Joint regression modeling for missing categorical covariates in generalized linear models

Missing covariates data is a common issue in generalized linear models (GLMs). A model-based procedure arising from properly specifying joint models for both the partially observed covariates and the corresponding missing indicator variables represents a sound and flexible methodology, which lends itself to maximum likelihood estimation as the likelihood function is available in computable form. In this paper, a novel model-based methodology is proposed for the regression analysis of GLMs when the partially observed covariates are categorical. Pair-copula constructions are used as graphical tools in order to facilitate the specification of the high-dimensional probability distributions of the underlying missingness components. The model parameters are estimated by maximizing the weighted log-likelihood function by using an EM algorithm. In order to compare the performance of the proposed methodology with other well-established approaches, which include complete-cases and multiple imputation, several simulation experiments of Binomial, Poisson and Normal regressions are carried out under both missing at random and non-missing at random mechanisms scenarios. The methods are illustrated by modeling data from a stage III melanoma clinical trial. The results show that the methodology is rather robust and flexible, representing a competitive alternative to traditional techniques.     

Recurrent events analysis in the presence of time-dependent covariates and dependent censoring

Summary.  Recurrent events models have had considerable attention recently. The majority of approaches show the consistency of parameter estimates under the assumption that censoring is independent of the recurrent events process of interest conditional on the covariates that are included in the model. We provide an overview of available recurrent events analysis methods and present an inverse probability of censoring weighted estimator for the regression parameters in the Andersen–Gill model that is commonly used for recurrent event analysis. This estimator remains consistent under informative censoring if the censoring mechanism is estimated consistently, and it generally improves on the naïve estimator for the Andersen–Gill model in the case of independent censoring. We illustrate the bias of ad hoc estimators in the presence of informative censoring with a simulation study and provide a data analysis of recurrent lung exacerbations in cystic fibrosis patients when some patients are lost to follow-up.     

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.     

Effect of censoring on adjusting for covariates in comparison of survival times

The increase in the variance of the estimate of treatment effect which results from omitting a dichotomous or continuous covariate is quantified as a function of censoring. The efficiency of not adjusting for a covariate is measured by the ratio of the variance obtained with and without adjustment for the covariate. The variance is derived using the Weibull proportional hazards model. Under random censoring, the efficiency of not adjusting for a continuous covariate is an increasing function of the percentage of censored observations.     

Inverse probability of censoring weighting for visual predictive checks of time-to-event models with time-varying covariates

When constructing models to summarize clinical data to be used for simulations, it is good practice to evaluate the models for their capacity to reproduce the data. This can be done by means of Visual Predictive Checks (VPC), which consist of several reproductions of the original study by simulation from the model under evaluation, calculating estimates of interest for each simulated study and comparing the distribution of those estimates with the estimate from the original study. This procedure is a generic method that is straightforward to apply, in general. Here we consider the application of the method to time-to-event data and consider the special case when a time-varying covariate is not known or cannot be approximated after event time. In this case, simulations cannot be conducted beyond the end of the follow-up time (event or censoring time) in the original study. Thus, the simulations must be censored at the end of the follow-up time. Since this censoring is not random, the standard KM estimates from the simulated studies and the resulting VPC will be biased. We propose to use inverse probability of censoring weighting (IPoC) method to correct the KM estimator for the simulated studies and obtain unbiased VPCs. For analyzing the Cantos study, the IPoC weighting as described here proved valuable and enabled the generation of VPCs to qualify PKPD models for simulations. Here, we use a generated data set, which allows illustration of the different situations and evaluation against the known truth.     

Distribution-free comparison of hazard rates of two distributions under progressive type-II censoring

Some distribution-free tests have been discussed in the literature with regard to the comparison of hazard rates of two distributions when the available samples are complete. We generalize here Kochar's [S.C. Kochar, A new distribution-free test for the equality of two failure rates, Biometrika 68 (1981), pp. 423–426] test statistic to the case when one available sample is progressively Type-II censored, and then derive its exact null distribution and examine its power properties by means of a Monte Carlo simulation study.     

