U.U.D.M. Project Report 2017:6
Examensarbete i matematik, 30 hpHandledare: Rolf Larsson Examinator: Denis GaidashevMaj 2017
Department of MathematicsUppsala University
Diagnosing Metastatic Prostate Cancer Using PSA:A Register-Based Cohort Study with Missing Data
Marcus Westerberg
Diagnosing Metastatic Prostate Cancer Using PSA:
A Register-Based Cohort Study with Missing Data
Marcus WesterbergUppsala University, Sweden
May 10, 2017
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Abstract
Serum levels of prostate specific antigen (PSA) is a cornerstone of the assessment of the prostate cancer(PCa) risk category, which, in turn, is central for treatment decisions and is a strong determinant ofoutcome of the PCa. Metastatic PCa is currently diagnosed with the use of a bone scan, indicated by thevariable M stage, or with a high PSA serum level (≥ 100 ng/ml). PSA has traditionally been consideredas a useful indicator for poorer PCa prognosis and also for predicting metastatic disease, although recentstudies have indicated that the usefulness of PSA for predicting metastatic disease is questionable. Theaim is to assess how well PSA testing can be used to predict distant metastatic PCa, evaluate how thecurrent threshold performs and potentially suggest an improved and more detailed threshold by stratifyingaccording to Gleason Grade Group (GGG).
In order to do this, the M stage must be known, but far from all men receive a bone scan sincethe harms of false positives often outweigh the benefits, and therefore there are men with a potentialmetastatic PCa who never underwent imaging. Missing data in clinical registers is common and thereare several ways of handling this, including complete case analysis and various imputation methods. Menwith unknown M stage is a big obstacle that needs to be tackled in order to answer the main question.
The data has been collected by several relevant institutions, including NPCR, and stored in a databasecalled PCBaSe. It contains comprehensive material of all men diagnosed with PCa between 2000-2012and their clinical data such as PSA, GGG, and TNM-stages, and socio-economic data such as educationallevel and civil status, and other relevant data.
To address the missing data issue, multiple imputation by chained equations was applied and theresults were compared with a complete case analysis. In addition, a sensitivity analysis was conductedto explore the effect of a nonignorable missing data mechanism on the results. The performance of PSAtesting was assessed using Receiver Operating Characteristic (ROC) curves and associated quantities liketrue and false positive fractions and likelihood ratios. Mortality stratified according to Gleason GradeGroup and intervals of PSA was analysed to compare the PCa prognosis between the strata, and also toevaluate the imputation model.
The results showed that men with and without metastatic PCa had considerably overlapping distri-butions of PSA in all Gleason grade groups, which was reflected by poor test performance in terms ofpositive fractions. For example, for false positive fractions below 10% then true positive fractions werebelow 50%. Survival curves showed fairly high agreement between imputed M stage and complete caseM stage, except for men with low PSA 0-50 where there were indications of an overestimation of theprevalence of metastases. The nonignorable imputation model partly accounted for this and managed tolower the prevalence of metastases, especially in this region of the data.
The main conclusion was that no PSA cut-off manages to distinguish between metastatic and non-metastatic PCa, in any GGG. The conclusions were consistent under various model assumptions governedby a parameter k, apart from the prevalence numbers of metastatic disease which were sensitive to thechoice of k. Therefore all men with high PSA should be imaged and the PSA threshold of 100 ng/ml asan indicator of metastatic disease should not be used.
Popularvetenskaplig Sammanfattning
Nar man bedommer hur allvarlig en prostatacancer ar sa anvander man en riskkategorisering. Det ar vik-tigt att bedomma riskkategorin eftersom den ligger till grund for behandlingsbeslut och ar starkt koppladtill utfallet av prostatacancern. Riskkategorin faststalls genom att man bland annat tar ett blodprov darman mater mangden prostataspecifikt antigen (PSA), bestammer Gleason-gruppnivan med hjalp av enbiopsi av prostatan, och bestammer T-stadiet genom en rektalundersokning. PSA provet har traditionelltansetts vara anvandbart for att upptacka varre prostatacancrar och aven metastaserad prostatacancer.Den senare formen diagnostiseras idag med hjalp av en skelettscintigrafi som bestammer M-stadiet hosprostatacancern, dar M0/M1 betyder inga metastaser pavisade/metastaser pavisade, eller med hjalp avPSA provet dar PSA-varden over 100 ng/ml indikerar metastaser. Pa senare tid har studier visat attPSA provet har en tveksam formaga att forutsaga metastaserad prostatacancer. Syftet med denna upp-
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sats ar darfor att avgora hur bra PSA provet ar pa att forutsaga metastaserad prostatacancer, om dennuvarande troskeln pa 100 ng/ml ar optimal eller kan justeras och forbattras genom att ocksa anvandaGleason-gruppnivan.
For att kunna ta reda pa detta sa maste M-stadiet vara kant, men langt ifran alla man med prosta-tacancer genomgar en skelettskintigrafi eftersom skadorna som kan folja vid felaktig positiv diagnos oftaovervager fordelarna, och darfor finns det man med en potentiell metastaserad prostatacancer som ejgenomgatt skelettscintigrafi. Detta resulterar i saknad data, vilket ar vanligt forekommande i kliniskaregister och innebar vissa problem vid statistiska analyser. Det finns flera satt att hantera detta pa, tillexempel sa kan man utesluta alla man med saknad data ur analysen, eller sa kan man anvanda sig avnagon typ av imputeringsmetod for att simulera troliga varden dar det saknas data. Man med okantM-stadium ar ett stort problem som maste hanteras for att besvara huvudfragan.
Data har samlats in av flera relevanta institutioner, inklusive Nationella Prostatacancerregistret(NPCR), och lagras i en databas som kallas PcBase. Den innehaller omfattande material av alla mansom diagnostiserats med prostatacancer mellan 2000-2012 och deras kliniska data sasom PSA, Gleason-gruppniva, och TNM-stadier, och socioekonomisk data som utbildningsniva och civilstand, och andrarelevanta uppgifter.
Den saknade datan imputerades med hjalp av en algoritm som kallas MICE. Denna algorithm gene-rerar flera fullstandiga dataset som sedan analyseras med vanliga statistiska metoder. Resultaten fran defullstandinga dataseten kombinerades sen till ett resultat, som sedan jamfordes med en analys dar allaman med saknad data uteslutits. En kanslighetsanalys utfordes for att undersoka effekten av en mekanismsom paverkar hur troligt det ar att en man med saknat M-stadium har metastaser jamfort med en manmed kant M-stadium. PSA testets formaga att urskilja man med M0 fran man med M1 bedomdes medhjalp av Receiver Operating Characteristic (ROC) kurvor och tillhorande storheter som sensitivitet ochspecificitet. Dessa storheter beskriver risken for felaktigt negativ diagnos och felaktigt positiv diagnos,dar storre varden betyder lagre risk. Prostatacancerdodligheten stratifierat enligt Gleason-gruppniva ochPSA analyserades for att jamfora prostatacancerprognosen mellan man med M0 och M1.
Resultaten visade att man med M1 och M0 hade ordentligt overlappade PSA-fordelningar i allaGleason-gruppnivaer, vilket aterspeglades av den laga sensitiviteten och specificiteten. Till exempel,nar specificiteten var over 90% sa var sensitiviteten under 50%. overlevnadskurvorna visade en gans-ka hog overensstammelse mellan imputerat M-stadium och icke imputerat M-stadium, med undantag forman med laga PSA-varden (0-50 ng/ml) dar det fanns indikationer pa en overskattning av forekomstenav metastaser. Utifran kanslighetsanalysen kunde man se att vissa imputeringsmodeller delvis lyckadeskompensera for detta. Detta betydde att det var svart att anvanda PSA-provet for att diagnostiserametastaserad prostatacancer korrekt.
Slutsatsen var att ingen PSA-troskel lyckades atskilja man med metastaserad och icke-metastaseradprostatacancer tillrackligt bra i nagon av Gleason-gruppnivaerna. Detta kunde ses i alla imputerings-modeller, aven om forekomsten av metastaserad prostatacancer skiljde sig beroende pa modell. Dettabetyder att alla man med hogt PSA borde genomga en skelettscint och att PSA-troskeln pa 100 ng/mlsom en indikator av metastaserad prostatacancer inte bor tillampas.
Foreword and Acknowledgements
Special acknowledgements to my supervisors Rolf Larsson, Uppsala University, guidance and advice,and Hans Garmo, Regional Cancer Centre Uppsala/Orebro, for leadership and for making this thesispossible. Additional thanks to Frederik B. Thomsen, Copenhagen Prostate Cancer Center, and ParStattin, Department of Surgical Sciences, Uppsala University Hospital for excellent input.
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Contents
1 Introduction 71.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Expectations and Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Delimitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Background 92.1 Medical Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Measures Used for Risk Categorisation . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.2 Other Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.1 PCBaSe and NPCR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.2 Data Retrieved from PCBaSe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.3 Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Previous Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Theory 133.1 Basic Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Survival Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.2 Asymptotic Distribution and Confidence Intervals . . . . . . . . . . . . . . . . . . 153.2.3 The Cox Proportional Hazards Model . . . . . . . . . . . . . . . . . . . . . . . . . 173.2.4 Restricted Mean Survival . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 Evaluation of Tests for Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3.1 Notation and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3.2 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4 Tests for Normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.5 Bayesian Inference for Missing Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.5.1 Notation and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.5.2 Multiple Imputations from the Posterior of the Missing Values . . . . . . . . . . . 263.5.3 Multiple Imputations Drawn from a Finite Number of Completed Datasets . . . . 273.5.4 Rubin’s Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.6 Multiple Imputation using Chained Equations . . . . . . . . . . . . . . . . . . . . . . . . . 313.6.1 Fully Conditional Specification and the MICE Algorithm . . . . . . . . . . . . . . 313.6.2 The Choice of Predictors and the Visit Sequence . . . . . . . . . . . . . . . . . . . 333.6.3 Univariate Imputation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.6.4 Nonignorable Missing Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.6.5 Derived Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.6.6 Parameter Settings and Convergence Diagnostics . . . . . . . . . . . . . . . . . . . 38
4 Statistical Analysis 394.1 Imputation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.1.1 Model Form and Derived Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2 Analysis of Imputed Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2.1 Model Convergence and Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . 41
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5 Results 425.1 Baseline Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.2 Imputation Model Convergence Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . 425.3 Study of Parameter Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.4 Analysis of Imputed Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.5 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6 Discussion and Conclusions 526.1 The Imputation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526.2 Diagnosing Metastatic Prostate Cancer using PSA . . . . . . . . . . . . . . . . . . . . . . 53
6.2.1 Defining a Good Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.2.2 Test Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6.3 Weaknesses and Strengths of the Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
7 Recommendations and Future Work 55
8 Appendix 568.1 Baseline Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568.2 Cox Proportional Hazards: Assessment of Model Assumption . . . . . . . . . . . . . . . . 578.3 Parameter Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588.4 Sensitivity Analysis on Prevalence of M1 Disease . . . . . . . . . . . . . . . . . . . . . . . 60
9 References 61
List of Figures
1 Relating pre-test and post-test probability through a likelihood ratio. . . . . . . . . . . . . 202 Visual representation of true positive and false negative fractions. . . . . . . . . . . . . . 203 Example of a ROC curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 Characteristics before and after imputation. . . . . . . . . . . . . . . . . . . . . . . . . . . 435 Convergence diagnostics for all variables with missing data. . . . . . . . . . . . . . . . . . 436 Distribution of prostate specific antigen (PSA). . . . . . . . . . . . . . . . . . . . . . . . . 457 Receiver Operating Characteristic (ROC) curves. . . . . . . . . . . . . . . . . . . . . . . . 458 Kaplan-Meier survival curves of Imputed and Complete cases. . . . . . . . . . . . . . . . . 479 Survival curves from Cox proportional hazards model . . . . . . . . . . . . . . . . . . . . 4910 Sensitivity Analysis on Survival Curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5111 Evaluation of proportional hazards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5712 Comparison of CI for positive fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5813 Comparison of CI at survival times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
List of Tables
1 Definition of GGG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Notation for cross-classified test results and disease status for a fixed threshold. . . . . . . 213 Proportion of complete data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 Predicator Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 Extract from baseline characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 Estimates and confidence intervals for positive fractions, predictive values and likelihood
ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 Confidence intervals for proportion of death at 10 years from diagnosis . . . . . . . . . . . 488 Comparison of curative and noncurative treatment . . . . . . . . . . . . . . . . . . . . . . 48
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9 Sensitivity Analysis on Positive Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . 5010 Baseline characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5711 Tests for normality of positive fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5812 Tests for normality of the proportion of death at 1, 5 and 10 years from diagnosis . . . . . 5913 Sensitivity Analysis on Prevalence of M1 Disease . . . . . . . . . . . . . . . . . . . . . . . 60
Nomenclature
Medical Terms
CCI Charlson Comorbidity Index
GGG Gleason Grade Group
LISA Longitudinal Integration Database for Health Insurance and Labour Market Studies
NPCR National Prostate Cancer Register of Sweden
PCa Prostate Cancer
PCBaSe Prostate Cancer Data Base Sweden
PSA Prostate-Specific Antigen
RP Radical Prostatectomy
RT Radio Therapy
Mathematical Notation
Pr(A) Probability of an event A
P (X) (Joint) density or probability mass function of a scalar (or vector) random variable X
Xi Explanatory variable
Yi Response variable
CI Confidence Interval
d−→ Convergence in distribution
P−→ Convergence in probability
TPF, FPF True and False Positive Fractions
PPV, NPV Positive and Negative Predictive Values
LHR+/- Positive and Negative Likelihood Ratios
ROC Curve Receiver Operating Characteristic Curve
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1 Introduction
Serum levels of prostate specific antigen (PSA) is a cornerstone of the assessment of prostate cancer (PCa)risk category, which, in turn, is central for treatment decisions and is a strong determinant of outcome ofthe PCa. Metastatic PCa is diagnosed with the use of a bone scan or with a high PSA serum level (≥ 100ng/ml). PSA has traditionally been considered as a useful indicator for poorer PCa prognosis and alsofor predicting metastatic disease (M stage), although recent studies have indicated that the usefulness ofPSA for predicting metastatic disease is questionable.
During the diagnostic procedure, far from all men undergo imaging/bone scan since the harms offalse positives often outweigh the benefits, and therefore there are men with a potential metastatic PCawho never underwent imaging. Missing data in clinical registers is common and there are several waysof handling this, including complete case analysis and various imputation methods.
1.1 Purpose
The aim is to assess the diagnostic test performance of classifying distant metastatic PCa using PSA andGleason Grade Group (GGG), to evaluate how the current threshold performs and potentially suggestan improved and more detailed threshold using GGG. In order to do this, the unknown M stages in thedata register need to be addressed. This issue will be approached with the use of multiple imputations,implemented using the MICE algorithm, and the results will be compared with a complete case analysisin order to assess potential bias, flaws and discrepancy in results, between the methods. In the end, thiswill justify the choice of methods for further analysis and increase the external validity.
1.2 Hypotheses
The null hypothesis is that the current PSA-threshold meets the requirements of a good test for predictingmetastatic disease, and that it is reasonable for all GGG. The alternative hypothesis is the contrapositive,that the current threshold is not satisfactory across all GGG.
1.3 Expectations and Challenges
In light of recent studies, it is expected that the current threshold of PSA≥100 ng/ml for classification ofmetastatic disease will not be uniformly optimal for all Gleason Grade Groups and that there could beroom for improvement. It can potentially prove to be difficult to settle for new threshold recommendationsif the test accuracy is poor. Also, the definition of a good test must first be discussed and may prove tobe a challenge.
Concerning the missing data issue, it is expected that the MICE algorithm will accommodate for mostof the missing information and generate reliable parameter estimates, if implemented carefully, based onthe flexibility of the algorithm and since it is known to perform well in various situations. One challengewill be to assess the validity of the imputation model, including model assumptions, in order to arguethat it generates parameter estimates for the analysis that are likely to be close to the unobserved ”true”values.
1.4 Delimitations
Men missing more than one of PSA, GGG and T stage were not considered in this study.
1.5 Method
To address the missing data issue, a multiple imputation technique based on the MICE algorithm willbe applied and, in parallel, a complete case analysis will be performed. The result will be compared ina sensitivity analysis under various assumptions on a nonignorable imputation model for imputing theindicator variable of metastatic PCa (M stage), depending on a parameter k.
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The performance of PSA testing will be assessed using Receiver Operating Characteristic (ROC)curves and associated quantities like true and false positive fractions and likelihood ratios. Survival curvesstratified according to Gleason Grade Group and intervals of PSA will be used to compare metastaticand nonmetastatic PCa mortality, and also to evaluate the imputation model. Restricted mean survivaland years of life lost will also be estimated for a relevant strata to assess consequences of a false positivemisclassification of metastatic disease. A Cox proportional hazards model will be fitted to explore thisin more detail.
The data has been collected by several relevant institutions and stored in a database called PCBaSe.The level of statistical significance is set to 0.05 and the statistical computer software used was R version3.3.2 1.
1.6 Conclusions
No PSA cutoff could separate men with M0 and M1 disease, and neither positive fractions or likelihoodratios were satisfactory for any cutoff, meaning all men with high PSA should be imaged. Metastaticdisease status for a large proportion of men was based on imputations rather than imaging, which wassuboptimal, but the imputations were performed using extensive data from NPCR combined with theMICE algorithm which has been shown to perform well in various scenarios. Although the results var-ied when comparing the complete case analysis with the various nonignorable imputation models, theconclusions were consistent; PSA should not be used as an indicator of metastatic disease, not evenin combination with GGG, meaning that the threshold of 100 ng/ml should be removed from currentguidelines.
1r-project.org
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2 Background
The risk stage classification of prostate cancer (PCa) is central for treatment decisions and is a strongdeterminant of outcome of the PCa. Serum levels of prostate specific antigen (PSA) is a cornerstone ofthe assessment of the risk stage, according to the Annual NPCR report 20142. High PSA is consideredas an indicator of poorer PCa prognosis, Cooperberg, Broering and Carroll (2009), and is recognized asuseful predictor of bone metastases, Lorente et al (1996). The definition of the most severe risk category,level 5, is based on the indicator M stage3 of present distant metastatic PCa, and a threshold of 100ng/ml for PSA. Formally, one is classified into the level 5 risk category (distant metastatic disease) if Mstage is M1 and/or PSA at least 100 ng/ml. Other variables used for risk stage classification are GleasonGrade Group (GGG), T stage and N stage.
Recently, the PSA threshold of 100 ng/ml has been questioned, Stattin et al (2015), and therefore itis of essence to thoroughly assess the diagnostic test performance of classifying distant metastatic PCausing PSA, and to evaluate how the current threshold performs.
In order to do this, the presence of distant metastases needs to be assessed, which is formally assessedby imaging using scintigraphy, Adami et al (2006), but far from all men receive imaging since the harmsof false positives often outweigh the benefits. The inappropriate PCa imaging has decreased since theyear 2000, with only a slight decrease in appropriate imaging in high-risk patients, Makarov et al (2013),and therefore there are men with a possibly metastatic PCa who never underwent imaging. On theother hand, missing data in clinical registers is common and there are several ways of treating missingdata, including complete case analysis (omitting all observations with at least one variable with missingdata) and various imputation methods. Each approach has its own benefits, limitations and assumptions,Newgard and Lewis (2015). This issue will be approached with the use of multiple imputations usingthe MICE algorithm and the results will be compared with a complete case analysis in order to assesspotential bias, flaws and discrepancy in results.
2.1 Medical Procedures
2.1.1 Measures Used for Risk Categorisation
Table 1: Definition ofGleason Grade Group.
GGG G1+G2
1 = 2-62 = 3 + 43 = 4 + 34 = 85 = 9-10
The Gleason Grade Group (GGG) is obtained with a needle core biopsy takenfrom the prostate. The procedure is performed by entering the rectum with atool equipped with 6-12 hollowed needles, and then inserting the needles intothe prostate to extract a tissue sample containing cells from the prostate. If thepatient has prostate cancer, then the sample will hopefully contain cells from apart of the prostate that contains cancer. Then the pathologist evaluates thelevel of cell mutations in the cancer cells and uses this to assess the severity of thecancer, which is measured in two Gleason grades on the scale 1-5. Then GGG iscalculated by summing the most common Gleason grade (G1) and the highestGleason grade (G2) according to table 1. The procedure might be painful andthere is a risk of side effects, of which the most common severe side effect isinfection, Sandin and Wigertz (2013).
To retain the serum level of prostate specific antigen (PSA), measured in ng/ml, an ordinary bloodsample is taken.
The three different stages T, N and M constitutes the TNM classification of malignant tumours.T-stage is the primary stage used in the risk categorisation of prostate cancer and is assessed by theurologist after a rectal examination of the prostate. Tumours found growing outside of the prostate andinto the vas deferens are classified as T4. If the tumour is outside the capsule it is classified as T3, ifthere are lumps that seem to be inside the capsule it is classified as T2, and if no lumps are found it isclassified as T1. T1a-b tumours are discovered by transurethral resection of the prostate (TURP), which
2http://npcr.se/in-english/3coded M1 for PCa with distant metastases and M0 for PCa without metastases
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basically is a reduction of the prostate by planing to simplify micturition, Sandin and Wigertz (2013),while T1c tumours are discovered by elevated PSA prior to a biopsy.
The condition of the regional lymph nodes is classified by N stage. N0 means that there were no signsof regional lymph node metastases during surgery and N1 corresponds to signs of region lymph nodemetastases during surgery. If the relevant surgery was not performed then the N stage is classified asNX. In some cases the N stage is unknown and there is no record of whether relevant surgery has beenconsidered or not, in which case the N stage is considered as missing.
Lastly, the distant metastases are described by the M stage, and is an indicator where M0/M1 meansno signs/signs of distant metastases, and MX indicates that the assessment was not performed (onlypossible before 2011), Stattin et al. (2013). The skeleton is the dominant locale for distant metastases ofprostate cancer. Therefore, the primary investigation is limited to the skeleton, unless symptoms indicatemetastasis to other organs. Bone imaging, mainly scintigraphy, is the standard method for investigatingbone metastases of prostate cancer. In case of unclear scintigraphy then magnetic resonance imaging(MRT), positron emission tomography (PET) or computed tomography (CT) is carried out if necessary.
2.1.2 Other Variables
There are several treatments for prostate cancer and the two main curative treatments are Radicalprostatectomy (RP) and Radio therapy (RT). RP is a surgery applied to any patient with clinicallylocalized prostate cancer that can be completely excised surgically, who has a life expectancy of at least10 years, and has no serious comorbid conditions that would contraindicate an elective operation, Mohleret al (2010). It is a common treatment for patients having clinically localized cancer, and the primarygoal of RP is cancer extirpation, where the prostate gland and seminal vesicles are removed, while thesecondary goals are preservation of urinary continence and potency. Urinary incontinence is one of themost common postoperative complications following RP, along with erectile dysfunction, Tewari et al.(2013). The incontinence is usually mild, but the risk of impotence is about 50%, Adami et al (2006).Radio therapy can be applied in several ways. Brachy therapy is a kind of RT that involves placingradioactive sources into the prostate tissue. External radio therapy uses external beams and is one ofthe principle treatment options for clinically localized prostate cancer, Mohler et al. (2010). Modern RToffers effective, durable treatment for all stages of prostate cancer with low levels of clinically significanttoxicity, Tewari et al. (2013). Side effects are mild forms of incontinence, gastrointestinal symptomsand impotence, Adami et al (2006). There are no randomized studies showing that men with metastaticprostate cancer that receive curative treatment by RP or RT are helped in terms of life expectancy andlife quality, Nationellt vardprogram for prostatacancer4 (2015).
Noncurative treatment includes hormonial treatments, or androgen deprivation therapy (ADT), whichmay include antiangrogens (AA) or Gonadotropin-releasing hormone (GnRH). These are used for retain-ing an incurable PCa, Mohler et al (2010). For example, ADT is the standard treatment for bonemetastases, and it has minimal benefit on survival meaning it is mainly palliative. Mild side effects ofADT can be for example hot flushes, libido, mood and cognitive changes. Severe side effects includeincreased cardiovascular risk and osteoporosis, increased risk of diabetes, coronary heart disease, myocar-dial infarction, sudden cardiac death and loss of bone mineral density leading to increased risk of clinicalfracture, Tewari et al (2013).
