Bivariate Modelling of Longitudinal Measurements of Two Human Immunodeficiency Type 1 Disease Progression Markers in the Presence of Informative Drop-Outs

Abstract

Summary The main statistical problem in many epidemiological studies which involve repeated measurements of surrogate markers is the frequent occurrence of missing data. Standard likelihood-based approaches like the linear random-effects model fail to give unbiased estimates when data are non-ignorably missing. In human immunodeficiency virus (HIV) type 1 infection, two markers which have been widely used to track progression of the disease are CD4 cell counts and HIV–ribonucleic acid (RNA) viral load levels. Repeated measurements of these markers tend to be informatively censored, which is a special case of non-ignorable missingness. In such cases, we need to apply methods that jointly model the observed data and the missingness process. Despite their high correlation, longitudinal data of these markers have been analysed independently by using mainly random-effects models. Touloumi and co-workers have proposed a model termed the joint multivariate random-effects model which combines a linear random-effects model for the underlying pattern of the marker with a log-normal survival model for the drop-out process. We extend the joint multivariate random-effects model to model simultaneously the CD4 cell and viral load data while adjusting for informative drop-outs due to disease progression or death. Estimates of all the model’s parameters are obtained by using the restricted iterative generalized least squares method or a modified version of it using the EM algorithm as a nested algorithm in the case of censored survival data taking also into account non-linearity in the HIV–RNA trend. The method proposed is evaluated and compared with simpler approaches in a simulation study. Finally the method is applied to a subset of the data from the ‘Concerted action on seroconversion to AIDS and death in Europe’ study.