We address several questions relating to the use of standard regression and causal approaches to analyzing how post-randomization factors may modify the intent-to-treat effects of randomized interventions. In assessing these questions, we present a causal linear rank preserving model (RPM) for analyzing the modification of a randomized baseline intervention's effect on a single endpoint outcome by post-randomization factors. Unlike standard interaction analyses in such a context, our approach does not assume that the post-randomization effect modifier is also randomly assigned to individuals in addition to the randomized baseline intervention (i.e., sequential ignorability). Our simulation results suggest that even without the sequential ignorability assumption, the standard regression method performs as well as the causal approach in terms of inference for the interaction term. Furthermore, additional conditions are needed for the standard regression approach to perform more poorly than the causal approach in estimating ITT effects stratified by the post randomization factor. An important condition may be a strong ITT effect on the post-randomization factor. These issues and methods are illustrated with application of the standard and causal methods to a randomized cognitive behavioral therapy (CBT) trial example. The behavioral theory of a common treatment effect motivates the focus on interactions between CBT and post-randomization behavioral factors such as negative problem solving and suicide ideation with depression as the outcome.
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