Conservative treatment (CT) means either active surveillance or watchful waiting. Active surveillanceinvolves actively monitoring the course of disease with the expectation to intervene with curative treat-ment if the cancer progresses, with a risk of missing the opportunity for cure and getting progressionand/or metastases. Similarly, watchful waiting is used with the intention of palliative (ADT) treatmentat progression in men with prostate cancer, Mohler et al (2010).
Mode of detection indicates the principal way of how the prostate cancer was revealed. Symptomaticmode of detection includes lower urinary tract symptoms (LUTS), other symptoms such as hematuria, orremote symptoms such as back pain when having distant metastases or other cancer. Non-symptomaticindicates health assessment including PSA testing and without urinary tract symptoms, Stattin et al(2013).
4www.cancercentrum.se
10
Charlson Comorbidity Index (CCI) is a measure of the comorbidity, which predicts the ten-yearmortality for a patient who may have a range of comorbid conditions. The CCI score is calculatedby assigning weights to 17 medical conditions, including diabetes and hypertension. Each condition isassigned a score of 1, 2, 3, or 6, and the final CCI score is given as the sum of these scores. Then CCIcategories are formed for final scores of 0, 1, 2, or 3+. The prostate cancer is not included, and one doesnot obtain a CCI score of 6 for metastatic PCa. The CCI scores indicate 0 = no comorbidity, 1 = mildcomorbidity, 2 = medium comorbidity, and 3+ = severe comorbidity, Charlson et al (1987).
2.2 Data Collection
2.2.1 PCBaSe and NPCR
The source of data, Prostate Cancer Data Base Sweden (PCBaSe), Hemelrijck et al (2013), is a nationwidepopulation-based research database consisting of men diagnosed with prostate cancer. For this study,records between 2000 and 2012, a total of 112 013 men, were retrieved. PCBaSe was created by recordlinkage between the National Prostate Cancer Register of Sweden (NPCR) and several of other highquality national registers, for example the Prescribed Drug Register, In-Patient Register, Cause of DeathRegister, National Population Register and the Longitudinal Integration Database for Health Insuranceand Labour Market Studies (LISA).
In particular, NPCR is a tool for documentation and assessment of quality of the health care of menwith prostate cancer. It is also used for research and to improve the treatment of the disease. Theregister contains data including waiting times, treatment and its outcome, spread and type of cancer,and diagnosis procedure. The participation in the register is voluntary and it is designed such that it isnot possible to track or identify specific individuals in the compiled material. Recent studies show thatNPCR has high capture rate, completeness and accuracy, Tomic et al (2015a, b). For an informative andconcise overview of NPCR, see Stattin et al (2017).
2.2.2 Data Retrieved from PCBaSe
The covariates age and calender year at the time of diagnosis, highest educational level5, martial status(married/partnership, unmarried, widow, divorced/separated), mode of detection, survival times andcensoring indication, were retrieved from PCBaSe.
In addition, clinical data were retrieved on serum levels of PSA, tumour differentiation by GleasonGrade Group, cancer stage according to the TNM classification, and primary treatment. The risk categoryis a composite variable based on these variables. The definition of the risk categories is similar tothe National Comprehensive Cancer Network (NCCN) guidelines, Mohler et al (2010) but altered todistinguish between regionally metastatic and distant metastatic disease.
- Low-risk localized prostate cancer was defined as clinical local stage T1/T2, GGG 1 and PSA lessthan 10 ng/ml
- Intermediate risk localized prostate cancer was defined as stage T1/T2, GGG6 2-3 and/or PSAlevel of 10 - 20 ng/ml
- High-risk localized prostate cancer was defined as stage T3 and/or GGG 4-5 and/or PSA levels of20 - 50 ng/ml
- Regionally metastatic or locally advanced prostate cancer was defined as stage T4 and/or N1 diseaseand/or PSA levels of 50 - 100 ng/ml without distant metastasis (M0 or MX disease)
- Distant metastatic disease was defined as M1 disease and/or PSA at least 100 ng/ml
5low (compulsory school, ≤9 years), middle (upper secondary school, 10 - 12 years), high (college and university, ≥13years)
6Beginning in 2007, data were available to further categorize GGG 2-3 cancers into 2 and 3 separately
11
The comorbidity burden was assessed using data from the National Patient Register and classified ac-cording to Charlsons comorbidity index (CCI). Data on treatment modalities included radical prostatec-tomy, radiotherapy, and hormonal therapy. Information on alternative treatment approaches like activesurveillance and watchful waiting was also obtained. Data on region where diagnosis was performed wasalso obtained, where Sweden is divided into six regions: North, Stockholm-Gotland, South, South-East,Uppsala-Orebro and West.
2.2.3 Costs
When evaluating a diagnostic test it is relevant to compare possible outcome scenarios and their impacton the patient and the society. The cost of certain treatments, their side effects and potential years oflife lost due to suboptimal treatment are some factors that need to be considered.
Estimates of costs have been obtained from Socialstyrelsen (2014) and are specified in Swedish kronoras of 2013. The estimated cost of a full RP treatment lies between 85 000 - 120 000 kr. An estimatedtotal treatment cost for antiandrogens is around 19 000 - 23 000 kr , for GnRH about 17 000 kr andpalliative treatment with Docetaxel and Prednison costs around 193 000 kr and with Cabazitaxel it costsaround 255 000 kr. Skeletal Scintigraphy of the whole body costs around 2600 kr while PET/CT costsaround 12 000-16 000 kr.
2.3 Previous Research
The association between PSA≥100 ng/ml and metastatic PCa was assessed in a population based registrystudy between 1998 and 2009, Stattin et al (2015). A total of 15 635 men in NPCR with PCa werecompared in three groups, men with PSA≥100 ng/ml and M0, men with PSA≥100 ng/ml and M1 andmen with PSA<100 ng/ml and M1. Results showed that a fourth of men who underwent imaging withPSA≥100 ng/ml had no distant metastases and had two to three times higher 5-year survival than menwith M1. The conclusion was that the threshold of PSA≥100 ng/ml as indicator of metastatic PCashould be reconsidered.
Moreover, the use of imaging to identify metastatic prostate cancer has changed over the years. In aretrospective cohort study performed on data from NPCR from 1998-2009 including 99 879 men diagnosedwith prostate cancer, Makarov et al (2013) reported that the use of imaging decreased over time and thatthe amount of inappropriate imaging has decreased from 45% to 3% among low-risk patients, while thenumber of appropriate imaging in high-risk patients has decreased from 63% to 47%. This was associatedwith the dissemination of imaging usage data and the latest imaging guidelines to urologists in Swedenled by NPCR. A total of 36% underwent imaging within 6 months of diagnosis, and these had generallyhigher GGG, PSA levels and clinical stages than men who did not receive imaging. This agrees withfindings made by Tomic et al (2016) where missing data for risk classification was assessed in NPCRbetween 1998 and 2012. In that study, it was concluded that the amount of missing information in orderto do a classification is generally low and that men with unknown risk classification most likely had alow risk prostate cancer.
The mortality of men in NPCR treated with noncurative intent was assessed in a retrospective cohortstudy of 76 437 cases between 1991-2009, Rider et al (2012). The prostate cancer mortality rates varied10-fold according to risk category, and Figure 1 in this article shows that there was a clear differencebetween high-risk patients without distant metastases and men with distant metastases, where the lattergroup had a substantially worse 15-year prognosis.
Lastly, in a Danish study, 207 men diagnosed in 1997 were compared with 316 men diagnosed between2007-2013, all with de novo metastatic prostate cancer, and the conclusion was that the significantlyimproved survival in men diagnosed with metastatic prostate cancer could partly be explained by theintroduction of new life-prolonging treatments, Berg et al (2007).
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3 Theory
3.1 Basic Inference
The Law of Large numbers and the Central Limit theorem are two fundamental theorems which, togetherwith the delta method and a few theorems from the second chapter of van der Vaart (1998), will playan important role when deriving asymptotic confidence intervals. This subsection is therefore devoted tosummarize vital material mainly from van der Vaart (1998) to be used later on. The first important toolis the law of large numbers, which is neatly formulated in Grimmet and Stirzaker (2001).
Theorem 1 (The (Weak) Law of Large Numbers). Let Xi, i = 1, . . . , n, be independent and identically
distributed (i.i.d.) random variables with EXi = µ, |µ| <∞, then∑ni=1Xi/n
P−→ µ
This theorem will be used when deriving asymptotic properties for estimators later on. An equallyimportant tool for this is the central limit theorem.
Theorem 2 (The Central Limit Theorem). Let Xi, i = 1, . . . , n, be i.i.d. mean zero, unit variance,
random variables, then√n∑ni=1Xi/n
d−→ X with X ∼ N(0, 1)
The continuous mapping theorem is a useful and simple tool and motivates the use of the plug-inprinciple. The stochastic convergence remains when applying a continuous function to an estimator witha priori known stochastic convergence.
Theorem 3 (Continuous Mapping). Let f : Rk 7→ Rl be continuous on C ⊂ Rk such that P (X ∈ C) = 1.
If Xnd−→ X then f(Xn)
d−→ f(X), and if XnP−→ X then f(Xn)
P−→ f(X).
From the continuous mapping theorem it is easy to prove Slutsky’s lemma, which is a useful tool forderiving asymptotic results for standardized random variables where the variance estimator is consistent,
since Ynd−→ c if and only if Yn
P−→ c for a constant c (van der Vaart, theorem 2.7 (iii)).
Lemma 1 (Slutsky). Let Xn, X and Yn be random variables such that Xnd−→ X and Yn
d−→ c ∈ R a
constant, then Xn + Ynd−→ X + c, YnXn
d−→ cX and Xn/Ynd−→ X/c as long as c 6= 0. �
The delta method is a basic but important tool used to derive asymptotic results for functions ofsequences of random variables. van der Vaart (1998) discusses a general version of the delta method, andbelow is a simplified version of it for the one dimensional case.
Proposition 1 (The Delta Method). Let φ : D ⊂ R −→ R, differentiable at θ, and Xn be a randomvariable taking values in D. Assume that there exists a sequence (rn) ⊂ R such that
rn(Xn − θ)d−→ X as rn −→∞
thenrn(φ(Xn)− φ(θ))
d−→ φ′θ(X) as rn −→∞and
rn(φ(Xn)− φ(θ))− φ′θ(rn(Xn − θ)) = op(rn)
where φ′θ(x) is a linear map identified with its matrix form φ′θ(x) = φ′θx = φ′(θ)x.
Remark 1. A special case of this, which will be used later, is when we have an asymptotic normaldistribution √
nXnd−→ X ∼ N(µ, σ2) as n −→∞
and φ be as above, then by the delta method
√nφ(Xn)
d−→ φ′(µ)X ∼ N(φ′(µ)µ, φ′(µ)2σ2) as n −→∞
and √nφ(Xn) = φ′(µ)
√nXn + op(
√n)
13
Example 1. Let Xn is a positive sequence of random variables such that√nXn
d−→ X ∼ N(µ, σ2).
(a) If φ(x) = log(x) on D = R+ with φ′(x) = 1/x, then Var(log(Xn)) = σ2
µ2 + oP (1),
(b) Var(exp(Xn)) = σ2 exp(µ)2 + oP (1)
(c) If Xn attains values in (0, 1) then Var(log(− log(Xn))) = σ2
log(µ)2µ2 + oP (1)
(d) If Xn attains values in (0, 1) then Var(log( Xn
1−Xn)) = σ2
µ2(1−µ)2 + oP (1)
Example 2. Following van der Vaart (1998). Let Xn be a sequence of estimators of µ such that√n(Xn − µ)
d−→ X ∼ N(0, σ2) and S2n be a sequence of estimators of σ2 such that S2
nP−→ σ2 > 0 then,
since convergence in probability implies convergence in distribution, we have by Slutsky’s lemma
√n(Xn − µ)/Sn
d−→ X/σ ∼ N(0, 1)
which can be used to construct an asymptotic confidence interval for µ: Xn±zα/2Sn/√n with asymptotic
significance level 1− α.
Example 3. A special case of example 2 is when we have Xi are independent ∼ Ber(p), so EXi =p,VarXi = p(1 − p). Then a natural consistent estimator of p is Xn = pn =
∑ni=1Xi/n. By the Law
of Large numbers pnP−→ p, so by continuous mapping pn(1− pn)
P−→ p(1− p). Thus an asymptoticallyconsistent variance estimator is S2
n = pn(1 − pn) and an an asymptotic confidence interval for p ispn ± zα/2
√pn(1− pn)/n with asymptotic significance level 1− α.
3.2 Survival Analysis
3.2.1 Definitions
The relation between survival time and cause of death can be described by survival curves. These show,quite intuitively, how members of the groups survived over time and may reveal a lot of information.Essentially, the survival curves estimate the probability of survival over the time period considered. Thefollowing is based on material from Moeschberger et al (1997).
From here on, let Pr(A) denote the probability of an event A, and P (X) denote the (joint) densityor probability mass function of a scalar (or vector) random variable X.
Definition 1. The function S(t) = Pr(T > t) describes the probability of an individual surviving beyondthe time t and is called the survivor function. When T is discrete, assuming it takes values tj , j = 1, 2, . . . ,having probability mass function p(tj) = Pr(T = tj), where t1 < t2 < . . . , then the survival function isdefined as
S(t) = Pr(T > t) =∑tj>t
p(tj)
Definition 2. The function
h(t) = limε→0
Pr(t ≤ T < t+ ε|T ≥ t)ε
is called the hazard rate. It is the conditional probability that an individual who survives just to priorto time t experiences the event at time t. When T is discrete the hazard function is given by
h(tj) = Pr(T = tj |T ≥ tj) =p(tj)
S(tj−1)
Remark 2. For discrete T the survival function may be written as the product of conditional survivalprobabilities
S(t) =∏j:tj≤t
S(tj)
S(tj−1)
14
and since p(tj) = S(tj−1)− S(tj) the survival function is related to the hazard function by the followingequality
S(t) =∏j:tj≤t
(1− h(tj))
Definition 3. Kaplan Meier Estimator Assume the events occur at n distinct times t1 < · · · < tnand that at time tf there are mf events, or deaths, and nf number of individuals at risk. The estimator
S(t) =
{1 if t1 > t∏f :tf≤t(1−
mf
nf) if t1 ≤ t
is called the Kaplan Meier or Product-limit estimator. It is a step function with jumps at the observedevent times. The quantity
mf
nfprovides an estimate of the hazard rate.
Remark 3. Hosmer, Lemeshow and May (2008) describe censoring mechanisms and how the Kaplan-Meier estimator handles this. Right censoring is dealt with in the following way: if time r is right censoredthen the corresponding individual has been counted towards the numbers at risk up until time r but willnot contribute to the numbers at risk after that, hence the number of individuals at risk after time r willbe reduced. The censoring will not contribute to the number of events or deaths. Had the individualexperienced death then the overall survival probability would have been smaller and had the individuallived the overall survival probability would have been larger, compared to the censored case.
3.2.2 Asymptotic Distribution and Confidence Intervals
In order to construct a confidence interval for the Kaplan-Meier estimator at a specific time, we needan asymptotic distribution. For a fixed time interval [0, T ] Kalbfleisch and Prentice (2002) show, usingcounting processes theory, that under mild assumptions about the processes involved, that the Kaplan-Meier estimator is uniformly consistent and, when scaled by the variance estimator, converges weakly toa normal distribution.
Theorem 4. Let t(f), f ∈ {1, . . . , n} be the ordered failure times where n is the total number of failuretimes, mf denote number of failures at t(f) and nf the number at risk at time t(f). Assume that theobservations of survival among the nf subjects at risk at time t(f) are independent Ber(pf ) distributed,then Greenwood’s formula for a variance estimator is given by
ˆV ar[Sn(t)] = Sn(t)2∑
f :t(f)≤t
mf
nf (nf −mf )
For a sample of n observations, this estimator is asymptotically consistent.Proof: We derive the variance estimator. By assumption, the observations of survival among the nf
subjects at risk at time t(f) are independent Ber(pf ) distributed, so the natural estimator is pf = 1− mf
nf
for t1 ≤ t with variance estimatorpf (1−pf )
nf. Set pf = (1 − mf
nf) for t1 ≤ t. The strategy is to first find
an estimator for the variance of log(S(t)) =∑f :tf≤t log(pf ), a sum instead of a product, using the delta
method and then back transform this estimator with another use of the delta method. Starting with thelogarithm
Var(log(pf )) =1
p2f
pf (1− pf )
nf=
mf
nf (nf −mf )
by example 1 part (a), and by independence we get
Var(log(S(t))) =∑f :tf≤t
mf
nf (nf −mf )
15
Another application of the delta method applied on the exponential function with X = log(S(t)), as inexample 1 part (b), renders
Var(S(t)) = Var(exp(log(S(t)))) = S(t)2Var(log(S(t))) = S(t)2∑f :tf≤t
mf
nf (nf −mf )
Remark 4. It is possible to show that the Kaplan-Meier estimator is uniformly asymptotically consistenton bounded intervals, i.e.
supt∈[0,T ]
|Sn(t)− S(t)| P−→ 0 n −→∞,
and, by the consistency of the Greenwood’s estimator, that
Sn(t)− S(t)√ˆV ar(Sn(t))
d−→ N(0, 1) as n −→∞.
The last display can be used to construct an approximate 100(1−α)% confidence interval, compare withexample 2.
Remark 5. Since a normally distributed random variable can assume any real value and S(t) ∈ [0, 1],then, for S(t) close to the endpoints, these intervals might stretch outside [0, 1] which is not desirable. Oneway to deal with this problem is for example to stretch [0, 1] to the whole real line by log-log-transformingS(t) and calculating an approximate confidence interval which then is back-transformed into [0, 1]. Thevariance of the log-log and logit transformations are given in the following proposition.
Proposition 2. Under the same assumptions as in theorem 4, if
i) L(t) = log(− log(S(t))), then
Var(L(t)) =1
log(S(t))2
∑f :t(f)≤t
mf
nf (nf −mf )
ii) L(t) = log(
S(t)
(1−S(t))
), then
Var(L(t)) =1
(1− S(t))2
∑f :t(f)≤t
mf
nf (nf −mf )
Proof: (i) The variance Var(L(t)) follows by yet another application of the delta method, as in example 1part (c),
Var(L(t)) =Var(S(t))
log(S(t))2S(t))2=
1
log(S(t))2
∑f :t(f)≤t
mf
nf (nf −mf )
(ii) As in (ii) variance Var(L(t)) follows by the delta method, as in example 1 part (d),
Var(L(t)) =Var(S(t))
(1− S(t))2S(t)2=
1
(1− S(t))2
∑f :t(f)≤t
mf
nf (nf −mf ).
�
Remark 6. Marshall et all (2009) performed a literature review and the log-log transformation of theKaplan-Meier estimator was mentioned as being the current practice the when combining estimates frommultiple imputations, compare with section 3.5.
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3.2.3 The Cox Proportional Hazards Model
A brief introduction to the relevant concepts of the Cox proportional hazards model is given below and isbased on material from Hosmer, Lemeshow and May (2008). The key point of using a Cox proportionalhazards model is to incorporate explanatory variables in a survival analysis setting and obtain hazardratios and adjusted survival curves.
Definition 4. Let t denote time, X = (X1, . . . , Xp) be a vector of explanatory variables that do notinvolve t, and β = (β1, . . . , βp). The function h(t,X) is called the hazard function and it is the hazardrate conditioned on X. The Cox proportional hazards model assumes that the hazard function can bewritten on the form
h(t,X) = h0(t) exp(XTβ),
where h0(t) is called the baseline hazard function. Furthermore, we call H(t,X) :=∫ t
0h(u,X)du the
cumulative hazard function and H0(t) :=∫ t
0h0(u)du the cumulative baseline hazard function.
The reason why h0(t) is called the baseline hazard function is because when X = 0, meaning noexplanatory variables are in the model, then h(t,X) = h0(t). The model is semiparametric because the
baseline hazard is unspecified. If X1 is a binary variable then eβ1 is the effect of exposure adjusted forother variables. The hazard function can be estimated from the model by maximum likelihood through apartial likelihood function in an iterative manner. The solution is in general not explicit and it is calledpartial since it only considers probabilities for those who fail and not for those observations that arecensored. From this the estimates h0(t) and β are obtained. An asymptotic confidence interval is given
by βi/sd(βi) ∼ N(0, 1).
Definition 5. The hazard ratio for two different individuals with explanatory variables X∗ and X isdefined as
HR(X∗, X, t) :=h(t,X∗)
h(t,X)
It is important to note that the model relies on the assumption that the hazard ratio is constantover time. The hazard ratio is estimated by plugging in the corresponding estimates from the model
HR(X∗, X, t) = h(t,X∗)
h(t,X)which simplifies to exp((X∗ −X)T β).
Definition 6. We call S(t,X) := exp(−H(t,X)) the adjusted survival function and S0(t) := exp(−H0(t))the baseline survival function.
Note that the cumulative hazard function can be written as∫ t
0h0(u)eX
T βdu = eXT βH0(t), which
implies that
S(t,X) = e− exp(XT β)H0(t) = [e−H0(t)]exp(XT β) = [S0(t)]exp(XT β).
The function S0(t) is called the baseline survival function since if there are no explanatory variables(X = 0), then S(t,X) = S0(t) and we obtain the usual survival function. If one plots the survival functionover time then one obtains the adjusted survival curve, meaning it has been adjusted for explanatory
variables. It has the estimate S(t,X) = (S0(t))exp(XT β) which is obtained during the estimation processmentioned above.
One may evaluate the proportional hazards assumption in several ways, and perhaps the most intuitiveway is to do it graphically. Note that if we log-log transform the adjusted survival function we obtain
log(− log(S(t,X))) = XTβ + ln(−ln(S0(t))),
and for two individuals with explanatory variables X∗ and X we have that
log(− log(S(t,X∗)))− log(− log(S(t,X∗))) = (X∗ −X)Tβ
is constant with respect to time. Therefore one may plot the log-log transformed adjusted survival curvesfor different individuals and see if they are parallel. If not, then the proportional hazards assumption isviolated.
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3.2.4 Restricted Mean Survival
We conclude by discussing the mean survival, which is defined below. The material of this section is fromSheldon (2013). The mean survival is a useful statistic to summarise survival data and can be used todescribe life years gained or lost. While the estimator is biased and underestimates the mean, it easy tointerpret in terms of expected survival time.
Definition 7. We call
µ := ET =
∫ ∞0
S(t)dt
the mean survival, and a natural estimator is
µ :=
∫ ∞0
S(t)dt =
n∑f=1
S(tf−1)(tf − tf−1), t0 = 0.
The problem with this definition is that it may not in general be useful in practice, since the lastobservation might be censored, in which case the integral is not defined since S never reaches zero. Oneway of solving this is to consider the restricted mean survival, meaning that we restrict the integral tothe interval [0, t∗], t∗ < tn and thus all later observations are disregarded.
Definition 8. We call
µ(t∗) := E min(T, t∗) =
∫ t∗
0
S(t)dt
the restricted mean survival, and a natural estimator is
µ(t∗) :=
∫ t∗
0
S(t)dt =∑
f≥1: tf≤t∗S(tf−1)(tf − tf−1),
and variance estimator
Var(µ(t∗)) := µ(t∗)2n∑f=1
mf
nf (nf −mf )
If the time is specified in years the restricted mean can be thought of as a t∗-year life expectancy, orin other words, the expected number of life years left within t∗ years.
To compare two groups, for example to compare treatment effects on survival through expected yearsof life lost, Karrison (1997) suggests that one may use the following asymptotically standard normalquantity
µ1(t∗)− µ2(t∗)√Var(µ1(t∗)) + Var(µ2(t∗))
∼ N(0, 1).
3.3 Evaluation of Tests for Classification
3.3.1 Notation and Definitions
One may evaluate medical tests with a continuous test result for classification of a binary variable byusing the Receiver Operating Characteristic curve and associated measures of accuracy. The purpose ofthe ROC curve is to visualize the discrepancy between two population densities for various thresholds ofthe test result, and is useful when looking for the optimal choice of threshold to separate the populations,comparing thresholds and to assess overall test performance.
