity score analysesrdquo European HeartJournal vol 32 no 14 Article ID ehr031 pp 1704ndash1708 2011
[55] L E Braitman and P R Rosenbaum ldquoRare outcomes commontreatments analytic strategies using propensity scoresrdquo Annalsof Internal Medicine vol 137 no 8 pp 693ndash695 2002
[56] D M Witten and R Tibshirani ldquoSurvival analysis withhigh-dimensional covariatesrdquo Statistical Methods in MedicalResearch vol 19 no 1 pp 29ndash51 2010
[57] A E Hoerl and R W Kennard ldquoRidge regression biasedestimation for nonorthogonal problemsrdquoTechnometrics vol 42no 1 pp 80ndash86 2000
[58] R Tibshirani ldquoRegression shrinkage and selection via the lassordquoJournal of the Royal Statistical Society Series B Methodologicalvol 58 no 1 pp 267ndash288 1996
[59] H Zou ldquoThe adaptive lasso and its oracle propertiesrdquo Journal ofthe American Statistical Association vol 101 no 476 pp 1418ndash1429 2006
[60] B Efron T Hastie I Johnstone and R Tibshirani ldquoLeast angleregressionrdquo The Annals of Statistics vol 32 no 2 pp 407ndash4992004
[61] T Hastie and B Efron ldquolars Least angle regression lasso andforward stagewiserdquo R package version 12 2013 httpcranr-projectorgwebpackageslarsindexhtml
[62] J Fan and R Li ldquoVariable selection via nonconcave penalizedlikelihood and its oracle propertiesrdquo Journal of the AmericanStatistical Association vol 96 no 456 pp 1348ndash1360 2001
[63] H Zou and T Hastie ldquoRegularization and variable selection viathe elastic netrdquo Journal of the Royal Statistical Society Series BStatistical Methodology vol 67 no 2 pp 301ndash320 2005
[64] N Kramer J Schafer and A-L Boulesteix ldquoRegularized esti-mation of large-scale gene association networks using graphicalgaussian modelsrdquo BMC Bioinformatics vol 10 no 1 article 3842009
[65] P Breheny and J Huang ldquoCoordinate descent algorithms fornonconvex penalized regression with applications to biologicalfeature selectionrdquo The Annals of Applied Statistics vol 5 no 1pp 232ndash253 2011
[66] S M Hammer D A Katzenstein M D Hughes et al ldquoA trialcomparing nucleoside monotherapy with combination therapyin HIV-infected adults with CD4 cell counts from 200 to 500per cubicmillimeterrdquoTheNew England Journal ofMedicine vol335 no 15 pp 1081ndash1090 1996
[67] H M Boslashvelstad S Nygard H L Stoslashrvold et al ldquoPredictingsurvival from microarray datamdasha comparative studyrdquo Bioinfor-matics vol 23 no 16 pp 2080ndash2087 2007
[68] V Kehl and K Ulm ldquoResponder identification in clinical trialswith censored datardquoComputational Statistics andDataAnalysisvol 50 no 5 pp 1338ndash1355 2006
[69] P Royston and M K Parmar ldquoThe use of restricted meansurvival time to estimate the treatment effect in randomizedclinical trials when the proportional hazards assumption is indoubtrdquo Statistics inMedicine vol 30 no 19 pp 2409ndash2421 2011
[70] P Royston andM K B Parmar ldquoRestrictedmean survival timean alternative to the hazard ratio for the design and analysis ofrandomized trials with a time-to-event outcomerdquo BMCMedicalResearch Methodology vol 13 no 1 article 152 2013
[71] E T Lee and J W Wang Statistical Methods for Survival DataAnalysis John Wiley amp Sons Hoboken NJ USA 2013
[72] Y Zhao D Zeng A J Rush and M R Kosorok ldquoEstimatingindividualized treatment rules using outcome weighted learn-ingrdquo Journal of the American Statistical Association vol 107 no499 pp 1106ndash1118 2012
