A hierarchical modelling approach to analysing longitudinal data with drop‐out and non‐compliance, with application to an equivalence trial in paediatric acquired immune deficiency syndrome

Longitudinal clinical trials with long follow‐up periods almost invariably suffer from a loss to follow‐up and non‐compliance with the assigned therapy. An example is protocol 128 of the AIDS Clinical Trials Group, a 5‐year equivalency trial comparing reduced dose zidovudine with the standard dose for treatment of paediatric acquired immune deficiency syndrome patients. This study compared responses to treatment by using both clinical and cognitive outcomes. The cognitive outcomes are of particular interest because the effects of human immunodeficiency virus infection of the central nervous system can be more acute in children than in adults. We formulate and apply a Bayesian hierarchical model to estimate both the intent‐to‐treat effect and the average causal effect of reducing the prescribed dose of zidovudine by 50%. The intent‐to‐treat effect quantifies the causal effect of assigning the lower dose, whereas the average causal effect represents the causal effect of actually taking the lower dose. We adopt a potential outcomes framework where, for each individual, we assume the existence of a different potential outcomes process at each level of time spent on treatment. The joint distribution of the potential outcomes and the time spent on assigned treatment is formulated using a hierarchical model: the potential outcomes distribution is given at the first level, and dependence between the outcomes and time on treatment is specified at the second level by linking the time on treatment to subject‐specific effects that characterize the potential outcomes processes. Several distributional and structural assumptions are used to identify the model from observed data, and these are described in detail. A detailed analysis of AIDS Clinical Trials Group protocol 128 is given; inference about both the intent‐to‐treat effect and average causal effect indicate a high probability of dose equivalence with respect to cognitive functioning.