Biometrics DOI: 10.1111/j.1541-0420.2011.01715.x

Two-Component Mixture Cure Rate Model with Spline EstimatedNonparametric Components

Lu Wang,1 Pang Du,2,∗ and Hua Liang3

1Novartis Oncology, One Health Plaza, East Hanover, New Jersey 07936, U.S.A.2Department of Statistics, Virginia Tech, Blacksburg, Virginia 24061, U.S.A.

3Department of Biostatistics and Computational Biology, University of Rochester Medical Center, Rochester,New York 14642, U.S.A.∗email: pangdu@vt.edu

Summary. In some survival analysis of medical studies, there are often long-term survivors who can be considered aspermanently cured. The goals in these studies are to estimate the noncured probability of the whole population and thehazard rate of the susceptible subpopulation. When covariates are present as often happens in practice, to understand covariateeffects on the noncured probability and hazard rate is of equal importance. The existing methods are limited to parametricand semiparametric models. We propose a two-component mixture cure rate model with nonparametric forms for both thecure probability and the hazard rate function. Identifiability of the model is guaranteed by an additive assumption that allowsno time–covariate interactions in the logarithm of hazard rate. Estimation is carried out by an expectation–maximizationalgorithm on maximizing a penalized likelihood. For inferential purpose, we apply the Louis formula to obtain point-wiseconfidence intervals for noncured probability and hazard rate. Asymptotic convergence rates of our function estimates areestablished. We then evaluate the proposed method by extensive simulations. We analyze the survival data from a melanomastudy and find interesting patterns for this study.

Key words: Confidence intervals; Convergence rates; Model selection; Nonparametric cure rate; Nonparametric hazard;Penalized EM algorithm; Smoothing spline ANOVA; Two-component mixture cure rate model.

1. IntroductionLifetime data are common in clinical trials and biomedicalstudies. In some survival studies the population under consid-eration consists of two groups of subjects, susceptible and non-susceptible individuals. All susceptible subjects would even-tually experience the failure if there is no censoring, whilenonsusceptible subjects are not at risk of developing suchevents and can be regarded as cured. One well-known exam-ple is cancer studies with long-term survivors. Modern can-cer treatments have substantially improved cure rates. Re-search on many types of cancer, for example breast cancer,non-Hodgkin’s lymphoma, and melanoma, has shown thata significant proportion of patients with these cancers arepermanently cured after therapy; see, e.g., Tai, Yu, Cserni,Vlastos, Royce, Kunkler, and Vinh-Hung (2005). These gen-erate a great interest in and need for proper statistical toolsto analyze cure rate data, with focus on the estimation ofboth the proportion and the failure time distribution of thesusceptible individuals.

Existing cure rate models can be roughly divided into twocategories. One category is promotion cure models first pro-posed in Yakovlev and Tsodikov (1996). Such models assumea survival function Spop(t,x) = G(−a(x, β)F (t)) for the wholepopulation, where x is the covariate vector, a(·, ·) is a knownlink function indexed by unknown parameters β, F (t) is an

unspecified distribution function, and G is a known transfor-mation function. When t is finite, the model gives a semipara-metric form for the survival function of susceptible subjects.When t is infinite, the proportion of nonsusceptible subjectsis given by G(−a(x, β)). Further details of promotion curemodels can be found in Zeng, Yin, and Ibrahim (2006) andreferences therein.

The model we consider belongs to the other category ofcure rate models called two-component mixture cure model.It is developed assuming the study population is a mixture oftwo subpopulations and has a survival function

Spop(t|x, z) = π(z)S(t|x) + 1 − π(z), (1)

where π(z) and S(t|x) are, respectively, the proportion andthe survival function of susceptible subjects. Here z and xare the covariates associated, respectively, with π and S.Two-component mixture cure model was first proposed inBerkson and Gage (1952), where π was simply an unknownconstant and S(t|x) assumed a parametric model. Farewell(1982) proposed an extension with a parametric logistic func-tion π(z). Kuk and Chen (1992) further extended Farewes(1982) work by considering a semiparametric Cox propor-tional hazards model for the hazard function of S(t|x). Theyapplied a marginal likelihood approach and used an estima-tion method involving Monte Carlo simulation. In Peng and

C© 2011, The International Biometric Society 1

2 Biomatrics

Dear (2000) and Sy and Taylor (2000), the model was similarin spirit to that of Kuk and Chen (1992), but the estima-tion was implemented through an expectation–maximization(EM) algorithm (Peng, 2003). The same model was consid-ered in Corbiere,Commenges,Taylor and Joly (2009). Theykept the parametric form of covariate effect in the relativerisk of Cox model and used splines to model the baseline haz-ard function. A direct optimization procedure was proposedthere to estimate the parameters. Lu and Ying (2004) pro-posed a class of semiparametric transformation models incor-porating cure fractions, which included the aforementionedmixture cure rate models as special cases. Othus, Li, andTiwari (2009) extended their model to allow for time-dependent covariates and dependent censoring. More recently,Lu (2010) proposed an accelerated failure time model withcure fraction, where the unknown error density was estimatedby the kernel method. All these papers only considered para-metric logistic regression model for π(z). Hence, the majordrawback of existing cure rate models is that they are limitedto parametric and semiparametric models. More specifically,the hazard function of susceptible subjects takes either a para-metric form such as Weibull distribution or a semiparametricform such as the Cox proportional hazards model, where thelog relative risk is linear in covariate. And the cure rate partoften uses a parametric logistic regression model with liner co-variate effects. In practice, such parametric or semiparametricassumptions may not hold and the analysis tools thus derivedmay not be valid.

