Multivariate survival analysis for case-control family data

Multivariate survival data arise from case-control family studies in which the ages at disease onset for family members may be correlated. In this paper, we consider a multivariate survival model with the marginal hazard function following the proportional hazards model. We use a frailty-based approach in the spirit of Glidden and Self (1999) to account for the correlation of ages at onset among family members. Specifically, we first estimate the baseline hazard function nonparametrically by the innovation theorem, and then obtain maximum pseudolikelihood estimators for the regression and correlation parameters plugging in the baseline hazard function estimator. We establish a connection with a previously proposed generalized estimating equation-based approach. Simulation studies and an analysis of case-control family data of breast cancer illustrate the methodology's practical utility.     

A Bayesian hierarchical approach for combining case-control and prospective studies

Motivated by the absolute risk predictions required in medical decision making and patient counseling, we propose an approach for the combined analysis of case-control and prospective studies of disease risk factors. The approach is hierarchical to account for parameter heterogeneity among studies and among sampling units of the same study. It is based on modeling the retrospective distribution of the covariates given the disease outcome, a strategy that greatly simplifies both the combination of prospective and retrospective studies and the computation of Bayesian predictions in the hierarchical case-control context. Retrospective modeling differentiates our approach from most current strategies for inference on risk factors, which are based on the assumption of a specific prospective model. To ensure modeling flexibility, we propose using a mixture model for the retrospective distributions of the covariates. This leads to a general nonlinear regression family for the implied prospective likelihood. After introducing and motivating our proposal, we present simple results that highlight its relationship with existing approaches, develop Markov chain Monte Carlo methods for inference and prediction, and present an illustration using ovarian cancer data.     

Two‐stage estimation for multivariate recurrent event data with a dependent terminal event

Recurrent event data arise in longitudinal follow‐up studies, where each subject may experience the same type of events repeatedly. The work in this article is motivated by the data from a study of repeated peritonitis for patients on peritoneal dialysis. Due to the aspects of medicine and cost, the peritonitis cases were classified into two types: Gram‐positive and non‐Gram‐positive peritonitis. Further, since the death and hemodialysis therapy preclude the occurrence of recurrent events, we face multivariate recurrent event data with a dependent terminal event. We propose a flexible marginal model, which has three characteristics: first, we assume marginal proportional hazard and proportional rates models for terminal event time and recurrent event processes, respectively; second, the inter‐recurrences dependence and the correlation between the multivariate recurrent event processes and terminal event time are modeled through three multiplicative frailties corresponding to the specified marginal models; third, the rate model with frailties for recurrent events is specified only on the time before the terminal event. We propose a two‐stage estimation procedure for estimating unknown parameters. We also establish the consistency of the two‐stage estimator. Simulation studies show that the proposed approach is appropriate for practical use. The methodology is applied to the peritonitis cohort data that motivated this study.     

Polynomial spline estimation and inference of proportional hazards regression models with flexible relative risk form

The Cox proportional hazards model usually assumes an exponential form for the dependence of the hazard function on covariate variables. However, in practice this assumption may be violated and other relative risk forms may be more appropriate. In this article, we consider the proportional hazards model with an unknown relative risk form. Issues in model interpretation are addressed. We propose a method to estimate the relative risk form and the regression parameters simultaneously by first approximating the logarithm of the relative risk form by a spline, and then employing the maximum partial likelihood estimation. An iterative alternating optimization procedure is developed for efficient implementation. Statistical inference of the regression coefficients and of the relative risk form based on parametric asymptotic theory is discussed. The proposed methods are illustrated using simulation and an application to the Veteran's Administration lung cancer data.     

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.     

Analysis of survival data from case-control family studies

In case-control family studies with survival endpoint, age of onset of diseases can be used to assess the familial aggregation of the disease and the relationship between the disease and genetic or environmental risk factors. Because of the retrospective nature of the case--control study, methods for analyzing prospectively collected correlated failure time data do not apply directly. In this article, we propose a semiparametric quasi-partial-likelihood approach to simultaneously estimate the effect of covariates on the age of onset and the association of ages of onset among family members that does not require specification of the baseline marginal distribution. We conducted a simulation study to evaluate the performance of the proposed approach and compare it with the existing semiparametric ones. Simulation results demonstrate that the proposed approach has better performance in terms of consistency and efficiency. We illustrate the methodology using a subset of data from the Washington Ashkenazi Study.     

