Semiparametric Analysis for Recurrent Event Data with Time-Dependent Covariates and Informative Censoring

Summary .  Recurrent event data analyses are usually conducted under the assumption that the censoring time is independent of the recurrent event process. In many applications the censoring time can be informative about the underlying recurrent event process, especially in situations where a correlated failure event could potentially terminate the observation of recurrent events. In this article, we consider a semiparametric model of recurrent event data that allows correlations between censoring times and recurrent event process via frailty. This flexible framework incorporates both time-dependent and time-independent covariates in the formulation, while leaving the distributions of frailty and censoring times unspecified. We propose a novel semiparametric inference procedure that depends on neither the frailty nor the censoring time distribution. Large sample properties of the regression parameter estimates and the estimated baseline cumulative intensity functions are studied. Numerical studies demonstrate that the proposed methodology performs well for realistic sample sizes. An analysis of hospitalization data for patients in an AIDS cohort study is presented to illustrate the proposed method.     

General right censoring and its impact on the analysis of survival data.

This paper concerns general right censoring and some of the difficulties it creates in the analysis of survival data. A general formulation of censored-survival processes leads to the partition of all models into those based on noninformative and informative censoring. Nearly all statistical methods for censored data assume that censoring is noninformative. Topics considered within this class include: the relationships between three models for noninformative censoring, the use of likelihood methods for inferences about the distribution of survival time, the effects of censoring on the K-sample problem, and the effects of censoring on model testing. Also considered are several topics which relate to informative censoring models. These include: problems of nonidentifiability that can be encountered when attempting to assess a set of data for the type of censoring in effect, the consequences of falsely assuming that censoring is noninformative, and classes of informative censoring models.     

Analysis of multivariate recurrent event data with time‐dependent covariates and informative censoring

Multivariate recurrent event data are usually encountered in many clinical and longitudinal studies in which each study subject may experience multiple recurrent events. For the analysis of such data, most existing approaches have been proposed under the assumption that the censoring times are noninformative, which may not be true especially when the observation of recurrent events is terminated by a failure event. In this article, we consider regression analysis of multivariate recurrent event data with both time‐dependent and time‐independent covariates where the censoring times and the recurrent event process are allowed to be correlated via a frailty. The proposed joint model is flexible where both the distributions of censoring and frailty variables are left unspecified. We propose a pairwise pseudolikelihood approach and an estimating equation‐based approach for estimating coefficients of time‐dependent and time‐independent covariates, respectively. The large sample properties of the proposed estimates are established, while the finite‐sample properties are demonstrated by simulation studies. The proposed methods are applied to the analysis of a set of bivariate recurrent event data from a study of platelet transfusion reactions.     

Weighted Estimation of the Accelerated Failure Time Model in the Presence of Dependent Censoring

Independent censoring is a crucial assumption in survival analysis. However, this is impractical in many medical studies, where the presence of dependent censoring leads to difficulty in analyzing covariate effects on disease outcomes. The semicompeting risks framework offers one approach to handling dependent censoring. There are two representative estimators based on an artificial censoring technique in this data structure. However, neither of these estimators is better than another with respect to efficiency (standard error). In this paper, we propose a new weighted estimator for the accelerated failure time (AFT) model under dependent censoring. One of the advantages in our approach is that these weights are optimal among all the linear combinations of the previously mentioned two estimators. To calculate these weights, a novel resampling-based scheme is employed. Attendant asymptotic statistical results for the estimator are established. In addition, simulation studies, as well as an application to real data, show the gains in efficiency for our estimator.     