To relax such limitations, we propose in this article a fam-ily of nonparametric smoothing spline analysis of variance (SSANOVA) models for cure rate data. In an SS ANOVA decom-position, a multivariate function is decomposed into sum oforthogonal components as main effects and interactions. Foranalysis of cure rate data, such functional ANOVA decom-position can be applied to the mean function in a nonpara-metric logistic regression model of π(z) and the log hazardfunction corresponding to survival function S(t|x). To ensuremodel identifiability, we assume a nonparametric proportionalhazards model for the hazard function, whose relative riskpart also takes a flexible nonparametric form, different fromthe traditional semiparametric proportional hazards model.In the functional ANOVA decomposition, this requires exclu-sion of interactions between time and any covariates, althoughany between-covariate interactions are still allowed.

Our smoothing spline function estimates are defined asthe minimizers of a penalized likelihood (PL), which consistsof the negative log likelihood representing the goodness offit, a roughness penalty enforcing smooth conditions, and asmoothing parameter balancing the tradeoff. In the absenceof penalty, the optimization of likelihood for cure rate data isalready difficult and has to rely on approaches like the EMalgorithm, see, e.g., Peng and Dear (2000), Peng (2003), andSy and Taylor (2000). To optimize the PL in our estima-tion procedure, we develop a penalized EM (PEM) algorithmextending the earlier versions in Green (1990) and Segal,Bacchetti, and Jewell (1994). The PEM algorithm inSegal et al.1994 concerned estimation of finite-dimensionalparameters. Green (1990) considered only estimation of dis-cretized functions, which are essentially finite-dimensional pa-rameters too. In contrast our PEM algorithm is designed for

nonparametric estimation of two smooth functions, both re-siding in infinite-dimension function spaces. So our algorithmcan be considered as a natural extension of these existingPEM algorithms to the case of estimating two smooth func-tions simultaneously. By introducing a latent cure statusvariable y, our PEM algorithm changes the problem to min-imizing a more optimization-friendly penalized complete loglikelihood. The new objective functional consists of two PLs,one involving only the noncure rate function and the otherinvolving only the log hazard function. In the E-step, the con-ditional expectations of yi ’s given the current function esti-mates is computed. In the M-step, the two PLs, both linear inyi ’s, are minimized to obtain the new function estimates. Ourempirical experience indicates a fast convergence of the algo-rithm and a good performance of the resulting estimates. Thisechoes a similar experience for the smoothed EM algorithm inSilverman, Jones, Wilson, and Nychka (1990), where an extrasmoothing step after each M-step yields faster convergence ofthe algorithm.

Point estimation alone is insufficient in practice, as it lacksan assessment of the estimation precision. Typically, point-wise confidence intervals for smoothing spline function esti-mate are derived from the Bayes model of PL as describedin Wahba (1983). However, this approach does not work herebecause our PLs at the M-step involve latent variables yi ’s.Instead, we extend the Louis formula (Louis, 1982), a clas-sical tool for computing observed information matrix whenusing the EM algorithm, to compute the variance estimatesfor our PEM algorithm. Variance estimation for PEM algo-rithm was also considered in Segal et al.1994. But their al-gorithm requires numerical differentiation with respect to theparameters, making it plausible only for finite-dimensionalparameters.

Asymptotic theory for cure rate models is generally diffi-cult and has not been studied rigorously only until the recentdecade. Examples are Fang, Li, and, Sun (2005) for propor-tional hazards mixture cure rate model, Ma (2010) for semi-parametric models with interval censored cure rate data, andthe aforementioned papers, Lu and Ying (2004), zeng et al.(2006), othus et al. (2009), and Lu (2010), for their cure ratemodels. Using the eigenvalue analysis tool developed in Coxand O’Sullivan (1990), we show that our nonparametric func-tion estimates are consistent and their convergence rates areoptimal for spline estimates.

As far as we know, this is the first purely nonparamet-ric method ever proposed for cure rate data. In additionalto its flexibility in modeling, our method offers smooth func-tion estimates that are appealing to practitioners especiallyat the exploratory stage of data analysis. The confidence in-tervals developed here also provide reliable inference toolsnecessary in public health studies. We also develop a sim-ple empirical model selection tool based on the Kullback–Leibler geometry introduced in Gu (2004). Combined with thefunctional ANOVA decomposition associated with smooth-ing splines, this allows flexible and well-informed specifica-tions of the model terms. Numerical studies demonstrateexcellent performance of the proposed method in both es-timation and inference. Illustrative application of the methodto a melanoma study reveals some interesting patterns in thedata.

