An evaluation of bias in propensity score-adjusted non-linear regression models. 
Propensity score methods are commonly used to adjust for observed confounding when estimating the conditional treatment effect in observational studies.  One popular method, covariate adjustment of the propensity score in a regression model, has been empirically shown to be biased in non-linear models.  However, no compelling underlying theoretical reason has been presented.  We propose a new framework to investigate bias and consistency of propensity score-adjusted treatment effects in non-linear models that uses a simple geometric approach to forge a link between the consistency of the propensity score estimator and the collapsibility of non-linear models.  Under this framework, we demonstrate that adjustment of the propensity score in an outcome model results in the decomposition of observed covariates into the propensity score and a remainder term.  Omission of this remainder term from a non-collapsible regression model leads to biased estimates of the conditional odds ratio and conditional hazard ratio, but not for the conditional rate ratio.  We further show, via simulation studies, that the bias in these propensity score-adjusted estimators increases with larger treatment effect size, larger covariate effects, and increasing dissimilarity between the coefficients of the covariates in the treatment model versus the outcome model.