[73] D B Rubin and M J van der Laan ldquoStatistical issues andlimitations in personalized medicine research with clinicaltrialsrdquoThe International Journal of Biostatistics vol 8 no 1 pp1ndash20 2012
[74] L Breiman ldquoRandom forestsrdquoMachine Learning vol 45 no 1pp 5ndash32 2001
[75] J Friedman T Hastie and R Tibshirani ldquoAdditive logisticregression a statistical view of boostingrdquo The Annals of Statis-tics vol 28 no 2 pp 337ndash407 2000
[76] C Cortes and V Vapnik ldquoSupport-vector networksrdquo MachineLearning vol 20 no 3 pp 273ndash297 1995
[77] L Breiman J Friedman C J Stone and R A Olshen Classi-fication and Regression Trees CRC Press New York NY USA1984
[78] S Dudoit J Fridlyand and T P Speed ldquoComparison of dis-criminationmethods for the classification of tumors using geneexpression datardquo Journal of the American Statistical Associationvol 97 no 457 pp 77ndash87 2002
[79] A Liaw and MWiener ldquoClassification and regression by rand-omforestrdquoRNews vol 2 no 3 pp 18ndash22 2002 httpCRANR-projectorgdocRnews
[80] T Therneau B Atkinson and B Ripley ldquorpart RecursivePartitioning and Regression Treesrdquo R package version 41-3httpcranr-projectorgwebpackagesrpartindexhtml
[81] A Kapelner J Bleich Z D Cohen R J DeRubeis and RBerk ldquoInference for treatment regime models in personalizedmedicinerdquo httparxivorgabs14047844
[82] X Song and M S Pepe ldquoEvaluating markers for selecting apatientrsquos treatmentrdquoBiometrics vol 60 no 4 pp 874ndash883 2004
[83] T Karrison ldquoRestricted mean life with adjustment for covari-atesrdquo Journal of the American Statistical Association vol 82 no400 pp 1169ndash1176 1987
[84] C Barker ldquoThe mean median and confidence intervals ofthe kaplan-meier survival estimatemdashcomputations and appli-cationsrdquo Journal of the American Statistical Association vol 63no 1 pp 78ndash80 2009
[85] B Efron and R J Tibshirani An Introduction to the Bootstrapvol 57 CRC Press 1994
[86] S Arlot andA Celisse ldquoA survey of cross-validation proceduresfor model selectionrdquo Statistics Surveys vol 4 pp 40ndash79 2010
[87] A Kapelner and J Bleich ldquoPTE Personalized Treatment Eva-luatorrdquo 2014 R package version 10 httpCRANR-projectorgpackage=PTE
[88] S A Murphy ldquoOptimal dynamic treatment regimesrdquo Journal ofthe Royal Statistical Society Series B StatisticalMethodology vol65 no 2 pp 331ndash355 2003
BioMed Research International 13
[89] J M Robins ldquoOptimal structural nested models for optimalsequential decisionsrdquo in Proceedings of the Second Seattle Sym-posium in Biostatistics vol 179 of Lecture Notes in Statistics pp189ndash326 Springer Berlin Germany 2004
[90] Y Zhao and D Zeng ldquoRecent development on statistical meth-ods for personalized medicine discoveryrdquo Frontiers of Medicinein China vol 7 no 1 pp 102ndash110 2013
the HOXB13IL17BR biomarkerrdquo Journal of the National CancerInstitute vol 105 no 14 pp 1036ndash1042 2013
[52] M S Cepeda R Boston J T Farrar and B L Strom ldquoCom-parison of logistic regression versus propensity score when thenumber of events is low and there are multiple confoundersrdquoThe American Journal of Epidemiology vol 158 no 3 pp 280ndash287 2003
[53] P C Austin ldquoAn introduction to propensity score methods forreducing the effects of confounding in observational studiesrdquoMultivariate Behavioral Research vol 46 no 3 pp 399ndash4242011