Nonparametric Mixture Cure Rate Model 3

The rest of the article is organized as follows. Section 2.1gives a detailed description of our nonparametric cure ratemodel and addresses the identifiability problem. Section 2.2introduces the SS ANOVA framework. Section 2.3 proposes aPEM algorithm to estimate the unknown functions. The ob-served information matrix and confidence interval are derivedin Section 2.4. An empirical model selection tool is derived inSection 2.5. Section 2.6 studies the asymptotic properties ofour estimates. We examine a melanoma study in Section 3,and present the simulation results in Section 4. Discussions inSection 5 conclude the article.

2. SS ANOVA Model for Cure Rate Data2.1 Two-Component Mixture Model and IdentifiabilityIn two-component mixture cure model (1), the event time isdecomposed as T = yT ∗ + (1 − y)∞, where T ∗ < ∞ denotesthe failure time of a susceptible subject, and y, generallyunobservable, is an indicator of being susceptible (y = 1) orcured (y = 0). Let z be the covariate associated with cure rateπ(·) and x be the covariate associated with survival compo-nent S(·, ·). The covariates z and x, though not necessarily so,can overlap with each other or even be the same. We assumethat each subject is subjected to random right censoring andthat the censoring time C is independent of T ∗ and y giventhe covariates z and x. Define T = min(T , C) and δ = IT ≤C .Then the observations are (ti , δi , zi ,xi ), i = 1, . . . , n, indepen-dent copies of the random vector (T, δ, z,x). Note that allthe cured subjects are censored and have δi = 0, but somecensored subjects may experience failures beyond the studyperiod. The observed likelihood function for our model is

lobs(π(·), S(·)) ∝n∏

i=1

{π(zi )f (ti ,xi )}δ i

×{1 − π(zi ) + π(zi )S(t,xi )}1−δ i, (2)

where f (t,x) and S(t,x) are, respectively, the probability den-sity function and the survival function of failure time t givencovariate x.Let us first clarify the model identifiability problem. A suf-ficient condition for model (1) to be identifiable is given inTheorem 2 of Li, Taylor, and Sy (2001) as: S(t|x) = {S(t)}r (x),with π(z) and S(t) unspecified, and r(x) > 0 for any x. Notethat this condition basically requires the hazard function forsusceptible subjects to have a proportional hazard structureh(t,x) = h0(t)r(x) with a general form of relative risk r(x), orequivalently, the log hazard function has an additive structurelog h(t,x) = log h0(t) + log r(x). In classic Cox proportionalhazards model, r(x) = exp(βT x) takes a parametric form. Inthis article, we lift such a restriction on r(x) and allow it tobe of a flexible nonparametric form.

Let ζ(z) = log{π(z)/(1 − π(z))} and η(t,x) = log h(t,x).From now on, we always assume η has the aforementionedadditive structure that involves no time-by-covariate inter-actions. Rewriting the observed likelihood (2) in terms of ζand η, the smoothing spline estimate of (ζ, η) is simply theminimizer of the PL

− 1n

log lobs(ζ, η) +β

2J1(ζ) +

λ

2J2(η), (3)

where the first term is negative log likelihood representing thegoodness of fit, J1 and J2 are roughness penalties enforcingcertain levels of smoothness on the functions ζ and η, andβ, λ > 0 are smoothing parameters controlling the tradeoff.

2.2 Smoothing Spline ANOVA

In this section, we give a short generic review of the SSANOVA models generated from a PL like (3). Given stochas-tic data “generated” according to an unknown “pattern” func-tion g0, the smoothing spline estimate of g0 is defined as theminimizer of the PL: L(g|data) + λ

2 J(g). Here L(g), usuallythe negative log likelihood, measures the goodness of fit ofg, J(g), the roughness penalty, measures the smoothness ofg, and the smoothing parameter λ(> 0) controls the trade-off. Through proper specifications of g and J(g) in a vari-ety of problem settings, the PL yields nonparametric modelsfor Gaussian and non-Gaussian regression, probability den-sity estimation, hazard rate estimation, etc. See Gu (2002)for examples.

We now describe how to construct smoothing SS ANOVAmodel for a generic function g(x), where x ∈ X . For exam-ple, x = z in a separate estimation of ζ(z) and x = (t,x) ina separate estimation of η(t,x), separate in the sense to beexplained in Section 2.3. The minimization of the PL is donein a reproducing kernel Hilbert space (RKHS) H of functionson the domain X . An RKHS is a function space where theevaluation functional [x](f ) ≡ f (x) is continuous. For a mul-tivariate x = (x1, . . . , xp ), the domain X = X1 × . . . ×Xp . LetH〈xj 〉 be the RKHSs on Xj , j = 1, . . . , p. Consider the ten-sor product RKHS space H = ⊗p

j=1H〈xj 〉. Suppose that eachcomponent space can be further decomposed as tensor sumsH〈xj 〉 = {1} ⊕H〈1,x j 〉, where {1} is the one-dimension spaceof constant functions. Then the tensor product space can bedecomposed into