Fundamentals of survival data

Survival data stand out as a special statistical field. This paper tries to describe what survival data is and what makes it so special. Survival data concern times to some events. A key point is the successive observation of time, which on the one hand leads to some times not being observed so that all that is known is that they exceed some given times (censoring), and on the other hand implies that predictions regarding the future course should be conditional on the present status (truncation). In the simplest case, this condition is that the individual is alive. The successive conditioning makes the hazard function, which describes the probability of an event happening during a short interval given that the individual is alive today (or more generally able to experience the event), the most relevant concept. Standard distributions available (normal, log-normal, gamma, inverse Gaussian, and so forth) can account for censoring and truncation, but this is cumbersome. Besides, they fit badly because they are either symmetric or right skewed, but survival time distributions can easily be left-skewed positive variables. A few distributions satisfying these requirements are available, but often nonparametric methods are preferable as they account better conceptually for truncation and censoring and give a better fit. Finally, we compare the proportional hazards regression models with accelerated failure time models.     

On the Fisher Information Matrix in Type II Censored Data from the Exponentiated Exponential Family

In this note, we express, in a general setting, the Fisher information matrix under Type II censoring in terms of the hazard function and then obtain the Fisher information matrix under Type II censoring as a single integral for the exponentiated exponential family, which can be easily evaluated. The Fisher information under Type II censoring can also be used to characterize the exponential distribution among the exponentiated exponential family.     

Muscle-Specific Splicing of a Heterologous Exon Mediated by a Single Muscle-Specific Splicing Enhancer from the Cardiac Troponin T Gene

The chicken cardiac troponin T (cTNT) gene contains a single 30-nucleotide alternative exon that is included in embryonic striated muscle and skipped in the adult. Transient-transfection analysis of cTNT minigenes in muscle and fibroblast cell cultures previously identified four muscle-specific splicing enhancers (MSEs) that promote exon inclusion specifically in embryonic striated muscle cultures. Three MSEs located in the intron downstream from the alternative exon were sufficient for muscle-specific exon inclusion. In the present study, the boundaries of these MSEs were defined by scanning mutagenesis, allowing analysis of individual elements in gain-of-function experiments. Concatamers of MSE2 were necessary and sufficient to promote muscle-specific inclusion of a heterologous exon, indicating that it is a target for muscle-specific regulation. Sequences present in MSE2 are also found in MSE4, suggesting that these two MSEs act in a similar manner. MSE3 appears to be different from MSE2 and MSE4 yet is able to functionally replace both of these elements, demonstrating functional redundancy of elements that are likely to bind different factors. MSE2 and MSE4 each contain a novel sequence motif that is found adjacent to a number of alternative exons that undergo regulated splicing in striated muscle, suggesting a common role for this element in muscle-specific regulation.     

Addressing an Idiosyncrasy in Estimating Survival Curves Using Double Sampling in the Presence of Self-Selected Right Censoring

We investigate the use of follow-up samples of individuals to estimate survival curves from studies that are subject to right censoring from two sources: (i) early termination of the study, namely, administrative censoring, or (ii) censoring due to lost data prior to administrative censoring, so-called dropout. We assume that, for the full cohort of individuals, administrative censoring times are independent of the subjects' inherent characteristics, including survival time. To address the loss to censoring due to dropout, which we allow to be possibly selective, we consider an intensive second phase of the study where a representative sample of the originally lost subjects is subsequently followed and their data recorded. As with double-sampling designs in survey methodology, the objective is to provide data on a representative subset of the dropouts. Despite assumed full response from the follow-up sample, we show that, in general in our setting, administrative censoring times are not independent of survival times within the two subgroups, nondropouts and sampled dropouts. As a result, the stratified Kaplan-Meier estimator is not appropriate for the cohort survival curve. Moreover, using the concept of potential outcomes, as opposed to observed outcomes, and thereby explicitly formulating the problem as a missing data problem, reveals and addresses these complications. We present an estimation method based on the likelihood of an easily observed subset of the data and study its properties analytically for large samples. We evaluate our method in a realistic situation by simulating data that match published margins on survival and dropout from an actual hip-replacement study. Limitations and extensions of our design and analytic method are discussed.     

