Identifiability assumptions for missing covariate data in failure time regression models

Methods in the literature for missing covariate data in survival models have relied on the missing at random (MAR) assumption to render regression parameters identifiable. MAR means that missingness can depend on the observed exit time, and whether or not that exit is a failure or a censoring event. By considering ways in which missingness of covariate X could depend on the true but possibly censored failure time T and the true censoring time C, we attempt to identify missingness mechanisms which would yield MAR data. We find that, under various reasonable assumptions about how missingness might depend on T and/or C, additional strong assumptions are needed to obtain MAR. We conclude that MAR is difficult to justify in practical applications. One exception arises when missingness is independent of T, and C is independent of the value of the missing X. As alternatives to MAR, we propose two new missingness assumptions. In one, the missingness depends on T but not on C; in the other, the situation is reversed. For each, we show that the failure time model is identifiable. When missingness is independent of T, we show that the naive complete record analysis will yield a consistent estimator of the failure time distribution. When missingness is independent of C, we develop a complete record likelihood function and a corresponding estimator for parametric failure time models. We propose analyses to evaluate the plausibility of either assumption in a particular data set, and illustrate the ideas using data from the literature on this problem.     

Joint inference for nonlinear mixed-effects models and time to event at the presence of missing data

In many longitudinal studies, the individual characteristics associated with the repeated measures may be possible covariates of the time to an event of interest, and thus, it is desirable to model the time-to-event process and the longitudinal process jointly. Statistical analyses may be further complicated in such studies with missing data such as informative dropouts. This article considers a nonlinear mixed-effects model for the longitudinal process and the Cox proportional hazards model for the time-to-event process. We provide a method for simultaneous likelihood inference on the 2 models and allow for nonignorable data missing. The approach is illustrated with a recent AIDS study by jointly modeling HIV viral dynamics and time to viral rebound.     

Maximum likelihood estimation in random effects cure rate models with nonignorable missing covariates

We introduce a method of parameter estimation for a random effects cure rate model. We also propose a methodology that allows us to account for nonignorable missing covariates in this class of models. The proposed method corrects for possible bias introduced by complete case analysis when missing data are not missing completely at random and is motivated by data from a pair of melanoma studies conducted by the Eastern Cooperative Oncology Group in which clustering by cohort or time of study entry was suspected. In addition, these models allow estimation of cure rates, which is desirable when we do not wish to assume that all subjects remain at risk of death or relapse from disease after sufficient follow-up. We develop an EM algorithm for the model and provide an efficient Gibbs sampling scheme for carrying out the E-step of the algorithm.     

Semiparametric regression estimation in the presence of dependent censoring

We propose a semiparametric estimation procedure for estimatingthe regression of an outcome Y, measured at the end of a fixedfollow-up period, on baseline explanatory variables X, measuredprior to start of follow-up, in the presence of dependent censoringgiven X. The proposed estimators are consistent when the dataare ‘missing at random’ but not ‘missing completelyat random’ (Rubin, 1976), and do not require full specificationof the complete data likelihood. Specifically, we assume thatthe probability of censoring at time t is independent of theoutcome Y conditional on the recorded history up to t of a vectorof time-dependent covariates that are correlated with Y. Ourestimators can be used to adjust for dependent censoring andnonrandom noncompliance in randomised trials studying the effectof a treatment on the mean of a response variable of interest.Even with independent censoring, our methods allow the investigatorto increase efficiency by exploiting the correlation of theoutcome with a vector of time-dependent covariates.     

A Monte Carlo EM algorithm for generalized linear mixed models with flexible random effects distribution

A popular way to represent clustered binary, count, or other data is via the generalized linear mixed model framework, which accommodates correlation through incorporation of random effects. A standard assumption is that the random effects follow a parametric family such as the normal distribution; however, this may be unrealistic or too restrictive to represent the data. We relax this assumption and require only that the distribution of random effects belong to a class of 'smooth' densities and approximate the density by the seminonparametric (SNP) approach of Gallant and Nychka (1987). This representation allows the density to be skewed, multi-modal, fat- or thin-tailed relative to the normal and includes the normal as a special case. Because an efficient algorithm to sample from an SNP density is available, we propose a Monte Carlo EM algorithm using a rejection sampling scheme to estimate the fixed parameters of the linear predictor, variance components and the SNP density. The approach is illustrated by application to a data set and via simulation.     

EM-REML estimation of covariance parameters in Gaussian mixed models for longitudinal data analysis

This paper presents procedures for implementing the EM algorithm to compute REML estimates of variance covariance components in Gaussian mixed models for longitudinal data analysis. The class of models considered includes random coefficient factors, stationary time processes and measurement errors. The EM algorithm allows separation of the computations pertaining to parameters involved in the random coefficient factors from those pertaining to the time processes and errors. The procedures are illustrated with Pothoff and Roy''s data example on growth measurements taken on 11 girls and 16 boys at four ages. Several variants and extensions are discussed.     

Imputation of missing covariate values in epigenome-wide analysis of DNA methylation data

DNA methylation is a widely studied epigenetic mechanism and alterations in methylation patterns may be involved in the development of common diseases. Unlike inherited changes in genetic sequence, variation in site-specific methylation varies by tissue, developmental stage, and disease status, and may be impacted by aging and exposure to environmental factors, such as diet or smoking. These non-genetic factors are typically included in epigenome-wide association studies (EWAS) because they may be confounding factors to the association between methylation and disease. However, missing values in these variables can lead to reduced sample size and decrease the statistical power of EWAS. We propose a site selection and multiple imputation (MI) method to impute missing covariate values and to perform association tests in EWAS. Then, we compare this method to an alternative projection-based method. Through simulations, we show that the MI-based method is slightly conservative, but provides consistent estimates for effect size. We also illustrate these methods with data from the Atherosclerosis Risk in Communities (ARIC) study to carry out an EWAS between methylation levels and smoking status, in which missing cell type compositions and white blood cell counts are imputed.     

Estimating disease prevalence in two-phase studies

Disease prevalence is ideally estimated using a 'gold standard' to ascertain true disease status on all subjects in a population of interest. In practice, however, the gold standard may be too costly or invasive to be applied to all subjects, in which case a two-phase design is often employed. Phase 1 data consisting of inexpensive and non-invasive screening tests on all study subjects are used to determine the subjects that receive the gold standard in the second phase. Naive estimates of prevalence in two-phase studies can be biased (verification bias). Imputation and re-weighting estimators are often used to avoid this bias. We contrast the forms and attributes of the various prevalence estimators. Distribution theory and simulation studies are used to investigate their bias and efficiency. We conclude that the semiparametric efficient approach is the preferred method for prevalence estimation in two-phase studies. It is more robust and comparable in its efficiency to imputation and other re-weighting estimators. It is also easy to implement. We use this approach to examine the prevalence of depression in adolescents with data from the Great Smoky Mountain Study.