Nonparametric Likelihood and Doubly Robust Estimating Equations for Marginal and Nested Structural Models

Drawing inferences about treatment effects is of interest in many fields. We consider Robins’s marginal and nested structural models and propose likelihood and regression estimators in the cross-sectional setting. First, we choose to retain a finite subset of all inherent and modelling constraints on the the joint distributions of potential outcomes and covariates under a correctly specified propensity score model. We derive a profile likelihood by maximizing the nonparametric likelihood over these joint distributions subject to the retained constraints. The maximum likelihood estimator is asymptotically equivalent to the optimal estimator in the class of estimators that are solutions to linear combinations of estimating equations based on the retained constraints. Second, we derive two regression estimators, named hat and tilde, as first-order approximations to the likelihood estimator under the propensity score model. The tilde regression estimator is intrinsically and weakly locally efficient and doubly robust. By intrinsic efficiency, it is asymptotically guaranteed to gain efficiency over the non-augmented inverse-probability-weighted or G estimator if the propensity score model is correctly specified and over the augmented inverse-probability-weighted or G estimator if the outcome regression model is misspecified but the propensity score model is correctly specified. We illustrate the methods by data analysis for an observational study on right heart catheterization.