Bias and Variance Trade-Offs When Combining Propensity Score Weighting and Regression

The quality of propensity scores is traditionally measured by assessing how well they make the distributions of covariates in the treatment and control groups match, which we refer to as "good balance". Good balance guarantees less biased estimates of the treatment effect. However, the cost of achieving good balance is that the variance of the estimates increases due to a reduction in effective sample size, either through the introduction of propensity score weights or dropping cases when propensity score matching. In this paper, we investigate whether it is best to optimize the balance or to settle for a less than optimal balance and use double robust estimation to adjust for remaining differences. We compare treatment effect estimates from regression, propensity score weighting, and double robust estimation with varying levels of effort expended to achieve balance using data from a study about the differences in outcomes by HIV status in heterosexually active homeless men residing in Los Angeles. Because of how costly data collection efforts are for this population, it is important to find an alternative estimation method that does not reduce effective sample size as much as methods that aggressively aim to optimize balance. Results from a simulation study suggest that there are instances in which we can obtain more precise treatment effect estimates without increasing bias too much by using a combination of regression and propensity score weights that achieve a less than optimal balance. There is a bias-variance tradeoff at work in propensity score estimation; every step toward better balance usually means an increase in variance and at some point a marginal decrease in bias may not be worth the associated increase in variance.