An Extended Hazard Model with Longitudinal Covariates

1 1234567891011121314151617181920212223242 5262728293031323334353637383940414243444 5464748 Biometrika(2014),xx, x,pp. 1 28C 2007 Biometrika TrustPrinted in Great BritainAn Extended Hazard Model with Longitudinal CovariatesBYY. K. TSENG,Graduate Institute of Statistics, National Central University, No. 300, Jhongda Rd., Jhong-Li,Taoyuan County 32049, R. SU,Division of Public Health Sciences, Fred Hutchinson Cancer Research Center, Seattle,Washington 98109, MAOANDJ. L. WANGD epartment of Statistics, University of California, Davis, California 95616, clinical trials and other medical studies, it has become increasingly common to observean event time of interest and Longitudinal Covariates simultaneously.

2 In the literature, joint mod-elling approaches have been employed to analyze both survival and Longitudinal processes and toinvestigate their association. Early attention has mostly been placed on developing adaptive andflexible Longitudinal processes based on a prespecified survival Model , most commonly the Coxproportional Model . In this paper, we propose a general class of semi-parametric Hazard regres-495051525354555657585960616263646 5666768697071727374757677787980818283848 586878889909192939495962Y. K. TSENG, Y. R. SU, M. MAO ANDJ. L. WANG sion models, termed Extended Hazard Model , for the survival component.

3 This class includes twopopular survival models, the Cox proportional hazards Model and the accelerated failure timemodel, as special cases. The proposed Model is flexible for modelling event data, and its nestedstructure facilitates Model selection for the survival component through likelihood ratio tests. Apseudo joint likelihood approach is proposed to estimate the unknown parameters and compo-nents through a Monte Carlo EM algorithm. Asymptotic theory for the estimators is developedtogether with theory for the semiparametric likelihood ratio tests.

4 The performance of the pro-cedure is demonstrated through simulation studies. A case study featuring data from a TaiwanHIV/AIDS cohort study further illustrates the usefulness of the Extended Hazard key words: Hazard smoothing; Joint modelling; Maximum likelihood estimation; Monte Carlo EM algorithm;Semiparametric likelihood ratio INTRODUCTIONIn many medical studies, Longitudinal biomarkers and the event time of interest are collectedsimultaneously in order to explore their association. One main interest of these studies is to de-tect any impact of biomarkers and treatments on the event time.

5 The patterns of the longitudinalbiomarkers are of additional interest. A well-known example is an AIDS clinical trial and cohortstudy where the disease marker CD4 counts and event time of patients are both recorded. Theprimary goal of such a study is to explore the association between Longitudinal CD4 counts andthe event time, and a secondary goal is to Model the patterns of the CD4 trajectories for a betterunderstanding of the Longitudinal time courses; see for instance Pawitan & Self (1993), Tsiatiset al. (1995), Wulfsohn & Tsiatis (1997), Bycott & Taylor (1998) and Wang & Taylor (2001).

6 Partial likelihood estimation (Cox, 1975) for the Cox Model encounters difficulties, because itrequires knowledge of the entire history of Longitudinal biomarkers and does not allow the longi-9798991001011021031041051061071081 0911011111211311411511611711811912012112 2123124125126127128129130131132133134135 136137138139140141142143144 Extended Hazard Model3tudinal Covariates to contain measurement errors. Both requirements might fail in such medicalstudies, thereby inducing biases. Moreover the Longitudinal processes are only accessible beforethe occurrence of the event, such as death, which results in informative missing or dropout.

7 Sev-eral solutions for these difficulties have been proposed in the literature; see Tsiatis & Davidian(2004), Verbeke & Davidian (2008) and Fitzmaurice et al. (2008). The most efficient solutionis the maximum likelihood approach to Model jointly the Longitudinal and event time involves selecting a Longitudinal Model for the biomarkers and a disease risk Model for theevent time this paper, the Longitudinal component is assumed to be a linear mixed effects Model withmeasurement errors,W(t) =X(t) +e(t),(1)X(t) =bT (t),(2)where the covariate processW(t)is observed intermittently with measurement errors or randomfluctuationse(t), soX(t)

8 Is the actual underlying covariate process that links the disease covariate processX(t)is modelled through known basis functions{ 1(t),.. p(t)}T= (t). The measurement errore(t)is independent ofbT= (b1,..,bp), which follows ap-dimensional multivariate normal distributionNp( , ). To accommodate various mechanismsthat generate such data, a number of mixed effects models (1) have been proposed in the liter-ature. Wulfsohn & Tsiatis (1997) considered one with{ 1(t), 2(t)}= (1,t), while Hendersonet al. (2000) and Wang & Taylor (2001) added an extra Gaussian process to explain additionalvariation that cannot be fully described by the random effects and the measurement errors.

9 Tsi-atis & Davidian (2001) and Song et al. (2002) also applied simple linear mixed effects modelsfor the Longitudinal process but relaxed the normality assumption, by invoking a conditional14514614714814915015115215315 4155156157158159160161162163164165166167 1681691701711721731741751761771781791801 811821831841851861871881891901911924Y. K. TSENG, Y. R. SU, M. MAO ANDJ. L. WANG score approach and a flexible class of parametric density functions, respectively. Another flex-ible Model by Brown et al. (2005) sets (t)to be a vector of B-spline basis functions with thenumber of knots determined by a Model selection procedure.

10 Ding & Wang (2008) simplifiedthis by proposing a multiplicative random effects Model with a single one-dimensional the survival component, the Cox proportional hazards Model is usually employed to de-scribe the risk at timet, stipulating {t| X(t)}= 0(t) exp{ X(t)},(3)where X(t) ={X(s) : 0 s < t}is the covariate history up to timet, is the regression pa-rameter, and 0(t)is the unspecified baseline Hazard rate function. To address the challenge thatthe proportional Hazard assumption may fail, Tseng et al. (2005) proposed a joint modellingapproach based on the accelerated failure time Model : {t| X(t)}= 0[ t0exp{ X(s)}ds]exp{ X(t)}.