H = ⊗pj=1({1} ⊕H〈1,x j 〉)

= H∅ ⊕(⊕p

j=1 Hj

)⊕

(⊕p

j,k=1 Hj k

)⊕H1···p , (4)

where H∅ = {1} ⊗ · · · ⊗ {1} ≡ {1}p , Hj = H〈1,x j 〉 ⊗ {1}p−1,Hj k = H〈1,x j 〉 ⊗ H〈1,xk 〉 ⊗ {1}p−2, . . . , and H1···p = ⊗p

j=1H〈1,x j 〉.In terms of the function, the decomposition in (4) is equivalentto decomposing g as

g(x) = g∅ +p∑

j=1

gj (xj )

+p∑

j,k=1

gjk (xj , xk ) + · · · + g1···p (x1, . . . , xp ), (5)

where g∅ is a constant, gj (xj )’s are the main effects of xj ’s,gjk (xj , xk )’s are two-way interaction effects, and so on. Suchinterpretation of the decomposition parallels that of classi-cal ANOVA linear models and thus earns itself the name SSANOVA decomposition. Any subset of the full SS ANOVAdecomposition may be considered as a model space for g andwill be referred to as an SS ANOVA model.

To overcome the curse of dimensionality, only some termsin (5), usually lower order terms, are included in the modelspace. For our cure rate model, the model space for log haz-ard function η(t,x) excludes any time–covariate interaction

4 Biomatrics

terms, in accordance with the identifiability condition inSection 2.1. Note that any between covariate interactions arestill allowed in η(t,x). After a model space H is chosen, it canbe represented as H = H0 ⊕

∑q

k=1 Hk , where H0 collects allthe unpenalized subspaces. With such a chosen model space,a general form of penalty is J(g) =

∑q

k=1 θ−1k ‖Pk g‖2, where

Pk is the orthogonal projector in H onto Hk , and θk ’s are ad-ditional smoothing parameters to allow for different degreesof smoothness on the components of g. See Wahba (1990)and Gu (2002) for more details. For simplicity of notation,we will assume our penalty functionals J1 and J2 in (3) im-plicitly include multiple smoothing parameters θjk ’s (j = 1, 2)throughout the article.

2.3 Penalized EM AlgorithmWe now introduce a PEM algorithm for the optimiza-tion of (3). Let yi be the unobservable susceptible indi-cator for the ith subject. Given y = (y1, . . . , yn ), the com-plete log likelihood can be written as Lc (ζ, η;y) = L1(ζ ;y) +L2(η;y), where L1(ζ ;y) =

∑n

i=1[yi ζ(zi ) − log{1 + eζ (zi )}] andL2(η;y) =

∑n

i=1{δiη(ti ,xi ) − yi

∫ t i

0 eη (t ,xi )dt}, noting thatδiyi = δi . Because both L1 and L2 involve only one unknownfunction, one can optimize with respect to ζ and η separatelywith y bridging the two parts.

In the E-step, one computes the conditional expectation ofLc with respect to the latent variable yi ’s given the currentestimates Θ(m ) = (ζ (m ), η(m )). Let

y(m )i = E[yi |Θ(m )] = δi

+ (1 − δi )exp

{−

∫ t i

0

eη (t ,xi )dt

}exp{−ζ(zi )} + exp

{−

∫ t i

0

eη (t ,xi )dt

}∣∣∣∣∣∣∣∣

Θ(m )

.

The M-step then minimizes

PL1(ζ |y(m )) ≡ − 1n

L1(ζ ;y(m )) +β

2J1(ζ) and

PL2(η|y(m )) ≡ − 1n

L2(η;y(m )) +λ

2J2(η) (6)

in their respective RKHSs Hζ = {g : J1(g) < ∞} and Hη ={k : J2(k) < ∞} to obtain Θ(m +1) = {ζ (m +1), η(m +1)}. Notethat both objective functions are convex in ζ and η, respec-tively. So the M-step computation can be handled by stan-dard Newton–Raphson procedure whose detail is given in WebAppendix A.

Our PEM algorithm usually converges in less than 10 steps.This fast convergence experience echoes well with the existingresults on a simpler version of PEM algorithm in Green (1990)and the smoothed EM algorithm in Silverman, Jones, Wilson,and Nychka (1990).

