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.     

Tobit regression for modeling mean survival time using data subject to multiple sources of censoring

Mean survival time is often of inherent interest in medical and epidemiologic studies. In the presence of censoring and when covariate effects are of interest, Cox regression is the strong default, but mostly due to convenience and familiarity. When survival times are uncensored, covariate effects can be estimated as differences in mean survival through linear regression. Tobit regression can validly be performed through maximum likelihood when the censoring times are fixed (ie, known for each subject, even in cases where the outcome is observed). However, Tobit regression is generally inapplicable when the response is subject to random right censoring. We propose Tobit regression methods based on weighted maximum likelihood which are applicable to survival times subject to both fixed and random censoring times. Under the proposed approach, known right censoring is handled naturally through the Tobit model, with inverse probability of censoring weighting used to overcome random censoring. Essentially, the re‐weighting data are intended to represent those that would have been observed in the absence of random censoring. We develop methods for estimating the Tobit regression parameter, then the population mean survival time. A closed form large‐sample variance estimator is proposed for the regression parameter estimator, with a semiparametric bootstrap standard error estimator derived for the population mean. The proposed methods are easily implementable using standard software. Finite‐sample properties are assessed through simulation. The methods are applied to a large cohort of patients wait‐listed for kidney transplantation.     

A likelihood based approach for joint modeling of longitudinal trajectories and informative censoring process

We propose a joint modeling likelihood-based approach for studies with repeated measures and informative right censoring. Joint modeling of longitudinal and survival data are common approaches but could result in biased estimates if proportionality of hazards is violated. To overcome this issue, and given that the exact time of dropout is typically unknown, we modeled the censoring time as the number of follow-up visits and extended it to be dependent on selected covariates. Longitudinal trajectories for each subject were modeled to provide insight into disease progression and incorporated with the number follow-up visits in one likelihood function.     

Joint Modelling of Survival and Longitudinal Data with Informative Observation Times

In this paper, we consider the joint modelling of survival and longitudinal data with informative observation time points. The survival model and the longitudinal model are linked via random effects, for which no distribution assumption is required under our estimation approach. The estimator is shown to be consistent and asymptotically normal. The proposed estimator and its estimated covariance matrix can be easily calculated. Simulation studies and an application to a primary biliary cirrhosis study are also provided.     

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.     

A general joint model for longitudinal measurements and competing risks survival data with heterogeneous random effects

This article studies a general joint model for longitudinal measurements and competing risks survival data. The model consists of a linear mixed effects sub-model for the longitudinal outcome, a proportional cause-specific hazards frailty sub-model for the competing risks survival data, and a regression sub-model for the variance–covariance matrix of the multivariate latent random effects based on a modified Cholesky decomposition. The model provides a useful approach to adjust for non-ignorable missing data due to dropout for the longitudinal outcome, enables analysis of the survival outcome with informative censoring and intermittently measured time-dependent covariates, as well as joint analysis of the longitudinal and survival outcomes. Unlike previously studied joint models, our model allows for heterogeneous random covariance matrices. It also offers a framework to assess the homogeneous covariance assumption of existing joint models. A Bayesian MCMC procedure is developed for parameter estimation and inference. Its performances and frequentist properties are investigated using simulations. A real data example is used to illustrate the usefulness of the approach.     

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.     

A Proportional Hazards Regression Model for the Subdistribution with Covariates‐adjusted Censoring Weight for Competing Risks Data

With competing risks data, one often needs to assess the treatment and covariate effects on the cumulative incidence function. Fine and Gray proposed a proportional hazards regression model for the subdistribution of a competing risk with the assumption that the censoring distribution and the covariates are independent. Covariate‐dependent censoring sometimes occurs in medical studies. In this paper, we study the proportional hazards regression model for the subdistribution of a competing risk with proper adjustments for covariate‐dependent censoring. We consider a covariate‐adjusted weight function by fitting the Cox model for the censoring distribution and using the predictive probability for each individual. Our simulation study shows that the covariate‐adjusted weight estimator is basically unbiased when the censoring time depends on the covariates, and the covariate‐adjusted weight approach works well for the variance estimator as well. We illustrate our methods with bone marrow transplant data from the Center for International Blood and Marrow Transplant Research. Here, cancer relapse and death in complete remission are two competing risks.     