The threshold is used to dichotomize the test result in order to classify members of the total populationinto one of the two subgroups. The threshold in figure 2 separates the two populations non-diseased anddiseased but introduces two kinds of errors. The first is the false negatives, which are diseased who areclassified as non-diseased by the dichotomized test, and the false positives, which are non-diseased whoare classified as diseased by the dichotomized test.
18
Definition 9. Let D denote true disease status, being a binary variable such that
D =
{1 for disease
0 for non-disease
Let Y be a continuous test result and c ∈ R a threshold such that a binary test can be constructed
T =
{positive if Y ≥ cnegative if Y ≤ c
Then the true and false positive fractions and positive and negative predictive values at threshold c aredefined as
TPF(c) = Pr(Y ≥ c|D = 1) , FPF(c) = Pr(Y ≥ c|D = 0)
PPV(c) = Pr(D = 1|Y ≥ c) , NPV(c) = Pr(D = 0|Y < c)
In biomedical research TPF is commonly referred to as sensitivity and 1− FPF as specificity. The falsepositive fraction describes how likely it is to get a positive test result when one is in fact free from disease,and 1 − TPF is equal to the false negative fraction, which describes how common it is to fail to detectdisease (by a negative test result) when disease is present. An ideal test at threshold c has FPF(c) = 0and TPF(c) = 1 while a useless test has FPF(c) = TPF(c).
The prevalence p of disease describes how common disease is in the population and is defined asp = Pr(D = 1). The positive and negative predictive values describe the usefulness of the test, are morerelevant to the patient and caregiver, and depend on the prevalence. A perfect test has PPV(c) = 1 =NPV(c) and a useless test (which does not contribute with any information about the true disease status)has
PPV(c) = Pr(D = 1|Y ≥ c) = Pr(D = 1) = p
NPV(c) = Pr(D = 0|Y < c) = Pr(D = 0) = 1− p
Another way of describing the prognostic or diagnostic value of a test are the likelihood ratios. In aBayesian testing context they correspond to the Bayes factor which, multiplied by the prior odds, givesthe posterior odds, compare with Robert (2007).
Definition 10. Using the same notation as above, the likelihood ratios are defined as
LR+(c) :=TPF(c)
FPF(c)=
Sensitivity
1-Specificity
LR-(c) :=1− TPF(c)
1− FPF(c)=
1− Sensitivity
Specificity
The likelihood ratios attain values in (0,∞) and describe how the pre-test odds Pr(D = 1)/Pr(D =0) = p/(1− p) is altered by the test result to obtain the post-test odds Pr(D = 1|Y )/(1−Pr(D = 1|Y )).
Proposition 3. The pretest and post-test odds can be related through the likelihood ratios in thefollowing way
(i) Pr(D=1|Y≥c)1−Pr(D=1|Y≥c) = LR+(c) Pr(D=1)
1−Pr(D=1) ,
(ii) Pr(D=1|Y <c)1−Pr(D=1|Y <c) = LR-(c) Pr(D=1)
1−Pr(D=1) .
Proof: we only show case (i) since the proof of the second is analogue. By the definition of conditionalprobability
Pr(D = 1|Y ≥ c)Pr(D = 0|Y ≥ c)
=Pr(D = 1, Y ≥ c))Pr(Y ≥ c)Pr(D = 0, Y ≥ c)Pr(Y ≥ c)
=Pr(Y ≥ c|D = 1))Pr(D = 1)
Pr(Y ≥ c|D = 0)Pr(D = 0)=
TPF(c)
FTF(c)
Pr(D = 1)
(1− Pr(D = 1))
�
19
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Relating Pre-test and Post-test Probability
Pre-test probability
Po
st-
test p
rob
ab
ility
d=0.01
d=0.1
d=0.5
d=1
d=2
d=10
d=100
Figure 1: Relating pre-test and post-test prob-ability through a likelihood ratio of size d.
The likelihood ratios describe the usefulness of atest, and a useless test has likelihood ratios equal to1 while a perfect test has LR+(c) =∞ and LR-(c) = 0.The further away from 1 in respective direction themore useful the test is. The relation between odds andprobability is one-to-one, meaning that we may derivethe post-test probability from the pre-test probabilityp through dp
1−p+dp where d is one of the likelihood ra-tios for a fixed threshold c. This relation for varioussizes of a likelihood ratio can be seen in 1 to the left.A large positive likelihood ratio means that the test isgood at ruling in disease and vice versa. Jaeschenke etal (1994) suggest thumb rules for the likelihood ratios:LHR+ ≥ 10 and LHR− ≤ 0.1 indicate a conclusivechange from pretest to post-test probability, 5-10 and0.1-0.2 for a moderate change, and 2-5 and 0.2-0.5 fora small change.
The positive fractions and the likelihood ratios caneasily be visualized for each threshold using the ReceiverOperating Characteristic (ROC) curve.
Definition 11. Using the same notation as above, then we let ROC curve be the curve created whentrue and false positive fractions are calculated for all thresholds c. Formally
ROC() = {(TPF(c),FPF(c)), c ∈ (−∞,∞)}
−4 −2 0 2 4 6
0.0
0.1
0.2
0.3
0.4
0.5
Y
Den
sity
Non−diseased Diseased
Threshold
TPF
FPF
Figure 2: Visual representation of true positiveand false negative fractions.
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
False Positive Fraction (1−Specificity)
True
Pos
itive
Fra
ctio
n (
Sen
sitiv
ity)
●x
Pos
itive
Lik
elih
ood
Rat
io
for
x
Negative Likelihood Ratio
for x
ROC curve
Useles
s tes
t
c − 8
c
8
Figure 3: Example of a ROC curve and theconcepts of likelihood ratios.
The ROC curve is a useful tool for assessing overall test performance or for finding an optimalthreshold. As the threshold c increases from −∞ to ∞ the ROC curve stretches from (1, 1) to the originand if the ROC curve is a straight line then the test is useless for all thresholds. The positive likelihoodratio for a threshold c corresponding to a point x = (FPF(c),TPF(c)) on the ROC curve can be visualized
as the slope of the line between x and the origin since ∆TPF∆FPF = TPF (x)−0
FPF (x)−0 = TPF (x)FPF (x) = LR+(x). Similarly,
20
the negative likelihood ratio is the slope of the line between x and (1, 1) since ∆TPF∆FPF = 1−TPF (x)
1−FPF (x) =
LR−(x), compare with fig. 3, Johnson (2004).
3.3.2 Estimation
Let N be the total number of observations and define nD=1 to be the number of observations with disease(D = 1) and nD=0 be the number of observations that are truly non-diseased (D = 0). Let n+ and n−,correspondingly, be the number of observations with positive and negative test results. Then the datacan be displayed according to the table below.
Table 2: Notation for cross-classified test results and disease status for a fixed threshold.
D=0 D=1Y=0 n−D=0 n−D=1 n−
Y=1 n+D=0 n+
D=1 n+
nD=0 nD=1
The prevalence p is the parameter of the binomial distribution governing the disease status nD=1 ∼Bin(p,N) and hence estimated simply as p = nD=1/N . The positive fractions and predictive values arejust marginal binomial distributions, so conditionally
n+D=1 ∼ Bin(TPF, nD=1) , n+
D=0 ∼ Bin(FPF, nD=0)
n+D=1 ∼ Bin(PPV, n+) , n−D=0 ∼ Bin(NPV, n−)
and the natural estimators are
ˆTPF = n+D=1/nD=1 , ˆFPF = n+
D=0/nD=0
ˆPPV = n+D=1/n
+ , ˆNPV = n−D=0/n−
Since the predicted values and positive fractions attain values in (0, 1) we may use the continuous mappingtheorem to obtain estimates of the likelihood ratios.
Remark 7. It is easy to see that the predictive values depend on the prevalence using table 2. Fix Nand change the proportion of diseased and non-diseased by changing the prevalence in the following way:multiply the quantities of the left column with v ∈ (0, 1) and the right column with r = N−vnD=0
nD=1. Now
the prevalence is pr, the predictive values are different but the positive fractions are unchanged,
ˆTPF =rn+
D
rnD=1, ˆFPF =
vn+D=0
vnD=0
ˆPPV =rn+
D=1
rn+D=0 + vn+
D=1
, ˆNPV =rn−D=0
vn−D=0 + rn−D=1
.
Since the positive fractions does not depend on the prevalence, the same applies to the likelihood ratiossince they are functions of the positive fractions.
Remark 8. By interpreting the binomial distributions as a sum of independent Bernoulli trials we mayuse example 3 to derive asymptotic confidence intervals, with p being one of TPF,FPF,PPV or NPV withits corresponding estimator. By assumption, we have that the positive fractions are independent (sincewe are given nD=1) and thus exact confidence rectangles can be computed. More precisely, given α, thena 1−α confidence rectangle can be constructed for (TPF,FPF) by finding corresponding 1− α =
√1− α
confidence intervals for TPF and FPF respectively.
21
3.4 Tests for Normality
Tests of normality test the null hypothesis H0 : data is normally distributed against the alternative H1 :data is not normally distributed. In the context of parameter pooling based on multiple imputations,these tests will be useful to assess whether a certain transformation yields a better or satisfactory normalapproximation. The Shapiro-Wilk, Shapiro-Francia, Anderson-Darling and Cramer-von Mises tests arefour examples of tests for normality. This section is a brief summary of material from Thode Jr (2002).
The first test is the Shapiro-Wilk test for normality and it is based on the test statistic
W =b2
(n− 1)s2,
where s2 is the usual variance estimator, n is the sample size and b is, up to a normalizing constant, ageneralized least squares estimate of the slope from a linear regression of the sample order statistics ontheir expected values. Shapiro and Wilk (1965) showed that the maximum value of W is 1, and, underH0, that the nominator and denominator are both, up to constant, estimating the variance. Thus onewould expect W to be close to 1 under normality and otherwise smaller. Therefore, one rejects H0 ifW ≤ c for some appropriate constant c, which means that a smaller W generally gives smaller p-values.
The Shapiro-Francia test for normality approximates W under the assumption that, for large samplesizes, the order statistics can be treated as independent. It is basically the squared correlation betweenthe approximated ordered quantiles from the standard normal distribution and the ordered sample values.
The Anderson-Darling and Cramer-von Mises tests for normality are goodness of fit tests based on theempirical distribution function Fn. They measure the weighted distance between the true distributionfunction F and Fn through
n
∫ ∞−∞
(Fn(x)− F (x))2w(x)dF (x),
for a weight function w.The Cramer-von Mises test is based on this distance with w(x) = 1 and uses the test statistic
W 2 =1
12n+
n∑i=1
(pi −
2i− 1
2n
)2
,
where pi = Φ(x(i)−xs ), where Φ is the standard normal cumulative distribution, and x(i) is the ordered
i:th observation. Under normality this distance should be small, and therefore one rejects H0 if W ≥ cfor some appropriate constant c.
The Anderson-Darling test is based on this distance with w(x) = (F (x)(1 − F (x))−1, thus puttingmore weight on the tails of the distribution compared to the Cramer-von Mises test. It uses the followingtest statistic
A2 = −n−∑ni=1(2i− 1)(log(pi) + log(pn−i+1))
n,
and one rejects H0 if A ≥ c for some appropriate constant c.
3.5 Bayesian Inference for Missing Data
Missing data can pose a challenge when performing statistical analyses and there are several ways tohandle the problem. This section is based on material from Rubin (2004) unless otherwise stated, and Ihave tried to fill in the gaps where Rubin chooses to be sketchy. The purpose of this section is describethe concepts involved when modelling missing data. This means that one in some clever way creates”reasonable guesses”, or imputations, for all the missing values in the data set using an appropriatemethod. The process is not deterministic given the data since each imputed value will be a draw fromits posterior distribution. The term multiple imputation means that this procedure is repeated multipletimes with different seeds sent to the random number generator, very much like how bootstrap replicationsare generated. This idea will be implemented using the MICE algorithm, and the following section willserve as a theoretical motivation for this.
First we begin by stating Bayes theorem, as in Robert (2007), which is central in Bayesian statistics.
22
Theorem 5 (Bayes Theorem). Let A,B be events such that Pr(B) > 0, then
Pr(A|B) = Pr(B|A)Pr(A)
Pr(B).
Furthermore, if X,Y are two random variables with conditional density P (X|Y ) and marginal P (Y ),then the conditional density of Y |X is
P (Y |X) =P (X|Y )P (Y )∫P (X|Y )P (Y )dY
.
Remark 9. When Bayes theorem is applied one might sometimes use the proportional sign ∝ to relatethe posterior distribution P (Y |X) with the prior P (Y ) and conditional distribution P (X|Y ) through
P (Y |X) ∝ P (X|Y )P (Y ).
This notation suggests that the right hand side is, up to a normalising constant, the marginal P (X) =∫P (X|Y )P (Y )dY , equal to the left hand side. The proportional sign will be used in more general settings
where densities or probability mass functions are, up to a normalising constant, equal, except for the caseof improper prior distributions, which do not integrate.
3.5.1 Notation and Definitions
Definition 12. Let Y be a n × p matrix containing data with missing values on p variables and nobservations and let X be a n × q matrix of fully observed variables. Furthermore, let the responseindicator R = {ri,j} be an n× p 0-1 matrix indicating the elements of Y that are missing
ri,j =
{1 if yi,j is observed
0 if yi,j is missing
This means that X and R are completely observed. Denote the observed data by Yobs, with obs ={(i, j) : ri,j = 1}, and the missing data by Ymis, with mis = {(i, j) : ri,j = 0}, respectively, and we writeY = (Yobs, Ymis) for simplicity, being aware of the abuse of notation.
Definition 13. Note that R is random and its distribution may depend on Y = (Yobs, Ymis). Thisdependency is described by the missing data model
P (R|X,Yobs, Ymis, φ)
where φ contains the parameters of the missing data model. The parameters φ are just used for modelling,have no intrinsic scientific value and are generally unknown.
Remark 10. In Rubin (2004) there is a sample process I, often involved in surveys, which indicates theobservations from the whole population that are included in the study. In this thesis the sample processwill not be considered and the indicator matrix I will be constant and constitute of only 1’s as everyoneis considered to be included.
Let us discuss some important concepts related to the missing data model.
Definition 14. The probability of an observation being missing is considered to be dictated by a processcalled the missing data mechanism or response mechanism P (R|X,Y ), and it can be described using amissing data model.
There are three different categories of missing data mechanisms, data being missing completely atrandom (MCAR), missing at random (MAR) and missing not at random (MNAR). When modellingmissing data, a certain missing data mechanism is a assumed and affects the choice of method to handlemissing data. If all observations are considered to have the same probability of missing data, implying
23
that the cause of missing data is unrelated to the data, then data is said to be MCAR. In this case, lotsof complexities occurring with missing data can be ignored, apart from the loss of information. Thisis in general an unrealistic assumption but may apply in some special situations. If the probability ofmissing data depends on the observed data then the data are said to be MAR and if the probability ofmissing data depend on observed and unobserved data, i.e on unknown reasons, such that neither MCARor MAR hold, then the missing data is said to be MNAR. In this latter case, special adjustments andadditional assumptions are needed in order to impute the missing data.
Example 4. Buuren (2012) gives a good example of these concepts. Consider a weighing scale whichhas run out of batteries during a weighting procedure. This will result in missing data, and would beMCAR if the order of items to be weighted is random. Placing the weighing scale on a soft surface forsome observations and on a hard surface for some, assuming it generates missing values differently on thetwo surfaces, and noting this, then we have an example of MAR. If, on the other hand, the weighing scalewears out and produces more missing data over time, we fail to note this and measure heavier objects atlater times, then the distribution of measurements will be distorted. This case is and example of MNAR.
We can restate the missing data patterns mentioned above as a definition using the missing datamodel.
Definition 15. Let
- MCAR P (R|X,Yobs, Ymis, φ) = P (R|φ)
- MAR P (R|X,Yobs, Ymis, φ) = P (R|X,Yobs, φ)
- MNAR P (R|X,Yobs, Ymis, φ) (no general simplification possible)
Example 5. A simple example from Buuren (2012) illustrates the difference between MCAR, MAR andMNAR and gives a concrete example of what the parameter φ may describe. Let Z = (X,Y ) be drawnfrom N2(µ,Σ) with the correlation coefficient ρ = 1
2 , and then generate missing data in Y according to
Pr(R = 0|Z, φ) = φ0 +eX
1 + eXφ1 +
eY
1 + eYφ2
with φ = (φ0, φ1, φ2). Then, in order, the different missing data mechanisms correspond to
φMCAR = (1/2, 0, 0), φMAR = (0, 1, 0), φMNAR = (0, 0, 1),
and gives different distributions of the response
Pr(R = 0|Z, φMCAR) =1
2, logit(Pr(R = 0|Z, φMAR)) = X and logit(Pr(R = 0|Z, φMNAR)) = Y.
These important concepts are closely related to the concept of an ignorable and non-ignorable missingdata mechanism.
Definition 16. The missing data mechanism P (R|X,Y ) is ignorable if P (R|X,Y ) = P (R|X,Yobs).
Lemma 2. An ignorable missing data mechanism is equivalent with
P (Ymis|X,Yobs, R) = P (Ymis|X,Yobs).
Proof: the first statement follows by a trivial application of theorem 5. �
24
Example 6. A simple example of a nonignorable missing data mechanism is given by Rubin (2004). LetYi > 0 be some i.i.d. measured quantity, for i = 1, . . . , n, R = (R1, . . . , Rn), and let
P (R|X,Y ) =
N∏i=1
(Yi
1 + Yi
)Ri(
1
1 + Yi
)1−Ri
be a Bernoulli distributed response mechanism that increases the probability of response with increasingYi, meaning that the equality in lemma 2 does not hold.
Lemma 3. Let f(X,Y |θ) be a model for the data given a parameter θ, for example a normal distributionwith mean vector µ and covariance matrix Σ, such that θ = (µ,Σ). Furthermore, let g(R|X,Y, φ) bea missing data model, with φ being a parameter for the model, compare with example 5. Then themissing data mechanism is ignorable if the MAR assumption holds and the parameters θ and φ are apriori independent.
Proof: start by rewriting
P (Ymis|X,Yobs) =P (X,Yobs, Ymis)∫
P (X,Yobs, Ymis)dYmis=
∫P (X,Yobs, Ymis|θ)P (θ)dθ∫ ∫
P (X,Yobs, Ymis|θ)P (θ)dθdYmis(1)
and similarly
P (Ymis|X,Yobs, R) =
∫ ∫P (X,Yobs, Ymis, R, θ, φ)dθdφ∫ ∫ ∫
P (X,Yobs, Ymis, R, θ, φ)dθdφdYmis. (2)
Observe that the integrand in eq. (2) can be written as
P (R|X,Yobs, Ymis, θ, φ)P (X,Yobs, Ymis, θ, φ) = g(R|X,Y, φ)P (X,Yobs, Ymis|θ, φ)P (θ, φ)
= g(R|X,Y, φ)f(X,Y |θ)P (θ, φ)
= g(R|X,Y, φ)f(X,Y |θ)P (θ)P (φ),
(3)
using Y = (Yobs, Ymis) and where the last equality follows by independence of θ and φ. Now, combiningeq. (2) and eq. (3) and using the MAR assumption, which implies that g(R|X,Y, φ) = g(R|X,Yobs, φ),then
P (Ymis|X,Yobs, R) =
∫ ∫g(R|X,Y, φ)f(X,Y |θ)P (θ)P (φ)dθdφ∫ ∫ ∫
g(R|X,Y, φ)f(X,Y |θ)P (θ)P (φ)dθdφdYmis
=
∫g(R|X,Yobs, φ)P (φ)dφ
∫f(X,Y |θ)P (θ)dθ∫
g(R|X,Yobs, φ)P (φ)dφ∫ ∫
f(X,Y |θ)P (θ)dθdYmis
=
∫f(X,Y |θ)P (θ)dθ∫ ∫
f(X,Y |θ)P (θ)dθdYmis
= P (Ymis|X,Yobs)
(4)
by eq. (1), which means that the response mechanism is ignorable, by lemma 2. �
Remark 11. Given a joint distribution P (X,Y ) then Rubin (2012) argues that we may choose a pa-rameter such that
P (X,Y ) =
∫P (X,Y, θ)dθ =
∫f(X,Y |θ)P (θ)dθ
and then we may choose a distribution for the response g(R|X,Y, φ), where φ is independent of θ andsuch that the MAR assumption holds. He stresses that the MAR requirement for an ignorable missingdata mechanism is generally the more important condition and that the condition of the parameters isintuitive, meaning that in practice, the MAR assumption and the ignorable assumption are effectivelythe same.
In the light of this, the two concepts will be used as if they are the same, and similarly for theircounterparts NMAR and nonignorable, as previously done in other studies, for example Siddique et al(2014).
25
3.5.2 Multiple Imputations from the Posterior of the Missing Values
Let Q = Q(X,Yobs, Ymis) be a quantity of interest, for example a survival function at a specific time or atrue positive fraction. We want to estimate Q but the problem is that Ymis is not observed. Bayesian in-ference for Q is based on the observed data through its posterior distribution, which density or probabilityfunction we denote by P (Q|X,Yobs, R) as before.
Definition 17. Let P (Q|X,Y,R) be the completed-data posterior distribution ofQ, where Y = (Yobs, Ymis).It need the unobserved Ymis to be known and therefore it is of only a theoretical concept.
Lemma 4. The posterior distribution of Q can be obtained by integrating the completed-data posteriorwith respect to Ymis. More specifically
P (Q|X,Yobs, R) =
∫P (Q|X,Y,R)P (Ymis|X,Yobs, R)dYmis.
Proof: follows directly since Y = (Yobs, Ymis). �
Remark 12. We may use lemma 4 to simulate the posterior distribution of Q using repeated draws Y ∗i ,i = 1, . . . ,m, from P (Ymis|X,Yobs, R). Given an interval or region C, we have by theorem 1 that
m∑i=1
Pr(Q ∈ C|X,Y ∗i , R)/mP−→ Pr(Q ∈ C|X,Yobs, R), as m −→∞.
Later we will discuss how to generate draws Y ∗i from P (Ymis|X,Yobs, R).Not only can we simulate the posterior distribution of Q but also its mean and variance, using the
completed-data posterior mean and variance.
Lemma 5. Let Q := E(Q|X,Y,R) and U(X,Y,R) := Var(Q|X,Y,R) denote the completed-data poste-rior mean and variance of Q, assuming that these exist. Then, the posterior distribution of Q has meanand variance
E(Q|X,Yobs, R) = E(Q|X,Yobs, R)
Var(Q|X,Yobs, R) = E(U |X,Yobs, R) + Var(Q|X,Yobs, R)
Proof: by the tower property of conditional expectation
E(Q|X,Yobs, R) = E(E(Q|X,Y,R)||X,Yobs, R) = E(Q|X,Yobs, R)
Claim: Given two random variables A,B we have that
Var(A) = Var(E(A|B)) + E(Var(A|B)).
Proof of claim:
Var(E(A|B)) = E[(E(A|B)− E(E(A|B)))2] = E((E(A|B))2)− E(A)2 (5)
Var(A|B) = E[(A|B − E(A|B))2] = E(A2|B)− (E(A|B))2
=⇒ E(Var(A|B)) = E(E(A2|B))− E((E(A|B))2) = E(A2)− E((E(A|B))2), (6)
and the claim follows. Now, apply the claim on Var(Q|X,Yobs, R), with A = (Q|X,Yobs, R) and B =(X,Y,R), to get
Var(Q|X,Yobs, R) = E(Var(Q|X,Y,R)|X,Yobs, R) + Var(E(Q|X,Y,R)|X,Yobs, R)
which gives the result by the tower property of conditional expectation. �
26
Remark 13. Using repeated draws Y ∗i , i = 1, . . . ,m, from P (Ymis|X,Yobs, R) we get m repeated values
of Q and U such that we may estimate the posterior mean and variance
Q∞ := limm−→∞
m∑i=1
Q∗i /mP= E(Q|X,Yobs, R) = E(Q|X,Yobs, R),
U∞ := limm−→∞
m∑i=1
U∗i /mP= E(U |X,Yobs, R),
B∞ := limm−→∞
m∑i=1
(Q∗i − Q∞)2/mP= Var(Q|X,Yobs, R),
by the Law of Large Numbers, theorem 1, with
Var(Q|X,Yobs, R) = U∞ +B∞ =: T∞,
by lemma 5, and where ”P=” means that the left hand side converges in probability to the right hand
side. The last display indicates that the total posterior variance T∞ consists of two components, thevariance U∞ within each repeated analysis, and the variance B∞ between the repeated draws. The latteris introduced by simulation and optimally there would be no between variance and no need for repeateddraws. Thus the between variance captures the uncertainty introduced by missing data.
