Model selection criterion for causal parameters in structural mean models based on a quasi-likelihood

Structural mean models (SMMs) have been proposed for estimating causal parameters in the presence of non-ignorable non-compliance in clinical trials. To obtain a valid causal estimate, we must impose several assumptions. One of these is the correct specification of the structural model. Building on Pan's work (2001, Biometrics 57, 120-125) on developing a model selection criterion for generalized estimating equations, we propose a new approach for model selection of SMMs based on a quasi-likelihood. We provide a formal model selection criterion that is an extension of Akaike's information criterion. Using subset selection of baseline covariates, our method allows us to understand whether the treatment effect varies across the available baseline covariate levels, and/or to quantify the treatment effect on a specific covariates level to target specific individuals to maximize treatment benefit. We present simulation results in which our method performs reasonably well compared to other testing methods in terms of both the probability of selecting the correct model and the predictive performances of the individual treatment effects. We use a large randomized clinical trial of pravastatin as a motivation.