Optimal balanced matching

New optimal balanced pair matching methods are proposed in an attempt to balance the marginal distributions of a multivariate pretreatment covariate vector between the treated and matched control groups. All matching methods are optimal in a sense that the total sum of distances between treated and control units in the matched sample is minimized. Integer linear programming is used to solve optimal balanced matchings. Since only exponential-time algorithms are available for solving integer programs, making matching methods practical is important. A penalty method is used in optimal balanced matchings to reduce the computer solving time. Simulation and real applications demonstrate the penalty method to be effective and practical. When the treatment and pretreatment covariate vectors are non-time dependent, simulated examples are used to compare the overt bias reduction of various optimal balanced matchings with that of optimal Mahalanobis metric matching. Simulated examples show that an optimal balanced matching achieves much better bias reduction with little sacrifice on the total Mahalanobis distance. When the treatment and some pretreatment covariates are time-dependent, optimal balanced risk set matchings are proposed. Balanced risk set matchings balance the marginal distributions of both time-dependent and baseline covariates. Two types of risk set matchings are proposed. In the first type, only never-treated patients can be selected as controls. In the second type, a patient treated at time t can be matched to any patient not-yet-treated prior and up to t. An important assumption in risk set matchings is that each patient is either treated once or never, although the treatment time varies among patients. Hence, in the case of repeated treatment, risk set matchings are only concerned with the effects of the first treatment. Risk set matchings are applied to a real clinical data set, namely, the Interstitial Cystitis data set. Permutation tests are applied to the resulting time-dependent matched sample to estimate the effect of the time-dependent treatment. Sensitivity to hidden bias is analyzed for the matched sample