Research Article Received 16 March 2010, Accepted 3 May 2011 Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/sim.4296 Causal models for randomized trials with two active treatments and continuous compliance ‡ Yan Ma, a * † Jason Roy b and Bess Marcus c 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 mea- sures 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. Copyright © 2011 John Wiley & Sons, Ltd. Keywords: copula model; Gauss–Hermite quadrature; partial compliance; potential outcomes; principal stratification 1. Introduction We consider the situation where subjects were randomized to one of two active treatments, and we measured compliance with each treatment on a continuous scale. Examples of continuous measures of compliance include the duration of compliance and the proportion of assigned treatment actually received. The motivating example for this research was a smoking cessation clinical trial. The Commit to Quit (CTQ) trials [1–3] comprise two longitudinal follow-up studies of supervised exercise to promote smok- ing cessation. One arm included cognitive–behavioral smoking cessation therapy (CBT) augmented by an individualized, supervised exercise program. In the control arm, CBT was augmented by a well- ness education program that included lectures, films, handouts, and discussions covering issues such as healthy eating and prevention of cardiovascular disease. Interest is in the comparison between stan- dard therapy augmented by wellness education and standard therapy augmented by an exercise regimen. However, many subjects only attended some of the exercise or wellness classes. One approach that has been proposed for inferring causal effects from trials with noncompliance is structural mean models [4–6]. A structural model must be specified that relates the mean of the outcome at the observed compliance level with the mean of the potential outcome at some reference level. For example, the reference outcome could be the treatment-free outcome. These methods have primarily been used for placebo-controlled trials. An appealing aspect of that approach is that causal effects can be inferred for the entire population at various compliance levels. However, for two-arm behavioral inter- vention trials, it is difficult to conceive of fixing compliance levels for the entire population. Whatever a Hospital for Special Surgery, Weill Cornell Medical College, New York, NY, USA b Center for Clinical Epidemiology and Biostatistics, University of Pennsylvania, Philadelphia, PA, USA c Department of Psychiatry and Human Behavior, Brown University, Providence, RI, USA *Correspondence to: Yan Ma, Hospital for Special Surgery, Weill Cornell Medical College, New York, NY, USA. † E-mail: yam2007@med.cornell.edu ‡ Supporting information may be found in the online version of this article. Copyright © 2011 John Wiley & Sons, Ltd. Statist. Med. 2011, 30 2349–2362 2349