Matching with Variable Set Sizes in Observational Studies

In observational studies, treated subjects and controls are often matched to remove bias in pre-treatment covariates. In fixed matching, each treated subject is matched to a fixed number of controls. In variable matching, the number of controls may vary. In full matching, a single control may also be matched to several treated subjects. In this dissertation, the optimal form of variable matching is determined. Variable matching may achieve substantially greater bias reduction than optimal fixed matching, even with the same number of controls. In certain cases, variable matching may remove 90% of the bias while optimal full matching can only remove 50%. In the population, or in large samples, full matching removes 100% of the bias, and variable matching removes more bias than fixed matching. If the matching covariate is categorical, zero bias may be possible for finite sample, and full matchings with certain matching patterns are determined to yield the minimum variance estimate of the treatment effect. In a matching with variable set sizes, a weighted estimate of the average treatment effect is needed. One choice of weights is unbiased, but another choice has smaller variance when the effect is constant, not varying from one set to another. A simple test for constant effect is proposed, along with a confidence interval, so the choice of weight may be informed by the available data. Matching with variable set sizes has substantially greater potential in removing pre-treatment bias in observational studies. In many studies, the common practice of matching with a fixed number of controls should be avoided, and appropriate variable matching or full matching may be employed to obtain treatment effect estimates with smaller bias