Joint modeling and estimation for longitudinal data with informative observation and terminal event times

ABSTRACT

In many longitudinal studies, there may exist informative observation times and a dependent terminal event that stops the follow-up. In this paper, we propose a joint model for analysis of longitudinal data with informative observation times and a dependent terminal event via two latent variables. Estimation procedures are developed for parameter estimation, and asymptotic properties of the proposed estimators are derived. Simulation studies demonstrate that the proposed method performs well for practical settings. An application to a bladder cancer study is illustrated.     

Analysis of longitudinal health-related quality of life data with terminal events

Longitudinal health-related quality of life data arise naturally from studies of progressive and neurodegenerative diseases. In such studies, patients’ mental and physical conditions are measured over their follow-up periods and the resulting data are often complicated by subject-specific measurement times and possible terminal events associated with outcome variables. Motivated by the “Predictor’s Cohort” study on patients with advanced Alzheimer disease, we propose in this paper a semiparametric modeling approach to longitudinal health-related quality of life data. It builds upon and extends some recent developments for longitudinal data with irregular observation times. The new approach handles possibly dependent terminal events. It allows one to examine time-dependent covariate effects on the evolution of outcome variable and to assess nonparametrically change of outcome measurement that is due to factors not incorporated in the covariates. The usual large-sample properties for parameter estimation are established. In particular, it is shown that relevant parameter estimators are asymptotically normal and the asymptotic variances can be estimated consistently by the simple plug-in method. A general procedure for testing a specific parametric form in the nonparametric component is also developed. Simulation studies show that the proposed approach performs well for practical settings. The method is applied to the motivating example.     

Semiparametric analysis of panel count data with correlated observation and follow-up times

This paper discusses regression analysis of panel count data that often arise in longitudinal studies concerning occurrence rates of certain recurrent events. Panel count data mean that each study subject is observed only at discrete time points rather than under continuous observation. Furthermore, both observation and follow-up times can vary from subject to subject and may be correlated with the recurrent events. For inference, we propose some shared frailty models and estimating equations are developed for estimation of regression parameters. The proposed estimates are consistent and have asymptotically a normal distribution. The finite sample properties of the proposed estimates are investigated through simulation and an illustrative example from a cancer study is provided.     

Joint modeling of longitudinal data with a dependent terminal event

ABSTRACT

Longitudinal data often arise in longitudinal follow-up studies, and there may exist a dependent terminal event such as death that stops the follow-up. In this article, we propose a new joint modeling for the analysis of longitudinal data with informative observation times via a dependent terminal event and two latent variables. Estimating equations are developed for parameter estimation, and asymptotic properties of the resulting estimators are established. In addition, a generalization of the joint model with time-varying coefficients for the longitudinal response variable is considered, and goodness-of-fit methods for assessing the adequacy of the model are also provided. The proposed method works well in our simulation studies, and is applied to a data set from a bladder cancer study.     

Non parametric analysis of gap times for bivariate recurrent event data with cure fraction

Recurrent event data often arise in longitudinal studies. In many applications, subjects may experience two different types of events alternatively over time or a pair of subjects may experience recurrent events of the same type. Medical advances have made it possible for some patients to be cured such that the disease of interest does not recur. In this article, we consider non parametric analysis of bivariate recurrent event data with cure fraction. Using the inverse-probability weighted (IPW) approach, we propose non parametric estimators for the proportion of cured patients and for the joint distribution functions of bivariate recurrence times of the uncured ones. The asymptotic properties of the proposed estimators are established. Simulation study indicates that the proposed estimators perform well in finite samples.     

Semiparametric analysis of longitudinal data with informative observation times and censoring times

We focus on regression analysis of irregularly observed longitudinal data which often occur in medical follow-up studies and observational investigations. The model for such data involves two processes: a longitudinal response process of interest and an observation process controlling observation times. Restrictive models and questionable assumptions, such as Poisson assumption and independent censoring time assumption, were posed in previous works for analysing longitudinal data. In this paper, we propose a more general model together with a robust estimation approach for longitudinal data with informative observation times and censoring times, and the asymptotic normalities of the proposed estimators are established. Both simulation studies and real data application indicate that the proposed method is promising.     

