Joint Models of Longitudinal Data and Recurrent Events with Informative Terminal Event

This article presents semiparametric joint models to analyze longitudinal data with recurrent events (e.g. multiple tumors, repeated hospital admissions) and a terminal event such as death. A broad class of transformation models for the cumulative intensity of the recurrent events and the cumulative hazard of the terminal event is considered, which includes the proportional hazards model and the proportional odds model as special cases. We propose to estimate all the parameters using the nonparametric maximum likelihood estimators (NPMLE). We provide the simple and efficient EM algorithms to implement the proposed inference procedure. Asymptotic properties of the estimators are shown to be asymptotically normal and semiparametrically efficient. Finally, we evaluate the performance of the method through extensive simulation studies and a real-data application.     

Measures of Departure from Constant Failure Rate Models and Proportional Hazards Rate Models for Grouped Data

Two measures are proposed to represent the degree of departure from the constant failure rate model of a system when data are grouped. Two measures are also proposed to represent the degree of departure from the proportional hazards rate model when two systems are present and grouped data are considered. In each case one measure is based on the Kullback-Leibler discrepancy and the other is based on the Pearson χ² type discrepancy using the failure rates. The usefulness of the proposed measures are discussed with applications. A simulation study shows that the proposed measures perform no worse than the goodness-of-fit tests when testing for the constant failure rate model.     

Bayesian semiparametric dynamic frailty models for multiple event time data

Many biomedical studies collect data on times of occurrence for a health event that can occur repeatedly, such as infection, hospitalization, recurrence of disease, or tumor onset. To analyze such data, it is necessary to account for within-subject dependency in the multiple event times. Motivated by data from studies of palpable tumors, this article proposes a dynamic frailty model and Bayesian semiparametric approach to inference. The widely used shared frailty proportional hazards model is generalized to allow subject-specific frailties to change dynamically with age while also accommodating nonproportional hazards. Parametric assumptions on the frailty distribution are avoided by using Dirichlet process priors for a shared frailty and for multiplicative innovations on this frailty. By centering the semiparametric model on a conditionally conjugate dynamic gamma model, we facilitate posterior computation and lack-of-fit assessments of the parametric model. Our proposed method is demonstrated using data from a cancer chemoprevention study.     

Shared frailty models for recurrent events and a terminal event

There has been an increasing interest in the analysis of recurrent event data (Cook and Lawless, 2002, Statistical Methods in Medical Research 11, 141-166). In many situations, a terminating event such as death can happen during the follow-up period to preclude further occurrence of the recurrent events. Furthermore, the death time may be dependent on the recurrent event history. In this article we consider frailty proportional hazards models for the recurrent and terminal event processes. The dependence is modeled by conditioning on a shared frailty that is included in both hazard functions. Covariate effects can be taken into account in the model as well. Maximum likelihood estimation and inference are carried out through a Monte Carlo EM algorithm with Metropolis-Hastings sampler in the E-step. An analysis of hospitalization and death data for waitlisted dialysis patients is presented to illustrate the proposed methods. Methods to check the validity of the proposed model are also demonstrated. This model avoids the difficulties encountered in alternative approaches which attempt to specify a dependent joint distribution with marginal proportional hazards and yields an estimate of the degree of dependence.     

Inverse probability weighting methods for Cox regression with right-truncated data

Right-truncated data arise when observations are ascertained retrospectively, and only subjects who experience the event of interest by the time of sampling are selected. Such a selection scheme, without adjustment, leads to biased estimation of covariate effects in the Cox proportional hazards model. The existing methods for fitting the Cox model to right-truncated data, which are based on the maximization of the likelihood or solving estimating equations with respect to both the baseline hazard function and the covariate effects, are numerically challenging. We consider two alternative simple methods based on inverse probability weighting (IPW) estimating equations, which allow consistent estimation of covariate effects under a positivity assumption and avoid estimation of baseline hazards. We discuss problems of identifiability and consistency that arise when positivity does not hold and show that although the partial tests for null effects based on these IPW methods can be used in some settings even in the absence of positivity, they are not valid in general. We propose adjusted estimating equations that incorporate the probability of observation when it is known from external sources, which results in consistent estimation. We compare the methods in simulations and apply them to the analyses of human immunodeficiency virus latency.     