Merits of Modelling Multivariate Survival Data Using Random Effects Proportional Odds Model

Recently, there has been a great deal of interest in the analysis of multivariate survival data. In most epidemiological studies, survival times of the same cluster are related because of some unobserved risk factors such as the environmental or genetic factors. Therefore, modelling of dependence between events of correlated individuals is required to ensure a correct inference on the effects of treatments or covariates on the survival times. In the past decades, extension of proportional hazards model has been widely considered for modelling multivariate survival data by incorporating a random effect which acts multiplicatively on the hazard function. In this article, we consider the proportional odds model, which is an alternative to the proportional hazards model at which the hazard ratio between individuals converges to unity eventually. This is a reasonable property particularly when the treatment effect fades out gradually and the homogeneity of the population increases over time. The objective of this paper is to assess the influence of the random effect on the within‐subject correlation and the population heterogeneity. We are particularly interested in the properties of the proportional odds model with univariate random effect and correlated random effect. The correlations between survival times are derived explicitly for both choices of mixing distributions and are shown to be independent of the covariates. The time path of the odds function among the survivors are also examined to study the effect of the choice of mixing distribution. Modelling multivariate survival data using a univariate mixing distribution may be inadequate as the random effect not only characterises the dependence of the survival times, but also the conditional heterogeneity among the survivors. A robust estimate for the correlation of the logarithm of the survival times within a cluster is obtained disregarding the choice of the mixing distributions. The sensitivity of the estimate of the regression parameter under a misspecification of the mixing distribution is studied through simulation. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A class of goodness of fit tests for a copula based on bivariate right-censored data

The copula of a bivariate distribution, constructed by making marginal transformations of each component, captures all the information in the bivariate distribution about the dependence between two variables. For frailty models for bivariate data the choice of a family of distributions for the random frailty corresponds to the choice of a parametric family for the copula. A class of tests of the hypothesis that the copula is in a given parametric family, with unspecified association parameter, based on bivariate right censored data is proposed. These tests are based on first making marginal Kaplan-Meier transformations of the data and then comparing a non-parametric estimate of the copula to an estimate based on the assumed family of models. A number of options are available for choosing the scale and the distance measure for this comparison. Significance levels of the test are found by a modified bootstrap procedure. The procedure is used to check the appropriateness of a gamma or a positive stable frailty model in a set of survival data on Danish twins.     

Smoothing spline ANOVA frailty model for recurrent event data

Gap time hazard estimation is of particular interest in recurrent event data. This article proposes a fully nonparametric approach for estimating the gap time hazard. Smoothing spline analysis of variance (ANOVA) decompositions are used to model the log gap time hazard as a joint function of gap time and covariates, and general frailty is introduced to account for between-subject heterogeneity and within-subject correlation. We estimate the nonparametric gap time hazard function and parameters in the frailty distribution using a combination of the Newton-Raphson procedure, the stochastic approximation algorithm (SAA), and the Markov chain Monte Carlo (MCMC) method. The convergence of the algorithm is guaranteed by decreasing the step size of parameter update and/or increasing the MCMC sample size along iterations. Model selection procedure is also developed to identify negligible components in a functional ANOVA decomposition of the log gap time hazard. We evaluate the proposed methods with simulation studies and illustrate its use through the analysis of bladder tumor data.     

Estimating haplotype relative risks on human survival in population-based association studies

Association-based linkage disequilibrium (LD) mapping is an increasingly important tool for localizing genes that show potential influence on human aging and longevity. As haplotypes contain more LD information than single markers, a haplotype-based LD approach can have increased power in detecting associations as well as increased robustness in statistical testing. In this paper, we develop a new statistical model to estimate haplotype relative risks (HRRs) on human survival using unphased multilocus genotype data from unrelated individuals in cross-sectional studies. Based on the proportional hazard assumption, the model can estimate haplotype risk and frequency parameters, incorporate observed covariates, assess interactions between haplotypes and the covariates, and investigate the modes of gene function. By introducing population survival information available from population statistics, we are able to develop a procedure that carries out the parameter estimation using a nonparametric baseline hazard function and estimates sex-specific HRRs to infer gene-sex interaction. We also evaluate the haplotype effects on human survival while taking into account individual heterogeneity in the unobserved genetic and nongenetic factors or frailty by introducing the gamma-distributed frailty into the survival function. After model validation by computer simulation, we apply our method to an empirical data set to measure haplotype effects on human survival and to estimate haplotype frequencies at birth and over the observed ages. Results from both simulation and model application indicate that our survival analysis model is an efficient method for inferring haplotype effects on human survival in population-based association studies.     

