Semiparametric approaches for joint modeling of longitudinal and survival data with time-varying coefficients

Summary .   We study joint modeling of survival and longitudinal data. There are two regression models of interest. The primary model is for survival outcomes, which are assumed to follow a time-varying coefficient proportional hazards model. The second model is for longitudinal data, which are assumed to follow a random effects model. Based on the trajectory of a subject's longitudinal data, some covariates in the survival model are functions of the unobserved random effects. Estimated random effects are generally different from the unobserved random effects and hence this leads to covariate measurement error. To deal with covariate measurement error, we propose a local corrected score estimator and a local conditional score estimator. Both approaches are semiparametric methods in the sense that there is no distributional assumption needed for the underlying true covariates. The estimators are shown to be consistent and asymptotically normal. However, simulation studies indicate that the conditional score estimator outperforms the corrected score estimator for finite samples, especially in the case of relatively large measurement error. The approaches are demonstrated by an application to data from an HIV clinical trial.     

Estimation in Semiparametric Transition Measurement Error Models for Longitudinal Data

Summary .  We consider semiparametric transition measurement error models for longitudinal data, where one of the covariates is measured with error in transition models, and no distributional assumption is made for the underlying unobserved covariate. An estimating equation approach based on the pseudo conditional score method is proposed. We show the resulting estimators of the regression coefficients are consistent and asymptotically normal. We also discuss the issue of efficiency loss. Simulation studies are conducted to examine the finite-sample performance of our estimators. The longitudinal AIDS Costs and Services Utilization Survey data are analyzed for illustration.     

A Bayesian Semiparametric Survival Model with Longitudinal Markers

Summary We consider inference for data from a clinical trial of treatments for metastatic prostate cancer. Patients joined the trial with diverse prior treatment histories. The resulting heterogeneous patient population gives rise to challenging statistical inference problems when trying to predict time to progression on different treatment arms. Inference is further complicated by the need to include a longitudinal marker as a covariate. To address these challenges, we develop a semiparametric model for joint inference of longitudinal data and an event time. The proposed approach includes the possibility of cure for some patients. The event time distribution is based on a nonparametric Pólya tree prior. For the longitudinal data we assume a mixed effects model. Incorporating a regression on covariates in a nonparametric event time model in general, and for a Pólya tree model in particular, is a challenging problem. We exploit the fact that the covariate itself is a random variable. We achieve an implementation of the desired regression by factoring the joint model for the event time and the longitudinal outcome into a marginal model for the event time and a regression of the longitudinal outcomes on the event time, i.e., we implicitly model the desired regression by modeling the reverse conditional distribution.     

A two-part joint model for the analysis of survival and longitudinal binary data with excess zeros

Summary .   Many longitudinal studies generate both the time to some event of interest and repeated measures data. This article is motivated by a study on patients with a renal allograft, in which interest lies in the association between longitudinal proteinuria (a dichotomous variable) measurements and the time to renal graft failure. An interesting feature of the sample at hand is that nearly half of the patients were never tested positive for proteinuria (≥1g/day) during follow-up, which introduces a degenerate part in the random-effects density for the longitudinal process. In this article we propose a two-part shared parameter model framework that effectively takes this feature into account, and we investigate sensitivity to the various dependence structures used to describe the association between the longitudinal measurements of proteinuria and the time to renal graft failure.     

度量误差对模型参数估计值的影响及参数估计方法的比较研究

基于模型V=aDb,首先在Matlab下用模拟实验的方法,研究了度量误差对模型参数估计的影响,结果表明:当V的误差固定而D的误差不断增大时,用通常最小二乘法对模型进行参数估计,参数a的估计值不断增大,参数b的估计值不断减小,参数估计值随着 D的度量误差的增大越来越远离参数真实值;然后对消除度量误差影响的参数估计方法进行研究,分别用回归校准法、模拟外推法和度量误差模型方法对V和D都有度量误差的数据进行参数估计,结果表明:回归校准法、模拟外推法和度量误差模型方法都能得到参数的无偏估计,克服了用通常最小二乘法进行估计造成的参数估计的系统偏差,结果进一步表明度量误差模型方法优于回归校准法和模拟外推法．     

Bayesian Semiparametric Models for Survival Data with a Cure Fraction

We propose methods for Bayesian inference for a new class of semiparametric survival models with a cure fraction. Specifically, we propose a semiparametric cure rate model with a smoothing parameter that controls the degree of parametricity in the right tail of the survival distribution. We show that such a parameter is crucial for these kinds of models and can have an impact on the posterior estimates. Several novel properties of the proposed model are derived. In addition, we propose a class of improper noninformative priors based on this model and examine the properties of the implied posterior. Also, a class of informative priors based on historical data is proposed and its theoretical properties are investigated. A case study involving a melanoma clinical trial is discussed in detail to demonstrate the proposed methodology.     