On a side note, in the Bayesian framework, under quadratic loss, the posterior mean is a Bayesestimator, compare with Robert (2007).
Remark 14. If we assume that, for large n, the posterior distribution of Q is approximately normal,then
(Q− Q∞|X,Yobs, R) ≈ N(0, T∞).
See Rubin (2004), and its references for a deeper motivation of why it is generally reasonable to assumeapproximate normality. The problem with this result is that in practice, it is only possible to perform afinite number of repeated draws Y ∗i , i = 1, . . . ,m, from P (Ymis|X,Yobs, R). The estimates of the posteriormean and variance need to take this into account, which is the topic of the next section.
3.5.3 Multiple Imputations Drawn from a Finite Number of Completed Datasets
For scalar Q we now derive what is commonly referred to as Rubin’s rules which are summarized inthe next section. These describe how to derive a parameter estimate, its approximate distribution andconfidence interval using a finite number of repeated draws from the posterior distribution of the missingdata. The idea is to perform a series of approximations under the assumption that Q given X,Y andR is distributed as N(Q, U). In practice, this assumption is only approximate and commonly motivatedby an argument using the Central Limit Theorem. Let Y ∗mis,i, i = 1, . . . ,m be repeated draws from
P (Ymis|X,Yobs, R), which gives us m repeated values of (Q, U) and let Sm = ((Q∗i , U∗i ), i = 1, . . . ,m) de-
note the completed-data statistics. The goal is to derive an approximation of the conditional distributionP (Q|Sm).
Remark 15. Note that Sm contains the completed-data statistics and are derived conditioned on R.The complete-data statistics are the corresponding statistics where R is ignored in the calculation, thustreating Yobs and Y ∗mis,i the same in the estimation process, just as if all data would be observed. It iseasy to see that under the ignorable assumption then these two concepts are the same, and Rubin (2004)argues that in most cases and for large samples it is appropriate to use the complete-data statistics.
To begin with, we will need two lemmas concerning samples from a normal distribution.
27
Lemma 6. Let (Zi|µ, σ2), i = 1, . . . ,m, be an i.i.d. N(µ, σ2) and assume that the prior P (µ, σ2) ∝ 1σ2 ,
then the posterior distribution of σ2 is given by
(σ2|Z1, . . . , Zm) ∼(
(m− 1)s2
χ2m−1
|Z1, . . . , Zm
)where s2 is the usual sample variance and 1
χ2m−1
is the inverted χ2 distribution.
Proof: first note that the density of(
(m−1)s2
χ2m−1
|Z1, . . . , Zm
)is proportional to 1
(σ2)(m+1)/2 exp(−(m−1)s2
2σ2 ).
By Bayes theorem, theorem 5
P (µ, σ2|Z1, . . . , Zm) ∝ P (Z1, . . . , Zm|µ, σ2)P (µ, σ2)
∝ P (Z1, . . . , Zm|µ, σ2)1
σ2
∝ (1
σ2)1+m/2 exp
(−∑mi=1(Zi − µ)2
2σ2
).
(7)
Now, add and subtract Z, the sample mean, in the kernel of the normal density, observing that∑mi=1(Zi−
Z)(Zµ) = 0, then
P (µ, σ2|Z1, . . . , Zm) ∝ (1
σ2)1+m/2 exp
(−∑mi=1(Zi − Z)2
2σ2− m(Z − µ)2
2σ2
)∝ (
1
σ2)(m+1)/2 exp
(−(m− 1)s2
2σ2
)(
1
σ2/m)1/2 exp
(−(Z − µ)2
2σ2/m
).
(8)
The last factor is a N(Z, σ2/m) density of µ, so if we integrate µ we get
P (σ2|Z1, . . . , Zm) =
∫P (µ, σ2|Z1, . . . , Zm)dµ
∝ (1
σ2)(m+1)/2 exp
(−(m− 1)s2
2σ2
),
(9)
which gives the result by our first observation. �
Lemma 7. Let (Zi|µ, σ2), i = 1, . . . ,m, be an i.i.d. N(µ, σ2) and assume that the prior P (µ|σ2) ∝ Cfor a constant C, then
P (µ|Z1, . . . , Zm, σ2) = N(Z,
σ2
m), with Z =
m∑i=1
Zi/m.
Proof: by Bayes theorem, theorem 5, we have
P (µ|Z1, . . . , Zm, σ2) ∝ P (Z1, . . . , Zn|µ, σ2)P (µ|σ2)
∝ Cm∏i=1
P (Zi|µ, σ2)
∝ exp
(− 1
2σ2
m∑i=1
(Zi − µ)2
)
= exp
(− 1
2σ2
[m∑i=1
(Z − µ)2 +
m∑i=1
(Zi − Z)2 − 2
m∑i=1
(Zi − Z)(Z − µ)
])
∝ exp
(− 1
2σ2
[m(Z − µ)2 − 0
])= exp
(− 1
2σ2/m(Z − µ)2
).
(10)
This is the kernel of a N(Z, σ2
m ) density, which concludes the proof. �
28
Remark 16. Let X be a positive random variable, Y = log(X) and assume that PY (y) ∝ C for aconstant C, then by the Jacobian rule
PY (y) = PX(x)ey = PX(x)x =⇒ PX(x) ∝ 1
x,
where PY (y) denotes the density or probability function of Y with argument y.
Now, note that if we observe S∞, meaning that we have access to an infinite number of simulations,then by the normality assumption and remark 13 we have that
P (Q|X,Yobs, R) = φQ∞,U∞+B∞(Q),
where φµ,σ2(X) denotes the normal density of a random variable X with mean µ and variance σ2. Thismeans that we may condition Q on its posterior mean and variance instead of X,Yobs and R. Sincenothing changes by conditioning on Sm, then
P (Q|Q∞, U∞, B∞) = P (Q|Q∞, U∞, B∞, Sm) = φQ∞,U∞+B∞(Q). (11)
To approximate P (Q|Sm) we will integrate (11) over the conditional distribution of Q∞, U∞, B∞ givenSm in several steps using approximations, under the following assumptions.
Assumption 1.
(Q∗i |X,Yobs, R) ∼ N(Q∞, B∞), or equivalently (Q∗i |Q∞, U∞, B∞) ∼ N(Q∞, B∞) (12)
Assumption 2.
(U∗i |X,Yobs, R)→ (U∞,� B∞), or equivalently (U∗i |Q∞, U∞, B∞)→ (U∞,� B∞), (13)
as n −→ ∞, where A → (B,� C) means that the distribution of A tends to a distribution centred atB with variance substantially smaller than C. See Rubin (2004), chapters 2 and 3, for more details andfurther motivation of these assumptions.
Assumption 3.P (Q∞|B∞) ∝ C, for a constant C (14)
Assumption 4.P (log(B∞)) ∝ D, for a constant D. (15)
Remark 17. Note that the first two assumptions indicate that Q∗i is conditionally independent (given
B∞) of U∞ and that, for large n, U∗i is conditionally independent of Q∞. This will then also hold inthe limit, such that Q∞ and U∞ are conditionally independent. Furthermore, assumption 3 is reasonablesince it corresponds to using Laplace’s noninformative uniform prior on R for the mean, and assumption 4is also reasonable since it corresponds to the scale-invariant noninformative prior P (σ) ∝ 1
σ of thevariance of a normal distribution, compare with Robert (2007). This follows since P (log(B∞)) ∝ C andP (Q∞|B∞) ∝ D implies that P (Q∞, B∞) ∝ 1
B∞by remark 16.
Now, we approximate the conditional distribution P (Q|Sm) in four steps. We start by deriving anapproximation for the distribution of (Q∞, U∞|Sm, B∞) to be used in steps 2 and 3.
Step 1. By the above remark it is enough to separately derive the distributions of Q∞ and U∞ given(Sm, B∞). Firstly, one may show that for any ”relatively diffuse” prior on U∞, compare with Rubin(2014), then
(U∞|Sm, B∞)→ (Um,� B∞/m), as n −→∞ (16)
where Um :=∑mi=1 U
∗i /m.
Secondly, by the normal sampling distribution in assumption 1, under assumption 3, we get that thedistribution of (Q∞|Sm, B∞) is normal by lemma 7, i.e.
(Q∞|Sm, B∞) ∼ N(Qm, B∞/m), (17)
29
where Qm :=∑mi=1 Q
∗i /m.
Step 2. Now we approximate the distribution of (Q|Sm, Q∞, B∞). By applying (16) to (11) we seethat we can simply replace U∞ with Um and approximately get
P (Q|Sm, Q∞, B∞) = φQ∞,Um+B∞(Q). (18)
This can be seen by writing
P (Q|Sm, Q∞, B∞) =
∫P (Q|Sm, Q∞, U∞, B∞)P (U∞|Sm, Q∞, B∞)dU∞
≈∫N(Q∞, U∞ +B∞)P (U∞|Sm, B∞)dU∞
≈ N(Q∞, Um +B∞)
by (11) and (16) since the approximate distribution of (U∞|Sm, B∞) puts a relatively large amount ofprobability mass at Um.
Step 3. Here we approximate the distribution of (Q|Sm, B∞). By combining (17) and (18) we get
P (Q|Sm, B∞) = N(Qm, Um + (1 +1
m)B∞). (19)
To see this, note that display (19) follows by the fact that the normal density is symmetric in its argumentand mean, φµ,σ2(x) = φx,σ2(µ), and so we write
P (Q|Sm, B∞) =
∫P (Q|Sm, Q∞, B∞)P (Q∞Sm, B∞)dQ∞
=
∫φQ∞,Um+B∞(Q)φQm,B∞/m(Q∞)dQ∞
=
∫φQ,Um+B∞(Q∞)φQm,B∞/m(Q∞)dQ∞
= φQm,Um+(1+ 1m )B∞(Q).
(20)
The last equality follows by simple but tedious algebra, see for example Bromiley (2014). The idea isthat we are integrating two normal densities over the same argument and the conclusion is that we addtheir corresponding variances and the form of the kernel consist of the difference of the correspondingmeans.
Step 4. Finally, the resulting distribution P (Q|Sm) will approximatively be a t-distribution
Q|Sm ≈ tv(Qm, Um + (1 +1
m)Bm) (21)
where Bm :=∑mi=1(Q∗i − Qm)/(m− 1).
To see this, one approximates the distribution of B∞|Sm and the distribution of Um+(1+ 1m )B∞|Sm.
Under assumptions 1, 2, 3 and 4 we may use lemma 6 with Zi = Q∗i , µ = Q∞, σ2 = B∞ and s2 = Bm.This gives the approximation
(m− 1)BmB∞
|Sm ∼ χ2m−1.
Now, referring to an argument including the Behrens-Fisher distribution, matching moments to obtainan approximation, see Rubin (2004), we may approximate
vUm + (1 + 1
m )Bm
Um + (1 + 1m )B∞
|Sm ∼ χ2v
where v = (m − 1)(1 + 1rm
)2 is the degrees of freedom and rm = (1 + 1m )Bm
Umis the relative increase in
conditional variance due to nonresponse, given B∞ = Bm. Since a normal random variable divided by
30
the square root of an independent χ2 is proportional to a t distributed random variable, Heumann et al(2016), we obtain the final result
Q− Qm√Um + (1 + 1
m )B∞
√Um+(1+ 1
m )Bm
Um+(1+ 1m )B∞
∣∣∣∣Sm ≈ tv.3.5.4 Rubin’s Rules
The results discussed in the previous section can be summarized as what is commonly referred to asRubin’s Rules. These describe how one can analyse parameters using multiple imputed data sets. Theaverage of the m complete-data estimates and the average of their variances
Qm =
m∑i=1
Q∗i /m , Um =
m∑i=1
U∗i /m
are estimators of Q and U , and the variance between the complete-data estimates is
Bm =
∑mi=1(Q∗i −Qm)2
m− 1
and the total variance of Q−Qm is
Tm = Um + (1 +1
m)Bm
An interval estimate using a t-distribution with v = (m− 1)(1 + 1rm
)2 degrees of freedom is given by
Qm ± tv(α
2)√Tm,
where rm = (1 + 1m )Bm
Umis the conditional variance given B∞ = Bm. These rules will be applied to
all approximately normal quantities used in this thesis, including positive fractions and Kaplan-Meiersurvival estimates, after appropriate transformations.
3.6 Multiple Imputation using Chained Equations
Multiple Imputation using Chained Equations (MICE) is an example of an implementation of the ideaspresented in the previous section. It is an algorithm based on a chain of equations, or marginal mod-els, that generate imputations in a variable-by-variable fashion, and needs a careful specification of itsparameters and assessment of convergence. We will follow Buuren (2012) throughout this section unlessotherwise stated.
3.6.1 Fully Conditional Specification and the MICE Algorithm
Fully conditional specification (FCS) means that the algorithm imputes missing data on a variable-by-variable basis using specified marginal imputation models for each incomplete variable. It is a Markovchain Monte Carlo (MCMC) method, where the state space of the Markov chain is the collection of allimputed values and its power lies in the fact that it does not require the user to specify a multivariatemodel. In other words, it specifies the multivariate distribution P (Y,X,R|θ) through a set of univariateconditional densities P (Yj , |X,Y−j , R, φj), where Y−j is the vector with all components of Y except forcomponent j. These are then used in an iterative way to impute Yj given X,Y−j and R.
Example 7. Assume Yj is a binary variable and the conditional or marginal density is specified through
a logistic regression model then φj = (β, V ) where β is the vector of estimated regression coefficients andV is the estimated covariance matrix.
31
If the conditional distributions are compatible, meaning that the joint distribution exists and fromwhich the conditional distributions can be derived, then it is easy to see that the MICE algorithm isa Gibbs sampler. In either case, MICE will still produce imputations, but one can minimize the in-compatibility issue by evaluating convergence and eliminating feedback loops introduced by for exampledeterministic functions of the data. For more information about the Gibbs sampler, see Robert (2007) forexample. Buuren et al (2006) explored practical consequences of incompatible conditional distributions.Since the parameters θ1, . . . , θp are specific to their respective marginal distribution they are not neces-sarily the product of some factorisation of the joint distribution. For example, if the space spanned bythese parameters have more dimensions than needed, then the implicit joint distribution does not exist.In any case, simulations showed that the performance of FSC can be quite good despite this issue. Inaddition, Buuren (2007) uses simulations to show that FCS behaves well, and concludes that FCS is auseful and flexible alternative to joint modelling, especially in the case where a joint model for the datais difficult to specify. For a more informative discussion about incompatible conditional distributions, seefor example Arnold and Press (1989).
Intuitively, the algorithm starts by noting all the positions which have missing data and proceeds byfilling all missing data points with random draws. Then, for each variable with missing data, it estimatesthe specified marginal model by drawing parameters φj using Y obs
j and a subset of the other variables,and then generates imputations which replace the initially missing data. One iteration is complete whenthe algorithm has done this for all variables with missing data, and then it iterates this procedure for afixed number of iterations. The whole procedure is repeated using different initial random seeds in orderto generate multiple imputations. It is important to note that in its standard form, it operates under theassumption of an ignorable missing data mechanism since, for each variable, the marginal model dependson other variables and model parameters will be estimated using the observed part of Yj .
The algorithm can be split into three stages, the first being the modelling stage where a specific modelis chosen for the data. Then, at the estimation stage, the posterior parameters distribution given themodel is formulated, and lastly, the imputation stage, where the missing data is completed by drawingsuccessively from parameter and data distributions.
The MICE algorithm is specified below, compare with Buuren (2012) where more details can befound. Let m denote the number of multiple imputations, T the number of iterations, p be the number ofvariables in the data with missing data and Y−j the vector with all components of Y except for componentj.
Algorithm 1 Multiple Imputation using Chained Equations (MICE)
1. Specify an imputation model P (Y misj |X,Y obs
j , Y−j , R, φj) for variable Yj , with j = 1, . . . , p
2. Repeat for i = 1, . . . ,m (generate imputations)
3. For each j, fill in starting imputations Y 0j by random draws from Y obs
j
4. Repeat for t = 1, . . . , T (begin iterations)
5. Repeat for j = 1, . . . , p (begin marginal imputation)
6. Define Y t−j = (Y t1 , . . . , Ytj−1, Y
t−1j+1 , . . . , Y
t−1p ) as the currently complete data except Yj
7. Draw φtj ∼ P (φtj |X,Y obsj , Y t−j , R)
8. Draw imputations Y tj ∼ P (Y misj |X,Y obs
j , Y t−j , R, φtj)
9. End repeat j (end marginal imputation)
10. End repeat t (end iterations)
11. End repeat i
32
3.6.2 The Choice of Predictors and the Visit Sequence
The predicator matrix is used by MICE to specify the subset of other the variables to be used as predictorswhen imputing each incomplete variable. It is a square matrix with size equal to the number of variablesin the data, filled with indicators 0 or 1. Each row j of the matrix constitutes a part of the correspondingimputation model P (Y mis
j |X,Y obsj , Y−j , R, φj) and identifies which predictors that will be used in the
imputation of Y misj by setting the corresponding column entries of the row to 1. If a variable has no
missing data then its row will only contain zeros. Van Buuren recommends that one should use as muchinformation as possible, i.e. include as many variables as possible, since it yields multiple imputationsthat have minimal bias and maximal efficiency. A large set of predictors is more likely to make theMAR assumption more plausible, which in turn will be reducing the need to make special adjustmentsfor MNAR mechanisms.
For each iteration of the MICE algorithm, the visit sequence specifies in what order the variablesshould be imputed. In theory it is irrelevant as long as each column is visited often enough, but inpractice some sequences can be more efficient than others.
3.6.3 Univariate Imputation Methods
There are several choices of methods to impute each variable with missing data. The choice of a specificmethod depends of the type of variable (continuous, categorical, etc.) and on the preference of the user.The choice of methods and the specification of the predicator matrix constitutes step 1 in the MICEalgorithm. Each method is univariate in the sense that Y = (Yobs, Ymis) is considered to be a univariatevariable with nonresponse and X a fully observed. A brief discussion on the theory behind univariateimputation methods will now be given, based on material from Rubin (2004).
We begin by assuming an ignorable response mechanism and that P (X,Y ) is modelled in i.i.d. form,i.e.
P (X,Y ) =
∫P (X,Y |θ)P (θ)dθ =
∫ n∏i=1
(fX,Y (Xi, Yi|θ))P (θ)dθ, (22)
where i = 1, . . . , n with n being the number of observations, and θ is a parameter for the model. If wewrite
fX,Y (Xi, Yi|θ) = fY |X(Yi|Xi, θY |X)fX(Xi|θX), (23)
where θY |X and θX are functions of θ, then fY |X(Yi|Xi, θY |X) may for example specify the density undera normal linear model for continuous Y or the density under a logistic regression model for binary Y ,with X being the explanatory variables and θY |X the model parameters.
Next, under the model (22), we want to draw Ymis from its posterior distribution, which can be writtenas
P (Ymis|X,Yobs) =
∫P (Ymis|X,Yobs, θ)P (θ|X,Yobs)dθ.
The idea is to first draw θ from its posterior P (θ|X,Yobs), which we denote θ∗, and then draw Ymis fromP (Ymis|X,Yobs, θ = θ∗). Then, repeating this procedure m times will generate m multiple imputations ordraws from the posterior distribution of (Ymis, θ). Under the assumption that the nonresponse is ignorableRubin (2004) shows that the components of Ymis|θ are a posteriori independent with a distributiondepending only on θ through θY |X . Therefore θY |X is the only function of θ needed for the imputationtask, and the next objective will therefore be to draw θY |X from its posterior distribution.
If, in addition to an ignorable missing data mechanism, we assume that θY |X and θX are a prioriindependent, then one can show that they are a posteriori independent, P (θY |X) does not involve fX andthat we do not need to specify fX or P (θX). Moreover, it also follows that the posterior distribution ofθY |X only involves fY |X , P (θY |X) and the respondents Yobs, and that the posterior distribution of Ymis|θcan simply be written as
P (Ymis|X,Yobs, θ) =∏mis
fY |X(Yi|Xi, θY |X).
This suggests a strategy for constructing univariate imputations. The most straight forward example isthe normal linear model, which also provides some intuition.
33
Example 8. Let Y be continuous and assume that it is reasonable to use a normal linear model
Yi ∼ N(Xiβ, σ2),
with θY |X = (β, log(σ)), such that fY |X(Yi|Xi, θY |X) = φβXi,σ2(Yi) is the corresponding normal density.If we assume the noninformative prior P (θY |X) ∝ C for a constant C and that the number of respondentsn1 is more than the number of regression parameters q, then the posterior distribution of σ2 is distributedas a σ2(n1 − q)/χ2
n1−q random variable, where β and σ2 are the usual least squares estimates
β =
(∑obs
XtiXi
)−1∑obs
XtiYi, σ2 =
∑obs
(Yi −Xiβ)2/(n1 − q),
and β|σ2, X, Y ∼ N(β, σ2 (
∑obsX
tiXi)
−1)
. This means that, to generate an imputation, one may
first draw a χ2n1−q random variable g and define σ2
∗ = σ2(n1 − q)/g, then draw q independent N(0, 1)
variables in a vector Z and set β∗ = β+σ∗ (∑
obsXtiXi)
−1/2Z, where the square root is computed using
Cholesky decomposition, see remark 18. Lastly, after having drawn the parameters of the model fromtheir posterior distribution, we draw the n0 = n − n1 values of Ymis using fY |X(Yi|Xi, θY |X = θ∗Y |X),
with θ∗Y |X = (β∗, σ2∗), by generating n0 draws zi from N(0, 1) and setting Yi,∗ = Xiβ∗ + ziσ∗, for i in the
index set that indexes the missing values of Y .
Remark 18. The Cholesky decomposition of a positive definite symmetric matrix A can be comparedwith LU factorization of A, but, instead of decomposing A into a product of a lower and an uppertriangular matrix, it can be written as A = LLt for a lower triangular matrix. Not surprisingly, it issometimes referred to as the square root of a matrix, compare with Press et al (1992).
Another example that is very relevant to this thesis, is under the logistic regression model.
Example 9. If Y is binary we may use the logistic regression model
f(Yi|Xi, θY |X) =
(exp(XiθY |X)
1 + exp(XiθY |X)
)Yi(
1−exp(XiθY |X)
1 + exp(XiθY |X)
)1−Yi
.
In this case, the posterior distribution of θY |X does not have a nice form for any reasonable priors, andthe common practice is to assume P (θY |X) ∝ C and use an asymptotic normal approximation. In thisway, the posterior mean and variance of θY |X are approximated with the maximum likelihood estimate
θY |X and
V (θY |X) = −
(d2
dθ2Y |X
log∏obs
f(Yi|Xi, θY |X)|θY |X=θY |X
).
To generate an imputation, one may first draw θY |X from N(θY |X , V (θY |X)), for each i, and then drawn0 independent U(0, 1) numbers ui and set Yi = 0 if 1/(1 + exp(Xiθ∗)) < ui, else set Yi = 1, for i in theindex set that indexes the missing values of Y .
These were two examples of univariate models but there many more, for example predictive meanmatching and multinomial logistic regression. The predictive mean matching algorithm is discussed below,and after that the algorithm for binary Y using a logistic regression model is summarized in algorithm 3.Note that all of these univariate imputation methods are specified via the method argument in MICE,but user specified and deterministic methods can also be used.
Intuitively, the predictive mean matching algorithm, algorithm 2, as defined in Buuren (2012), sub-samples from the observed data and then uses a specified imputation model to calculate the predictedvalue of target variable Y , which has missing data. For each missing entry, the algorithm generates a setof suggested donors (candidates) from all complete cases that have predicted values close to the predictedvalue for the missing data point. One of the candidates is randomly chosen to fill the spot of the missing
34
value. The underlying assumption made is that the distributions of the candidate’s and receivers’ dataare the same within each of the constructed sets.
