Methods for Estimating Center Effects on Recurrent Events

In this article, we develop methods for quantifying center effects with respect to recurrent event data. In the models of interest, center effects are assumed to act multiplicatively on the recurrent event rate function. When the number of centers is large, traditional estimation methods that treat centers as categorical variables have many parameters and are sometimes not feasible to implement, especially with large numbers of distinct recurrent event times. We propose a new estimation method for center effects which avoids including indicator variables for centers. We then show that center effects can be consistently estimated by the center-specific ratio of observed to expected cumulative numbers of events. We also consider the case where the recurrent event sequence can be stopped permanently by a terminating event. Large-sample results are developed for the proposed estimators. We assess the finite-sample properties of the proposed estimators through simulation studies. The methods are then applied to national hospital admissions data for end stage renal disease patients.     

Predicting events in clinical trials using two time‐to‐event outcomes

In clinical trials with time‐to‐event outcomes, it is of interest to predict when a prespecified number of events can be reached. Interim analysis is conducted to estimate the underlying survival function. When another correlated time‐to‐event endpoint is available, both outcome variables can be used to improve estimation efficiency. In this paper, we propose to use the convolution of two time‐to‐event variables to estimate the survival function of interest. Propositions and examples are provided based on exponential models that accommodate possible change points. We further propose a new estimation equation about the expected time that exploits the relationship of two endpoints. Simulations and the analysis of real data show that the proposed methods with bivariate information yield significant improvement in prediction over that of the univariate method.     

Explained variation and degrees of necessity and of sufficiency for competing risks survival data

In this contribution, the Schemper–Henderson measure of explained variation for survival outcomes is extended to accommodate competing events (CEs) in addition to events of interest. The extension is achieved by moving from the unconditional and conditional survival functions of the original measure to unconditional and conditional cumulative incidence functions, the latter obtained, for example, from Fine and Gray models. In the absence of CEs, the original measure is obtained as a special case. We define explained variation on the population level and provide two different types of estimates. Recently, the authors have achieved a multiplicative decomposition of explained variation into degrees of necessity and degrees of sufficiency. These measures are also extended to the case of competing risks survival data. A SAS macro and an R function are provided to facilitate application. Interesting empirical properties of the measures are explored on the population level and by an extensive simulation study. Advantages of the approach are exemplified by an Austrian study of breast cancer with a high proportion of CEs.     

Nonparametric estimation of the concordance correlation coefficient under univariate censoring

Assessing agreement is often of interest in clinical studies to evaluate the similarity of measurements produced by different raters or methods on the same subjects. Lin's (1989, Biometrics 45, 255-268) concordance correlation coefficient (CCC) has become a popular measure of agreement for correlated continuous outcomes. However, commonly used estimation methods for the CCC do not accommodate censored observations and are, therefore, not applicable for survival outcomes. In this article, we estimate the CCC nonparametrically through the bivariate survival function. The proposed estimator of the CCC is proven to be strongly consistent and asymptotically normal, with a consistent bootstrap variance estimator. Furthermore, we propose a time-dependent agreement coefficient as an extension of Lin's (1989) CCC for measuring the agreement between survival times among subjects who survive beyond a specified time point. A nonparametric estimator is developed for the time-dependent agreement coefficient as well. It has the same asymptotic properties as the estimator of the CCC. Simulation studies are conducted to evaluate the performance of the proposed estimators. A real data example from a prostate cancer study is used to illustrate the method.     

A Semiparametric Joint Model for Longitudinal and Survival Data with Application to Hemodialysis Study

Summary .  In many longitudinal clinical studies, the level and progression rate of repeatedly measured biomarkers on each subject quantify the severity of the disease and that subject's susceptibility to progression of the disease. It is of scientific and clinical interest to relate such quantities to a later time-to-event clinical endpoint such as patient survival. This is usually done with a shared parameter model. In such models, the longitudinal biomarker data and the survival outcome of each subject are assumed to be conditionally independent given subject-level severity or susceptibility (also called frailty in statistical terms). In this article, we study the case where the conditional distribution of longitudinal data is modeled by a linear mixed-effect model, and the conditional distribution of the survival data is given by a Cox proportional hazard model. We allow unknown regression coefficients and time-dependent covariates in both models. The proposed estimators are maximizers of an exact correction to the joint log likelihood with the frailties eliminated as nuisance parameters, an idea that originated from correction of covariate measurement error in measurement error models. The corrected joint log likelihood is shown to be asymptotically concave and leads to consistent and asymptotically normal estimators. Unlike most published methods for joint modeling, the proposed estimation procedure does not rely on distributional assumptions of the frailties. The proposed method was studied in simulations and applied to a data set from the Hemodialysis Study.     