Multiple-Imputation-Based Residuals and Diagnostic Plots for Joint Models of Longitudinal and Survival Outcomes

Summary .  The majority of the statistical literature for the joint modeling of longitudinal and time-to-event data has focused on the development of models that aim at capturing specific aspects of the motivating case studies. However, little attention has been given to the development of diagnostic and model-assessment tools. The main difficulty in using standard model diagnostics in joint models is the nonrandom dropout in the longitudinal outcome caused by the occurrence of events. In particular, the reference distribution of statistics, such as the residuals, in missing data settings is not directly available and complex calculations are required to derive it. In this article, we propose a multiple-imputation-based approach for creating multiple versions of the completed data set under the assumed joint model. Residuals and diagnostic plots for the complete data model can then be calculated based on these imputed data sets. Our proposals are exemplified using two real data sets.     

Approximate nonparametric corrected-score method for joint modeling of survival and longitudinal data measured with error

We consider the problem of jointly modeling survival time and longitudinal data subject to measurement error. The survival times are modeled through the proportional hazards model and a random effects model is assumed for the longitudinal covariate process. Under this framework, we propose an approximate nonparametric corrected-score estimator for the parameter, which describes the association between the time-to-event and the longitudinal covariate. The term nonparametric refers to the fact that assumptions regarding the distribution of the random effects and that of the measurement error are unnecessary. The finite sample size performance of the approximate nonparametric corrected-score estimator is examined through simulation studies and its asymptotic properties are also developed. Furthermore, the proposed estimator and some existing estimators are applied to real data from an AIDS clinical trial.     

Semiparametric estimation for joint modeling of colorectal cancer risk and functional biomarkers measured with errors

This research is motivated by a pilot colorectal adenoma study, where the outcome of interest is the presence of colorectal adenoma representing risk for colorectal cancer, and the predictors of interest are protein biomarkers that are repeatedly measured with errors along the length of a microscopic structure in the human colon, the colon crypt. Biomarkers of this type are referred to as functional biomarkers. The investigators are interested in identifying features of functional biomarkers that are associated with risk for colorectal cancer. In this paper, we investigate a joint modeling approach, where the binary clinical outcome is modeled using a logistic regression model with the unobserved true functional biomarkers as the predictors. Most existing methods are developed either for linear models or for functional biomarkers measured without errors and cannot be directly applied to our data. The applicable methods include a two-step method and a maximum likelihood method, which have some limitations. We propose a robust semiparametric method to overcome the limitations of the existing methods. We study the properties of the proposed method, and show in simulations that it compares favorably with other methods and also offers significant savings in CPU time. We analyze the pilot colorectal adenoma data and show that expression levels of AFC, a tumor suppressor gene, in the transitional area from the proliferation zone to the differentiation zone of colon crypts are likely to be associated with risk for colorectal cancer. Given the relatively small sample size in the pilot study, our results need to be validated in the future full-scale studies.     

On Estimating Stratified PH Model with Single Covariate from Sparse Data with Application to Brain Metastases Study

The estimation of the unknown parameters in the stratified Cox's proportional hazard model is a typical example of the trade‐off between bias and precision. The stratified partial likelihood estimator is unbiased when the number of strata is large but suffer from being unstable when many strata are non‐informative about the unknown parameters. The estimator obtained by ignoring the heterogeneity among strata, on the other hand, increases the precision of estimates although pays the price for being biased. An estimating procedure, based on the asymptotic properties of the above two estimators, serving to compromise between bias and precision is proposed. Two examples in a radiosurgery for brain metastases study provide some interesting demonstration of such applications.     

Survival Analysis with Error‐Prone Time‐Varying Covariates: A Risk Set Calibration Approach

Summary Occupational, environmental, and nutritional epidemiologists are often interested in estimating the prospective effect of time‐varying exposure variables such as cumulative exposure or cumulative updated average exposure, in relation to chronic disease endpoints such as cancer incidence and mortality. From exposure validation studies, it is apparent that many of the variables of interest are measured with moderate to substantial error. Although the ordinary regression calibration (ORC) approach is approximately valid and efficient for measurement error correction of relative risk estimates from the Cox model with time‐independent point exposures when the disease is rare, it is not adaptable for use with time‐varying exposures. By recalibrating the measurement error model within each risk set, a risk set regression calibration (RRC) method is proposed for this setting. An algorithm for a bias‐corrected point estimate of the relative risk using an RRC approach is presented, followed by the derivation of an estimate of its variance, resulting in a sandwich estimator. Emphasis is on methods applicable to the main study/external validation study design, which arises in important applications. Simulation studies under several assumptions about the error model were carried out, which demonstrated the validity and efficiency of the method in finite samples. The method was applied to a study of diet and cancer from Harvard's Health Professionals Follow‐up Study (HPFS).     

