Effectiveness of potent antiretroviral therapy on progression of human immunodeficiency virus: Bayesian modelling and model checking via counterfactual replicates

Summary.  We analyse data from a seroincident cohort of 457 homosexual men who were infected with the human immunodeficiency virus, followed within the multicentre Italian Seroconversion Study. These data include onset times to acquired immune deficiency syndrome (AIDS), longitudinal measurements of CD4⁺ T-cell counts taken on each subject during the AIDS-free period of observation and the period of administration of a highly active antiretro- viral therapy (HAART), for the subset of individuals who received it. The aim of the study is to assess the effect of HAART on the course of the disease. We analyse the data by a Bayesian model in which the sequence of longitudinal CD4⁺ cell count observations and the associated time to AIDS are jointly modelled at an individual subject's level as depending on the treatment. We discuss the inferences obtained about the efficacy of HAART, as well as modelling and computation difficulties that were encountered in the analysis. These latter motivate a model criticism stage of the analysis, in which the model specification of CD4⁺ cell count progression and of the effect of treatment are checked. Our approach to model criticism is based on the notion of a counterfactual replicate data set Z ^c . This is a data set with the same shape and size as the observed data, which we might have observed by rerunning the study in exactly the same conditions as the actual study if the treated patients had not been treated at all. We draw samples of Z ^c from a null model M ₀, which assumes absence of treatment effect, conditioning on data collected in each subject before initiation of treatment. Model checking is performed by comparing the observed data with a set of samples of Z ^c drawn from M ₀.