Bayesian models for multivariate current status data with informative censoring

Multivariate current status data, consist of indicators of whether each of several events occur by the time of a single examination. Our interest focuses on inferences about the joint distribution of the event times. Conventional methods for analysis of multiple event-time data cannot be used because all of the event times are censored and censoring may be informative. Within a given subject, we account for correlated event times through a subject-specific latent variable, conditional upon which the various events are assumed to occur independently. We also assume that each event contributes independently to the hazard of censoring. Nonparametric step functions are used to characterize the baseline distributions of the different event times and of the examination times. Covariate and subject-specific effects are incorporated through generalized linear models. A Markov chain Monte Carlo algorithm is described for estimation of the posterior distributions of the unknowns. The methods are illustrated through application to multiple tumor site data from an animal carcinogenicity study.     

Directly parameterized regression conditioning on being alive: analysis of longitudinal data truncated by deaths

For observational longitudinal studies of geriatric populations, outcomes such as disability or cognitive functioning are often censored by death. Statistical analysis of such data may explicitly condition on either vital status or survival time when summarizing the longitudinal response. For example a pattern-mixture model characterizes the mean response at time t conditional on death at time S = s (for s > t), and thus uses future status as a predictor for the time t response. As an alternative, we define regression conditioning on being alive as a regression model that conditions on survival status, rather than a specific survival time. Such models may be referred to as partly conditional since the mean at time t is specified conditional on being alive (S > t), rather than using finer stratification (S = s for s > t). We show that naive use of standard likelihood-based longitudinal methods and generalized estimating equations with non-independence weights may lead to biased estimation of the partly conditional mean model. We develop a taxonomy for accommodation of both dropout and death, and describe estimation for binary longitudinal data that applies selection weights to estimating equations with independence working correlation. Simulation studies and an analysis of monthly disability status illustrate potential bias in regression methods that do not explicitly condition on survival.     

Semiparametric Transformation Rate Model for Recurrent Event Data

In this article, we propose a class of semiparametric transformation rate models for recurrent event data subject to right censoring and potentially stopped by a terminating event (e.g., death). These transformation models include both additive rates model and proportional rates model as special cases. Respecting the property that no recurrent events can occur after the terminating event, we model the conditional recurrent event rate given survival. Weighted estimating equations are constructed to estimate the regression coefficients and baseline rate function. In particular, the baseline rate function is approximated by wavelet function. Asymptotic properties of the proposed estimators are derived and a data-dependent criterion is proposed for selecting the most suitable transformation. Simulation studies show that the proposed estimators perform well for practical sample sizes. The proposed methods are used in two real-data examples: a randomized trial of rhDNase and a community trial of vitamin A.     

Measuring agreement of multivariate discrete survival times using a modified weighted kappa coefficient

Summary .  Assessing agreement is often of interest in clinical studies to evaluate the similarity of measurements produced by different raters or methods on the same subjects. We present a modified weighted kappa coefficient to measure agreement between bivariate discrete survival times. The proposed kappa coefficient accommodates censoring by redistributing the mass of censored observations within the grid where the unobserved events may potentially happen. A generalized modified weighted kappa is proposed for multivariate discrete survival times. We estimate the modified kappa coefficients nonparametrically through a multivariate survival function estimator. The asymptotic properties of the kappa estimators are established and the performance of the estimators are examined through simulation studies of bivariate and trivariate survival times. We illustrate the application of the modified kappa coefficient in the presence of censored observations with data from a prostate cancer study.     

Estimation of Bayesian confidence intervals for the product of conditional survival probabilities

A model is derived to estimate the survival probability of a time interval when censorings occur. The time interval is divided into partial intervals in order to obtain the conditional survival probabilities, each of which is a parameter of a Binomial distributed random variable. To allow for the dependence between the events in the different intervals these parameters are transformed. Corresponding a priori density functions are formulated regarding both the Bayesian uniform distribution and the special model. The a posteriori density function is derived for the product of the conditional survival probabilities, and formulae for the BAYE sian confidence interval and the expectation are given. Lower and upper bounds for the confidence interval and the expectation are derived. Some examples are given to compare the results with other methods.     