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.     

Analysis of Gamma and Weibull lifetime data under a general censoring scheme and in the presence of covariates

We consider the problem of estimating the lifetime distributions of survival times subject to a general censoring scheme called “middle censoring”. The lifetimes are assumed to follow a parametric family of distributions, such as the Gamma or Weibull distributions, and is applied to cases when the lifetimes come with covariates affecting them. For any individual in the sample, there is an independent, random, censoring interval. We will observe the actual lifetime if the lifetime falls outside of this censoring interval, otherwise we only observe the interval of censoring. This censoring mechanism, which includes both right- and left-censoring, has been called “middle censoring” (see Jammalamadaka and Mangalam, 2003 Jammalamadaka, S. Rao, Mangalam, V. (2003). Nonparametric estimation for middle censored data. J. Nonparamet. Stat. 15(2):253–265.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]). Maximum-likelihood estimation of the parameters as well as their large-sample properties are studied under this censoring scheme, including the case when covariates are available. We conclude with an application to a dataset from Environmental Economics dealing with ContingentValuation of natural resources.     

Slope estimation of covariates that influence renal outcome following renal transplant adjusting for informative right censoring

A new statistical model is proposed to estimate population and individual slopes that are adjusted for covariates and informative right censoring. Individual slopes are assumed to have a mean that depends on the population slope for the covariates. The number of observations for each individual is modeled as a truncated discrete distribution with mean dependent on the individual subjects’ slopes. Our simulation study results indicated that the associated bias and mean squared errors for the proposed model were comparable to those associated with the model that only adjusts for informative right censoring. The proposed model was illustrated using renal transplant dataset to estimate population slopes for covariates that could impact the outcome of renal function following renal transplantation.     

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.     

Jointly modeling skew longitudinal survival data with missingness and mismeasured covariates

Jointly modeling longitudinal and survival data has been an active research area. Most researches focus on improving the estimating efficiency but ignore many data features frequently encountered in practice. In the current study, we develop the joint models that concurrently accounting for longitudinal and survival data with multiple features. Specifically, the proposed model handles skewness, missingness and measurement errors in covariates which are typically observed in the collection of longitudinal survival data from many studies. We employ a Bayesian inferential method to make inference on the proposed model. We applied the proposed model to an real data study. A few alternative models under different conditions are compared. We conduct extensive simulations in order to evaluate how the method works.     

Estimating the hazard functions of two alternating recurrent events in the presence of covariates

The motivation for this paper is a cystic fibrosis data which records a patient’s times to relapse and times to cure under several recurrences of the disease. The idea is to study the impact of covariates on the hazard rates of two alternately occurring events. The dependence between the times to the two events over the different cycles is modeled through an autoregressive-type setup. The partial likelihood function is then derived and the estimators obtained. The estimators are shown to be consistent and asymptotically normal. The technique is applied to study the motivating data. A simulation study is also conducted to corroborate the results.     

Estimation of a change point in the hazard rate of Lindley model under right censoring

In this article, we consider a single change point model for a sudden change in the hazard rate of Lindley distribution to model right-censored survival data. We derive the quantile function to generate random numbers from the proposed distribution by using the Lambert function. The maximum likelihood estimation method is used to estimate parameters of the change point model. A simulation study is also carried out to analyze the performance of the estimators. To validate our findings, a dataset on bone marrow transplant for patients of acute lymphoblastic leukemia is analyzed using the proposed model and is compared with the existing exponential single change point model.     

Nonparametric estimation of survival functions for doubly censored observations when the lifetime hazard rate is proportionally related to the censoring hazard rates

Let T, X and Y be non-negative random variables, where T is the time of occurrence of an event of interest, X and Y being the lefl and right censoring variables respectively.

In this paper we propose a nonparametric estimator of the survival function, ST, when T, X and Y are supposed to be independent and their corresponding hazard rates are proportionally related. In this way, our results extend Ebrahimi's work (1985) to the doubly censored data case.     

