Simultaneously modelling censored survival data and repeatedly measured covariates: a Gibbs sampling approach

Recent methodologic developments in the analysis of longitudinal data have typically addressed one of two aspects: (i) the modelling of repeated measurements of a covariate as a function of time or other covariates, or (ii) the modelling of the effect of a covariate on disease risk. In this paper, we address both of these issues in a single analysis by modelling a continuous covariate over time and simultaneously relating the covariate to disease risk. We use the Markov chain Monte Carlo technique of Gibbs sampling to estimate the joint posterior distribution of the unknown parameters of the model. Simulation studies showed that jointly modelling survival and covariate data reduced bias in parameter estimates due to covariate measurement error and informative censoring. We illustrate the methodology by application to a data set that consists of repeated measurements of the immunologic marker CD4 and times of diagnosis of AIDS for a cohort of anti-HIV-1 positive recipients of anti-HIV-1 positive blood transfusions. We assume a linear random effects model with subject-specific intercepts and slopes and normal errors for the true log and square root CD4 counts, and a proportional hazards model for AIDS-free survival time expressed as a function of current true CD4 value. On the square root scale, the joint approach yielded a mean slope for CD4 that was 7 per cent steeper and a log relative risk of AIDS that was 35 per cent larger than those obtained by analysis of the component sub-models separately.