2.4 Confidence IntervalsIn this section, we extend the Louis formula to derive point-wise confidence intervals for the functions ζ(z) and η(t,x).In Web Appendix A, we write ζ(z) = ψζ (z)T bζ and η(t,x) =ψη (t,x)T bη , where ψζ , ψη are chosen spline basis functionsand bζ , bη are coefficient vectors. Recall that Θ = (ζ, η),or essentially (bζ ,bη ). The Louis formula for computing

the observed information matrix is Iobs (Θ) = EΘ[B(y; Θ)] −EΘ[G(y; Θ)G(y; Θ)T ]. Here G and B are, respectively, thegradient vector and the negative second derivative matrix ofthe penalized complete log likelihood L(y; (ζ, η)) = L1(ζ,y) +n β2 J(ζ) + L2(η,y) + n λ

2 J(η),After obtaining Iobs following the steps in Web Appendix

B, we can compute the 100(1 − α)% confidence intervals forζ(z0) and η(t0,x0) at given points z0 and (t0,x0) by(

ζ(z0)

η(t0,x0)

)± zα/2Diag

{(ψζ (z0)T 0

0 ψη (t0,x0)T

)I−1obs

×(

ψζ (z0) 0

0 ψη (t0,x0)

)},

where (ζ , η) are the estimates obtained at the end of the PEMalgorithm.

2.5 Model SelectionIn this section, we develop an empirical model selection pro-cedure based on the Kullback–Leibler geometry introducedin Gu (2004). The procedure can be used to detect negligibleterms in the SS ANOVA decompositions of ζ(z) and η(t,x).

Let us consider ζ first. For two estimates ζ1 and ζ2 of thetrue function ζ0, define the Kullback–Leibler distance betweenζ1 and ζ2 as

KL(ζ1, ζ2) =1n

n∑i=1

[exp{ζ1(zi )}

1 + exp{ζ1(zi )}{ζ1(zi ) − ζ2(zi )}

− {log(1 + ζ1(zi )) − log(1 + ζ2(zi ))}]

.

Suppose the estimation of ζ0 has been done in a space of H1

and one wants to assess the possibility of reducing the modelspace to a subspace H2 ⊂ H1. Let ζ be the estimate of ζ0 in H1.Let ζ∗ be the Kullback–Leibler projection of ζ in H2, that is,the minimizer of KL(ζ , ζ) for ζ ∈ H2. Let ζc be the estimatefrom the constant model. Then straightforward calculationyields the following triangle equality:

KL(ζ , ζc ) = KL(ζ , ζ∗) + KL(ζ∗, ζc ), (7)

where KL(ζ , ζc ) is the “total entropy” and KL(ζ∗, ζc ) isthe “preserved entropy” by the subspace H2. Thus ρζ =KL(ζ , ζ∗)/KL(ζ , ζc ) can be used to identify negligible terms,with a small value of ρζ favoring the reduced model.

Similarly, define the Kullback–Leibler distance between twoestimates η1 and η2 as

KL(η1, η2) =1n

n∑i=1

∫ t i

0

[eη 1(t ,xi ){η1(t,xi ) − η2(t,xi )}

−{eη 1(t ,xi ) − eη 2(t ,xi )}]dt.

Then a triangle equality similar to (7) holds and we can usethe ratio ρη = KL(η, η∗)/KL(η, ηc ) to identify negligible termsin the SS ANOVA decomposition of η.

Nonparametric Mixture Cure Rate Model 5

A small threshold of 0.05 has been used in Gu (2004). Ourempirical study in Section 4.2 indicates that this threshold,favoring reduced model when ρζ < 0.05 (or ρη < 0.05), alsoworks well for selecting models of ζ and η in our problemsetting.

2.6 Asymptotic PropertiesIn this section, we will present the convergence rates of ourfunction estimates ζ and η. Its technical proof is in WebAppendix D. Let π0(z), ζ0(z), S0(t,x), and η0(t,x) be thetrue functions, and r1 and r2 be the constants associated withHζ and Hη that measures the smoothness levels enforced bythese two function spaces. A typical value for r1 and r2 are2m when order-m splines are used for modeling ζ and η. Thenwe have the following theorem.

Theorem 2.1: Under Conditions A1–A6, we have ‖ζ −ζ0‖2

2 = Op (n−r / (r+1)), ‖η − η0‖22 = Op (n−r / (r+1)), where ‖ · ‖2 is

the L2-norm and r = min(r1, r2).

Note that this is the optimal convergence rate of spline es-timates when splines of order r/2 are used. If r1 = r2, bothfunction estimates ζ and η obtain their optimal convergencerates. Otherwise, only one of them can achieve the optimalrate, and the other cannot, suffering from entangled jointestimation.

3. Analysis of Melanoma Cancer DataWe now examine a data set from a melanoma study byusing of the proposed method. The data set is availablefrom the Surveillance Epidemiology and End Results (SEER)(www.seer.cancer.gov) database released in 2008. Specifi-cally, we looked at patients diagnosed with melanoma from allthe nine registered metropolitan areas whose cancer stage wasclassified as local or regional. To avoid potential confoundingrelated to previous cancer diagnoses, we restricted our sampleto the patients whose melanoma was their first cancer diag-nosis. Our “failure time” of interest was time from diagnosisof melanoma to death from melanoma. A question of interestwas whether survival or cure fractions differed in this data setby gender, tumor size, and age. Due to the small number ofsubjects in other racial groups, we restricted our attention towhite patients. Melanoma is often followed by routine treat-ments including surgery and radiotherapy for almost all thepatients but those with certain medical conditions that pre-vent such routine treatment. We focused only on the patientswho received routine treatment. A total of 637 cases in theSEER database met all of our selection criteria. The covari-ates considered in our example were age at diagnosis (range: 5to 101 years), gender (M or F), and tumor size (big or small).