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)     

Regression analysis of doubly censored failure time data with frailty

In doubly censored failure time data, the survival time of interest is defined as the elapsed time between an initial event and a subsequent event, and the occurrences of both events cannot be observed exactly. Instead, only right- or interval-censored observations on the occurrence times are available. For the analysis of such data, a number of methods have been proposed under the assumption that the survival time of interest is independent of the occurrence time of the initial event. This article investigates a different situation where the independence may not be true with the focus on regression analysis of doubly censored data. Cox frailty models are applied to describe the effects of covariates and an EM algorithm is developed for estimation. Simulation studies are performed to investigate finite sample properties of the proposed method and an illustrative example from an acquired immune deficiency syndrome (AIDS) cohort study is provided.     

Maximum likelihood estimation of marginal pairwise associations with multiple source predictors

Researchers interested in the association of a predictor with an outcome will often collect information about that predictor from more than one source. Standard multiple regression methods allow estimation of the effect of each predictor on the outcome while controlling for the remaining predictors. The resulting regression coefficient for each predictor has an interpretation that is conditional on all other predictors. In settings in which interest is in comparison of the marginal pairwise relationships between each predictor and the outcome separately (e.g., studies in psychiatry with multiple informants or comparison of the predictive values of diagnostic tests), standard regression methods are not appropriate. Instead, the generalized estimating equations (GEE) approach can be used to simultaneously estimate, and make comparisons among, the separate pairwise marginal associations. In this paper, we consider maximum likelihood (ML) estimation of these marginal relationships when the outcome is binary. ML enjoys benefits over GEE methods in that it is asymptotically efficient, can accommodate missing data that are ignorable, and allows likelihood-based inferences about the pairwise marginal relationships. We also explore the asymptotic relative efficiency of ML and GEE methods in this setting.     

Correlated versus uncorrelated frailty Cox models: A comparison of different estimation procedures

In many studies in medicine, including clinical trials and epidemiological investigations, data are clustered into groups such as health centers or herds in veterinary medicine. Such data are usually analyzed by hierarchical regression models to account for possible variation between groups. When such variation is large, it is of potential interest to explore whether additionally the effect of a within‐group predictor varies between groups. In survival analysis, this may be investigated by including two frailty terms at group level in a Cox proportional hazards model. Several estimation methods have been proposed to estimate this type of frailty Cox models. We review four of these methods, apply them to real data from veterinary medicine, and compare them using a simulation study.     

Robust tests for treatment effects based on censored recurrent event data observed over multiple periods

We derive semiparametric methods for estimating and testing treatment effects when censored recurrent event data are available over multiple periods. These methods are based on estimating functions motivated by a working "mixed-Poisson" assumption under which conditioning can eliminate subject-specific random effects. Robust pseudoscore test statistics are obtained via "sandwich" variance estimation. The relative efficiency of conditional versus marginal analyses is assessed analytically under a mixed time-homogeneous Poisson model. The robustness and empirical power of the semiparametric approach are assessed through simulation. Adaptations to handle recurrent events arising in crossover trials are described and these methods are applied to data from a two-period crossover trial of patients with bronchial asthma.     

Analysis of secondary phenotypes in multigroup association studies

Although case-control association studies have been widely used, they are insufficient for many complex diseases, such as Alzheimer's disease and breast cancer, since these diseases may have multiple subtypes with distinct morphologies and clinical implications. Many multigroup studies, such as the Alzheimer's Disease Neuroimaging Initiative (ADNI), have been undertaken by recruiting subjects based on their multiclass primary disease status, while extensive secondary outcomes have been collected. The aim of this paper is to develop a general regression framework for the analysis of secondary phenotypes collected in multigroup association studies. Our regression framework is built on a conditional model for the secondary outcome given the multigroup status and covariates and its relationship with the population regression of interest of the secondary outcome given the covariates. Then, we develop generalized estimation equations to estimate the parameters of interest. We use both simulations and a large-scale imaging genetic data analysis from the ADNI to evaluate the effect of the multigroup sampling scheme on standard genome-wide association analyses based on linear regression methods, while comparing it with our statistical methods that appropriately adjust for the multigroup sampling scheme. Data used in preparation of this article were obtained from the ADNI database.     

