Part III begins with another pair of back-to-back Biometrics articles, Rubin (1976b, c), which were written shortly after completing my PhD, and were focused entirely on analytic results concerning multivariate matching. Obviously, for practice the multivariate case is the norm; in fact, often there are very many matching variables.

Chapter 6, Rubin (1976b), defines a class of matching methods called “Equal Percent Bias Reducing” (EPBR). EPBR methods have the property that the percent reduction in bias due to the matching is the same for each of the matching variables. There are always linear combinations of the covariates that have the same means in the treatment and control groups before matching, and if a method is not EPBR, some of these will have different means after matching, implying that the matching infinitely increases the bias for them! This is not an attractive feature of a matching method – to increase bias in some directions, especially in the context of outcomes, Y, that are commonly assumed to be approximately linearly related to the X variables.

Chapter 6 goes on to describe classes of matching methods (e.g., caliper methods, inner-product methods) and corresponding distributional conditions on X (e.g., exchangeable, ellipsoidally symmetric) that lead to the methods being EPBR for those distributions. The most generally useful of these methods have turned out to be inner-product methods, including Mahalanobis-metric matching and discriminant matching, which can be a special case of the now widely used propensity score matching.