A plot of the Kaplan–Meier curves for four patient groupsstratified on age is provided in Figure 1. The age cutoffs werechosen to yield similar group sizes. The plateau shown at theend of each curve suggests the possible existence of a subpop-ulation of cured subjects in the study. Thus a cure rate dataanalysis is appropriate here.

We started with the model including all the main and inter-action effects for covariates z = x = (age, gender, size) such

0.0

0.2

0.4

0.6

0.8

1.0

time after diagnosis (months)

surv

ival

pro

babi

lity

0 50 100 150 200

Figure 1. Melanoma example: Kaplan–Meier estimates ofpatient groups stratified on age. The lines, from thinner tothicker, correspond to the patient groups younger than 40(149 patients), between 40 and 50 (174 patients), between 55and 70 (172 patients), and older than 70(142 patients).

that

ζ(age, gender, size) = ζ∅ + ζa (age) + ζg (gender) + ζs (size)

+ ζag(age, gender) + ζas(age, size)

+ ζgs(gender , size)

+ ζags(age, gender , size), (8)

and η(t, (age, gender, size)) excludes only the time–covariateinteractions to ensure model identifiability. We then did astepwise backward selection using the model selection proce-dure with a threshold of 0.05 for ρζ and ρη . The proceduresuggested no possible reduction for the model of ζ but themodel of η can be reduced to

η(t, age) = η0 + ηt (t) + ηa (age). (9)

Several conclusions can be drawn from the final models (8)and (9). First, all the covariates, age, gender, and tumor size,were associated with the curing status of a patient, with eachgender-size group having a different trend of cure rate versusage. These distinct trends are clear in Figure 2. Second, al-though gender and tumor size were important in determininga patient’s curing status, they had no effects on a patient’ssurvival time once the patient was deemed to fail. Instead,age of such a patient was important for the patient’s survival.

We analyzed the data using the final models (8) for ζ and(9) for η. The corresponding fits together with their point-wise confidence intervals, are plotted in Figures 2 and 3. InFigure 2, we see that male patients with big tumors gener-ally had higher noncure rate than male patients with smalltumors. Also, the noncure rate for both male patient groupshad a common pattern: the noncure rate kept on increasingwith age until around age 65, where the noncure rate leveled

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0 20 40 60 80 100

−4

−3

−2

−1

01

23

male, small tumor

age (years)

logi

t(π(z

))

0.0

0.2

0.4

0.6

0.8

1.0

y’s

0 20 40 60 80 100

−4

−3

−2

−1

01

23

female, small tumor

age (years)

logi

t(π(z

))

0.0

0.2

0.4

0.6

0.8

1.0

y’s

0 20 40 60 80 100

−4

−3

−2

−1

01

23

male, big tumor

age (years)

logi

t(π(z

))

0.0

0.2

0.4

0.6

0.8

1.0

y’s

0 20 40 60 80 100

−4

−3

−2

−1

01

23

female, big tumor

age (years)

logi

t(π(z

))

0.0

0.2

0.4

0.6

0.8

1.0

y’s

Figure 2. Estimated logit noncure rates and their confidence intervals against age for the four patient groups determined bygender and tumor size (big or small). Superimposed are true data points with positions determined by age and converged y’s.

off. Interestingly, such a pattern did not appear in the femalegroups, both of which showed steady increasing noncure rateagainst age. Although female patients with bigger tumors stillshowed higher noncure rates than those with smaller tumors,the noncure rate of the latter group increased in a faster pacealong with age and caught up quickly with the noncure rateof the former group. Figure 3 revealed the hazard patternfor noncured patients. Although the plotted log hazards werefixed at certain age or time after diagnosis, the trend revealedcan be generalized to any fixed age or time after diagnosisdue to the additive structure in (9). For any fixed-age pa-tient group, the hazard of failure first increased up to about30 months after diagnosis, then started to decrease up to

about 60 months after diagnosis, and increased steadily af-terward. This seems to coincide with the popular treatmentprocedure for melanoma patients. In general, a patient isfirst treated with common procedures following the diagno-sis, whose hazard increases as time moves on. After a while, ifcommon procedures do not work, more aggressive procedureswill then be adopted and can indeed reduce the patient’s haz-ard for a period of time. If the aggressive procedures stillcould not cure the patient, the failure hazard of the patientwill eventually pick up again. When looked at a cross sectionof certain time point after diagnosis, the failure hazard of apatient was constant until around 60 years old, and increasedsteadily afterward.

Nonparametric Mixture Cure Rate Model 7

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Figure 3. Estimated log hazard and confidence intervals against time at age = 55 years and against age at time = 28 months.