[54] G Heinze and P Juni ldquoAn overview of the objectives of andthe approaches to propensity score analysesrdquo European HeartJournal vol 32 no 14 Article ID ehr031 pp 1704ndash1708 2011
[55] L E Braitman and P R Rosenbaum ldquoRare outcomes commontreatments analytic strategies using propensity scoresrdquo Annalsof Internal Medicine vol 137 no 8 pp 693ndash695 2002
[56] D M Witten and R Tibshirani ldquoSurvival analysis withhigh-dimensional covariatesrdquo Statistical Methods in MedicalResearch vol 19 no 1 pp 29ndash51 2010
[57] A E Hoerl and R W Kennard ldquoRidge regression biasedestimation for nonorthogonal problemsrdquoTechnometrics vol 42no 1 pp 80ndash86 2000
[58] R Tibshirani ldquoRegression shrinkage and selection via the lassordquoJournal of the Royal Statistical Society Series B Methodologicalvol 58 no 1 pp 267ndash288 1996
[59] H Zou ldquoThe adaptive lasso and its oracle propertiesrdquo Journal ofthe American Statistical Association vol 101 no 476 pp 1418ndash1429 2006
[60] B Efron T Hastie I Johnstone and R Tibshirani ldquoLeast angleregressionrdquo The Annals of Statistics vol 32 no 2 pp 407ndash4992004
[61] T Hastie and B Efron ldquolars Least angle regression lasso andforward stagewiserdquo R package version 12 2013 httpcranr-projectorgwebpackageslarsindexhtml
[62] J Fan and R Li ldquoVariable selection via nonconcave penalizedlikelihood and its oracle propertiesrdquo Journal of the AmericanStatistical Association vol 96 no 456 pp 1348ndash1360 2001
[63] H Zou and T Hastie ldquoRegularization and variable selection viathe elastic netrdquo Journal of the Royal Statistical Society Series BStatistical Methodology vol 67 no 2 pp 301ndash320 2005
[64] N Kramer J Schafer and A-L Boulesteix ldquoRegularized esti-mation of large-scale gene association networks using graphicalgaussian modelsrdquo BMC Bioinformatics vol 10 no 1 article 3842009
[65] P Breheny and J Huang ldquoCoordinate descent algorithms fornonconvex penalized regression with applications to biologicalfeature selectionrdquo The Annals of Applied Statistics vol 5 no 1pp 232ndash253 2011
[66] S M Hammer D A Katzenstein M D Hughes et al ldquoA trialcomparing nucleoside monotherapy with combination therapyin HIV-infected adults with CD4 cell counts from 200 to 500per cubicmillimeterrdquoTheNew England Journal ofMedicine vol335 no 15 pp 1081ndash1090 1996
[67] H M Boslashvelstad S Nygard H L Stoslashrvold et al ldquoPredictingsurvival from microarray datamdasha comparative studyrdquo Bioinfor-matics vol 23 no 16 pp 2080ndash2087 2007
[68] V Kehl and K Ulm ldquoResponder identification in clinical trialswith censored datardquoComputational Statistics andDataAnalysisvol 50 no 5 pp 1338ndash1355 2006
[69] P Royston and M K Parmar ldquoThe use of restricted meansurvival time to estimate the treatment effect in randomizedclinical trials when the proportional hazards assumption is indoubtrdquo Statistics inMedicine vol 30 no 19 pp 2409ndash2421 2011
[70] P Royston andM K B Parmar ldquoRestrictedmean survival timean alternative to the hazard ratio for the design and analysis ofrandomized trials with a time-to-event outcomerdquo BMCMedicalResearch Methodology vol 13 no 1 article 152 2013
[71] E T Lee and J W Wang Statistical Methods for Survival DataAnalysis John Wiley amp Sons Hoboken NJ USA 2013