Estimating regression parameters and degree of dependence for multivariate failure time data

Multivariate failure time data are frequently encountered in longitudinal studies when subjects may experience several events or when there is a grouping of individuals into a cluster. To take into account the dependence of the failure times within the unit (the individual or the cluster) as well as censoring, two multivariate generalizations of the Cox proportional hazards model are commonly used. The marginal hazard model is used when the purpose is to estimate mean regression parameters, while the frailty model is retained when the purpose is to assess the degree of dependence within the unit. We propose a new approach based on the combination of the two aforementioned models to estimate both these quantities. This two-step estimation procedure is quicker and more simple to implement than the EM algorithm used in frailty models estimation. Simulation results are provided to illustrate robustness, consistency, and large-sample properties of estimators. Finally, this method is exemplified on a diabetic retinopathy study in order to assess the effect of photocoagulation in delaying the onset of blindness as well as the dependence between the two eyes blindness times of a patient.     

Estimating Subject‐Specific Dependent Competing Risk Profile with Censored Event Time Observations

Summary In a longitudinal study, suppose that the primary endpoint is the time to a specific event. This response variable, however, may be censored by an independent censoring variable or by the occurrence of one of several dependent competing events. For each study subject, a set of baseline covariates is collected. The question is how to construct a reliable prediction rule for the future subject's profile of all competing risks of interest at a specific time point for risk‐benefit decision making. In this article, we propose a two‐stage procedure to make inferences about such subject‐specific profiles. For the first step, we use a parametric model to obtain a univariate risk index score system. We then estimate consistently the average competing risks for subjects who have the same parametric index score via a nonparametric function estimation procedure. We illustrate this new proposal with the data from a randomized clinical trial for evaluating the efficacy of a treatment for prostate cancer. The primary endpoint for this study was the time to prostate cancer death, but had two types of dependent competing events, one from cardiovascular death and the other from death of other causes.     

Marginal hazard ratio estimates in joint frailty models for heart failure trials

This work is motivated by clinical trials in chronic heart failure disease, where treatment has effects both on morbidity (assessed as recurrent non‐fatal hospitalisations) and on mortality (assessed as cardiovascular death, CV death). Recently, a joint frailty proportional hazards model has been proposed for these kind of efficacy outcomes to account for a potential association between the risk rates for hospital admissions and CV death. However, more often clinical trial results are presented by treatment effect estimates that have been derived from marginal proportional hazards models, that is, a Cox model for mortality and an Andersen–Gill model for recurrent hospitalisations. We show how these marginal hazard ratios and their estimates depend on the association between the risk processes, when these are actually linked by shared or dependent frailty terms. First we derive the marginal hazard ratios as a function of time. Then, applying least false parameter theory, we show that the marginal hazard ratio estimate for the hospitalisation rate depends on study duration and on parameters of the underlying joint frailty model. In particular, we identify parameters, for example the treatment effect on mortality, that determine if the marginal hazard ratio estimate for hospitalisations is smaller, equal or larger than the conditional one. How this affects rejection probabilities is further investigated in simulation studies. Our findings can be used to interpret marginal hazard ratio estimates in heart failure trials and are illustrated by the results of the CHARM‐Preserved trial (where CHARM is the ‘Candesartan in Heart failure Assessment of Reduction in Mortality and morbidity’ programme).     

Hazard function for cancer patients and cancer cell dynamics

The aim of the paper is to develop a procedure for an estimate of an analytical form of a hazard function for cancer patients. Although a deterministic approach based on cancer cell population dynamics yields the analytical expression, it depends on several parameters which should be estimated. On the other hand, a kernel estimate is an effective nonparametric method for estimating hazard functions. This method provides the pointwise estimate of the hazard function. Our procedure consists of two steps: in the first step we find the kernel estimate of the hazard function and in the second step the parameters in the deterministic model are obtained by the least squares method. A simulation study with different types of censorship is carried out and the developed procedure is applied to real data.     

Design Aspects and Odds Ratio Estimation in Multicenter Case-Control Studies

In this paper we propose a method to be used in the planning stage of a case-control study. An allocation rule for controls in multicenter case-control studies is proposed which would assure a simple, efficient and unbiased estimation of the odds ratio in the pooled data. It is shown that the efficiency of the design increases with increasing correlation between study center and risk factor. Sources of bias and their implications for relative risk estimation are discussed. The method is demonstrated with data from a case-control study.     