Regression Analysis of Longitudinal Data with Time‐Dependent Covariates and Informative Observation Times

Abstract. Longitudinal data frequently occur in many studies, and longitudinal responses may be correlated with observation times. In this paper, we propose a new joint modelling for the analysis of longitudinal data with time‐dependent covariates and possibly informative observation times via two latent variables. For inference about regression parameters, estimating equation approaches are developed and asymptotic properties of the proposed estimators are established. In addition, a lack‐of‐fit test is presented for assessing the adequacy of the model. The proposed method performs well in finite‐sample simulation studies, and an application to a bladder tumour study is provided.     

Comparison of Selected Methods for Modeling of Multi-State Disease Progression Processes: A Simulation Study

Prognostic studies are essential to understand the role of particular prognostic factors and, thus, improve prognosis. In most studies, disease progression trajectories of individual patients may end up with one of mutually exclusive endpoints or can involve a sequence of different events.

One challenge in such studies concerns separating the effects of putative prognostic factors on these different endpoints and testing the differences between these effects.

In this article, we systematically evaluate and compare, through simulations, the performance of three alternative multivariable regression approaches in analyzing competing risks and multiple-event longitudinal data. The three approaches are: (1) fitting separate event-specific Cox's proportional hazards models; (2) the extension of Cox's model to competing risks proposed by Lunn and McNeil; and (3) Markov multi-state model.

The simulation design is based on a prognostic study of cancer progression, and several simulated scenarios help investigate different methodological issues relevant to the modeling of multiple-event processes of disease progression. The results highlight some practically important issues. Specifically, the decreased precision of the observed timing of intermediary (non fatal) events has a strong negative impact on the accuracy of regression coefficients estimated with either the Cox's or Lunn-McNeil models, while the Markov model appears to be quite robust, under the same circumstances. Furthermore, the tests based on both Markov and Lunn-McNeil models had similar power for detecting a difference between the effects of the same covariate on the hazards of two mutually exclusive events. The Markov approach yields also accurate Type I error rate and good empirical power for testing the hypothesis that the effect of a prognostic factor on changes after an intermediary event, which cannot be directly tested with the Lunn-McNeil method. Bootstrap-based standard errors improve the coverage rates for Markov model estimates. Overall, the results of our simulations validate Markov multi-state model for a wide range of data structures encountered in prognostic studies of disease progression, and may guide end users regarding the choice of model(s) most appropriate for their specific application.     

Quantile regression methods for censored gap time data

ABSTRACT

In longitudinal studies, subjects may potentially undergo a series of sequentially ordered events. The gap times, which are the times between two serial events, are often the outcome variables of interest. This study considers quantile regression models of gap times for censored serial-event data and adapts a weighted version of the estimating equation for regression coefficients. The resulting estimators are uniformly consistent and asymptotically normal. Extensive simulation studies are presented to evaluate the finite-sample performance of the proposed methods. An analysis of the tumor recurrence data for bladder cancer patients is also provided to illustrate our proposed methods.     

Regression analysis of longitudinal data with time-dependent covariates in the presence of informative observation and censoring times

In longitudinal observational studies, repeated measures are often correlated with observation times as well as censoring time. This article proposes joint modeling and analysis of longitudinal data with time-dependent covariates in the presence of informative observation and censoring times via a latent variable. Estimating equation approaches are developed for parameter estimation and asymptotic properties of the proposed estimators are established. In addition, a generalization of the semiparametric model with time-varying coefficients for the longitudinal response is considered. Furthermore, a lack-of-fit test is provided for assessing the adequacy of the model, and some tests are presented for investigating whether or not covariate effects vary with time. The finite-sample behavior of the proposed methods is examined in simulation studies, and an application to a bladder cancer study is illustrated.     