Attributable risk function in the proportional hazards model for censored time-to-event

Time-to-event endpoints are often used in clinical and epidemiological studies to evaluate disease association with hazardous exposures. In the statistical literature of time-to-event analysis, such association is usually measured by the hazard ratio in the proportional hazards model. In public health, it is also of important interest to assess the excess risk attributable to an exposure in a given population. In this article, we extend the notion of 'population attributable fraction' for the binary outcomes to the attributable risk function for the event times in prospective studies. A simple estimator of the time-varying attributable risk function is proposed under the proportional hazards model. Its inference procedures are established. Monte-Carlo simulation studies are conducted to evaluate its validity and performance. The proposed methodology is motivated and demonstrated by the data collected in a multicenter acquired immunodeficiency syndrome (AIDS) cohort study to estimate the attributable risk of human immunodeficiency virus type 1 (HIV-1) infections due to several potential risk factors.     

Tests of proportional hazards and proportional odds models for grouped survival data

In this paper, we derive score test statistics to discriminate between proportional hazards and proportional odds models for grouped survival data. These models are embedded within a power family transformation in order to obtain the score tests. In simple cases, some small-sample results are obtained for the score statistics using Monte Carlo simulations. Score statistics have distributions well approximated by the chi-squared distribution. Real examples illustrate the proposed tests.     

Bayesian cure rate frailty models with application to a root canal therapy study

Due to natural or artificial clustering, multivariate survival data often arise in biomedical studies, for example, a dental study involving multiple teeth from each subject. A certain proportion of subjects in the population who are not expected to experience the event of interest are considered to be "cured" or insusceptible. To model correlated or clustered failure time data incorporating a surviving fraction, we propose two forms of cure rate frailty models. One model naturally introduces frailty based on biological considerations while the other is motivated from the Cox proportional hazards frailty model. We formulate the likelihood functions based on piecewise constant hazards and derive the full conditional distributions for Gibbs sampling in the Bayesian paradigm. As opposed to the Cox frailty model, the proposed methods demonstrate great potential in modeling multivariate survival data with a cure fraction. We illustrate the cure rate frailty models with a root canal therapy data set.     

Score tests for homogeneity of regression effect in the proportional hazards model

A simple model, containing the proportional hazards regression model as a special case, is presented. The purpose of the model is to provide a framework in which specific alternatives to the proportional hazards assumption may be tested. Rank-invariant score tests for linear, quadratic, or exponential trends can, for instance, all be undertaken within this framework. In the case of the two-sample problem the required calculations are shown to take a particularly simple form. Special consideration is given to the two-sample case in which there is an inversion of the regression effect, i.e., where the hazard functions cross at some given point. Both of the motivating examples are concerned with this problem. Computational aspects are relatively straightforward and some discussion on this is provided.     

Regression analysis of doubly censored failure time data with applications to AIDS studies

In many epidemiological studies, the survival time of interest is the elapsed time between two related events, the originating event and the failure event, and the times of the occurrences of both events are right or interval censored. We discuss the regression analysis of such studies and a simple estimating equation approach is proposed under the proportional hazards model. The method can easily be implemented and does not involve any iteration among unknown parameters, as full likelihood approaches proposed in the literature do. The asymptotic properties of the proposed regression coefficient estimates are derived and an AIDS cohort study is analyzed to illustrate the proposed approach.     

A Joint Modeling Approach for Longitudinal Data with Informative Observation Times and a Terminal Event

In this article, we propose a new joint modeling approach for the analysis of longitudinal data with informative observation times and a dependent terminal event. We specify a semiparametric mixed effects model for the longitudinal process, a proportional rate frailty model for the observation process, and a proportional hazards frailty model for the terminal event. The association among the three related processes is modeled via two latent variables. Estimating equation approaches are developed for parameter estimation, and the asymptotic properties of the proposed estimators are established. The finite sample performance of the proposed estimators is examined through simulation studies, and an application to a medical cost study of chronic heart failure patients is illustrated.     