4. Empirical Studies4.1 Estimation and Coverage PropertiesThis section presents some simulations to evaluate the esti-mation performance of the proposed method and the coverageproperties of the confidence intervals derived in Section 2.4.

We considered the following simulation setting of test curerate and hazard functions.

π1(z) = 0.1722 + 0.7 sin{2(z + 0.6)},

h1(t, x) =2.5t1.5

{1 + 0.5 sin(2πx)}2.5 ,

where z and x are continuous covariates. The constants inπ1 were chosen to yield 20% overall cure probability of allsubjects. Note that the hazard function has a logarithm freeof time–covariate interactions to ensure model identifiability.One more function setting was considered and each func-tion setting was studied with two sample sizes n = 400 and800. Due to the similarity of the simulation results, we onlypresent the results of the above function setting with n = 400here, and put the details of other simulations into WebAppendix C.

The covariates zk = xk , k = 1, . . . , 20 were generated as agrid of 20 equally spaced values over the range [−0.4, 0.4].Then 20 observations were generated at each zk = xk : first,20 binary values were generated from the Bernoulli distribu-tion with probability πj (zk ); then the 20 observations wereclassified as either cured or not cured based on these binaryoutputs; failure times were randomly generated for the non-cured observations from the distribution with hazard functionhl (t, xk ); finally, for all the observations, censoring times weregenerated from Weibull distributions and the censoring sta-tus indicators were recorded. Note that all the cured sampleswere recorded as being censored.

One hundred replicates were generated. The point-wise95% confidence intervals were calculated for logit noncure rateζ(z) on a z grid of size 100 equally spaced on [−0.4, 0.4], forlog hazard η(t, xtest ) with x fixed at certain point xtest from[−0.4, 0.4], and t on 100 equally spaced grid points on thecommon range of t for all the hundred replicates, and for loghazard η(ttest , x) with t fixed at certain point ttest within the

above common range of t, and x on 100 equally spaced gridpoints on [−0.4, 0.4].

Figure 4 plots the simulation results. The rows from topto bottom represent functions ζ(·), η(·, xtest ) and η(ttest , ·), theleft frames show point-wise coverage of the 95% interval esti-mates of the three functions at the selected grid points, theright frames plot the true test functions (dash–dotted), theaverages of point-wise function estimates (solid), the averagesof point-wise 95% CIs (dashed), and the empirical 2.5% and97.5% percentiles of point-wise function estimates (dotted).Also superimposed in the left frames are the magnitudes ofthe curvatures of the corresponding curves.

The estimation performance is very good with the meanfunction estimates close to the true functions. For the confi-dence intervals, the mean interval estimates are close to theempirical percentiles of the 100 function estimates, so our in-terval estimates are of proper magnitude. It is reassuring tosee the widening of the intervals at the ends of the curveswhere information from the data is vanishing. The point-wisecoverage is generally close to the nominal level 0.95 with a bitundercoverage in certain areas. A couple of factors may havecontributed here. First, Wahba (1983) observed association oflow coverage with high curvature in regression settings. Thisappears to be the case in our simulations as well, as shown inthe left frames of Figure 4. In general, higher curvature impliesa rougher curve which is more difficult to be recovered fullyby nonparametric smoothing methods. Second, low coveragealso seemed to happen at both ends of the data range wheredata are sparse and information for nonparametric method isdwindling.

4.2 Model SelectionWe considered three settings to assess the model selectionprocedure in Section 2.5. We used 0.05 as the cutoff valuefor ρζ and ρη , which corresponds to 5% of entropy loss mea-sured by the Kullback–Leibler distance if reduced model isadopted. All settings were iterated for 100 times with samplesize n = 400. We focused on model selection for log hazard ηin the first and second settings and for ζ in the third setting.These simulations provide empirical bases for a backward

8 Biomatrics

−0.4 −0.2 0.0 0.2 0.4

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haza

rd (

time=

0.9)

Figure 4. Simulation results for test functions π1(z), h1(t, x) and n = 400 (Section 4.1). Left column: point-wise coverages(stepped lines). Superimposed are nominal coverage (dotted lines) and scaled |ζ ′′(z)| (dashed lines). Right column: truefunctions (dash–dotted) and their estimates, including averages of point-wise function estimates (solid), averages of point-wise 95% CIs (dashed), and empirical 2.5 and 97.5 percentiles of point-wise function estimates (dotted), all based on 100 datareplicates.

application of the model selection procedure to the examplein Section 3.

In the first setting, we introduced a binary variable xc =0 or 1. The true functions were ζ(z) = logit(π1(z)) andη(t, (x, xc )) = log(h1(t, x)), where logit(p) = log(p/(1 − p)).

Note that the log hazard η(t, (x, xc )) does not depend on xc

and its components of t and x are separate from each other.Hence, the true model of η is additive with only main effectsof time and continuous covariate x, or (t, x) using short-handnotation. We considered two scenarios: models (t, x) versus

Nonparametric Mixture Cure Rate Model 9

Table 1Simulation result for model selection

Proportion of

True Underfit Correct OverfitSetting Function model (%) fit(%) (%)

I η (t, x) 0 94 6II η (t, xc ) 0 94 6III ζ (z, zc ) 3 93 4

(t, x, xc , x ∗ xc ) and models (t) versus (t, x). The percentagesof selecting the correct model (t, x) were 94% and 100%, re-spectively, in these two scenarios.