Buckley–James-type estimation of quantile regression with recurrent gap time data

In longitudinal studies, an individual may potentially undergo a series of repeated recurrence events. The gap times, which are referred to as the times between successive recurrent events, are typically the outcome variables of interest. Various regression models have been developed in order to evaluate covariate effects on gap times based on recurrence event data. The proportional hazards model, additive hazards model, and the accelerated failure time model are all notable examples. Quantile regression is a useful alternative to the aforementioned models for survival analysis since it can provide great flexibility to assess covariate effects on the entire distribution of the gap time. In order to analyze recurrence gap time data, we must overcome the problem of the last gap time subjected to induced dependent censoring, when numbers of recurrent events exceed one time. In this paper, we adopt the Buckley–James-type estimation method in order to construct a weighted estimation equation for regression coefficients under the quantile model, and develop an iterative procedure to obtain the estimates. We use extensive simulation studies to evaluate the finite-sample performance of the proposed estimator. Finally, analysis of bladder cancer data is presented as an illustration of our proposed methodology.     

Modeling marginal features in studies of recurrent events in the presence of a terminal event

We study models for recurrent events with special emphasis on the situation where a terminal event acts as a competing risk for the recurrent events process and where there may be gaps between periods during which subjects are at risk for the recurrent event. We focus on marginal analysis of the expected number of events and show that an Aalen–Johansen type estimator proposed by Cook and Lawless is applicable in this situation. A motivating example deals with psychiatric hospital admissions where we supplement with analyses of the marginal distribution of time to the competing event and the marginal distribution of the time spent in hospital. Pseudo-observations are used for the latter purpose.

     

A Semi‐parametric Transformation Frailty Model for Semi‐competing Risks Survival Data

In the analysis of semi‐competing risks data interest lies in estimation and inference with respect to a so‐called non‐terminal event, the observation of which is subject to a terminal event. Multi‐state models are commonly used to analyse such data, with covariate effects on the transition/intensity functions typically specified via the Cox model and dependence between the non‐terminal and terminal events specified, in part, by a unit‐specific shared frailty term. To ensure identifiability, the frailties are typically assumed to arise from a parametric distribution, specifically a Gamma distribution with mean 1.0 and variance, say, σ². When the frailty distribution is misspecified, however, the resulting estimator is not guaranteed to be consistent, with the extent of asymptotic bias depending on the discrepancy between the assumed and true frailty distributions. In this paper, we propose a novel class of transformation models for semi‐competing risks analysis that permit the non‐parametric specification of the frailty distribution. To ensure identifiability, the class restricts to parametric specifications of the transformation and the error distribution; the latter are flexible, however, and cover a broad range of possible specifications. We also derive the semi‐parametric efficient score under the complete data setting and propose a non‐parametric score imputation method to handle right censoring; consistency and asymptotic normality of the resulting estimators is derived and small‐sample operating characteristics evaluated via simulation. Although the proposed semi‐parametric transformation model and non‐parametric score imputation method are motivated by the analysis of semi‐competing risks data, they are broadly applicable to any analysis of multivariate time‐to‐event outcomes in which a unit‐specific shared frailty is used to account for correlation. Finally, the proposed model and estimation procedures are applied to a study of hospital readmission among patients diagnosed with pancreatic cancer.     

Regression analysis of panel count data with covariate-dependent observation and censoring times

Panel count data often occur in a long-term study where the primary end point is the time to a specific event and each subject may experience multiple recurrences of this event. Furthermore, suppose that it is not feasible to keep subjects under observation continuously and the numbers of recurrences for each subject are only recorded at several distinct time points over the study period. Moreover, the set of observation times may vary from subject to subject. In this paper, regression methods, which are derived under simple semiparametric models, are proposed for the analysis of such longitudinal count data. Especially, we consider the situation when both observation and censoring times may depend on covariates. The new procedures are illustrated with data from a well-known cancer study.     