Cause‐Specific Cumulative Incidence Estimation and the Fine and Gray Model Under Both Left Truncation and Right Censoring

Summary The standard estimator for the cause‐specific cumulative incidence function in a competing risks setting with left truncated and/or right censored data can be written in two alternative forms. One is a weighted empirical cumulative distribution function and the other a product‐limit estimator. This equivalence suggests an alternative view of the analysis of time‐to‐event data with left truncation and right censoring: individuals who are still at risk or experienced an earlier competing event receive weights from the censoring and truncation mechanisms. As a consequence, inference on the cumulative scale can be performed using weighted versions of standard procedures. This holds for estimation of the cause‐specific cumulative incidence function as well as for estimation of the regression parameters in the Fine and Gray proportional subdistribution hazards model. We show that, with the appropriate filtration, a martingale property holds that allows deriving asymptotic results for the proportional subdistribution hazards model in the same way as for the standard Cox proportional hazards model. Estimation of the cause‐specific cumulative incidence function and regression on the subdistribution hazard can be performed using standard software for survival analysis if the software allows for inclusion of time‐dependent weights. We show the implementation in the R statistical package. The proportional subdistribution hazards model is used to investigate the effect of calendar period as a deterministic external time varying covariate, which can be seen as a special case of left truncation, on AIDS related and non‐AIDS related cumulative mortality.     

Comparison of procedures to assess non-linear and time-varying effects in multivariable models for survival data

The focus of many medical applications is to model the impact of several factors on time to an event. A standard approach for such analyses is the Cox proportional hazards model. It assumes that the factors act linearly on the log hazard function (linearity assumption) and that their effects are constant over time (proportional hazards (PH) assumption). Variable selection is often required to specify a more parsimonious model aiming to include only variables with an influence on the outcome. As follow-up increases the effect of a variable often gets weaker, which means that it varies in time. However, spurious time-varying effects may also be introduced by mismodelling other parts of the multivariable model, such as omission of an important covariate or an incorrect functional form of a continuous covariate. These issues interact. To check whether the effect of a variable varies in time several tests for non-PH have been proposed. However, they are not sufficient to derive a model, as appropriate modelling of the shape of time-varying effects is required. In three examples we will compare five recently published strategies to assess whether and how the effects of covariates from a multivariable model vary in time. For practical use we will give some recommendations.     

Semiparametric bayes' proportional odds models for current status data with underreporting

Current status data are a type of interval-censored event time data in which all the individuals are either left or right censored. For example, our motivation is drawn from a cross-sectional study, which measured whether or not fibroid onset had occurred by the age of an ultrasound exam for each woman. We propose a semiparametric Bayesian proportional odds model in which the baseline event time distribution is estimated nonparametrically by using adaptive monotone splines in a logistic regression model and the potential risk factors are included in the parametric part of the mean structure. The proposed approach has the advantage of being straightforward to implement using a simple and efficient Gibbs sampler, whereas alternative semiparametric Bayes' event time models encounter problems for current status data. The model is generalized to allow systematic underreporting in a subset of the data, and the methods are applied to an epidemiologic study of uterine fibroids.     

Robust combination of multiple diagnostic tests for classifying censored event times

Recent advancement in technology promises to yield a multitude of tests for disease diagnosis and prognosis. When there are multiple sources of information available, it is often of interest to construct a composite score that can provide better classification accuracy than any individual measurement. In this paper, we consider robust procedures for optimally combining tests when test results are measured prior to disease onset and disease status evolves over time. To account for censoring of disease onset time, the most commonly used approach to combining tests to detect subsequent disease status is to fit a proportional hazards model (Cox, 1972) and use the estimated risk score. However, simulation studies suggested that such a risk score may have poor accuracy when the proportional hazards assumption fails. We propose the use of a nonparametric transformation model (Han, 1987) as a working model to derive an optimal composite score with theoretical justification. We demonstrate that the proposed score is the optimal score when the model holds and is optimal "on average" among linear scores even if the model fails. Time-dependent sensitivity, specificity, and receiver operating characteristic curve functions are used to quantify the accuracy of the resulting composite score. We provide consistent and asymptotically Gaussian estimators of these accuracy measures. A simple model-free resampling procedure is proposed to obtain all consistent variance estimators. We illustrate the new proposals with simulation studies and an analysis of a breast cancer gene expression data set.     

