Structural mean effects of noncompliance: Estimating interaction with baseline prognosis and selection effects

A randomized, placebo-controlled blood pressure (BP) reduction trial has recorded compliance measures by electronic monitoring. Causal questions regarding dose effects are of interest, but one often wants to avoid the assumption that compliance quantiles on both randomized arms are comparable. Structural mean models (SMMs) do avoid this assumption; however, they do not allow for interaction effects between variables observed on different randomized arms. Such an interaction between observed exposure on the treatment arm and latent treatment-free response (i.e., placebo response) was suggested by the data. Building on structural models, we propose an approach that makes this possible. To allow for a structural interaction effect with potential placebo response, we invoke a compliance selection model. It turns out that SMMs imply identifiable selection models; hence the secondary model is testable. Next, to assess the amount of variation explained by the structural model, we identify separate variance components for different error types. This yields a measure of the variation in treatment effects over individuals, In addition, it allows us to compare the fit of different plausible structural models. On the BP data, we find a significant interaction effect: Higher exposure levels are estimated to have more effect in subgroups with poorer placebo response. Finite-sarnple properties of the proposed estimators are verified through simulation.