Model misspecification and robust analysis for outcome-dependent sampling designs under generalized linear models

Outcome-dependent sampling (ODS) is a commonly used class of sampling designs to increase estimation efficiency in settings where response information (and possibly adjuster covariates) is available, but the exposure is expensive and/or cumbersome to collect. We focus on ODS within the context of a two-phase study, where in Phase One the response and adjuster covariate information is collected on a large cohort that is representative of the target population, but the expensive exposure variable is not yet measured. In Phase Two, using response information from Phase One, we selectively oversample a subset of informative subjects in whom we collect expensive exposure information. Importantly, the Phase Two sample is no longer representative, and we must use ascertainment-correcting analysis procedures for valid inferences. In this paper, we focus on likelihood-based analysis procedures, particularly a conditional-likelihood approach and a full-likelihood approach. Whereas the full-likelihood retains incomplete Phase One data for subjects not selected into Phase Two, the conditional-likelihood explicitly conditions on Phase Two sample selection (ie, it is a "complete case" analysis procedure). These designs and analysis procedures are typically implemented assuming a known, parametric model for the response distribution. However, in this paper, we approach analyses implementing a novel semi-parametric extension to generalized linear models (SPGLM) to develop likelihood-based procedures with improved robustness to misspecification of distributional assumptions. We specifically focus on the common setting where standard GLM distributional assumptions are not satisfied (eg, misspecified mean/variance relationship). We aim to provide practical design guidance and flexible tools for practitioners in these settings.