An additive‐multiplicative mean residual life model for right‐censored data

Many studies have focused on determining the effect of the body mass index (BMI) on the mortality in different cohorts. In this article, we propose an additive‐multiplicative mean residual life (MRL) model to assess the effects of BMI and other risk factors on the MRL function of survival time in a cohort of Chinese type 2 diabetic patients. The proposed model can simultaneously manage additive and multiplicative risk factors and provide a comprehensible interpretation of their effects on the MRL function of interest. We develop an estimation procedure through pseudo partial score equations to obtain parameter estimates. We establish the asymptotic properties of the proposed estimators and conduct simulations to demonstrate the performance of the proposed method. The application of the procedure to a study on the life expectancy of type 2 diabetic patients reveals new insights into the extension of the life expectancy of such patients.     

Estimating marginal and incremental effects on health outcomes using flexible link and variance function models

We propose an extension to the estimating equations in generalized linear models to estimate parameters in the link function and variance structure simultaneously with regression coefficients. Rather than focusing on the regression coefficients, the purpose of these models is inference about the mean of the outcome as a function of a set of covariates, and various functionals of the mean function used to measure the effects of the covariates. A commonly used functional in econometrics, referred to as the marginal effect, is the partial derivative of the mean function with respect to any covariate, averaged over the empirical distribution of covariates in the model. We define an analogous parameter for discrete covariates. The proposed estimation method not only helps to identify an appropriate link function and to suggest an underlying distribution for a specific application but also serves as a robust estimator when no specific distribution for the outcome measure can be identified. Using Monte Carlo simulations, we show that the resulting parameter estimators are consistent. The method is illustrated with an analysis of inpatient expenditure data from a study of hospitalists.     

A Procedure for the Common Estimation of Parameters Corresponding to Several Treatment Groups

In this article we give a procedure for the common estimation of parameters corresponding to several treatment groups. Thereby we assume that the distribution functions of the groups belong to the same family and differ only in the parameter values. The procedure allows the common estimation of some of these parameters. The parameters themselves will be estimated by the maximum likelihood method; the estimators will be calculated iteratively by the Newton-Raphson method. To prove if the common estimation is possible, we propose as a suitable test the maximum likelihood ratio test. Finally we show the application of our procedure in the case of the probit analysis.     

Estimation of multivariate frailty models using penalized partial likelihood

There exists a growing literature on the estimation of gamma distributed multiplicative shared frailty models. There is, however, often a need to model more complicated frailty structures, but attempts to extend gamma frailties run into complications. Motivated by hip replacement data with a more complicated dependence structure, we propose a model based on multiplicative frailties with a multivariate log-normal joint distribution. We give a justification and an estimation procedure for this generally structured frailty model, which is a generalization of the one presented by McGilchrist (1993, Biometrics 49, 221-225). The estimation is based on Laplace approximation of the likelihood function. This leads to estimating equations based on a penalized fixed effects partial likelihood, where the marginal distribution of the frailty terms determines the penalty term. The tuning parameters of the penalty function, i.e., the frailty variances, are estimated by maximizing an approximate profile likelihood. The performance of the approximation is evaluated by simulation, and the frailty model is fitted to the hip replacement data.     

Boosting distributional copula regression

Capturing complex dependence structures between outcome variables (e.g., study endpoints) is of high relevance in contemporary biomedical data problems and medical research. Distributional copula regression provides a flexible tool to model the joint distribution of multiple outcome variables by disentangling the marginal response distributions and their dependence structure. In a regression setup, each parameter of the copula model, that is, the marginal distribution parameters and the copula dependence parameters, can be related to covariates via structured additive predictors. We propose a framework to fit distributional copula regression via model-based boosting, which is a modern estimation technique that incorporates useful features like an intrinsic variable selection mechanism, parameter shrinkage and the capability to fit regression models in high-dimensional data setting, that is, situations with more covariates than observations. Thus, model-based boosting does not only complement existing Bayesian and maximum-likelihood based estimation frameworks for this model class but rather enables unique intrinsic mechanisms that can be helpful in many applied problems. The performance of our boosting algorithm for copula regression models with continuous margins is evaluated in simulation studies that cover low- and high-dimensional data settings and situations with and without dependence between the responses. Moreover, distributional copula boosting is used to jointly analyze and predict the length and the weight of newborns conditional on sonographic measurements of the fetus before delivery together with other clinical variables.