The set of candidate donors is, in the MICE version of PPM, set to the size d = 3 for continuous data.The reason to why it is not set to d = 1 is that the algorithm may reselect the same donor repeatedly,and it has been shown that it performs badly when d is small and there is a high number of ties for thepredictors among the observations to be imputed. In the latter case, an unrealistic result can follow sinceseveral observations can end up with the same value. If d is set too high it might introduce bias since theprobability of bad matches increases, i.e. candidate donors with observed values far away from the mostrealistic or true value. The algorithm is specified below, similarly as by Buuren (2012) where a moredetailed discussion of the method can be found. Let n1 be the number of observations where the targetvariable Y is observed, n0 be the number of observations where Y is missing, and d be the number ofcandidate donors. Let Xobs be a n1 × q matrix of predictors of rows with observed data in Y and Xmis
be a n0 × q matrix of predictors of rows with missing data in Y , assuming that both matrices containno missing data. The dot over a parameter indicates it being drawn from the corresponding posteriordistribution. κ is a ridge parameter used to evade problems with singular matrices. It should be smalland positive.
Algorithm 2 Imputation by Predictive Mean Matching (PPM)
1. Calculate S = XtobsXobs
2. Calculate V = (S + diag(S)κ)−1 for some small κ
3. Calculate regression weights β = V XtobsYobs
4. Draw a random variable g ∼ χ2v with v = n1 − q
5. Calculate σ2 = (Yobs −Xobsβ)t(Yobs −Xobsβ)/g
6. Draw q independent N(0, 1) variables in a vector z1
7. Calculate V 1/2 by Cholesky decomposition
8. Calculate β = β + σz1V1/2
9. Calculate η(i, j) = |Xobs,[i]β −Xmis,[j]β| for i = 1, . . . , n1, j = 1, . . . , n0
10. Construct n0 sets Zj , each containing d candidate donors, from Yobs, such that∑di=1 η(i, j) is
minimum for all j = 1, . . . , n0. Break ties randomly.
11. Draw one donor ij from Zj randomly for j = 1, . . . , n0
12. Calculate imputations Yj = Yij for j = 1, . . . , n0
For incomplete binary variables, logistic regression can be used to impute missing data in Y . Theoutcome probability is modelled as
P (Yi = 1|Xi, β) =exp(Xiβ)
1 + exp(Xiβ)
In a similar way, a categorical variable with K unordered categories is imputed under the multinomiallogit model
P (Yi = k|Xi, β) =exp(Xiβk)∑Kk=1 exp(Xiβk)
for k = 1, . . . ,K. Here β1 = 0 to identify the model. A categorical variable with K ordered categories
35
can be imputed by the ordered logit model
P (Yi ≤ k|Xi, β, τk) =exp(τk −Xiβ)
1 +∑Kk=1 exp(τk −Xiβ)
Here the intercepts τk differ but β is constant across categories. More about generalized linear modelscan be found in for example McCullagh and Nelder (1989).
Below, algorithm 3 provides steps for an approximate Bayesian imputation method using logisticregression, analogously stated as in Buuren (2012). Intuitively, it estimates the regression coefficientsβ from the observed data and uses the covariance matrix V of the coefficients just as in example 9to generate a draw from the posterior distribution of the same parameter. These coefficients are thenplugged in to the logistic model to generate predicted probabilities of the missing variable being a 1 or a0. It is approximate Bayesian since it does not draw the covariance matrix V and since it assumes thatthe parameter vector β follows a multivariate normal distribution, which might not be the case in allsituations. Recall that n0 is the number of observations where Y is missing.
Algorithm 3 Imputation by Logistic Regression
1. Estimate regression weights β from (Yobs, Xobs) by iteratively reweighted least squares.
2. Obtain V , the unscaled estimated covariance matrix of β
3. Draw q independent N(0, 1) variates in vector z1
4. Calculate V 1/2 by Cholesky decomposition
5. Calculate β = β + z1V1/2
6. Calculate n0 predicted probabilities p = 1/(1 + exp(−Xmisβ))
7. Draw n0 random variates from the uniform distribution U(0, 1) in the vector u
8. Calculate imputations: set yj = 1 if uj ≤ pj and yj = 0 else, with j = 1, . . . , n0
In mice the algorithm is implemented as function mice.impute.logreg(). The algorithms for imputationof variables with more than two categories are analogously structured. In mice the multinomial logit modelis estimated by the multinom() function in the nnet package, and the ordered logit model is estimatedby the polr() function of the MASS package. In cases where polr() failes to converge, multinom() is usedinstead.
3.6.4 Nonignorable Missing Data
This section will be devoted to one way of modelling a nonignorable missing data mechanism for aunivariate outcome variable Y , and we will discuss in more detail the special case when Y is binary.First, similarly as in the previous section, we begin by assuming that P (X,Y,R) is modelled in i.i.d.form, i.e.
P (X,Y,R) =
∫P (X,Y,R|θ)P (θ)dθ =
∫ n∏i=1
(fX,Y,R(Xi, Yi, Ri|θ))P (θ)dθ, (24)
Since X and R are fully observed we may repeat the arguments in the previous section with Y,R replacingY and θY,R|X replacing θY |X , and write
fX,Y,R(Xi, Yi, Ri|θ) = fY,R|X(Yi, Ri|Xi, θY,R|X)fX(Xi|θX)
= fY |X,R(Yi|Xi, Ri, θY |X,R)fR|X(Ri|Xi, θR|X)fX(Xi|θX),(25)
36
where θY |X,R and θR|X are functions of θY,R|X , which is a function of θ, just as θX . Under a nonignor-able missing data mechanism the components of Ymis|θY |X,R are a posteriori independent and if θY |X,Rand (θR|X , θX) are a priori independent, then they are a posteriori independent and that the posteriordistribution of θY |X,R is defined only through fY |X,R, P (θY |X,R) and observed data. The implicationsare the same as in the ignorable case in terms of how to construct imputations by first drawing θY |X,Rfrom its posterior distribution and then drawing Ymis from is posterior distribution.
Note that under a nonignorable missing data mechanism we see we need two models fY |X,R(Yi|Xi, Ri =1, θY |X,R) and fY |X,R(Yi|Xi, Ri = 0, θY |X,R), meaning that we need to specify one model for Yobs andone for Ymis, and this is called a pattern-mixture model, compare with Buuren (2012).
Now, if Y is binary, what remains is to explicitly specify fY |X,R. Under an ignorable missing datamechanism we would use fY |X,R(Yi|Xi, Ri = 1, θY |X,R) to estimate the parameter and to generate impu-tations, since conditioning on Ri would have no effect, but now we will instead use fY |X,R(Yi|Xi, Ri =0, θY |X,R) to generate imputations. Following Siddique et al (2014), we define πingr,i := Pr(Yi =1|Xi, Ri = 1, θY |X,R) under a logistic regression model on Y . Then we define πnonignr,i := Pr(Yi =1|Xi, Ri = 0, θY |X,R) by the following equation
πnonignr,i/(1− πnonignr,i)
πignr,i/(1− πignr,i)= k, (26)
for some k > 0. This means that the parameter k governs the odds between the ignorable and nonignorablemodels and can be seen as an adjustment of the ignorable model, in terms of odds, by a constant, sincewe can rewrite the above expression using the logarithm
logit(πnonignr,i) = log(k) + logit(πignr,i).
The parameter k is not estimable from the data and has to be specified based on a priori knowledgeabout the missing data process. When k = 1 then πnonignr = πignr and we obtain the ignorable model,and for k < 1 the a priori knowledge indicates that the probability of the event is in reality lower thanunder the ignorable assumption.
The effect of k on πnonignr can be understood from the example below.
Example 10. Solving for πnonignr, we get πnonignr =kπignr
1−πignr+kπignrwhich can be used understand the
effect of k. When πnonignr is 0.1, 0.5 and 0.9 and k = 0.7 then πignr is about 0.07, 0.41 and 0.86 respectively,and when k = 0.1 then πignr is about 0.01, 0.09 and 0.47 respectively, meaning that k = 0.1 significantlychanges the probability compared to k = 0.7, especially for probabilities that are already small. Theeffect is also illustrated in fig. 1 where the pre-test probability is πignr, the post-test probability is πnonignr
and d = k.
The implementation of this method is straightforward. Given a k we modify step 6 in algorithm 3by calculating pnonignr using p and k just as indicated before. Then the algorithm proceeds as usual butwith pnonignr instead of p. This means that we draw the model parameters θY |X,R from their posteriordistribution just as if the missing data mechanism was ignorable, and then we define the model fromwhich we draw Ymis using eq. (26).
3.6.5 Derived Variables
Variables which are deterministically calculated based on other variables are called derived variables.These can be handled in two ways, either by omitting them from the imputation and deriving themafterwards, or, if they are needed in the imputation stage, they can be imputed using passive imputation.Passive imputation of a variable means that MICE ignores the corresponding row of the predicator andimputes the variable using a deterministic function of other variables specified by the method argument.One might need to alter the predicator matrix to break feedback loops between derived variables andoriginal variables, which can occur when derived variables are used to impute original variables forexample. If the predicator matrix is kept unchanged then one may get nonsensical imputations andissues with convergence.
37
3.6.6 Parameter Settings and Convergence Diagnostics
Choosing the number of imputations m is important and higher numbers decrease the effect of thesimulation error but increase the computation time. As discussed in section 3.5.3 the total estimatedvariance Tm from a simulation with m multiple imputations approximates the variance T∞ withoutsimulation error. The question is when Tm is close enough to T∞.
Example 11. Rubin showed that the two variances are related by
Tm = (1 +γ0
m)T∞
where γ0 is the true population fraction of missing information. This quantity is equal to the expectedfraction of observations missing if Y is a single variable without covariates, and commonly less than thisif there are covariates that predict Y . If we set γ0 = 0.3 and m = 5 then Tm is 1.06 times the idealvariance T∞, and the corresponding confidence interval would be
√1.06 = 1.03 times larger. Increasing
m =10 or 20 would reduce the factor to 1.5% and 0.7% respectively.
Theoretically it is better to choose large m, but, according to Buuren, the substantive conclusions areunlikely to change as a result of raising m beyond 5. Most recommendations land at around m = 20,depending on the proportion of missing data, and a rule of thumb for up to 50% of missing data is tochoose m similar to the percentage of incomplete cases.
To determine convergence of the algorithm one may plot parameters against iteration number for eachimputation. This can quickly become cumbersome if the number of interesting parameters or the numberof multiple imputations is high. On convergence, the different streams should be freely intermingled withone another without showing any definite trends. The variance between different sequences should be nolarger than the variance within each individual sequence. In practice, a low number of iterations appearsto be enough according to Buuren, and it is suggested to use around 5-20.
38
4 Statistical Analysis
The statistical computer software used was R version 3.3.2 7. The level of statistical significance is set toα = 0.05. Missing data was imputed for a sequence of missing data models and analysed as described indetail below.
4.1 Imputation
The imputation were performed using the R package mice8. The proportions of missing data acrossvariables was generally low, most variables included in the study had less than 10 % missing data, asseen in table 3. Time-to-event and cause of death were considered to be completely observed even thoughthere was censoring. Other variables that were completely observed were age at diagnosis and year ofdiagnosis. On the other hand, M stage, which was of high interest in this study, had as much as 69.5%missing data which could pose a problem. Therefore the modelling was primarily focused on optimallyimputing M stage. This included the modelling of a nonignorable missing data mechanism on M stage,and a sensitivity analysis on the parameter k since there was not enough a priori information availableto determine a prior distribution on k.
Table 3: Proportion of complete data
PSA Civil Status Educational Level Mode of Detection T stage N stage99.3 % 99.9 % 99 % 96.2 % 99.1 % 99.3 %M stage Gleason 1 Gleason 2 Gleason Sum GGG Treatment30.5 % 83.9 % 83.7 % 93.7 % 90.6 % 96.6 %
The normality in assumption 1 was investigated with a series of normality tests, Shapiro-Wilks,Shapiro-Francia, Andersson-Darling and Cramer-von Mises using complete-data statistics for log-log,logit and no transformation of different survival times and positive fractions. The last three normalitytests are contained in the R package nortest9. The transformation that yielded the largest p-values,or correspondingly, the smallest or largest test statistics, would then be chosen for further analysis. Abootstrap confidence interval was included for comparison and the widths of the confidence intervals werecompared, using the non-transformed confidence interval as reference.
4.1.1 Model Form and Derived Variables
Educational level, Gleason 1, Gleason sum and T stage were imputed using ordinal logistic regression whilePSA was imputed using predictive mean matching. Gleason 2 was imputed as a derived variable usingthe formula Gleason sum - Gleason 1, and GGG was successively imputed using the scheme describedin the Background section. Civil status, N stage and treatment were imputed using multinomial logisticregression since there is no natural order of the levels of these factors. M stage was imputed using logisticregression but with a modified algorithm to model a nonignorable missing data mechanism.
In general, for each marginal imputation, all available variables were included as predictors, with theexception of when imputing Gleason 1 and Gleason sum, where the Gleason sum is imputed without GGGand Gleason 1 and 2, and Gleason 1 is imputed without Gleason 2 and GGG, as seen in table 4. This wasdone to avoid feedback loops between the derived and non-derived Gleason variables. The visit sequencewas set to 1. Civil status, 2. Educational Level, 3. T stage, 4. N stage, 5. PSA, 6. Treatment, 7. Modeof Detection, 8. Gleason sum, 9. Gleason 1, 10. Gleason 2, 11. GGG, 12. M stage. The main argumentfor this choice was to order the variables according to their percent of missing data and imputing themfrom low to high proportions, keeping in mind the logical structure of the derived variables.
7r-project.org8cran.r-project.org/web/packages/mice9cran.r-project.org/web/packages/nortest
39
Table 4: Predicator Matrix. Variables PSA, civil status, educational level, mode of detection, M stage,Gleason 1, Gleason 2, Gleason sum, GGG, T stage, N stage and treatment. Age at diagnosis, year ofdiagnosis, Region, CCI, censoring and survival time have been excluded since these had no missing data,yet they were used as predictors in all marginal models.
PSA Civ Edu MoD M G1 G2 GS GGG T N TrPSA 0 1 1 1 1 1 1 1 1 1 1 1Civ 1 0 1 1 1 1 1 1 1 1 1 1Edu 1 1 0 1 1 1 1 1 1 1 1 1MoD 1 1 1 0 1 1 1 1 1 1 1 1M 1 1 1 1 0 1 1 1 1 1 1 1G1 1 1 1 1 1 0 0 1 0 1 1 1G2 1 1 1 1 1 1 0 1 1 1 1 1GS 1 1 1 1 1 0 0 0 0 1 1 1GGG 1 1 1 1 1 1 1 1 0 1 1 1T 1 1 1 1 1 1 1 1 1 0 1 1N 1 1 1 1 1 1 1 1 1 1 0 1Tr 1 1 1 1 1 1 1 1 1 1 1 0
4.2 Analysis of Imputed Data
The main analysis was conducted using the nonignorable imputation model with k = 0.7. In all analysesthe complete-data statistics were used, compare with remark 15. Kernel densities were calculated usingthe density function with default settings. These together with histograms were used to display thedistribution of PSA between men with M0 and M1 stratified across GGG. The histograms were averagedacross the imputed data sets and the kernel densities were drawn as a polygon, where for each point themaximum and minimum across the imputed data sets constituted the boundaries.
ROC curves and corresponding positive fractions, predictive values and likelihood ratios were com-puted using Rubin’s rules after a log-log transformation, including 95% confidence intervals and confidencerectangles for the positive fractions.
Survival curves were estimated using the cpmrsk10 package, where the 10 year net PCa cause spe-cific survival was computed, meaning that other causes of death were treated as censored. These wereillustrated as mortality curves (1-survival) and 95% confidence intervals of the proportion of death werecomputed after 10 years of diagnosis. Restricted mean survival time and years of life lost (YLL) com-paring RT/RT against ADT treatment, for men with M0 between 60-70 years old at diagnosis with PSA50-150 ng/ml and GGG 4-5, was computed using the survival11. The reason of the restriction to menbetween 60 and 70 years old at diagnosis was to choose a representative group with high risk of dyingof PCa compared to other causes of death, compare with Figure 118 in Regionalt cancercentrum Upp-sala/Orebro (2015). This is also why only men with GGG 4-5 and PSA ≥ 50 ng/ml were considered,since that indicates high risk PCa. Since the current threshold for PSA is 100 ng/ml for indicatingmetastatic PCa it was reasonable to restrict the analysis to men with PSA < 150 ng/ml, resulting ina symmetric interval around 100 ng/ml, as these would be most relevant to the discussion about YLLdue to a false positive metastatic PCa diagnose. These restrictions were also intended to minimize thepotential effect of confounding variables. A Cox proportional hazards model was fitted on complete casesand imputed data, stratified by treatment, age at diagnosis, GGG and PSA and again, the restrictedmean survival and YLL were computed. Model assessment was done in both cases, where p-values ofcoefficients were examined and along with a graphical evaluation of the proportional hazards assumption.To ease readability, only the model assessment of the complete cases was displayed.
10cran.r-project.org/web/packages/cmprsk11cran.r-project.org/web/packages/survival
40
4.2.1 Model Convergence and Sensitivity Analysis
The number of imputations was set to 200 for the main analysis and 20 for the sensitivity analysis, tokeep the running time feasible, although the rules of thumb would suggest about 70 multiple imputations,and the number of iterations was limited to 10. Model convergence was assessed by plotting boxplots ofmeans and variances of each variable with missing data from each completed dataset against the iterationnumber. The individual streams were also analysed but are not shown.
A sensitivity analysis on the parameter k was conducted since there was reason to believe that theprevalence of M1 disease would be to high under an ignorable model. Mortality, prevalence of metastaticPCa and positive fractions were therefore considered for the nonignorable imputation model with k ∈{1, 0.7, 0.85, 0, 6, 0.55, 0.4, 0.25, 0.1} and compared with the complete cases. The reason to include k wasto see how sensitive the analysis would be to a model misspecification in relation to prior informationabout k.
41
5 Results
5.1 Baseline Characteristics
The baseline characteristics table 5 below, or the full version table 10 in the appendix, of the men in thedataset suggest that those who did not receive a bone scan (MX) have similar educational level and civilstatus as those who underwent a bone scan and were diagnosed as M0, or are more similar to men withM0 than M1. For example, the distribution of the educational level was almost identical between menwith MX and M0 while clearly different for men with M1, the latter with a higher proportion of lowereducated men, while the age at diagnosis of men with MX was more or less in between men with M0 andM1, perhaps more similar to men with M0. In addition, men with M0 were generally younger than menwith M1, and men with MX had an age distribution somewhere in between.
On the other hand, the clinical data, for example the Gleason Grade Group, suggest that men withMX had even less severe prostate cancer than men with M0. This tendency can also be seen whenlooking at T and N stages and PSA where men with MX generally had lower PSA levels with a medianPSA of 9 (IQR=6-18), compared with men with M0 who had a median PSA of 16 (IQR=9-34) and 158(IQR=45-565) for men with M1.
M0 M1 MX All
n (%) n (%) n (%) n (%)
Total 24665 (100) 9496 (100) 77852 (100) 112013 (100)Age at diagnosis, years
0-64 7687 (31) 1731 (18) 23427 (30) 32845 (29)65-80 14967 (61) 5524 (58) 43009 (55) 63500 (57)81+ 2011 (8) 2241 (24) 11416 (15) 15668 (14)
Educational levelLow 9881 (40) 4670 (49) 30741 (39) 45292 (40)Middle 9154 (37) 3206 (34) 28834 (37) 41194 (37)High 5433 (22) 1487 (16) 17505 (22) 24425 (22)Missing 197 (1) 133 (1) 772 (1) 1102 (1)
Gleason Grade Group1 7448 (30) 642 (7) 39375 (51) 47465 (42)2 4925 (20) 783 (8) 14250 (18) 19958 (18)3 3941 (16) 1186 (12) 7252 (9) 12379 (11)4 3690 (15) 2158 (23) 6077 (8) 11925 (11)5 2447 (10) 2744 (29) 4615 (6) 9806 (9)Missing 2214 (9) 1983 (21) 6283 (8) 10480 (9)
Table 5: Extract from baseline characteristics found in full version in appendix as table 10.
5.2 Imputation Model Convergence Diagnostics
The data set contained 112 013 men of which 77 852 (69.5%) had unknown M stage, as seen in table 5.After imputation using the nonignorable model with k = 0.7, a minority, 11.5% (95% CI 11.1-12%), ofmen with MX were classified as having M1, and the rest were classified as M0, yielding a total prevalenceof 16.5 (95% CI 16.2-16.8%) of M1 disease. Moreover, the baseline characteristics of complete casescompared with each completed dataset were very similar, compare with fig. 4 below, although clinicaldata on the PCa differs slightly. For example, the imputations had generally lower T stage and GGGthan the complete cases for both men with M0 and M1.
While the variance between the imputed datasets is almost unnoticeable in fig. 4, convergence is moreaccurately diagnosed in fig. 5 below, where we see that the distribution of each stream stabilized after afew iterations. Individual streams for parameters also indicated satisfying convergence (not shown).
42
0-64 65-70 70+
0.0
0.3
0.6
M0A
ge a
t d
iag
no
sis
Complete cases
Imputations
0-64 65-70 70+
M1
2000-2003 2004-2007 2008-2012
0.0
0.2
0.4
Ye
ar
of
dia
gn
osis
2000-2003 2004-2007 2008-2012
Low Middle High
0.0
0.2
0.4
Ed
uca
tion
al
leve
l
Low Middle High
Widow
Married/
partnership Unmarried
Divorced/
separated
0.0
0.3
0.6
Civ
il
sta
tus
Widow
Married/
partnership Unmarried
Divorced/
separated
0 1 2 3+
0.0
0.4
0.8
CC
I
0 1 2 3+
Asymptomatic Symptomatic
0.0
0.4
0.8
M0
Mo
de o
f d
ete
ction
Asymptomatic Symptomatic
M1
T1a/b T1c T2 T3 T4
0.0
0.2
0.4
T s
tage
T1a/b T1c T2 T3 T4
N0 N1 NX
0.0
0.4
0.8
N s
tage
N0 N1 NX
1 2 3 4 5
0.0
0.3
0.6
GG
G
1 2 3 4 5
RT ADT RP CT
0.0
0.4
0.8
Tre
atm
ent
RT ADT RP CT
Figure 4: Characteristics before and after imputation, based on the first 10 datasets. Green indicatescomplete cases and blue indicates imputed data.