Analysis of times to repeated events in two-arm randomized trials with noncompliance and dependent censoring

This article develops randomization-based methods for times to repeated events in two-arm randomized trials with noncompliance and dependent censoring. Structural accelerated failure time models are assumed to capture causal effects on repeated event times and dependent censoring time, but the dependence structure among repeated event times and dependent censoring time is unspecified. Artificial censoring techniques to accommodate nonrandom noncompliance and dependent censoring are proposed. Estimation of the acceleration parameters are based on rank-based estimating functions. A simulation study is conducted to evaluate the performance of the developed methods. An illustration of the methods using data from an acute myeloid leukemia trial is provided.     

Semiparametric analysis of recurrent events data in the presence of dependent censoring

Dependent censoring occurs in longitudinal studies of recurrent events when the censoring time depends on the potentially unobserved recurrent event times. To perform regression analysis in this setting, we propose a semiparametric joint model that formulates the marginal distributions of the recurrent event process and dependent censoring time through scale-change models, while leaving the distributional form and dependence structure unspecified. We derive consistent and asymptotically normal estimators for the regression parameters. We also develop graphical and numerical methods for assessing the adequacy of the proposed model. The finite-sample behavior of the new inference procedures is evaluated through simulation studies. An application to recurrent hospitalization data taken from a study of intravenous drug users is provided.     

Nonparametric comparison of two survival-time distributions in the presence of dependent censoring

When testing the null hypothesis that treatment arm-specific survival-time distributions are equal, the log-rank test is asymptotically valid when the distribution of time to censoring is conditionally independent of randomized treatment group given survival time. We introduce a test of the null hypothesis for use when the distribution of time to censoring depends on treatment group and survival time. This test does not make any assumptions regarding independence of censoring time and survival time. Asymptotic validity of this test only requires a consistent estimate of the conditional probability that the survival event is observed given both treatment group and that the survival event occurred before the time of analysis. However, by not making unverifiable assumptions about the data-generating mechanism, there exists a set of possible values of corresponding sample-mean estimates of these probabilities that are consistent with the observed data. Over this subset of the unit square, the proposed test can be calculated and a rejection region identified. A decision on the null that considers uncertainty because of censoring that may depend on treatment group and survival time can then be directly made. We also present a generalized log-rank test that enables us to provide conditions under which the ordinary log-rank test is asymptotically valid. This generalized test can also be used for testing the null hypothesis when the distribution of censoring depends on treatment group and survival time. However, use of this test requires semiparametric modeling assumptions. A simulation study and an example using a recent AIDS clinical trial are provided.     

Nonparametric estimation of a multivariate distribution in the presence of censoring

This paper presents examples of situations in which one wishes to estimate a multivariate distribution from data that may be right-censored. A distinction is made between what we term 'homogeneous' and 'heterogeneous' censoring. It is shown how a multivariate empirical survivor function must be constructed in order to be considered a (nonparametric) maximum likelihood estimate of the underlying survivor function. A closed-form solution, similar to the product-limit estimate of Kaplan and Meier, is possible with homogeneous censoring, but an iterative method, such as the EM algorithm, is required with heterogeneous censoring. An example is given in which an anomaly is produced if censored multivariate data are analyzed as a series of univariate variables; this anomaly is shown to disappear if the methods of this paper are used.     

On the Information Matrix of Exponential Mixture Models with Long-term Survivors

The aim of this paper is to study the properties of the asymptotic variances of the maximum likelihood estimators of the parameters of the exponential mixture model with long-term survivors for randomly censored data. In addition, we study the asymptotic relative efficiency of these estimators versus those which would be obtained with complete follow-up. It is shown that fixed censoring at time T produces higher precision as well as higher asymptotic relative efficiency than those obtainable under uniform and uniform-exponential censoring distributions over (0, T). The results are useful in planning the size and duration of survival experiments with long-term survivors under random censoring schemes.     

