Matching using estimated propensity scores: relating theory to practice

Matched sampling is a standard technique in the evaluation of treatments in observational studies. Matching on estimated propensity scores comprises an important class of procedures when there are numerous matching variables. Recent theoretical work (Rubin, D. B. and Thomas, N., 1992, The Annals of Statistics 20, 1079-1093) on affinely invariant matching methods with ellipsoidal distributions provides a general framework for evaluating the operating characteristics of such methods. Moreover, Rubin and Thomas (1992, Biometrika 79, 797-809) uses this framework to derive several analytic approximations under normality for the distribution of the first two moments of the matching variables in samples obtained by matching on estimated linear propensity scores. Here we provide a bridge between these theoretical approximations and actual practice. First, we complete and refine the nomal-based analytic approximations, thereby making it possible to apply these results to practice. Second, we perform Monte Carlo evaluations of the analytic results under normal and nonnormal ellipsoidal distributions, which confirm the accuracy of the analytic approximations, and demonstrate the predictable ways in which the approximations deviate from simulation results when normal assumptions are violated within the ellipsoidal family. Third, we apply the analytic approximations to real data with clearly nonellipsoidal distributions, and show that the theoretical expressions, although derived under artificial distributional conditions, produce useful guidance for practice. Our results delineate the wide range of settings in which matching on estimated linear propensity scores performs well, thereby providing useful information for the design of matching studies. When matching with a particular data set, our theoretical approximations provide benchmarks for expected performance under favorable conditions, thereby identifying matching variables requiring special treatment. After matching is complete and data analysis is at hand, our results provide the variances required to compute valid standard errors for common estimators.