100
140
180
220
mean Civil Status
100
140
180
220
100
140
180
220
100
140
180
220
100
140
180
220
100
140
180
220
100
140
180
220
100
140
180
220
100
140
180
220
100
140
180
220
100
140
180
220
0500000
1500000
variance Civil Status
0500000
1500000
0500000
1500000
0500000
1500000
0500000
1500000
0500000
1500000
0500000
1500000
0500000
1500000
0500000
1500000
0500000
1500000
0500000
1500000
1.65
1.70
1.75
mean Educational Level
1.65
1.70
1.75
1.65
1.70
1.75
1.65
1.70
1.75
1.65
1.70
1.75
1.65
1.70
1.75
1.65
1.70
1.75
1.65
1.70
1.75
1.65
1.70
1.75
1.65
1.70
1.75
1.65
1.70
1.75
0.52
0.56
0.60
variance Educational Level
0.52
0.56
0.60
0.52
0.56
0.60
0.52
0.56
0.60
0.52
0.56
0.60
0.52
0.56
0.60
0.52
0.56
0.60
0.52
0.56
0.60
0.52
0.56
0.60
0.52
0.56
0.60
0.52
0.56
0.60
1.110
1.115
1.120
1.125
mean Treatment
1.110
1.115
1.120
1.125
1.110
1.115
1.120
1.125
1.110
1.115
1.120
1.125
1.110
1.115
1.120
1.125
1.110
1.115
1.120
1.125
1.110
1.115
1.120
1.125
1.110
1.115
1.120
1.125
1.110
1.115
1.120
1.125
1.110
1.115
1.120
1.125
1.110
1.115
1.120
1.125
0.096
0.100
0.104
0.108
variance Treatment
0.096
0.100
0.104
0.108
0.096
0.100
0.104
0.108
0.096
0.100
0.104
0.108
0.096
0.100
0.104
0.108
0.096
0.100
0.104
0.108
0.096
0.100
0.104
0.108
0.096
0.100
0.104
0.108
0.096
0.100
0.104
0.108
0.096
0.100
0.104
0.108
0.096
0.100
0.104
0.108
2.0
2.2
2.4
2.6
2.8
mean Mode of Detection
2.0
2.2
2.4
2.6
2.8
2.0
2.2
2.4
2.6
2.8
2.0
2.2
2.4
2.6
2.8
2.0
2.2
2.4
2.6
2.8
2.0
2.2
2.4
2.6
2.8
2.0
2.2
2.4
2.6
2.8
2.0
2.2
2.4
2.6
2.8
2.0
2.2
2.4
2.6
2.8
2.0
2.2
2.4
2.6
2.8
2.0
2.2
2.4
2.6
2.8
0.2
0.4
0.6
0.8
1.0
variance Mode of Detection
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
2.002.02
2.042.06
mean PSA
2.002.02
2.042.06
2.002.02
2.042.06
2.002.02
2.042.06
2.002.02
2.042.06
2.002.02
2.042.06
2.002.02
2.042.06
2.002.02
2.042.06
2.002.02
2.042.06
2.002.02
2.042.06
2.002.02
2.042.06
0.71
0.73
0.75
variance PSA
0.71
0.73
0.75
0.71
0.73
0.75
0.71
0.73
0.75
0.71
0.73
0.75
0.71
0.73
0.75
0.71
0.73
0.75
0.71
0.73
0.75
0.71
0.73
0.75
0.71
0.73
0.75
0.71
0.73
0.75
3.25
3.30
3.35
3.40
mean Gleason Sum
3.25
3.30
3.35
3.40
3.25
3.30
3.35
3.40
3.25
3.30
3.35
3.40
3.25
3.30
3.35
3.40
3.25
3.30
3.35
3.40
3.25
3.30
3.35
3.40
3.25
3.30
3.35
3.40
3.25
3.30
3.35
3.40
3.25
3.30
3.35
3.40
3.25
3.30
3.35
3.40
0.45
0.50
0.55
0.60
variance Gleason Sum
0.45
0.50
0.55
0.60
0.45
0.50
0.55
0.60
0.45
0.50
0.55
0.60
0.45
0.50
0.55
0.60
0.45
0.50
0.55
0.60
0.45
0.50
0.55
0.60
0.45
0.50
0.55
0.60
0.45
0.50
0.55
0.60
0.45
0.50
0.55
0.60
0.45
0.50
0.55
0.60
3.30
3.35
3.40
3.45
mean Gleason 1
3.30
3.35
3.40
3.45
3.30
3.35
3.40
3.45
3.30
3.35
3.40
3.45
3.30
3.35
3.40
3.45
3.30
3.35
3.40
3.45
3.30
3.35
3.40
3.45
3.30
3.35
3.40
3.45
3.30
3.35
3.40
3.45
3.30
3.35
3.40
3.45
3.30
3.35
3.40
3.45
0.60
0.650.700.75
0.80
variance Gleason 1
0.60
0.650.700.75
0.80
0.60
0.650.700.75
0.80
0.60
0.650.700.75
0.80
0.60
0.650.700.75
0.80
0.60
0.650.700.75
0.80
0.60
0.650.700.75
0.80
0.60
0.650.700.75
0.80
0.60
0.650.700.75
0.80
0.60
0.650.700.75
0.80
0.60
0.650.700.75
0.80
6.15
6.20
6.25
6.30
6.35
mean Gleason 2
6.15
6.20
6.25
6.30
6.35
6.15
6.20
6.25
6.30
6.35
6.15
6.20
6.25
6.30
6.35
6.15
6.20
6.25
6.30
6.35
6.15
6.20
6.25
6.30
6.35
6.15
6.20
6.25
6.30
6.35
6.15
6.20
6.25
6.30
6.35
6.15
6.20
6.25
6.30
6.35
6.15
6.20
6.25
6.30
6.35
6.15
6.20
6.25
6.30
6.35
1.6
1.7
1.8
1.9
variance Gleason 2
1.6
1.7
1.8
1.9
1.6
1.7
1.8
1.9
1.6
1.7
1.8
1.9
1.6
1.7
1.8
1.9
1.6
1.7
1.8
1.9
1.6
1.7
1.8
1.9
1.6
1.7
1.8
1.9
1.6
1.7
1.8
1.9
1.6
1.7
1.8
1.9
1.6
1.7
1.8
1.9
2.65
2.70
2.75
2.80
mean GGG
2.65
2.70
2.75
2.80
2.65
2.70
2.75
2.80
2.65
2.70
2.75
2.80
2.65
2.70
2.75
2.80
2.65
2.70
2.75
2.80
2.65
2.70
2.75
2.80
2.65
2.70
2.75
2.80
2.65
2.70
2.75
2.80
2.65
2.70
2.75
2.80
2.65
2.70
2.75
2.80
1.55
1.60
1.65
1.70
1.75
variance GGG
1.55
1.60
1.65
1.70
1.75
1.55
1.60
1.65
1.70
1.75
1.55
1.60
1.65
1.70
1.75
1.55
1.60
1.65
1.70
1.75
1.55
1.60
1.65
1.70
1.75
1.55
1.60
1.65
1.70
1.75
1.55
1.60
1.65
1.70
1.75
1.55
1.60
1.65
1.70
1.75
1.55
1.60
1.65
1.70
1.75
1.55
1.60
1.65
1.70
1.75
4.10
4.20
4.30
mean T stage
4.10
4.20
4.30
4.10
4.20
4.30
4.10
4.20
4.30
4.10
4.20
4.30
4.10
4.20
4.30
4.10
4.20
4.30
4.10
4.20
4.30
4.10
4.20
4.30
4.10
4.20
4.30
4.10
4.20
4.30
1.1
1.2
1.3
1.4
variance T stage
1.1
1.2
1.3
1.4
1.1
1.2
1.3
1.4
1.1
1.2
1.3
1.4
1.1
1.2
1.3
1.4
1.1
1.2
1.3
1.4
1.1
1.2
1.3
1.4
1.1
1.2
1.3
1.4
1.1
1.2
1.3
1.4
1.1
1.2
1.3
1.4
1.1
1.2
1.3
1.4
2.70
2.75
2.80
mean N stage
2.70
2.75
2.80
2.70
2.75
2.80
2.70
2.75
2.80
2.70
2.75
2.80
2.70
2.75
2.80
2.70
2.75
2.80
2.70
2.75
2.80
2.70
2.75
2.80
2.70
2.75
2.80
2.70
2.75
2.80
0.30
0.35
0.40
0.45
0.50
variance N stage
0.30
0.35
0.40
0.45
0.50
0.30
0.35
0.40
0.45
0.50
0.30
0.35
0.40
0.45
0.50
0.30
0.35
0.40
0.45
0.50
0.30
0.35
0.40
0.45
0.50
0.30
0.35
0.40
0.45
0.50
0.30
0.35
0.40
0.45
0.50
0.30
0.35
0.40
0.45
0.50
0.30
0.35
0.40
0.45
0.50
0.30
0.35
0.40
0.45
0.50
5.4
5.6
5.8
6.0
mean M stage
5.4
5.6
5.8
6.0
5.4
5.6
5.8
6.0
5.4
5.6
5.8
6.0
5.4
5.6
5.8
6.0
5.4
5.6
5.8
6.0
5.4
5.6
5.8
6.0
5.4
5.6
5.8
6.0
5.4
5.6
5.8
6.0
5.4
5.6
5.8
6.0
5.4
5.6
5.8
6.0
9.5
10.0
10.5
11.0
11.5
variance M stage
9.5
10.0
10.5
11.0
11.5
9.5
10.0
10.5
11.0
11.5
9.5
10.0
10.5
11.0
11.5
9.5
10.0
10.5
11.0
11.5
9.5
10.0
10.5
11.0
11.5
9.5
10.0
10.5
11.0
11.5
9.5
10.0
10.5
11.0
11.5
9.5
10.0
10.5
11.0
11.5
9.5
10.0
10.5
11.0
11.5
9.5
10.0
10.5
11.0
11.5
IterationFigure 5: Convergence diagnostics for all variables with missing data.
43
5.3 Study of Parameter Transformations
The three parameter transformations (basic, loglog and logit) for the quantities true positive and falsepositive fractions (TPF and FPF) were compared by calculating the p-values and test statistics for thefour tests for normality, see table 11 in appendix. Here, we look for small p-values (rejecting normality)and compare p-values and test statistics between the transformations for each threshold and quantity.
The first thing to note in table 11 is that almost all p-values were large, apart from perhaps the CMVtest for false positive fraction at PSA threshold 10 where all transformations yielded a p-value of 7%.Secondly, all transformations produced almost identical values of the test statistics within each quantityand threshold. Similarly, the p-values within each quantity and threshold were similar and there was noclear tendency that a certain transformation gave better normality approximation.
When comparing the confidence intervals produced by the three transformations, with the additionof a bootstrap CI, fig. 12 in appendix, it is clear that all transformations produced essentially the samepoint estimate and the only thing that differed was the widths of the CIs and their centre. While thebootstrap confidence interval produced the smallest width across all quantities (between 16.3-76% of thebasic CI width), the basic, log-log and logit transformation confidence intervals were almost identicalapart from some cases where the log-log and logit CIs were shifted in one direction or the other.
Looking at the same comparisons but for the proportion of death at 0.5, 5, 10 and 14.5 years fromdiagnosis for men with M0 and M1, table 12 in appendix, we see the same pattern, or rather, the absenceof a pattern. All p-values were fairly large, the test statistics were almost identical across all time pointsand for both men with M0 and M1. There was no clear indication that any transformation yielded asuperior normality approximation.
Analogously, fig. 13 in appendix show that all transformations gave almost identical point estimates,that the bootstrap CIs were narrow (between 15.5-65.3% of the basic CI width).
In light of this, the log-log transformation was used in all subsequent analyses.
5.4 Analysis of Imputed Datasets
In fig. 6 below we see that, for both the complete cases and the imputed data, men with M1 diseasehad a flatter PSA distribution shifted towards higher PSA levels, compared with men with M0. Mostimportantly, the overlap of the two distributions was substantial in all GGG, and increased with decreasingGGG, especially for imputed data. Note that for both M0 and M1 disease the PSA distributions shiftedto higher PSA levels with increasing GGG.
This was reflected in the ROC curves, fig. 7 below, where the overall ability of separating men withM0 and M1 disease using a PSA cutoff is diagnosed by how far up towards the top-right corner the ROCcurve stretches. Since the overlap of the PSA distributions was considerable it is not surprising that thetest performance was poor, which was especially evident for GGG 1. Although the true positive fractionsfor a specific threshold increased with increasing GGG, the same occurred for the false positive fractions,and the highest true positive fraction for the tested thresholds was 85% for GGG 4 and 5 at a thresholdof 20 ng/ml, compare with table 6 below. The corresponding false positive fractions were 48 and 57%.For false positive fractions below 10% the true positive fractions were at best around 50%.
Consequently, no threshold managed to simultaneously produce a large positive likelihood ratio anda small negative likelihood ratio, meaning that they did not even moderately change the pre-test topost-test probability, with for example no negative likelihood ratios below 0.2. In some strata, only smallchanges in pretest to post-test probability was observed, at PSA 20 ng/ml and GGG 2-5, and PSA 50ng/ml and GGG 3-5. From a diagnostic point of view, the prevalence of M1 disease was high even forlower PSA thresholds, for example the positive predictive value was at 70% for GGG 5 at 50 ng/ml, andincreased with increasing PSA. The negative predictive value decreases with increasing GGG and wasaround or above 90% for GGG 1-2 at all thresholds, while for GGG 5 it drops from 74% at 20 ng/mldown to about 54% at 1000 ng/ml.
44
0.1 1 10 100 1000 10000
M0
M1
GGG 1
0.1 1 10 100 1000 10000
GGG 2
0.1 1 10 100 1000 10000
GGG 3
0.1 1 10 100 1000 10000
GGG 4
0.1 1 10 100 1000 10000
GGG 5
Imp
ute
d d
ata
0.1 1 10 100 1000 10000 0.1 1 10 100 1000 10000 0.1 1 10 100 1000 10000 0.1 1 10 100 1000 10000 0.1 1 10 100 1000 10000
PSA concentration (ng/ml)
Co
mp
lete
ca
se
s
Figure 6: Distribution of prostate specific antigen (PSA) stratified by Gleason Grade Groups (GGG) forimputed data (top) and complete cases (bottom).
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
PSA=50
PSA=100PSA=200
PSA=500
GGG 1
Imp
ute
d d
ata
Se
nsitiv
ity
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
PSA=50
PSA=100
PSA=200
PSA=500
GGG 2
Imp
ute
d d
ata
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
PSA=50
PSA=100
PSA=200
PSA=500
GGG 3
Imp
ute
d d
ata
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
PSA=50
PSA=100
PSA=200
PSA=500
GGG 4
Imp
ute
d d
ata
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
PSA=50
PSA=100
PSA=200
PSA=500
GGG 5
Imp
ute
d d
ata
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
PSA=50
PSA=100
PSA=200
PSA=500
1-Specificity
Co
mp
lete
ca
se
sS
en
sitiv
ity
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
PSA=50
PSA=100
PSA=200
PSA=500
1-Specificity
Co
mp
lete
ca
se
s
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
PSA=50
PSA=100
PSA=200
PSA=500
1-Specificity
Co
mp
lete
ca
se
s
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
PSA=50
PSA=100
PSA=200
PSA=500
1-Specificity
Co
mp
lete
ca
se
s
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
PSA=50
PSA=100
PSA=200
PSA=500
1-Specificity
Co
mp
lete
ca
se
s
Figure 7: Receiver Operating Characteristic (ROC) curves stratified by GGG for imputed data (top) andcomplete cases (bottom).
45
Table 6: Estimated positive fractions with jointly 95% confidence intervals, predictive values and likelihood ratios with 95% confidence intervals.Based on imputed data. All values are given in percent apart from likelihood ratios. PSA=Prostate Specific Antigen, GGG=Gleason GradeGroup, TPF/FPF=True/False Positive Fraction, PPV/NPV=Positive/Negative Predictive Value, LHR+/-= Positive/Negative Likelihood Ratio.
PSA GGG TPF CI FPF CI PPV CI NPV CI LRH+ CI LHR- CI
20ng/ml 1 39 (35 - 42) 12 (11 - 14) 16 (15 - 18) 96 (95 - 96) 3 (3 - 4) 0.7 (0.66 - 0.74)
2 71 (68 - 74) 26 (24 - 28) 27 (25 - 28) 95 (95 - 96) 3 (3 - 3) 0.39 (0.35 - 0.43)3 79 (77 - 81) 39 (36 - 41) 32 (31 - 34) 93 (92 - 93) 2 (2 - 2) 0.34 (0.31 - 0.38)4 85 (84 - 86) 48 (47 - 50) 47 (46 - 49) 87 (86 - 88) 2 (2 - 2) 0.29 (0.26 - 0.32)5 85 (84 - 86) 57 (55 - 58) 59 (58 - 61) 74 (73 - 76) 1 (1 - 2) 0.35 (0.33 - 0.38)
50ng/ml 1 23 (20 - 26) 3 (2 - 4) 33 (30 - 36) 95 (95 - 96) 8 (6 - 10) 0.79 (0.76 - 0.82)
2 52 (48 - 56) 9 (8 - 10) 44 (41 - 47) 94 (93 - 94) 6 (5 - 7) 0.53 (0.49 - 0.57)3 59 (56 - 62) 15 (14 - 17) 47 (45 - 50) 90 (89 - 91) 4 (3 - 4) 0.49 (0.46 - 0.52)4 69 (67 - 70) 23 (22 - 25) 60 (58 - 62) 83 (82 - 84) 3 (3 - 3) 0.41 (0.39 - 0.43)5 68 (67 - 70) 28 (27 - 30) 70 (69 - 71) 70 (68 - 71) 2 (2 - 3) 0.44 (0.42 - 0.47)
100ng/ml 1 15 (13 - 18) 1 (0.5 - 1) 54 (49 - 59) 95 (94 - 95) 19 (12 - 29) 0.85 (0.83 - 0.87)
2 38 (35 - 42) 3 (2 - 4) 61 (57 - 65) 92 (91 - 93) 12 (10 - 15) 0.64 (0.61 - 0.67)3 44 (41 - 46) 6 (5 - 7) 63 (60 - 66) 88 (87 - 89) 7 (6 - 8) 0.6 (0.57 - 0.63)4 54 (52 - 56) 11 (10 - 12) 71 (69 - 73) 79 (78 - 80) 5 (4 - 5) 0.52 (0.5 - 0.54)5 53 (52 - 55) 14 (12 - 15) 79 (78 - 81) 65 (64 - 67) 4 (4 - 4) 0.54 (0.52 - 0.56)
200ng/ml 1 11 (9 - 13) 0.3 (0.1 - 1) 72 (66 - 78) 95 (94 - 95) 41 (20 - 87) 0.89 (0.88 - 0.91)
2 28 (25 - 31) 1 (1 - 2) 75 (71 - 79) 91 (90 - 92) 23 (16 - 34) 0.73 (0.7 - 0.76)3 31 (29 - 34) 2 (2 - 3) 75 (72 - 79) 86 (85 - 87) 13 (10 - 17) 0.7 (0.68 - 0.73)4 40 (39 - 42) 5 (4 - 6) 80 (78 - 82) 76 (75 - 77) 8 (7 - 9) 0.63 (0.61 - 0.65)5 40 (38 - 41) 6 (5 - 7) 87 (85 - 89) 62 (60 - 63) 7 (6 - 8) 0.64 (0.62 - 0.66)
500ng/ml 1 7 (5 - 8) 0.1 (0.0 - 0.3) 87 (80 - 92) 94 (94 - 95) 106 (24 - 466) 0.93 (0.92 - 0.95)
2 17 (14 - 20) 0.3 (0.1 - 1) 89 (85 - 92) 90 (89 - 91) 63 (29 - 135) 0.83 (0.81 - 0.86)3 18 (16 - 20) 1 (0.3 - 1) 87 (83 - 91) 84 (83 - 85) 30 (18 - 50) 0.82 (0.8 - 0.84)4 26 (24 - 28) 1 (1 - 2) 91 (88 - 92) 72 (71 - 73) 19 (14 - 25) 0.75 (0.74 - 0.77)5 24 (22 - 25) 1 (1 - 2) 94 (93 - 96) 57 (56 - 58) 17 (13 - 22) 0.77 (0.76 - 0.79)
1000ng/ml 1 4 (3 - 5) 0.0 (0.0 - 0.3) 94 (86 - 98) 94 (94 - 95) 268 (11 - 6323) 0.96 (0.95 - 0.97)
2 9 (6 - 12) 0.1 (0.0 - 0.3) 94 (90 - 97) 89 (88 - 90) 128 (29 - 571) 0.91 (0.89 - 0.93)3 10 (8 - 12) 0.2 (0.1 - 1) 92 (87 - 95) 83 (82 - 84) 49 (20 - 120) 0.9 (0.89 - 0.92)4 15 (14 - 17) 0.4 (0.2 - 1) 96 (93 - 97) 70 (69 - 71) 42 (24 - 72) 0.85 (0.84 - 0.86)5 14 (12 - 15) 0.4 (0.2 - 1) 97 (95 - 98) 54 (53 - 56) 33 (20 - 52) 0.87 (0.85 - 0.88)
46
The mortality curves, displayed in fig. 8 below, may be used to pinpoint certain strata where thereare more distinct differences in mortality between men with M0 and M1. More specifically, if, in a certainstratum, men with M0 and M1 had much more different mortality, then this would indicate that it isimportant to accurately identify metastatic disease for men in this stratum. On the other hand, if themortality would be less different then it would indicate that the metastatic disease has less influence onthe mortality and the consequences of misclassification of M stage are milder.
40751 28942 10549
1764 598 1015216 4388 2638
208 63 31
0 ≤ PSA < 20
0 2 4 6 8 10
Years since
diagnosis:
A.I. M0 no. at risk:
A.I. M1 no. at risk:C.C. M0 no. at risk:
C.C. M1 no. at risk:
Pro
port
ion
of death
GG
G 1
00.2
0.4
0.6
0.8
1
A.I. M0A.I. M1C.C. M0C.C. M1
4365 3040 1183
437 160 341711 1368 665
129 62 21
20 ≤ PSA < 50
0 2 4 6 8 10
1020 664 234
233 75 15388 292 121
79 29 9
50 ≤ PSA < 100
0 2 4 6 8 10
253 149 43
128 38 1087 54 19
59 18 7
100 ≤ PSA < 200
0 2 4 6 8 10
124 72 21
315 70 2239 32 9
154 36 13
PSA ≥ 200
0 2 4 6 8 10
15299 8981 2335
777 209 242885 2113 881
139 41 8
0 2 4 6 8 10
Years since
diagnosis:
A.I. M0 no. at risk:
A.I. M1 no. at risk:C.C. M0 no. at risk:
C.C. M1 no. at risk:
Pro
port
ion
of death
GG
G 2
00.2
0.4
0.6
0.8
1
3489 2127 610
511 143 181362 1008 348
139 51 12
0 2 4 6 8 10
1160 657 162
369 88 11447 300 81
112 35 5
0 2 4 6 8 10
416 195 45
271 54 7124 73 21
90 24 3
0 2 4 6 8 10
245 105 21
773 123 1467 36 6
294 58 6
0 2 4 6 8 10
7352 4257 1094
587 152 182130 1486 474
150 45 8
0 2 4 6 8 10
Years since
diagnosis:
A.I. M0 no. at risk:
A.I. M1 no. at risk:C.C. M0 no. at risk:
C.C. M1 no. at risk:
Pro
port
ion
of death
GG
G 3
00.2
0.4
0.6
0.8
1
2803 1530 385
559 138 171115 711 188
186 57 11
0 2 4 6 8 10
1098 550 141
427 95 10406 254 71
169 45 5
0 2 4 6 8 10
438 199 37
344 70 6165 86 16
174 40 3
0 2 4 6 8 10
280 124 27
882 149 15108 54 11
499 88 7
0 2 4 6 8 10
4542 2521 632
670 145 121843 1218 376
208 52 7
0 2 4 6 8 10
Years since
diagnosis:
A.I. M0 no. at risk:
A.I. M1 no. at risk:C.C. M0 no. at risk:
C.C. M1 no. at risk:
Pro
port
ion
of death
GG
G 4
00.2
0.4
0.6
0.8
1
2219 1117 260
744 149 17965 585 156
293 69 11
0 2 4 6 8 10
1055 478 126
660 120 9439 242 74
304 65 4
0 2 4 6 8 10
536 232 41
603 95 8236 116 25
331 50 5
0 2 4 6 8 10
436 176 38
1816 273 35178 88 22
1005 171 24
0 2 4 6 8 10
2470 1088 228
851 113 81040 577 137
319 36 5
0 2 4 6 8 10
Years since
diagnosis:
A.I. M0 no. at risk:
A.I. M1 no. at risk:C.C. M0 no. at risk:
C.C. M1 no. at risk:
Pro
port
ion
of death
GG
G 5
00.2
0.4
0.6
0.8
1
1604 644 117
917 113 8702 355 74
390 46 5
0 2 4 6 8 10
852 302 52
836 88 6360 156 34
396 38 3
0 2 4 6 8 10
453 138 21
752 74 6198 72 12
398 38 4
0 2 4 6 8 10
326 97 20
2201 175 13124 51 12
1214 108 7
0 2 4 6 8 10
Figure 8: Kaplan-Meier survival curves of Imputed and Complete cases.
47
Table 7: Confidence intervals for proportion of deathat 10 years from diagnosis, given in percentages, cor-responding to fig. 8 below. Based on imputed data.
