Affinely Invariant Matching Methods with Ellipsoidal Distributions
Paul R. Rosenbaum · Cambridge University Press eBooks · 2006
: Matched sampling is a common technique used for controlling bias in observational studies. We present a general theoretical framework for studying the performance of such matching methods. Specifically, results are obtained concerning the performance of affinely invariant matching methods with ellipsoidal distributions, which extend previous results on equal percent bias reducing methods. Additional extensions cover conditionally affinely invariant matching methods for covariates with conditionally ellipsoidal distributions. These results decompose the effects of matching into one subspace containing the best linear discriminant, and the subspace of variables uncorrelated with the discriminant. This characterization of the effects of matching provides a theoretical foundation for understanding the performance of specific methods such as matched sampling using estimated propensity scores. Calculations for such methods are given in subsequent articles. BACKGROUND Matched sampling is a popular and important technique for controlling bias in observational studies. It has received increasing attention in the statistical literature in recent years [Cochran (1968a); Cochran and Rubin (1973); Rubin (1973a, b), (1976b, c), (1979b); Carpenter (1977); and Rosenbaum and Rubin (1983a, 1985a)]. The basic situation has two populations of units, treated (e.g., smokers) and control (e.g., nonsmokers), and a set of observed matching variables X = ( X 1 , …, X p ) (e.g., age, gender, weight). The objective is to compare the distributions of the outcome variables having adjusted for differences in the distributions of X in the two populations. Matched sampling is a way of adjusting for X through data collection.