Double propensity-score adjustment: A solution to design bias or bias due to incomplete matching

Propensity-score matching is frequently used to reduce the effects of confounding when using observational data to estimate the effects of treatments. Matching allows one to estimate the average effect of treatment in the treated. Rosenbaum and Rubin coined the term "bias due to incomplete matching" to describe the bias that can occur when some treated subjects are excluded from the matched sample because no appropriate control subject was available. The presence of incomplete matching raises important questions around the generalizability of estimated treatment effects to the entire population of treated subjects. We describe an analytic solution to address the bias due to incomplete matching. Our method is based on using optimal or nearest neighbor matching, rather than caliper matching (which frequently results in the exclusion of some treated subjects). Within the sample matched on the propensity score, covariate adjustment using the propensity score is then employed to impute missing potential outcomes under lack of treatment for each treated subject. Using Monte Carlo simulations, we found that the proposed method resulted in estimates of treatment effect that were essentially unbiased. This method resulted in decreased bias compared to caliper matching alone and compared to either optimal matching or nearest neighbor matching alone. Caliper matching alone resulted in design bias or bias due to incomplete matching, while optimal matching or nearest neighbor matching alone resulted in bias due to residual confounding. The proposed method also tended to result in estimates with decreased mean squared error compared to when caliper matching was used.