Joint modelling of longitudinal outcome and interval-censored competing risk dropout in a schizophrenia clinical trial

The 'Clinical antipsychotic trials in intervention effectiveness' study, was designed to evaluate whether there were significant differences between several antipsychotic medications in effectiveness, tolerability, cost and quality of life of subjects with schizophrenia. Overall, 74 % of patients discontinued the study medication for various reasons before the end of 18 months in phase I of the study. When such a large percentage of study participants fail to complete the study schedule, it is not clear whether the apparent profile in effectiveness reflects genuine changes over time or is influenced by selection bias, with participants with worse (or better) outcome values being more likely to drop out or to discontinue. To assess the effect of dropouts for different reasons on inferences, we construct a joint model for the longitudinal outcome and cause-specific dropouts that allows for interval-censored dropout times. Incorporating the information regarding the cause of dropout improves inferences and provides better understanding of the association between cause-specific dropout and the outcome process. We use simulations to demonstrate the advantages of the joint modelling approach in terms of bias and efficiency.     

Joint modeling of degradation and failure time data

This paper surveys some approaches to model the relationship between failure time data and covariate data like internal degradation and external environmental processes. These models which reflect the dependency between system state and system reliability include threshold models and hazard-based models. In particular, we consider the class of degradation–threshold–shock models (DTS models) in which failure is due to the competing causes of degradation and trauma. For this class of reliability models we express the failure time in terms of degradation and covariates. We compute the survival function of the resulting failure time and derive the likelihood function for the joint observation of failure times and degradation data at discrete times. We consider a special class of DTS models where degradation is modeled by a process with stationary independent increments and related to external covariates through a random time scale and extend this model class to repairable items by a marked point process approach. The proposed model class provides a rich conceptual framework for the study of degradation–failure issues.     

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.

     

Comparison of different control charts for a Weibull process with type-I censoring

Some control charts have been proposed to monitor the mean of a Weibull process with type-I censoring. One type of control charts is to monitor changes in the scale parameter because it indicates changes in the mean. With this approach, we compare different control charts such as Shewhart-type and exponentially weighted moving average (EWMA) charts based on conditional expected value (CEV) and cumulative sum (CUSUM) chart based on likelihood-ratio. A simulation approach is employed to compute control limits and average run lengths. The results show that the CUSUM chart has the best performance. However, the EWMA-CEV chart is recommendable for practitioners with its competitive performance and ease of use advantage. An illustrative example is also provided.     

Segmental modeling of changing immunologic response for CD4 data with skewness,missingness and dropout

In clinical practice, the profile of each subject's CD4 response from a longitudinal study may follow a ‘broken stick’ like trajectory, indicating multiple phases of increase and/or decline in response. Such multiple phases (changepoints) may be important indicators to help quantify treatment effect and improve management of patient care. Although it is a common practice to analyze complex AIDS longitudinal data using nonlinear mixed-effects (NLME) or nonparametric mixed-effects (NPME) models in the literature, NLME or NPME models become a challenge to estimate changepoint due to complicated structures of model formulations. In this paper, we propose a changepoint mixed-effects model with random subject-specific parameters, including the changepoint for the analysis of longitudinal CD4 cell counts for HIV infected subjects following highly active antiretroviral treatment. The longitudinal CD4 data in this study may exhibit departures from symmetry, may encounter missing observations due to various reasons, which are likely to be non-ignorable in the sense that missingness may be related to the missing values, and may be censored at the time of the subject going off study-treatment, which is a potentially informative dropout mechanism. Inferential procedures can be complicated dramatically when longitudinal CD4 data with asymmetry (skewness), incompleteness and informative dropout are observed in conjunction with an unknown changepoint. Our objective is to address the simultaneous impact of skewness, missingness and informative censoring by jointly modeling the CD4 response and dropout time processes under a Bayesian framework. The method is illustrated using a real AIDS data set to compare potential models with various scenarios, and some interested results are presented.