A Positive Stable Frailty Model for Clustered Failure Time Data with Covariate‐Dependent Frailty

Summary In this article, we propose a positive stable shared frailty Cox model for clustered failure time data where the frailty distribution varies with cluster‐level covariates. The proposed model accounts for covariate‐dependent intracluster correlation and permits both conditional and marginal inferences. We obtain marginal inference directly from a marginal model, then use a stratified Cox‐type pseudo‐partial likelihood approach to estimate the regression coefficient for the frailty parameter. The proposed estimators are consistent and asymptotically normal and a consistent estimator of the covariance matrix is provided. Simulation studies show that the proposed estimation procedure is appropriate for practical use with a realistic number of clusters. Finally, we present an application of the proposed method to kidney transplantation data from the Scientific Registry of Transplant Recipients.     

Score Test for Conditional Independence Between Longitudinal Outcome and Time to Event Given the Classes in the Joint Latent Class Model

Summary .  Latent class models have been recently developed for the joint analysis of a longitudinal quantitative outcome and a time to event. These models assume that the population is divided in  G  latent classes characterized by different risk functions for the event, and different profiles of evolution for the markers that are described by a mixed model for each class. However, the key assumption of conditional independence between the marker and the event given the latent classes is difficult to evaluate because the latent classes are not observed. Using a joint model with latent classes and shared random effects, we propose a score test for the null hypothesis of independence between the marker and the outcome given the latent classes versus the alternative hypothesis that the risk of event depends on one or several random effects from the mixed model in addition to the latent classes. A simulation study was performed to compare the behavior of the score test to other previously proposed tests, including situations where the alternative hypothesis or the baseline risk function are misspecified. In all the investigated situations, the score test was the most powerful. The methodology was applied to develop a prognostic model for recurrence of prostate cancer given the evolution of prostate-specific antigen in a cohort of patients treated by radiation therapy.     

Joint modelling of longitudinal and time-to-event data with application to predicting abdominal aortic aneurysm growth and rupture

Shared random effects joint models are becoming increasingly popular for investigating the relationship between longitudinal and time‐to‐event data. Although appealing, such complex models are computationally intensive, and quick, approximate methods may provide a reasonable alternative. In this paper, we first compare the shared random effects model with two approximate approaches: a naïve proportional hazards model with time‐dependent covariate and a two‐stage joint model, which uses plug‐in estimates of the fitted values from a longitudinal analysis as covariates in a survival model. We show that the approximate approaches should be avoided since they can severely underestimate any association between the current underlying longitudinal value and the event hazard. We present classical and Bayesian implementations of the shared random effects model and highlight the advantages of the latter for making predictions. We then apply the models described to a study of abdominal aortic aneurysms (AAA) to investigate the association between AAA diameter and the hazard of AAA rupture. Out‐of‐sample predictions of future AAA growth and hazard of rupture are derived from Bayesian posterior predictive distributions, which are easily calculated within an MCMC framework. Finally, using a multivariate survival sub‐model we show that underlying diameter rather than the rate of growth is the most important predictor of AAA rupture.     

Joint models for a primary endpoint and multiple longitudinal covariate processes

Summary .   Joint models are formulated to investigate the association between a primary endpoint and features of multiple longitudinal processes. In particular, the subject-specific random effects in a multivariate linear random-effects model for multiple longitudinal processes are predictors in a generalized linear model for primary endpoints. Li, Zhang, and Davidian (2004, Biometrics 60 , 1–7) proposed an estimation procedure that makes no distributional assumption on the random effects but assumes independent within-subject measurement errors in the longitudinal covariate process. Based on an asymptotic bias analysis, we found that their estimators can be biased when random effects do not fully explain the within-subject correlations among longitudinal covariate measurements. Specifically, the existing procedure is fairly sensitive to the independent measurement error assumption. To overcome this limitation, we propose new estimation procedures that require neither a distributional or covariance structural assumption on covariate random effects nor an independence assumption on within-subject measurement errors. These new procedures are more flexible, readily cover scenarios that have multivariate longitudinal covariate processes, and can be implemented using available software. Through simulations and an analysis of data from a hypertension study, we evaluate and illustrate the numerical performances of the new estimators.