Causal models for randomized trials with two active treatments and continuous compliance

In many clinical trials, compliance with assigned treatment could be measured on a continuous scale (e.g., the proportion of assigned treatment actually taken). In general, inference about principal causal effects can be challenging, particularly when there are two active treatments; the problem is exacerbated for continuous measures of compliance. We address this issue by first proposing a structural model for the principal effects. We then specify compliance models within each arm of the study. These marginal models are identifiable. The joint distribution of the observed and counterfactual compliance variables is assumed to follow a Gaussian copula model, which links the two marginal models and includes a dependence parameter that cannot be identified. This dependence parameter can be varied as part of a sensitivity analysis. We illustrate the methodology with an analysis of data from a smoking cessation trial. As part of the analysis, we estimate causal effects at particular levels of the compliance variables and within subpopulations that have similar compliance behavior.