The second setting still focused on model selection for thehazard part. The true cure rate probability function π(z) wasthe same as in the first setting but the true log hazard functionwas η(t, (x, xc )) = log xc + 1.5 log t, where xc = 2.5 or 0.44 isa categorical variable with two levels. Thus the true modelfor log hazard η is (t, xc ). We again considered two scenarios:models (t, xc ) versus (t, x, xc , x ∗ xc ) and models (t) versus(t, xc ). The percentages of selecting the correct model (t, xc )were 94% and 100%, respectively, in these two scenarios.

The third setting focused on model selection for the curerate component and introduced an additional categoricalvariable zc with two levels. The true functions were ζ(z, zc ) =3.2zc + logit[−0.0278 + 0.7 sin{2(−z + 0.6)}] and η(t, x) =log{h1(t, x)}, where zc is a categorical variable with 2 levels0 and 1 Thus the true model for function ζ was (z, zc ). Weconsidered three scenarios: models (z, zc ) versus (z, zc , z ∗ zc ),models (z) versus (z, zc ) and models (zc ) versus (z, zc ). Thepercentages of correct selection were 96%, 100%, and 97%,respectively.

Table 1 summarizes the model selection results in termsof proportions of underfit, where some true effect(s) is notselected, correct fit, and overfit, where some noise effect is se-lected. Overall the results are very good with high percentages(> 90%) of correct fit in all the three settings. Hence, we willrecommend a threshold of 0.05 to use in practice.

5. DiscussionThis article proposes a family of nonparametric models forcure rate data based on the smoothing SS ANOVA frame-work. Both the probability function of being susceptible andthe hazard function of susceptible subjects assume flexiblenonparametric forms and are shown to have consistent es-timates. Such flexibility is particularly important at the ex-ploratory stage of data analysis. While a misspecified para-metric model may yield inaccurate prediction, nonparametricmodels like the proposed functional ANOVA models can offermore reliable information for disease prognosis.

Inference tools we develop include point-wise confidenceintervals derived from the Louis formula for our PEM al-gorithm, and a simple empirical model selection tool basedon the Kullback–Leibler geometry. More interesting inferencetools are simultaneous confidence bands. However, they arehard to establish with rigorous theory due to the lack ofsampling distributions for the estimates. Alternatively, onemay extend the bootstrap hypothesis test procedures in Liu,

Meiring, and Wang (2005) for regression settings to our curerate model setting.

Some other extensions merit future investigation. Incor-poration of time-dependent covariates is straightforward fol-lowing othus et al.2009 but needs nontrivial changes in thecomputing program. Based on latent activation schemes,Cooner,Banerjee,Carlin and Sinha (2007) proposed a class ofcure rate models that include both the mixture-componentmodels and the promotion time models as special cases, andGu, Sinha, and Banerjee (2011) considered a proportionalodds model with a parametric Weibull model for the sur-vival function of latent activation times. Extensions in thisdirection requires tricky treatment of nonidentifiability, espe-cially when some covariates affect both the cure probabilityand the event time. Last, the noncured subpopulation mightconsist of two subgroups, one aggressive and the other mild.The inclusion of this additional mixing dimension would re-quire modeling a latent trinomial response and two distincthazard functions. To ensure identifiability, one may need torestrict the number of new parameters to be introduced inthe additional component.

The sample sizes used in the numerical studies are oftenconsidered medium in general survival studies. Estimationwith smaller sample sizes is possible but practical identifi-ability may affect the estimation stability. Because cure ratedata often requires relatively bigger sample sizes to claim adisease “curable,” this should not limit the practical use ofthe proposed method too much.

Under parametric and semiparametric mixture cure ratemodels, numerical nonidentifiability often occurs when cen-soring after the largest failure time is low. One remedy is toenforce the zero-tail constraint in Taylor (1995), which es-sentially assumes all the observations after the largest failuretime are cured. Although our simulations do not show sucha problem in the proposed nonparametric cure rate model, itdoes not completely rule it out. In case it happens, one canincorporate the zero-tail constraint into our estimation pro-cedure by zeroing out all the yi ’s of the censored observationsafter the largest failure time.

6. Supplementary MaterialsThe Web Appendices referenced in Sections 2.3, 2.4, 2.6, and4.1 are available under the Paper Information link at the Bio-metrics website http://www.biometrics.tibs.org/.

Acknowledgements

The authors thank the editor, the associate editor, and twoanonymous referees for very useful comments that improvedthe presentation of the article. The research of PD is sup-ported by NSF DMS-1007126. HL’s research is supported byNSF DMS-0806097 and DMS-1007167.

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Received February 2011. Revised September 2011.Accepted September 2011.