Developing and Applying the Propensity Score to Make Causal Inferences: Variable Selection and Stratification.

This Monte Carlo simulation examined the effects of variable selection (combinations of confounders with four patterns of relationships to outcome and assignment to treatment) and number of strata (5, 10, or 20) in propensity score analyses. The focus was on how the variations affected the average effect size compared to quasi-assignment without adjustment for bias. Results indicate that if a propensity score model does not include variables strongly related to both outcome and assignment, not only will bias not decrease, but it may possibly increase. Furthermore, models that include a variable highly related to assignment to treatment but do not also include a variable highly related to the outcome could increase bias. In regards to the number of strata, results varied depending on the propensity score model and sample size. In 75% of the models that resulted in a significant reduction in bias, quintiles outperformed the other stratification schemes. In fact, the richer that the propensity score model was (i.e., including multiple covariates of varying relationships to the outcome and to assignment to treatment), the more likely that the model required fewer strata to balance the covariates. In models without that same richness, additional strata were necessary. Finally, the study suggests that when developing a rich propensity score model with stratification, it is crucial to examine the strata for overlap.