PSA M0 CI M1 CI
GGG
1
0-20 4 (4 - 4) 22 (17 - 27)20-50 16 (15 - 16) 56 (48 - 64)50-100 26 (24 - 28) 70 (62 - 77)100-200 36 (30 - 42) 72 (61 - 81)200+ 48 (39 - 57) 79 (74 - 84)
GGG
2
0-20 10 (9 - 10) 51 (42 - 59)20-50 25 (23 - 26) 72 (65 - 78)50-100 38 (35 - 41) 84 (78 - 88)100-200 49 (44 - 54) 86 (80 - 90)200+ 64 (56 - 71) 93 (90 - 95)
GGG
3
0-20 16 (15 - 18) 57 (48 - 67)20-50 32 (30 - 34) 74 (66 - 80)50-100 38 (34 - 41) 81 (77 - 85)100-200 53 (47 - 58) 86 (80 - 90)200+ 56 (49 - 64) 91 (88 - 93)
GGG
4
0-20 24 (23 - 25) 78 (71 - 84)20-50 40 (38 - 42) 80 (76 - 84)50-100 48 (44 - 51) 86 (82 - 90)100-200 59 (54 - 64) 91 (88 - 94)200+ 64 (58 - 69) 90 (88 - 91)
GGG
5
0-20 46 (44 - 49) 90 (85 - 94)20-50 56 (54 - 59) 91 (88 - 93)50-100 61 (57 - 65) 95 (92 - 97)100-200 74 (70 - 78) 92 (89 - 95)200+ 72 (67 - 77) 96 (94 - 97)
In fig. 8 we see that for GGG 2-5 the separa-tion between the mortality curves of M0 and M1decreased slightly with increasing PSA, while forGGG 1 the effect of PSA seemed more inconclu-sive, for example, for PSA 0-20 ng/ml and GGG 1the mortality curves based on imputed data werevery low and quite close.
In addition, mortality in general increased withincreasing GGG and PSA levels, and, apart fromGGG 1 and 2, the imputed and complete case mor-tality were almost identical. For PSA below 50ng/ml, and especially below 20 ng/ml, there was atendency of imputed M1 having a lower mortalitythan complete case M1. The prognostic useful-ness of PSA depended on GGG and was betterfor lower GGG. For example, if we consider menwith GGG 5, then the effect of increasing PSA wassmall for men with M1, with a 10 year mortalityof 90% PSA 0-20, 96% PSA > 200, compare withtable 7, and slightly larger for M0 46% PSA 0-20,72% PSA > 200, while for men with GGG 1 theeffect of increasing PSA was profound for both M1and M0 with 22% PSA 0-20, 79% PSA > 200, and4% PSA 0-20, 48% PSA > 200 correspondingly.
In fig. 8 we also see that a large proportion ofmen with M1 disease had GGG 4-5 and a PSAabove 200 ng/ml, about 1815 and 2207 men, butalso a considerable amount of men, about 1807,
with GGG 1 and PSA below 20 ng/ml. In all other strata the amount of men with M1 disease was below900. The majority of men with M0 had a PSA below 50 ng/ml and most of these had GGG 1-2.
Table 8: Comparison of curative and noncurative treatment by years of life lost (YLL) and mean survival(MS), restricted to the first 15 years from diagnosis, stratified by age at diagnosis, GGG and PSA.
Complete Cases Imputed Data
Treatment Age GGG PSA MS CI YLL YLLRP/RT 60 4 50 13.52 (13.45-13.58)
3.97 3.02ADT 60 4 50 9.66 (9.63-9.7)RP/RT 60 5 50 12.59 (12.53-12.65)
5.01 4.16ADT 60 5 50 7.65 (7.62-7.67)RP/RT 70 4 50 13.42 (13.36-13.49)
4.15 3.66ADT 70 4 50 9.43 (9.39-9.46)RP/RT 70 5 50 12.45 (12.39-12.51)
5.14 4.74ADT 70 5 50 7.4 (7.38-7.43)RP/RT 60 4 150 13.47 (13.4-13.53)
4.05 3.11ADT 60 4 150 9.53 (9.49-9.57)RP/RT 60 5 150 12.52 (12.46-12.57)
5.07 4.25ADT 60 5 150 7.51 (7.49-7.54)RP/RT 70 4 150 13.37 (13.3-13.44)
4.23 3.76ADT 70 4 150 9.29 (9.26-9.33)RP/RT 70 5 150 12.37 (12.31-12.43)
5.19 4.82ADT 70 5 150 7.27 (7.24-7.29)
Altogether, the difference in mortality between men with M0 and M1 was profound in all strata andmildly depends on PSA and GGG while the mortality in general increased with both PSA and GGG.
48
0 5 10 15
0.0
0.2
0.4
0.6
0.8
1.0
Years since diagnosis
Su
rviv
al p
rob
ab
ility
RPRT, 60, 4, 50
ADT, 60, 4, 50
RPRT, 60, 5, 50
ADT, 60, 5, 50
RPRT, 70, 4, 50
ADT, 70, 4, 50
RPRT, 70, 5, 50
ADT, 70, 5, 50
RPRT, 60, 4, 150
ADT, 60, 4, 150
RPRT, 60, 5, 150
ADT, 60, 5, 150
RPRT, 70, 4, 150
ADT, 70, 4, 150
RPRT, 70, 5, 150
ADT, 70, 5, 150(Treatment, Age, GGG, PSA)
Figure 9: Survival curves from Cox proportional haz-ards model. RPRT=Radical Prostatectomy or RadioTherapy, ADT=Androgen Deprivation Therapy.
For men between 60 and 70 years old withM0, PSA 50-150 ng/ml and GGG 4-5, the com-plete case mean survival was 9 years when re-ceiving ADT treatment and 12.1 years whenreceiving RP or RT treatment, meaning thatthe expected years of life lost was 3.1 (95% CI1.9-4.3) when ADT is given instead of RP orRT. The corresponding imputed mean survivalwas 8.9 and 12.3 years, with an expected yearsof life lost of 3.4 (95% CI 2.4-3.5). When in-stead comparing the survival curves from theCox proportional hazards model in fig. 9 wesee that for fixed covariates (Age at diagnosis,GGG and PSA) the survival curve was pro-foundly lower for a man treated with ADT com-pared with RP or RT. The effect when keepingall other covariates constant and increasing ageat diagnosis from 60 to 70 years or PSA from50-150 ng/ml was small, while increasing GGGfrom 4 to 5 elevated the survival curve notably.
This was reflected in the correspondingmean survival in each strata, compare with ta-ble 8, where the years of life lost in each strata,restricted to the first 15 years from diagnosis,is also displayed. Here we see that the com-plete cases had higher YLL compared with theimputation data, but in any case, there wereseveral strata where the YLL was between 4and 5 years, especially for GGG 5.
Generally, the fit of the Cox regressionmodel was good where all coefficients had p-values < 0.001 and the proportional hazardsassumption seemed valid, see fig. 11 in ap-pendix, where the lines in each strata were par-allel. This analysis was also conducted for eachmodel based on imputed data and showed verysimilar results, and therefore it is not shown.
5.5 Sensitivity Analysis
The effect of the parameter k on the results will be studied below. Recall that k was the parameter of thenonignorable missing data model used to impute M stage, were k < 1 corresponds to a lower probabilityof being classified as M1 compared to the ignorable model and k = 1 corresponds to the ignorablemodel. First, the mortality was assessed and compared with complete cases over different k, and thenthe prevalence of M1 disease and positive fractions were compared across the different imputation modelsdepending on k.
In fig. 10 we see that a smaller k tended to increase the overlap of survival curves of complete caseswith imputed for men with M1 disease while only moderately increasing the difference in survival betweenimputed and complete cases for men with M0. The overall difference between different k’s was moderate.For men with PSA 0-20 ng/ml, the distances between survival curves for M1 were the largest for k = 1and we see a considerable increase in overlap of their survival for decreasing k, whereas for men with
49
PSA≥ 200ng/ml and M0 the difference between imputed and complete case survival increased withdecreasing k.
Furthermore, in table 13 in appendix, we see that the prevalence of M1 disease was clearly dependenton k, where a smaller k leads to a lower prevalence, for example consider men with GGG 1 and PSA50-100 ng/ml, then the prevalence was 21% for k = 1 and 15% for k = 0.4.
Table 9: Sensitivity Analysis on Positive Fractions, all given in percent. PSA=Prostate Specific Antigen,GGG=Gleason Grade Group, CC=Complete Cases, TPF/FPF=True/False Positive Fraction.
k PSA cutoff 20 ng/ml PSA cutoff 50 ng/ml PSA cutoff 100 ng/ml
TPF CI FPF CI TPF CI FPF CI TPF CI FPF CI
GG
G1
1 35 (31 - 38) 12 (12 - 13) 20 (18 - 22) 3 (3 - 3) 13 (11 - 14) 1 (1 - 1)0.85 36 (33 - 39) 12 (12 - 13) 21 (19 - 24) 3 (3 - 3) 14 (12 - 16) 1 (1 - 1)0.7 38 (34 - 42) 12 (12 - 13) 23 (21 - 26) 3 (3 - 3) 15 (14 - 17) 1 (1 - 1)0.55 41 (38 - 45) 12 (12 - 13) 26 (24 - 28) 3 (3 - 3) 17 (15 - 19) 1 (1 - 1)0.4 44 (40 - 48) 13 (12 - 13) 29 (26 - 32) 3 (3 - 3) 19 (17 - 22) 1 (1 - 1)0.25 50 (46 - 54) 13 (11 - 15) 34 (30 - 37) 3 (2 - 4) 24 (21 - 27) 1 (0.5 - 2)0.1 59 (54 - 63) 13 (11 - 15) 41 (37 - 46) 3 (2 - 5) 30 (26 - 35) 1 (0.5 - 2)CC 67 (65 - 69) 30 (29 - 30) 46 (44 - 48) 7 (7 - 7) 34 (32 - 36) 2 (2 - 2)
GG
G2-3
1 73 (71 - 75) 30 (29 - 31) 53 (51 - 55) 11 (10 - 12) 38 (36 - 40) 4 (3 - 5)0.85 74 (72 - 76) 30 (29 - 32) 54 (52 - 56) 11 (10 - 12) 39 (38 - 41) 4 (4 - 5)0.7 75 (73 - 77) 30 (29 - 32) 55 (53 - 57) 11 (10 - 12) 41 (39 - 43) 4 (4 - 5)0.55 77 (75 - 78) 31 (29 - 32) 58 (56 - 60) 11 (10 - 12) 43 (41 - 45) 4 (4 - 5)0.4 78 (77 - 80) 31 (30 - 33) 60 (58 - 62) 12 (11 - 13) 45 (43 - 47) 5 (4 - 5)0.25 81 (79 - 83) 32 (30 - 33) 63 (61 - 65) 12 (11 - 13) 49 (46 - 51) 5 (4 - 6)0.1 84 (82 - 86) 32 (30 - 34) 68 (66 - 70) 13 (11 - 14) 54 (51 - 56) 5 (4 - 6)CC 85 (85 - 86) 43 (43 - 44) 69 (68 - 69) 15 (15 - 15) 54 (53 - 55) 5 (5 - 5)
GG
G4-5
1 84 (83 - 85) 51 (50 - 52) 67 (65 - 68) 25 (24 - 26) 52 (50 - 53) 12 (11 - 12)0.85 84 (83 - 85) 51 (50 - 52) 67 (66 - 69) 25 (24 - 26) 52 (51 - 54) 12 (11 - 13)0.7 85 (84 - 86) 52 (50 - 53) 68 (67 - 70) 25 (24 - 26) 53 (52 - 55) 12 (11 - 13)0.55 85 (85 - 86) 52 (51 - 53) 69 (68 - 71) 26 (25 - 27) 55 (53 - 56) 12 (12 - 13)0.4 86 (85 - 87) 53 (52 - 54) 71 (70 - 72) 26 (25 - 28) 56 (55 - 58) 13 (12 - 14)0.25 87 (86 - 88) 54 (52 - 55) 73 (71 - 74) 27 (26 - 28) 58 (57 - 60) 14 (13 - 15)0.1 89 (88 - 90) 55 (54 - 56) 75 (74 - 76) 29 (28 - 30) 61 (60 - 63) 15 (14 - 16)CC 89 (89 - 89) 53 (52 - 53) 75 (75 - 76) 25 (25 - 26) 61 (60 - 61) 12 (12 - 12)
PSA cutoff 200 ng/ml PSA cutoff 500 ng/ml PSA cutoff 1000 ng/ml
TPF CI FPF CI TPF CI FPF CI TPF CI FPF CI
GG
G1
1 9 (8 - 10) 0.2 (0.2 - 0.3) 5 (4 - 7) 0.1 (0.0 - 0.1) 3 (2 - 4) 0.0 (0.0 - 0.0)0.85 10 (8 - 11) 0.3 (0.2 - 0.3) 6 (5 - 7) 0.1 (0.0 - 0.1) 3 (3 - 4) 0.0 (0.0 - 0.0)0.7 11 (9 - 13) 0.3 (0.2 - 0.3) 7 (6 - 8) 0.1 (0.0 - 0.1) 4 (3 - 5) 0.0 (0.0 - 0.0)0.55 12 (11 - 14) 0.3 (0.2 - 0.3) 7 (7 - 8) 0.1 (0.1 - 0.1) 4 (4 - 5) 0.0 (0.0 - 0.0)0.4 14 (12 - 16) 0.3 (0.2 - 0.3) 9 (7 - 11) 0.1 (0.1 - 0.1) 5 (4 - 7) 0.0 (0.0 - 0.0)0.25 18 (15 - 21) 0.3 (0.1 - 1) 11 (9 - 14) 0.1 (0.0 - 0.4) 7 (5 - 9) 0 (0.0 - 0.5)0.1 23 (19 - 27) 0.3 (0.1 - 1) 15 (12 - 19) 0.1 (0.0 - 1) 9 (6 - 12) 0 (0.0 - 1)CC 24 (23 - 26) 1 (1 - 1) 16 (15 - 17) 0.1 (0.1 - 0.1) 9 (8 - 10) 0.0 (0.0 - 0.0)
GG
G2-3
1 28 (26 - 29) 2 (1 - 2) 16 (15 - 18) 0.4 (0.2 - 1) 9 (7 - 10) 0.1 (0.0 - 0.3)0.85 29 (27 - 30) 2 (1 - 2) 17 (15 - 18) 0.4 (0.2 - 1) 9 (8 - 10) 0.1 (0.0 - 0.3)0.7 30 (28 - 32) 2 (1 - 2) 18 (16 - 19) 0.4 (0.2 - 1) 9 (8 - 11) 0.1 (0.0 - 0.3)0.55 32 (30 - 34) 2 (1 - 2) 19 (17 - 21) 0.4 (0.2 - 1) 10 (9 - 12) 0.1 (0.0 - 0.3)0.4 34 (32 - 35) 2 (1 - 2) 20 (18 - 22) 0.4 (0.3 - 1) 11 (9 - 12) 0.1 (0.0 - 0.3)0.25 37 (34 - 39) 2 (1 - 2) 22 (20 - 25) 0.5 (0.3 - 1) 12 (10 - 14) 0.1 (0.0 - 0.3)0.1 41 (39 - 44) 2 (2 - 3) 26 (23 - 28) 1 (0.4 - 1) 14 (12 - 17) 0.2 (0.1 - 0.4)CC 41 (40 - 42) 2 (2 - 2) 24 (23 - 25) 0.5 (0.5 - 0.5) 13 (12 - 13) 0.2 (0.2 - 0.2)
GG
G4-5
1 38 (37 - 39) 5 (4 - 6) 23 (22 - 25) 1 (1 - 2) 14 (12 - 15) 0.4 (0.3 - 1)0.85 39 (38 - 40) 5 (5 - 6) 24 (23 - 25) 1 (1 - 2) 14 (13 - 15) 0.4 (0.3 - 1)0.7 40 (39 - 41) 5 (5 - 6) 25 (23 - 26) 1 (1 - 2) 14 (13 - 16) 0.4 (0.3 - 1)0.55 41 (40 - 42) 5 (5 - 6) 26 (24 - 27) 1 (1 - 2) 15 (14 - 16) 0.4 (0.3 - 1)0.4 43 (41 - 44) 6 (5 - 6) 27 (26 - 28) 2 (1 - 2) 16 (15 - 17) 0.4 (0.3 - 1)0.25 45 (43 - 46) 6 (6 - 7) 28 (27 - 30) 2 (1 - 2) 17 (16 - 18) 0.5 (0.3 - 1)0.1 48 (46 - 49) 7 (6 - 8) 31 (29 - 32) 2 (2 - 3) 19 (17 - 20) 1 (0.4 - 1)CC 46 (45 - 46) 5 (5 - 5) 29 (28 - 29) 1 (1 - 2) 17 (16 - 17) 1 (1 - 1)
More importantly, the false positive fractions were barely affected by the choice of k, as can be seenin table 9, but the true positive fractions increased with decreasing k. Although true positive fractionswere notably larger in most strata for complete cases compared to the imputations, the same appliedto false positive fractions. Also, for all k, when the false positive fractions were small the true positivefractions were also small.
In summary, the test performance using PSA was mainly affected by decreasing k through increasingpositive fractions, and while the prevalence of metastatic was strongly affected and decreased with k, thesurvival curves were overall only moderately affected.
50
0 ≤ PSA < 20
0 2 4 6 8 10
Pro
port
ion
of death
k=
1
00.2
0.6
1
A.I. M0A.I. M1C.C. M0C.C. M1
20 ≤ PSA < 200
0 2 4 6 8 10
PSA ≥ 200
0 2 4 6 8 10
0 ≤ PSA < 20
0 2 4 6 8 10
20 ≤ PSA < 200
0 2 4 6 8 10
PSA ≥ 200
0 2 4 6 8 10
GGG 1-3 GGG 4-5
Years since
diagnosis:
0 ≤ PSA < 20
0 2 4 6 8 10
Pro
port
ion
of death
k=
0.8
5
00.2
0.6
1
20 ≤ PSA < 200
0 2 4 6 8 10
PSA ≥ 200
0 2 4 6 8 10
0 ≤ PSA < 20
0 2 4 6 8 10
20 ≤ PSA < 200
0 2 4 6 8 10
PSA ≥ 200
0 2 4 6 8 10
Years since
diagnosis:
0 ≤ PSA < 20
0 2 4 6 8 10
Pro
port
ion
of death
k=
0.7
00.2
0.6
1
20 ≤ PSA < 200
0 2 4 6 8 10
PSA ≥ 200
0 2 4 6 8 10
0 ≤ PSA < 20
0 2 4 6 8 10
20 ≤ PSA < 200
0 2 4 6 8 10
PSA ≥ 200
0 2 4 6 8 10
Years since
diagnosis:
0 ≤ PSA < 20
0 2 4 6 8 10
Pro
port
ion
of death
k=
0.5
5
00.2
0.6
1
20 ≤ PSA < 200
0 2 4 6 8 10
PSA ≥ 200
0 2 4 6 8 10
0 ≤ PSA < 20
0 2 4 6 8 10
20 ≤ PSA < 200
0 2 4 6 8 10
PSA ≥ 200
0 2 4 6 8 10
Years since
diagnosis:
0 ≤ PSA < 20
0 2 4 6 8 10
Pro
port
ion
of death
k=
0.4
00.2
0.6
1
20 ≤ PSA < 200
0 2 4 6 8 10
PSA ≥ 200
0 2 4 6 8 10
0 ≤ PSA < 20
0 2 4 6 8 10
20 ≤ PSA < 200
0 2 4 6 8 10
PSA ≥ 200
0 2 4 6 8 10
Years since
diagnosis:
0 ≤ PSA < 20
0 2 4 6 8 10
Pro
port
ion
of death
k=
0.2
5
00.2
0.6
1
20 ≤ PSA < 200
0 2 4 6 8 10
PSA ≥ 200
0 2 4 6 8 10
0 ≤ PSA < 20
0 2 4 6 8 10
20 ≤ PSA < 200
0 2 4 6 8 10
PSA ≥ 200
0 2 4 6 8 10
Years since
diagnosis:
0 ≤ PSA < 20
0 2 4 6 8 10
Pro
port
ion
of death
k=
0.1
00.2
0.6
1
20 ≤ PSA < 200
0 2 4 6 8 10
PSA ≥ 200
0 2 4 6 8 10
0 ≤ PSA < 20
0 2 4 6 8 10
20 ≤ PSA < 200
0 2 4 6 8 10
PSA ≥ 200
0 2 4 6 8 10
Years since
diagnosis:
Figure 10: Sensitivity Analysis on Kaplan-Meier survival curves of Imputed and Complete cases, stratifiedaccording to Gleason Grade Group (GGG) and PSA.
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6 Discussion and Conclusions
6.1 The Imputation
The amount of missing data was low for all variables except M stage, and these were satisfactory im-puted. The number of iterations seemed to be sufficient and the number of multiple imputations seemedsatisfactory. The effect of m > 20 was negligible and resulted in longer computation time. The ignorableimputation model for M stage with k = 0.7 showed a low proportion of M1 among men with unknown Mstage, which agrees with findings in Makarov et al (2013) and Tomic et al (2016). In table 10 we see thatmost men with unknown M stage had similar general characteristics but different clinical properties ofthe PCa compared to men with known M stage, indicating relatively lower risk PCa. A consequence ofthis is that most unknown M stages classified by the imputation model have lower GGG and PSA levels,which also agrees with the low proportion of M1. It also means that a potential violation of the MARassumption made during the modelling of missing data would have had the most significant impact onresults in this region of the data, which was reflected in the sensitivity analysis. The variable mode ofdetection which indicates how the PCa was revealed does not specify whether symptoms originated fromthe skeleton, and this unmeasured information could potentially be of great value when classifying un-known M stages. Therefore, a nonignorable missing data mechanism was considered, and the hypothesiswas that an ignorable model would overestimate the prevalence of M1 disease in the region of low PSAand GGG.
Following Siddique et al (2014), a simple nonignorable model parametrised by k was applied, and asensitivity analysis was conducted accordingly. No prior information on k was available, other than thatk ∈ (0, 1). The reason to why k < 1 was that metastatic PCa is rare and that the indicator of symptomsin the skeleton would likely correlate well with metastatic PCa and render less men with MX imputed asM1. There was no reason to believe that survival curves for men with imputed M stage and men withknown M stage should differ significantly, although no arguments could be found to motivate why theyshould be exactly the same. Therefore, these were used carefully to assess reasonable values for k, wherea good separation of imputed survival curves and reasonable agreement with complete cases was desired.Smaller k had perhaps too great influence on the survival curves, affecting the survival curves of menwith M0 and higher PSA to an unreasonable degree. On the other hand, k closer to 1 rendered resultsalmost indistinguishable from the ignorable model (k = 1).
One could possibly imagine more complicated nonignorable models for imputing M stage, for examplea more extensive pattern-mixture model or a selection model, compare with Rubin (2004) or Buuren(2012). These would likely include more parameters and due to the lack of precise prior knowledge aboutthese the modelling would have been less feasible and more arbitrary. The choice of k for the more detailedanalysis landed at k = 0.7 with the argument that it mainly led to an improvement of the model for menwith lower PSA and GGG while virtually not affecting the other. Interestingly, the sensitivity analysissubsequently conducted indicated that the choice of k did not seem to influence the main conclusions ontest performance using PSA, see below, although the prevalence of M1 disease was sensitive to the choiceof k, meaning that the prevalence numbers reported could be larger than a priori expected, maybe evenof the same magnitude as complete cases. Since the predictive values depend on the prevalence, theseshould also be handled with care.
The different parameter transformations explored for the Kaplan-Meier estimate and quantities relatedto the ROC curve seemed to be equally valid in terms of normal approximation, and all gave virtually thesame point estimates. A reasonable guess to why the bootstrap CIs were narrower is because they onlyaccount for the between variance and not the within variance of the parameter estimate. Therefore thebootstrap CIs were not considered subsequently. The only difference between the different transformationswas the length of the CIs therefore the log-log transformation was chosen for survival probabilities sinceit seems to be the current practice, compare with Marshall et al (2009), and also for other probabilitieslike positive fractions, since they are also bounded to the interval [0, 1].
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6.2 Diagnosing Metastatic Prostate Cancer using PSA
6.2.1 Defining a Good Test
In order to define what acceptable test performance means one must discuss the consequences of misclas-sification. Let us say a threshold c for PSA is used such that men with PSA ≥ c are considered to havemetastatic PCa.
