Estimating the causal effect of compliance on binary outcome in randomized controlled trials.

We examine likelihood based methods aimed at analysing the causal effect of actual exposure to drug treatment on a (repeated) binary outcome in two randomized trials with partial compliance. Starting with the univariate compliance summary 'total treatment dose history', we apply a method for ordinal compliance and monotone dose response, proposed by Goetghebeur and Molenberghs. In a short duration trial of blood pressure reduction, this summary leads to meaningful effect estimators. However, in the analysis of a vitamin A trial, this method reaches a boundary solution; the estimated possible benefit from vitamin A for children who did not receive any pills on the treatment arm is zero. In our formulation the number of pills that were taken captures part of the outcome, and the corresponding effect parameters suffer from this confounding. To gain additional insight, we account explicitly for the temporal structure of compliance. We extend the likelihood based methodology for univariate ordered compliance to more dimensional compliance with only a partial order structure on exposure. The randomization assumptions in the causal formulation of Rubin are translated to this setting. We motivate a set of parametric assumptions on the joint distribution of potential outcomes and observed compliance levels and reanalyse the vitamin A trial. Our findings suggest that one capsule of vitamin A had a large impact on mortality during the first 4 months. The greatest reduction in risk was estimated amongst children who received two doses. This supports findings from a vitamin A trial in Ghana and in Nepal. Finally, we discuss extensions of this method, covering uncensored and censored grouped survival data.