We show via simulation and counterexamples that the Mantel-Haenszel estimator of a common odds ratio, adapted for complex survey designs using survey weights, is inconsistent for sparse-data limiting models. We also propose an alternative estimator that is consistent for sparse-data limiting...

We show how to use generalized linear mixed models to adjust for confounding by cluster of the effect of a within-cluster covariate. We derive estimators for both a cluster-specific causal effect and a population-averaged causal effect.

We provide simple intuition why using the estimated versus known propensity score tends to increase, and never decreases, asymptotic efficiency. When a covariate is independent of response conditional on treatment, using the known score can have greater finite-sample efficiency.