Joint frailty models for recurring events and death using maximum penalized likelihood estimation: application on cancer events

The observation of repeated events for subjects in cohort studies could be terminated by loss to follow-up, end of study, or a major failure event such as death. In this context, the major failure event could be correlated with recurrent events, and the usual assumption of noninformative censoring of the recurrent event process by death, required by most statistical analyses, can be violated. Recently, joint modeling for 2 survival processes has received considerable attention because it makes it possible to study the joint evolution over time of 2 processes and gives unbiased and efficient parameters. The most commonly used estimation procedure in the joint models for survival events is the expectation maximization algorithm. We show how maximum penalized likelihood estimation can be applied to nonparametric estimation of the continuous hazard functions in a general joint frailty model with right censoring and delayed entry. The simulation study demonstrates that this semiparametric approach yields satisfactory results in this complex setting. As an illustration, such an approach is applied to a prospective cohort with recurrent events of follicular lymphomas, jointly modeled with death.     

Consistent estimation of the expected Brier score in general survival models with right-censored event times

In survival analysis with censored data the mean squared error of prediction can be estimated by weighted averages of time-dependent residuals. Graf et al. (1999) suggested a robust weighting scheme based on the assumption that the censoring mechanism is independent of the covariates. We show consistency of the estimator. Furthermore, we show that a modified version of this estimator is consistent even when censoring and event times are only conditionally independent given the covariates. The modified estimators are derived on the basis of regression models for the censoring distribution. A simulation study and a real data example illustrate the results.     

A Likelihood-Based Approach with Shared Latent Random Parameters for the Longitudinal Binary and Informative Censoring Processes

Longitudinal studies with binary outcomes characterized by informative right censoring are commonly encountered in clinical, basic, behavioral, and health sciences. Approaches developed to analyze data with binary outcomes were mainly tailored to clustered or longitudinal data with missing completely at random or at random. Studies that focused on informative right censoring with binary outcomes are characterized by their imbedded computational complexity and difficulty of implementation. Here we present a new maximum likelihood-based approach with repeated binary measures modeled in a generalized linear mixed model as a function of time and other covariates. The longitudinal binary outcome and the censoring process determined by the number of times a subject is observed share latent random variables (random intercept and slope) where these subject-specific random effects are common to both models. A simulation study and sensitivity analysis were conducted to test the model under different assumptions and censoring settings. Our results showed accuracy of the estimates generated under this model when censoring was fully informative or partially informative with dependence on the slopes. A successful implementation was undertaken on a cohort of renal transplant patients with blood urea nitrogen as a binary outcome measured over time to indicate normal and abnormal kidney function until the emanation of graft rejection that eventuated in informative right censoring. In addition to its novelty and accuracy, an additional key feature and advantage of the proposed model is its viability of implementation on available analytical tools and widespread application on any other longitudinal dataset with informative censoring.

     

A Model for the Censoring Process in Incomplete Repeated Measurement Experiments

Premature terminations or dropouts occur often in repeated measurement experiments. A number of methods have been proposed to analyze such data but most of them assume that the censoring mechanism is, within each group, unaffected by the mechanism generating the response variables. In this paper, we propose a model for the censoring mechanism that generates dropouts. We then show how this model can be used to check whether the censoring mechanism is affected by the response variables and other covariates. Finally, the methods of the paper are applied to the “Halothane” data set.     

Estimation and comparison of changes in the presence of informative right censoring: conditional linear model

A general linear regression model for the usual least squares estimated rate of change (slope) on censoring time is described as an approximation to account for informative right censoring in estimating and comparing changes of a continuous variable in two groups. Two noniterative estimators for the group slope means, the linear minimum variance unbiased (LMVUB) estimator and the linear minimum mean squared error (LMMSE) estimator, are proposed under this conditional model. In realistic situations, we illustrate that the LMVUB and LMMSE estimators, derived under a simple linear regression model, are quite competitive compared to the pseudo maximum likelihood estimator (PMLE) derived by modeling the censoring probabilities. Generalizations to polynomial response curves and general linear models are also described.