A Profile Conditional Likelihood Approach for the Semiparametric Transformation Regression Model with Missing Covariates

We propose a profile conditional likelihood approach to handle missing covariates in the general semiparametric transformation regression model. The method estimates the marginal survival function by the Kaplan-Meier estimator, and then estimates the parameters of the survival model and the covariate distribution from a conditional likelihood, substituting the Kaplan-Meier estimator for the marginal survival function in the conditional likelihood. This method is simpler than full maximum likelihood approaches, and yields consistent and asymptotically normally distributed estimator of the regression parameter when censoring is independent of the covariates. The estimator demonstrates very high relative efficiency in simulations. When compared with complete-case analysis, the proposed estimator can be more efficient when the missing data are missing completely at random and can correct bias when the missing data are missing at random. The potential application of the proposed method to the generalized probit model with missing continuous covariates is also outlined.     

Regression analysis of informative current status data with the semiparametric linear transformation model

Many methods have been developed in the literature for regression analysis of current status data with noninformative censoring and also some approaches have been proposed for semiparametric regression analysis of current status data with informative censoring. However, the existing approaches for the latter situation are mainly on specific models such as the proportional hazards model and the additive hazard model. Corresponding to this, in this paper, we consider a general class of semiparametric linear transformation models and develop a sieve maximum likelihood estimation approach for the inference. In the method, the copula model is employed to describe the informative censoring or relationship between the failure time of interest and the censoring time, and Bernstein polynomials are used to approximate the nonparametric functions involved. The asymptotic consistency and normality of the proposed estimators are established, and an extensive simulation study is conducted and indicates that the proposed approach works well for practical situations. In addition, an illustrative example is provided.     

AN APPROACH FOR JOINTLY MODELING MULTIVARIATE LONGITUDINAL MEASUREMENTS AND DISCRETE TIME-TO-EVENT DATA

In many medical studies, patients are followed longitudinally and interest is on assessing the relationship between longitudinal measurements and time to an event. Recently, various authors have proposed joint modeling approaches for longitudinal and time-to-event data for a single longitudinal variable. These joint modeling approaches become intractable with even a few longitudinal variables. In this paper we propose a regression calibration approach for jointly modeling multiple longitudinal measurements and discrete time-to-event data. Ideally, a two-stage modeling approach could be applied in which the multiple longitudinal measurements are modeled in the first stage and the longitudinal model is related to the time-to-event data in the second stage. Biased parameter estimation due to informative dropout makes this direct two-stage modeling approach problematic. We propose a regression calibration approach which appropriately accounts for informative dropout. We approximate the conditional distribution of the multiple longitudinal measurements given the event time by modeling all pairwise combinations of the longitudinal measurements using a bivariate linear mixed model which conditions on the event time. Complete data are then simulated based on estimates from these pairwise conditional models, and regression calibration is used to estimate the relationship between longitudinal data and time-to-event data using the complete data. We show that this approach performs well in estimating the relationship between multivariate longitudinal measurements and the time-to-event data and in estimating the parameters of the multiple longitudinal process subject to informative dropout. We illustrate this methodology with simulations and with an analysis of primary biliary cirrhosis (PBC) data.     

A semiparametric stochastic mixed effects model for bivariate cyclic longitudinal data

We propose a flexible semiparametric stochastic mixed effects model for bivariate cyclic longitudinal data. The model can handle either single cycle or, more generally, multiple consecutive cycle data. The approach models the mean of responses by parametric fixed effects and a smooth nonparametric function for the underlying time effects, and the relationship across the bivariate responses by a bivariate Gaussian random field and a joint distribution of random effects. The proposed model not only can model complicated individual profiles, but also allows for more flexible within-subject and between-response correlations. The fixed effects regression coefficients and the nonparametric time functions are estimated using maximum penalized likelihood, where the resulting estimator for the nonparametric time function is a cubic smoothing spline. The smoothing parameters and variance components are estimated simultaneously using restricted maximum likelihood. Simulation results show that the parameter estimates are close to the true values. The fit of the proposed model on a real bivariate longitudinal dataset of pre-menopausal women also performs well, both for a single cycle analysis and for a multiple consecutive cycle analysis. The Canadian Journal of Statistics 48: 471–498; 2020 © 2020 Statistical Society of Canada     

Maximum likelihood analysis of semicompeting risks data with semiparametric regression models

The “semicompeting risks” include a terminal event and a non-terminal event. The terminal event may censor the non-terminal event but not vice versa. Because times to the two events are usually correlated, the non-terminal event is subject to dependent/informative censoring by the terminal event. We seek to conduct marginal regressions and joint association analyses for the two event times under semicompeting risks. The proposed method is based on the modeling setup where the semiparametric transformation models are assumed for marginal regressions, and a copula model is assumed for the joint distribution. We propose a nonparametric maximum likelihood approach for inferences, which provides a martingale representation for the score function and an analytical expression for the information matrix. Direct theoretical developments and computational implementation are allowed for the proposed approach. Simulations and a real data application demonstrate the utility of the proposed methodology.     