Firstly, men with metastatic PCa with PSA< c will be treated as men without metastases, meaningthat they will receive some form of curative treatment, like RP or RT. The time and resources spenton trying to cure these men will be wasted since there is no research indicating that curative treatmenthas effect on metastatic PCa, and the patient will likely die by the PCa. In addition, the side effects ofcurative treatment would negatively affect the life quality and one can imagine that performing surgeryon the prostate could potentially allow for some cancer to spread to other parts of the body and increasingthe rate of spreading of metastases, thus shortening the already short life expectancy. Moreover, Berg etal (2007) concluded that there is a slightly increased survival benefit for men with metastatic disease whoreceive new life-prolonging treatments, meaning that it is important to identify metastatic PCa. On thepositive side, the patient will experience that efforts have been made to help him and therefore it mightbe easier to accept the tragic fate. Possibly, during the treatment process, hopefully before an invasivesurgery, symptoms of metastatic disease will show and appropriate actions can be made, such as imagingand/or change of treatment to ADT. Therefore, a false negative classification is undesirable and shouldbe minimized but does not significantly affect life expectancy.
On the contrary, men with non-metastatic PCa with PSA≥ c will be non-curatively treated, experienceside effects of ADT treatment, and most importantly, lose years of life since they likely have a curablePCa. Rider et al showed that men treated with non-curative intent but with metastases (or PSA ≥ 100ng/ml) had significantly worse 15 year PCa prognosis compared to men without metastases, meaningthat it is important to correctly diagnose metastatic PCa as these could benefit from curative treatment.This agreed with findings in fig. 8, where the mortality was lower for men with M0 than men with M1and the difference remained profound for higher PSA levels in all GGG. The expected years of life lostwithin 15 years from diagnosis was around three to five years when comparing ADT to curative treatment,compare with table 8, although one must note that these numbers are not based on a randomised trialand there might be confounding effects from the selection mechanism that determines the treatment. Theactual unrestricted years of life lost could be significantly larger depending on circumstances, but thereis no reason to believe that ADT treatment of men without metastases would increase life expectancycompared to curative treatment. Therefore, the amount of years of life lost is by far the most severeconsequence of misclassification, and any diagnostic test for metastatic PCa should keep the risk of falsepositives to a minimum.
To conclude, the costs for different treatments, see the Background section, are relatively similar andtherefore not very important when comparing the effects of false positives versus false negatives. Sincethere is no evidence implying that curative treatment of metastatic PCa improves survival, the benefitsof curative treatment to men with metastatic disease are small. Men without metastatic PCa who receiveADT will not cost much but will suffer the side effects of ADT treatment and the patient will not begiven a chance of being cured, ultimately dying earlier than necessary, possibly around three to five ormore years of life lost. The bottom line is that false negatives are significantly less severe comparedto false positives. Therefore, keeping false positives to a minimum and minimizing false negatives is areasonable strategy. For a PSA based test this would mean that a good threshold c is a threshold thatgives a low false positive fraction (for example ≤ 5-10%) and a satisfyingly high true positive fraction.Since the prevalence of M1 disease is high for very high PSA levels it is reasonable to perform imaging formen with high PSA levels to rule out metastases. Therefore, it is reasonable to accept lower true positivefractions, for example below 90%, since metastatic PCa is incurable, while requiring them to be at leastaround 70%, if not higher, depending on how small the false positive fraction is for that threshold, dueto the side effects of a false negative result.
53
6.2.2 Test Performance
Now, the ROC curves, fig. 7, and their corresponding sensitivities and specificities, table 11, show thatthe performance of PSA testing varies between GGG, where GGG 1 stands out by being closer to thediagonal line and thus indicating a worse overall performance. The ROC curves based on imputeddata were similar to those based on complete cases for higher GGG, but differed for lower GGG. Thisis perhaps not unexpected in light of the above discussion, and when looking at fig. 6 we see the samepattern. From fig. 6 we can also see how difficult it was to separate the two populations using a threshold.This is reflected in table 6, which shows that no threshold in any GGG managed to reach reasonableperformance requirements in terms of TPF and FPF, for both complete cases and all choices of k. Thelikelihood ratios did not reach conclusive nor moderate changes from pretest to post-test probability, withonly small changes in some GGG and lower PSA (20 and 50 ng/ml respectively). Since the imaging costis quite low this means all men who have high PSA and/or are under consideration of receiving palliativetreatment for potential metastases should be imaged.
An important remark concerns the use, or more specifically, the over-use, of bone-scans. One argumentto not use bone-scans on men with low PSA is that the prevalence of M1 disease is very low in this groupof men, and it is undesirable to expose a large proportion of men to unnecessary radiation. On the otherhand, this study gave indications that the prevalence numbers in this category of men might be higherthan expected, although this is unlikely with respect to previous findings, for example Makarov et al(2013).
6.3 Weaknesses and Strengths of the Study
The main weakness of this study is evidently the amount of missing data on M stage. But the potentialof the MICE algorithm speaks in favour of the choice of carefully imputing missing data, Sterne et al(2009), and simulation studies have shown that the algorithm performs well in various settings, Buuren etal (2006) and Buuren (2007). Had more data closely correlated with metastatic PCa been available thenan ignorable imputation model might have been sufficient. The chosen model of an nonignorable missingdata mechanism is simple and could possibly have been improved at the expense of complexity, althougha common attitude is to keep the model simple, Saddique et al (2014). In addition, the sensitivity analysisshowed that the conclusions were the same under different choices of k.
One strength of the study is that it was based on comprehensive and generalizable data from NPCR,which has been shown to have high capture rate, accuracy and completeness, Tomic et al (2015a, b), andfrom other high quality health care registers and demographic databases, Hemelrijck et al (2013).
6.4 Conclusions
In this thesis, the performance of the diagnostic test for metastatic PCa using PSA and GGG wasevaluated on complete cases and imputed data. The nonignorable imputation model governed by k,which affected the probability of being classified as M1, generally performed satisfactory. It is unclearwhether the choice of k for further analysis was optimal or if the nonignorable model used to impute Mstage was too simple, although the lack of prior information on parameters would have made a complicatedmodel less feasible. In any case, the sensitivity analysis showed that these conclusions were consistentunder a sequence of different k and with the complete case analysis.
Most importantly, no PSA cutoff could satisfactory separate men with M0 and M1 disease, and neitherpositive fractions or likelihood ratios were adequate for any cutoff when compared to standard rules ofthumb. The arguments for the test performance requirements above contained a subjective componentand one can of course argue differently, but I believe that, in retrospect, this would have little effect onconclusions since the performance in general was quite poor. Also, men with and without metastaseshad distinctively different 10 year mortality in most PSA and GGG strata, meaning that it is importantto correctly diagnose metastatic PCa. The years of life lost (restricted to the first 15 years) due to afalse positive diagnosis also showed the importance of high precision and argues in favour of the chosenrequirements. Indications of relatively high prevalence of metastatic disease for lower PSA levels disagree
54
with previous findings and prevalence numbers were sensitive to imputation model specification, meaningthat these should be handled with care. Lastly, in agreement with Stattin et al (2015), there were noindications that a PSA cutoff of 100 ng/ml is reasonable for indicating metastatic PCa. Also, no PSAcutoff managed to neither rule in disease, meaning all men with high PSA should be imaged, nor ruleout disease, meaning men with moderate PSA levels should be imaged.
7 Recommendations and Future Work
A recommendation based on findings in this thesis is that the PSA threshold of 100 ng/ml as an indicatorof metastatic disease should be removed from current guidelines. If accurate numbers on test performanceof metastatic PCa using PSA is of essence, then it would be desirable to perform a randomized studyinvolving bone scanning and PSA testing, where the prevalence of metastatic PCa in different strata isexamined and the test performance using PSA, GGG and other potential markers is evaluated.
55
8 Appendix
8.1 Baseline Characteristics
M0 M1 MX All
n (%) n (%) n (%) n (%)
Total 24665 (100) 9496 (100) 77852 (100) 112013 (100)Age at diagnosis, years
0-64 7687 (31) 1731 (18) 23427 (30) 32845 (29)65-80 14967 (61) 5524 (58) 43009 (55) 63500 (57)81+ 2011 (8) 2241 (24) 11416 (15) 15668 (14)
Year of diagnosis2000-2003 9887 (40) 3360 (35) 16740 (22) 29987 (27)2004-2007 8344 (34) 2757 (29) 25168 (32) 36269 (32)2008-2012 6434 (26) 3379 (36) 35944 (46) 45757 (41)
Educational levelLow 9881 (40) 4670 (49) 30741 (39) 45292 (40)Middle 9154 (37) 3206 (34) 28834 (37) 41194 (37)High 5433 (22) 1487 (16) 17505 (22) 24425 (22)Missing 197 (1) 133 (1) 772 (1) 1102 (1)
Civil statusWidow 1791 (7) 1111 (12) 6638 (9) 9540 (9)Married/partnership 17362 (70) 5989 (63) 53113 (68) 76464 (68)Unmarried 2218 (9) 1119 (12) 7533 (10) 10870 (10)Divorced/separated 3282 (13) 1271 (13) 10537 (14) 15090 (13)Missing 12 (0) 6 (0) 31 (0) 49 (0)
CCI0 19061 (77) 6375 (67) 58067 (75) 83503 (75)1 3152 (13) 1572 (17) 10307 (13) 15031 (13)2 1635 (7) 892 (9) 5860 (8) 8387 (7)3+ 817 (3) 657 (7) 3618 (5) 5092 (5)
Mode of detectionAsymptomatic 7519 (30) 826 (9) 28084 (36) 36429 (33)Symptomatic 16204 (66) 8367 (88) 46803 (60) 71374 (64)Missing 942 (4) 303 (3) 2965 (4) 4210 (4)
T stageTX 190 (1) 345 (4) 507 (1) 1042 (1)T1a-b 540 (2) 101 (1) 4802 (6) 5443 (5)T1c/abc missing 7452 (30) 698 (7) 36415 (47) 44565 (40)T2 9262 (38) 1813 (19) 23341 (30) 34416 (31)T3 6746 (27) 4650 (49) 11146 (14) 22542 (20)T4 475 (2) 1889 (20) 1641 (2) 4005 (4)
N stageN0 6972 (28) 301 (3) 4793 (6) 12066 (11)N1 1057 (4) 598 (6) 482 (1) 2137 (2)NX 16636 (67) 8597 (91) 72577 (93) 97810 (87)
Gleason Grade Group1 7448 (30) 642 (7) 39375 (51) 47465 (42)2 4925 (20) 783 (8) 14250 (18) 19958 (18)3 3941 (16) 1186 (12) 7252 (9) 12379 (11)4 3690 (15) 2158 (23) 6077 (8) 11925 (11)5 2447 (10) 2744 (29) 4615 (6) 9806 (9)Missing 2214 (9) 1983 (21) 6283 (8) 10480 (9)
56
M0 M1 MX All
n (%) n (%) n (%) n (%)
Total 24665 (100) 9496 (100) 77852 (100) 112013 (100)PSAMedian (Q1-Q3) 16 (9-34) 158 (45-565) 9 (6-18) 11 (6-29)
0-4 1765 (7) 205 (2) 15135 (19) 17105 (15)5-9 5372 (22) 364 (4) 28345 (36) 34081 (30)10-19 6869 (28) 622 (7) 15943 (20) 23434 (21)20-49 6533 (26) 1324 (14) 9642 (12) 17499 (16)50-99 2357 (10) 1312 (14) 3965 (5) 7634 (7)100-199 990 (4) 1294 (14) 1862 (2) 4146 (4)200+ 662 (3) 4272 (45) 2374 (3) 7308 (7)Missing 117 (0) 103 (1) 586 (1) 806 (1)
TreatmentRT 6865 (28) 67 (1) 8431 (11) 15363 (14)ADT 8203 (33) 8961 (94) 20693 (27) 37857 (34)RP 5703 (23) 58 (1) 20766 (27) 26527 (24)CT 3206 (13) 210 (2) 25090 (32) 28506 (25)Unspecified 688 (3) 200 (2) 2872 (4) 3760 (3)
Table 10: Baseline characteristics of men with M stage 0, 1 and unknown (X). RT=Radio therapy,ADT=Adrogen deprivation therapy, RP=Radical prostatectomy, CT=Conservative therapy. Unspecifiedtreatment includes missing, curative therapy other/missing, and death before treatment decision.
8.2 Cox Proportional Hazards: Assessment of Model Assumption
-2 -1 0 1 2 3
-10
-8-6
-4-2
02 Age: 60
GGG: 4
PSA: 50
RP/RT
ADT
-2 -1 0 1 2 3
-10
-8-6
-4-2
02 Age: 60
GGG: 5
PSA: 50
-2 -1 0 1 2 3
-10
-8-6
-4-2
02 Age: 70
GGG: 4
PSA: 50
-2 -1 0 1 2 3
-10
-8-6
-4-2
02 Age: 70
GGG: 5
PSA: 50
-2 -1 0 1 2 3
-10
-8-6
-4-2
02 Age: 60
GGG: 4
PSA: 150
-2 -1 0 1 2 3
-10
-8-6
-4-2
02 Age: 60
GGG: 5
PSA: 150
-2 -1 0 1 2 3
-10
-8-6
-4-2
02 Age: 70
GGG: 4
PSA: 150
-2 -1 0 1 2 3
-10
-8-6
-4-2
02 Age: 70
GGG: 5
PSA: 150
log years since diagnosis
Figure 11: Evaluation of proportional hazards assumption of a Cox proportional hazards model.
57
8.3 Parameter Transformations
Table 11: Tests for normality of positive fractions. TPF=true positive fraction, FPF=false positivefraction. All values are given in percentage. SW=Shapiro-Wilk, AD=Andersson-Darling, CVM=Cramer-von Mises, SF=Shapiro-Francia
P-value (%) Test statistic (%)
CI type Threshold SW AD CVM SF SW AD CVM SF
TPF
basic 10 39 26 24 43 95 44 7 96loglog 41 30 28 45 95 42 7 96logit 41 29 27 45 95 42 7 96basic 100 69 73 79 74 97 24 3 97loglog 69 74 79 75 97 24 3 97logit 68 73 79 74 97 24 3 97basic 500 21 39 49 19 94 37 5 94loglog 20 38 49 18 94 37 5 94logit 20 37 48 17 94 38 5 94basic 1000 98 89 78 94 98 19 3 98loglog 98 89 78 94 98 19 3 98logit 97 89 77 93 98 19 3 98
FPF
basic 10 22 10 7 19 94 60 11 94loglog 22 10 7 19 94 60 11 94logit 22 10 7 19 94 60 11 94basic 100 62 36 25 58 96 39 7 97loglog 61 35 25 58 96 39 7 97logit 61 35 25 58 96 39 7 97basic 500 94 81 73 85 98 22 4 98loglog 93 80 73 83 98 22 4 98logit 92 80 73 82 98 22 4 98basic 1000 49 55 50 59 96 30 5 97loglog 40 48 42 47 95 33 6 96logit 39 47 41 45 95 34 6 96
0.8384 0.8568 0.8753
loglog 100.1%
logit 100.2%
bootstrap 83.2%
basic 100%
PSA = 10
TP
F
0.4246 0.439 0.4533
loglog 100%
logit 100%
bootstrap 68%
basic 100%
PSA = 100
0.1896 0.1984 0.2072
loglog 100%
logit 100%
bootstrap 47.8%
basic 100%
PSA = 500
0.1064 0.1126 0.1188
loglog 100%
logit 100%
bootstrap 37.8%
basic 100%
PSA = 1000
0.4662 0.4783 0.4904
loglog 100%
logit 100%
bootstrap 15.8%
basic 100%
FP
F
0.0343 0.0376 0.0408
loglog 100.1%
logit 100.1%
bootstrap 22.7%
basic 100%
0.0028 0.0039 0.0049
loglog 101%
logit 101%
bootstrap 29%
basic 100%
6e-04 0.0011 0.0017
loglog 103.4%
logit 103.4%
bootstrap 21.9%
basic 100%
Figure 12: Comparison of 95% confidence intervals for positive fractions. The width of a confidenceinterval is given in percent of non-transformed (basic) confidence interval width.
58
Table 12: Tests for normality of Kaplan-Meier estimate at 1, 5 and 10 years from diagnosis. All val-ues are given in percentage. SW=Shapiro-Wilk, AD=Andersson-Darling, CVM=Cramer-von Mises,SF=Shapiro-Francia
P-value (%) Test statistic (%)
CI type Year SW AD CVM SF SW AD CVM SF
M0
basic 0.5 100 99 100 100 99 10 1 99loglog 100 100 100 100 99 10 1 99logit 100 100 100 100 99 10 1 99basic 5 61 46 50 34 96 34 5 95loglog 60 47 51 33 96 34 5 95logit 60 47 51 33 96 34 5 95basic 10 28 38 39 45 94 38 6 96loglog 28 38 39 45 94 38 6 96logit 28 38 39 45 94 38 6 96basic 14.5 24 21 20 35 94 48 8 95loglog 24 21 20 35 94 48 8 95logit 24 21 20 35 94 48 8 95
M1
basic 0.5 70 66 76 47 97 26 4 96loglog 71 66 75 49 97 26 4 96logit 71 66 75 49 97 26 4 96basic 5 27 29 26 42 94 42 7 96loglog 27 29 26 42 94 42 7 96logit 27 29 26 41 94 42 7 96basic 10 79 54 43 67 97 31 5 97loglog 80 55 45 68 97 30 5 97logit 80 56 46 68 97 30 5 97basic 14.5 49 61 60 72 96 28 4 97loglog 51 65 64 74 96 27 4 97logit 53 67 67 76 96 26 4 97
0.0018 0.0022 0.0026
loglog 100.5%
logit 100.5%
bootstrap 51.5%
basic 100%
0.5 years
M0
0.0533 0.0558 0.0583
loglog 100%
logit 100%
bootstrap 51.7%
basic 100%
5 years
0.1438 0.1489 0.154
loglog 100%
logit 100%
bootstrap 39.1%
basic 100%
10 years
0.2343 0.2461 0.2579
loglog 100%
logit 100%
bootstrap 18.1%
basic 100%
14.5 years
0.0554 0.0598 0.0641
loglog 100%
logit 100%
bootstrap 33.1%
basic 100%
M1
0.5669 0.5849 0.603
loglog 100%
logit 100%
bootstrap 61.9%
basic 100%
0.779 0.8012 0.8234
loglog 100%
logit 100.2%
bootstrap 66%
basic 100%
0.844 0.8784 0.9129
loglog 100.6%
logit 101.5%
bootstrap 48.7%
basic 100%
Figure 13: Comparison of confidence intervals at survival times. The width of a confidence interval isgiven in percent of non-transformed (basic) confidence interval width.
59
8.4 Sensitivity Analysis on Prevalence of M1 Disease
Table 13: Sensitivity Analysis on Prevalence of M1 Disease. CC=Complete Cases.
k PSA 0-10 10-20 20-50 50-100 100-200 200-500 500-1000 1000+
GG
G1
1 Total 32006 10518 4800 1251 381 213 109 115M1 1739 (5%) 638 (6%) 527 (11%) 268 (21%) 140 (36%) 127 (60%) 88 (81%) 109 (95%)
0.85 Total 32003 10518 4794 1250 379 214 109 116M1 1532 (5%) 573 (5%) 483 (10%) 253 (20%) 133 (35%) 124 (58%) 87 (81%) 109 (94%)
0.7 Total 32004 10516 4798 1248 381 215 109 118M1 1308 (4%) 499 (5%) 435 (9%) 233 (19%) 129 (34%) 122 (57%) 86 (80%) 111 (95%)
0.55 Total 32004 10520 4799 1253 379 215 109 111M1 1072 (3%) 422 (4%) 388 (8%) 218 (17%) 122 (33%) 118 (55%) 85 (78%) 104 (94%)
0.4 Total 32003 10517 4793 1248 376 212 106 113M1 832 (3%) 340 (3%) 330 (7%) 191 (15%) 113 (30%) 109 (51%) 80 (76%) 106 (93%)
0.25 Total 32006 10520 4801 1250 381 216 108 119M1 579 (2%) 254 (2%) 269 (6%) 164 (13%) 105 (28%) 103 (47%) 80 (74%) 110 (93%)
0.1 Total 32003 10520 4800 1250 380 212 106 114M1 320 (1%) 163 (2%) 200 (4%) 127 (10%) 90 (24%) 90 (42%) 72 (68%) 103 (91%)
CC Total 3063 2361 1840 467 146 85 49 59M1 127 (4%) 81 (3%) 129 (7%) 79 (17%) 59 (40%) 55 (65%) 43 (88%) 56 (95%)
GG
G2-3
1 Total 15160 8852 7374 3058 1466 1080 547 568M1 870 (6%) 819 (9%) 1239 (17%) 878 (29%) 663 (45%) 702 (65%) 465 (85%) 532 (94%)
0.85 Total 15163 8845 7362 3050 1465 1071 542 563M1 783 (5%) 756 (8%) 1158 (16%) 834 (27%) 638 (43%) 683 (64%) 456 (84%) 526 (93%)
0.7 Total 15161 8854 7364 3053 1465 1073 543 550M1 699 (5%) 685 (8%) 1075 (15%) 792 (26%) 616 (42%) 671 (62%) 453 (84%) 511 (93%)
0.55 Total 15164 8852 7373 3056 1469 1079 546 564M1 597 (4%) 603 (7%) 971 (13%) 743 (24%) 585 (40%) 658 (61%) 449 (82%) 524 (93%)
0.4 Total 15166 8857 7368 3058 1467 1077 543 542M1 498 (3%) 516 (6%) 862 (12%) 685 (22%) 550 (38%) 635 (59%) 437 (80%) 499 (92%)
0.25 Total 15160 8849 7361 3052 1469 1073 542 543M1 385 (3%) 413 (5%) 728 (10%) 598 (20%) 501 (34%) 598 (56%) 421 (78%) 496 (91%)
0.1 Total 15164 8846 7363 3053 1463 1073 542 558M1 255 (2%) 297 (3%) 561 (8%) 494 (16%) 429 (29%) 545 (51%) 391 (72%) 497 (89%)
CC Total 1475 1549 1501 559 214 183 94 84M1 61 (4%) 78 (5%) 139 (9%) 112 (20%) 90 (42%) 126 (69%) 87 (93%) 81 (96%)
GG
G4-5
1 Total 4304 4221 5477 3400 2346 2098 1179 1490M1 763 (18%) 956 (23%) 1833 (33%) 1606 (47%) 1423 (61%) 1581 (75%) 1050 (89%) 1438 (96%)
0.85 Total 4308 4228 5491 3409 2352 2105 1184 1493M1 722 (17%) 905 (21%) 1757 (32%) 1560 (46%) 1396 (59%) 1566 (74%) 1047 (88%) 1438 (96%)
0.7 Total 4308 4221 5483 3408 2349 2103 1183 1505M1 677 (16%) 842 (20%) 1659 (30%) 1497 (44%) 1354 (58%) 1537 (73%) 1037 (88%) 1449 (96%)
0.55 Total 4306 4219 5474 3401 2345 2098 1180 1498M1 629 (15%) 771 (18%) 1551 (28%) 1421 (42%) 1307 (56%) 1499 (72%) 1021 (87%) 1438 (96%)
0.4 Total 4303 4225 5484 3405 2347 2098 1185 1518M1 561 (13%) 694 (16%) 1419 (26%) 1333 (39%) 1248 (53%) 1452 (69%) 1008 (85%) 1452 (96%)
0.25 Total 4307 4223 5485 3405 2349 2100 1184 1511M1 486 (11%) 594 (14%) 1251 (23%) 1215 (36%) 1164 (50%) 1380 (66%) 980 (83%) 1436 (95%)
0.1 Total 4312 4224 5484 3406 2353 2106 1184 1499M1 384 (9%) 461 (11%) 1018 (19%) 1033 (30%) 1038 (44%) 1256 (60%) 916 (77%) 1400 (93%)
CC Total 2732 2958 3651 2074 1502 1325 789 1014M1 305 (11%) 372 (13%) 869 (24%) 869 (42%) 903 (60%) 1037 (78%) 714 (90%) 967 (95%)
60
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