Semiparametric approaches to inference in joint models for longitudinal and time-to-event data

In many longitudinal studies, it is of interest to characterize the relationship between a time-to-event (e.g. survival) and time-dependent and time-independent covariates. Time-dependent covariates are generally observed intermittently and with error.For a single time-dependent covariate, a popular approach is to assume a joint longitudinal data-survival model, where the time-dependent covariate follows a linear mixed effects model and the hazard of failure depends on random effects and time-independent covariates via a proportional hazards relationship. Interest may focus on inference on the longitudinal data process, which is informatively censored by death or withdrawal, or on the hazard relationship. Several methods for fitting such models have been proposed, including regression calibration and likelihood or Bayesian methods. However, most approaches require a parametric distributional assumption (normality) on the random effects. In addition, generalization to more than one time-dependent covariate may become prohibitive. For a single time-dependent covariate, Tsiatis and Davidian (2001) have proposed an approach that is easily implemented and does not require an assumption on the distribution of the random effects. We extend this technique to multiple, possibly correlated,time-dependent covariates. This approach is easy to compute.However, the conditional score approach might be less efficient relative to the likelihood approaches. In addition, inference on the longitudinal data process is not available. To improve the efficiency and meanwhile obtain an estimator for the random effects distribution, we propose to approximate the random effects distribution by the seminonparametric (SNP) densities of Gallant and Nychka (1987), which requires only the assumption that the random effects have a "smooth" density, and take a semiparametric likelihood approach. The EM algorithm is used for implementation. We demonstrate the approaches via simulations and apply them to data from an HIV clinical trial.