Simultaneous inference of a misclassified outcome and competing risks failure time data

Ipsilateral breast tumor relapse (IBTR) often occurs in breast cancer patients after their breast conservation therapy. The IBTR status' classification (true local recurrence versus new ipsilateral primary tumor) is subject to error and there is no widely accepted gold standard. Time to IBTR is likely informative for IBTR classification because new primary tumor tends to have a longer mean time to IBTR and is associated with improved survival as compared with the true local recurrence tumor. Moreover, some patients may die from breast cancer or other causes in a competing risk scenario during the follow-up period. Because the time to death can be correlated to the unobserved true IBTR status and time to IBTR (if relapse occurs), this terminal mechanism is non-ignorable. In this paper, we propose a unified framework that addresses these issues simultaneously by modeling the misclassified binary outcome without a gold standard and the correlated time to IBTR, subject to dependent competing terminal events. We evaluate the proposed framework by a simulation study and apply it to a real data set consisting of 4477 breast cancer patients. The adaptive Gaussian quadrature tools in SAS procedure NLMIXED can be conveniently used to fit the proposed model. We expect to see broad applications of our model in other studies with a similar data structure.     

Joint modeling approach for semicompeting risks data with missing nonterminal event status

Semicompeting risks data, where a subject may experience sequential non-terminal and terminal events, and the terminal event may censor the non-terminal event but not vice versa, are widely available in many biomedical studies. We consider the situation when a proportion of subjects’ non-terminal events is missing, such that the observed data become a mixture of “true” semicompeting risks data and partially observed terminal event only data. An illness–death multistate model with proportional hazards assumptions is proposed to study the relationship between non-terminal and terminal events, and provide covariate-specific global and local association measures. Maximum likelihood estimation based on semiparametric regression analysis is used for statistical inference, and asymptotic properties of proposed estimators are studied using empirical process and martingale arguments. We illustrate the proposed method with simulation studies and data analysis of a follicular cell lymphoma study.     

Bayesian hierarchical joint modeling using skew-normal/independent distributions

The multiple longitudinal outcomes collected in many clinical trials are often analyzed by multilevel item response theory (MLIRT) models. The normality assumption for the continuous outcomes in the MLIRT models can be violated due to skewness and/or outliers. Moreover, patients’ follow-up may be stopped by some terminal events (e.g., death or dropout), which are dependent on the multiple longitudinal outcomes. We proposed a joint modeling framework based on the MLIRT model to account for three data features: skewness, outliers, and dependent censoring. Our method development was motivated by a clinical study for Parkinson’s disease.     

Marginal Regression Analysis for Semi-Competing Risks Data Under Dependent Censoring

Abstract.  Multiple events data are commonly seen in medical applications. There are two types of events, namely terminal and non-terminal. Statistical analysis for non-terminal events is complicated due to dependent censoring. Consequently, joint modelling and inference are often needed to avoid the problem of non-identifiability. This article considers regression analysis for multiple events data with major interest in a non-terminal event such as disease progression. We generalize the technique of artificial censoring, which is a popular way to handle dependent censoring, under flexible model assumptions on the two types of events. The proposed method is applied to analyse a data set of bone marrow transplantation.     

Modelling and analysis of recall-based competing risks data

In this paper we consider the analysis of recall-based competing risks data. The chance of an individual recalling the exact time to event depends on the time of occurrence of the event and time of observation of the individual. In particular, it is assumed that the probability of recall depends on the time elapsed since the occurrence of an event. In this study we consider the likelihood-based inference for the analysis of recall-based competing risks data. The likelihood function is constructed by incorporating the information about the probability of recall. We consider the maximum likelihood estimation of parameters. Simulation studies are carried out to examine the performance of the estimators. The proposed estimation procedure is applied to a real life data set.