The statistical study of the utility of general methods of matched sampling starts with the simplest setting, that with only one matching variable, X, as in Cochran (1968a). When attempting to control for bias in X, a prominent competitor to matching is regression adjustment, also called covariance adjustment (ANCOVA = analysis of covariance). Therefore, it is important when studying matching to compare the relative merits of the methods, including their combination, that is, regression adjustment on the matched samples. In fact, this topic was the focus of my thesis.

Chapters 3 and 4, Rubin (1973a, b), were originally published back-to-back in Biometrics, and were improved versions of the material in my PhD thesis. The first considered matching on a single normally distributed covariate, X, where the outcome variable, Y, was monotonely but possibly nonlinearly (e.g., exponentially) related to X with possibly nonparallel regressions in the treatment and control groups. The estimand was the average causal effect for the treated group, and the estimator was the simple difference in Y means. The criterion for comparing methods was “percent reduction in the bias” due to selecting samples by pair matching or mean matching on X, rather than random sampling. The conditions considered involved various treatment sample sizes and ratios of control to treatment sample sizes, as well as various mean differences on X between treatment and control groups, and ratios of variance of X in the treatment and control groups.