Joint modelling of longitudinal measurements and event time data

This paper formulates a class of models for the joint behaviour of a sequence of longitudinal measurements and an associated sequence of event times, including single-event survival data. This class includes and extends a number of specific models which have been proposed recently, and, in the absence of association, reduces to separate models for the measurements and events based, respectively, on a normal linear model with correlated errors and a semi-parametric proportional hazards or intensity model with frailty. Special cases of the model class are discussed in detail and an estimation procedure which allows the two components to be linked through a latent stochastic process is described. Methods are illustrated using results from a clinical trial into the treatment of schizophrenia.     

Hazard rate models with covariates.

Many problems, particularly in medical research, concern the relationship between certain covariates and the time to occurrence of an event. The hazard or failure rate function provides a conceptually simple representation of time to occurrence data that readily adapts to include such generalizations as competing risks and covariates that vary with time. Two partially parametric models for the hazard function are considered. These are the proportional hazards model of Cox (1972) and the class of log-linear or accelerated failure time models. A synthesis of the literature on estimation from these models under prospective sampling indicates that, although important advances have occurred during the past decade, further effort is warranted on such topics as distribution theory, tests of fit, robustness, and the full utilization of a methodology that permits non-standard features. It is further argued that a good deal of fruitful research could be done on applying the same models under a variety of other sampling schemes. A discussion of estimation from case-control studies illustrates this point.     

A General Class of Semiparametric Transformation Frailty Models for Nonproportional Hazards Survival Data

Summary We propose a semiparametrically efficient estimation of a broad class of transformation regression models for nonproportional hazards data. Classical transformation models are to be viewed from a frailty model paradigm, and the proposed method provides a unified approach that is valid for both continuous and discrete frailty models. The proposed models are shown to be flexible enough to model long‐term follow‐up survival data when the treatment effect diminishes over time, a case for which the PH or proportional odds assumption is violated, or a situation in which a substantial proportion of patients remains cured after treatment. Estimation of the link parameter in frailty distribution, considered to be unknown and possibly dependent on a time‐independent covariates, is automatically included in the proposed methods. The observed information matrix is computed to evaluate the variances of all the parameter estimates. Our likelihood‐based approach provides a natural way to construct simple statistics for testing the PH and proportional odds assumptions for usual survival data or testing the short‐ and long‐term effects for survival data with a cure fraction. Simulation studies demonstrate that the proposed inference procedures perform well in realistic settings. Applications to two medical studies are provided.     

Analyses of two-way chronic studies

This paper proposes a method for analyzing tumor data from chronic studies when the experimental design includes combinations of two factors, for example, sex and dose. Both main effects and combined-effect (interaction) hypotheses are considered. A stratified log-rank statistic is presented for tests of no column or row (main) effects. The paper shows that when the numbers of animals in the cells are unequal and disproportional, the null distribution of the unstratified log-rank statistic does not have a chi-square distribution. Two simple models, additive and multiplicative, for representing the combined effect of row and column are considered under the proportional hazards model. A simple conservative statistic is proposed for testing the additivity of the row and column effects. A simulation experiment to examine the behavior of the null distribution of the combined-effect test statistic under the additive model and the power of the test against the multiplicative model is reported. The procedure is illustrated by analyzing mammary tumors induced by 7,12-dimethylbenz[a]anthracene (DMBA) in yellow and agouti F1 female mice from a laboratory experiment.     

Regression analysis of grouped survival data with application to breast cancer data

Use of the proportional hazards regression model (Cox 1972) substantially liberalized the analysis of censored survival data with covariates. Available procedures for estimation of the relative risk parameter, however, do not adequately handle grouped survival data, or large data sets with many tied failure times. The grouped data version of the proportional hazards model is proposed here for such estimation. Asymptotic likelihood results are given, both for the estimation of the regression coefficient and the survivor function. Some special results are given for testing the hypothesis of a zero regression coefficient which leads, for example, to a generalization of the log-rank test for the comparison of several survival curves. Application to breast cancer data, from the National Cancer Institute-sponsored End Results Group, indicates that previously noted race differences in breast cancer survival times are explained to a large extent by differences in disease extent and other demographic characteristics at diagnosis.