Multiple event times in the presence of informative censoring: modeling and analysis by copulas

Motivated by a breast cancer research program, this paper is concerned with the joint survivor function of multiple event times when their observations are subject to informative censoring caused by a terminating event. We formulate the correlation of the multiple event times together with the time to the terminating event by an Archimedean copula to account for the informative censoring. Adapting the widely used two-stage procedure under a copula model, we propose an easy-to-implement pseudo-likelihood based procedure for estimating the model parameters. The approach yields a new estimator for the marginal distribution of a single event time with semicompeting-risks data. We conduct both asymptotics and simulation studies to examine the proposed approach in consistency, efficiency, and robustness. Data from the breast cancer program are employed to illustrate this research.

     

Non parametric hazard estimation with dependent censoring using penalized likelihood and an assumed copula

This article introduces a novel non parametric penalized likelihood hazard estimation when the censoring time is dependent on the failure time for each subject under observation. More specifically, we model this dependence using a copula, and the method of maximum penalized likelihood (MPL) is adopted to estimate the hazard function. We do not consider covariates in this article. The non negatively constrained MPL hazard estimation is obtained using a multiplicative iterative algorithm. The consistency results and the asymptotic properties of the proposed hazard estimator are derived. The simulation studies show that our MPL estimator under dependent censoring with an assumed copula model provides a better accuracy than the MPL estimator under independent censoring if the sign of dependence is correctly specified in the copula function. The proposed method is applied to a real dataset, with a sensitivity analysis performed over various values of correlation between failure and censoring times.     

Empirical likelihood inference for the regression model of mean quality‐adjusted lifetime with censored data

The authors consider the empirical likelihood method for the regression model of mean quality‐adjusted lifetime with right censoring. They show that an empirical log‐likelihood ratio for the vector of the regression parameters is asymptotically a weighted sum of independent chi‐squared random variables. They adjust this empirical log‐likelihood ratio so that the limiting distribution is a standard chi‐square and construct corresponding confidence regions. Simulation studies lead them to conclude that empirical likelihood methods outperform the normal approximation methods in terms of coverage probability. They illustrate their methods with a data example from a breast cancer clinical trial study.     

Likelihood estimation for a longitudinal negative binomial regression model with missing outcomes

Joint damage in psoriatic arthritis can be measured by clinical and radiological methods, the former being done more frequently during longitudinal follow-up of patients. Motivated by the need to compare findings based on the different methods with different observation patterns, we consider longitudinal data where the outcome variable is a cumulative total of counts that can be unobserved when other, informative, explanatory variables are recorded. We demonstrate how to calculate the likelihood for such data when it is assumed that the increment in the cumulative total follows a discrete distribution with a location parameter that depends on a linear function of explanatory variables. An approach to the incorporation of informative observation is suggested. We present analyses based on an observational database from a psoriatic arthritis clinic. Although the use of the new statistical methodology has relatively little effect in this example, simulation studies indicate that the method can provide substantial improvements in bias and coverage in some situations where there is an important time varying explanatory variable.     

Smooth Plug‐in Inverse Estimators in the Current Status Continuous Mark Model

Abstract. We consider the problem of estimating the joint distribution function of the event time and a continuous mark variable when the event time is subject to interval censoring case 1 and the continuous mark variable is only observed in case the event occurred before the time of inspection. The non‐parametric maximum likelihood estimator in this model is known to be inconsistent. We study two alternative smooth estimators, based on the explicit (inverse) expression of the distribution function of interest in terms of the density of the observable vector. We derive the pointwise asymptotic distribution of both estimators.     

Consistency of the Generalized MLE with Interval-Censored and Masked Competing Risks Data

We consider nonparametric estimation based on interval-censored competing risks data with masked failure cause. The generalized maximum likelihood estimator of the joint survival function of the failure time and the failure cause is studied under mixed case interval censorship and random partition masking. Strong consistency in the L ₁(μ)-topology is established for some finite measure μ which is derived from the joint censoring and masking distribution. Under additional regularity assumptions we also establish the strong consistencies in the topologies of weak convergence, point-wise convergence, and uniform convergence.