[72] Y Zhao D Zeng A J Rush and M R Kosorok ldquoEstimatingindividualized treatment rules using outcome weighted learn-ingrdquo Journal of the American Statistical Association vol 107 no499 pp 1106ndash1118 2012
[73] D B Rubin and M J van der Laan ldquoStatistical issues andlimitations in personalized medicine research with clinicaltrialsrdquoThe International Journal of Biostatistics vol 8 no 1 pp1ndash20 2012
[74] L Breiman ldquoRandom forestsrdquoMachine Learning vol 45 no 1pp 5ndash32 2001
[75] J Friedman T Hastie and R Tibshirani ldquoAdditive logisticregression a statistical view of boostingrdquo The Annals of Statis-tics vol 28 no 2 pp 337ndash407 2000
[76] C Cortes and V Vapnik ldquoSupport-vector networksrdquo MachineLearning vol 20 no 3 pp 273ndash297 1995
[77] L Breiman J Friedman C J Stone and R A Olshen Classi-fication and Regression Trees CRC Press New York NY USA1984
[78] S Dudoit J Fridlyand and T P Speed ldquoComparison of dis-criminationmethods for the classification of tumors using geneexpression datardquo Journal of the American Statistical Associationvol 97 no 457 pp 77ndash87 2002
[79] A Liaw and MWiener ldquoClassification and regression by rand-omforestrdquoRNews vol 2 no 3 pp 18ndash22 2002 httpCRANR-projectorgdocRnews
[80] T Therneau B Atkinson and B Ripley ldquorpart RecursivePartitioning and Regression Treesrdquo R package version 41-3httpcranr-projectorgwebpackagesrpartindexhtml
[81] A Kapelner J Bleich Z D Cohen R J DeRubeis and RBerk ldquoInference for treatment regime models in personalizedmedicinerdquo httparxivorgabs14047844
[82] X Song and M S Pepe ldquoEvaluating markers for selecting apatientrsquos treatmentrdquoBiometrics vol 60 no 4 pp 874ndash883 2004
[83] T Karrison ldquoRestricted mean life with adjustment for covari-atesrdquo Journal of the American Statistical Association vol 82 no400 pp 1169ndash1176 1987
[84] C Barker ldquoThe mean median and confidence intervals ofthe kaplan-meier survival estimatemdashcomputations and appli-cationsrdquo Journal of the American Statistical Association vol 63no 1 pp 78ndash80 2009
[85] B Efron and R J Tibshirani An Introduction to the Bootstrapvol 57 CRC Press 1994
[86] S Arlot andA Celisse ldquoA survey of cross-validation proceduresfor model selectionrdquo Statistics Surveys vol 4 pp 40ndash79 2010
[87] A Kapelner and J Bleich ldquoPTE Personalized Treatment Eva-luatorrdquo 2014 R package version 10 httpCRANR-projectorgpackage=PTE
[88] S A Murphy ldquoOptimal dynamic treatment regimesrdquo Journal ofthe Royal Statistical Society Series B StatisticalMethodology vol65 no 2 pp 331ndash355 2003
BioMed Research International 13
[89] J M Robins ldquoOptimal structural nested models for optimalsequential decisionsrdquo in Proceedings of the Second Seattle Sym-posium in Biostatistics vol 179 of Lecture Notes in Statistics pp189ndash326 Springer Berlin Germany 2004
[90] Y Zhao and D Zeng ldquoRecent development on statistical meth-ods for personalized medicine discoveryrdquo Frontiers of Medicinein China vol 7 no 1 pp 102ndash110 2013
[89] J M Robins ldquoOptimal structural nested models for optimalsequential decisionsrdquo in Proceedings of the Second Seattle Sym-posium in Biostatistics vol 179 of Lecture Notes in Statistics pp189ndash326 Springer Berlin Germany 2004
[90] Y Zhao and D Zeng ldquoRecent development on statistical meth-ods for personalized medicine discoveryrdquo Frontiers of Medicinein China vol 7 no 1 pp 102ndash110 2013