Estimating survival and association in a semicompeting risks model

Summary .   In many follow-up studies, patients are subject to concurrent events. In this article, we consider semicompeting risks data as defined by Fine, Jiang, and Chappell (2001, Biometrika 88 , 907–919) where one event is censored by the other but not vice versa. The proposed model involves marginal survival functions for the two events and a parametric family of copulas for their dependency. This article suggests a general method for estimating the dependence parameter when the dependency is modeled with an Archimedean copula. It uses the copula-graphic estimator of Zheng and Klein (1995, Biometrika 82 , 127–138) for estimating the survival function of the nonterminal event, subject to dependent censoring. Asymptotic properties of these estimators are derived. Simulations show that the new methods work well with finite samples. The copula-graphic estimator is shown to be more accurate than the estimator proposed by Fine et al. (2001) ; its performances are similar to those of the self-consistent estimator of Jiang, Fine, Kosorok, and Chappell (2005, Scandinavian Journal of Statistics 33, 1–20). The analysis of a data set, emphasizing the estimation of characteristics of the observable region, is presented as an illustration.     

Joint inference for competing risks data using multiple endpoints

Competing risks data are commonly encountered in randomized clinical trials and observational studies. This paper considers the situation where the ending statuses of competing events have different clinical interpretations and/or are of simultaneous interest. In clinical trials, often more than one competing event has meaningful clinical interpretations even though the trial effects of different events could be different or even opposite to each other. In this paper, we develop estimation procedures and inferential properties for the joint use of multiple cumulative incidence functions (CIFs). Additionally, by incorporating longitudinal marker information, we develop estimation and inference procedures for weighted CIFs and related metrics. The proposed methods are applied to a COVID-19 in-patient treatment clinical trial, where the outcomes of COVID-19 hospitalization are either death or discharge from the hospital, two competing events with completely different clinical implications.     

Dimension reduction in survival regressions with censored data via an imputed spline approach

Dimension reduction methods have been proposed for regression analysis with predictors of high dimension, but have not received much attention on the problems with censored data. In this article, we present an iterative imputed spline approach based on principal Hessian directions (PHD) for censored survival data in order to reduce the dimension of predictors without requiring a prespecified parametric model. Our proposal is to replace the right-censored survival time with its conditional expectation for adjusting the censoring effect by using the Kaplan-Meier estimator and an adaptive polynomial spline regression in the residual imputation. A sparse estimation strategy is incorporated in our approach to enhance the interpretation of variable selection. This approach can be implemented in not only PHD, but also other methods developed for estimating the central mean subspace. Simulation studies with right-censored data are conducted for the imputed spline approach to PHD (IS-PHD) in comparison with two methods of sliced inverse regression, minimum average variance estimation, and naive PHD in ignorance of censoring. The results demonstrate that the proposed IS-PHD method is particularly useful for survival time responses approximating symmetric or bending structures. Illustrative applications to two real data sets are also presented.     

Nonparametric inference for median costs with censored data

Summary Increasingly, estimations of health care costs are used to evaluate competing treatments or to assess the expected expenditures associated with certain diseases. In health policy and economics, the primary focus of these estimations has been on the mean cost, because the total cost can be derived directly from the mean cost, and because information about total resources utilized is highly relevant for policymakers. Yet, the median cost also could be important, both as an intuitive measure of central tendency in cost distribution and as a subject of interest to payers and consumers. In many prospective studies, cost data collection is sometimes incomplete for some subjects due to right censoring, which typically is caused by loss to follow-up or by limited study duration. Censoring poses a unique challenge for cost data analysis because of so-called induced informative censoring, in that traditional methods suited for survival data generally are invalid in censored cost estimation. In this article, we propose methods for estimating the median cost and its confidence interval (CI) when data are subject to right censoring. We also consider the estimation of the ratio and difference of two median costs and their CIs. These methods can be extended to the estimation of other quantiles and other informatively censored data. We conduct simulation and real data analysis in order to examine the performance of the proposed methods.