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In observational studies designed to estimate the effects of interventions orexposures, such as cigarette smoking, it is desirable to try to controlbackground differences between the treated group (e.g., current smokers) andthe control group (e.g., never smokers) on covariates $X$ (e.g., age,education). Matched sampling attempts to effect this control by selectingsubsets of the treated and control groups with similar distributions of suchcovariates. This paper examines the consequences of matching using affinelyinvariant methods when the covariate distributions are ``discriminant mixturesof proportional ellipsoidally symmetric'' (DMPES) distributions, a class hereindefined, which generalizes the ellipsoidal symmetry class of Rubin and Thomas[Ann. Statist. 20 (1992) 1079--1093]. The resulting generalized results helpindicate why earlier results hold quite well even when the simple assumption ofellipsoidal symmetry is not met [e.g., Biometrics 52 (1996) 249--264].Extensions to conditionally affinely invariant matching with conditionallyDMPES distributions are also discussed.

Affinely invariant matching methods with discriminant mixtures of proportional ellipsoidally symmetric distributions