Imposing Minimax and Quantile Constraints on Optimal Matching in Observational Studies

Modern methods construct a matched sample by minimizing the total cost of a flow in a network, finding a pairing of treated and control individuals that minimizes the sum of within-pair covariate distances subject to constraints that ensure distributions of covariates are balanced. In aggregate, these methods work well; however, they can exhibit a lack of interest in a small number of pairs with large covariate distances. Here, a new method is proposed for imposing a minimax constraint on a minimum total distance matching. Such a match minimizes the total within-pair distance subject to various constraints including the constraint that the maximum pair difference is as small as possible. In an example with 1391 matched pairs, this constraint eliminates dozens of pairs with moderately large differences in age, but otherwise exhibits the same excellent covariate balance found without this additional constraint. A minimax constraint eliminates edges in the network, and can improve the worst-case time bound for the performance of the minimum cost flow algorithm, that is, a better match from a practical perspective may take less time to construct. The technique adapts ideas for a different problem, the bottleneck assignment problem, whose sole objective is to minimize the maximum within-pair difference; however, here, that objective becomes a constraint on the minimum cost flow problem. The method generalizes. Rather than constrain the maximum distance, it can constrain an order statistic. Alternatively, the method can minimize the maximum difference in propensity scores, and subject to doing that, minimize the maximum robust Mahalanobis distance. An example from labor economics is used to illustrate. Supplementary materials for this article are available online.主流方法通过最小化流网络（flow network）中流的总成本来构建匹配样本（matched sample），即寻找处理组与对照组个体的配对方案，在确保协变量分布均衡的约束条件下，最小化组内配对的协变量距离（covariate distances）总和。整体而言，这类方法表现优异，但往往会忽视少量协变量距离较大的配对。本文提出一种全新方法，可在最小总距离匹配（minimum total distance matching）中引入极小极大约束（minimax constraint）：该匹配方案在多种约束条件下最小化组内配对的总距离，其中包含将最大配对差异控制在尽可能小的范围内这一约束。在一项包含1391个匹配对的案例中，该约束剔除了数十个年龄差异中等偏大的配对，但其余协变量的均衡性仍与未添加该额外约束时的最优结果保持一致。极小极大约束会对网络中的边进行筛选，同时可优化最小费用流算法（minimum cost flow algorithm）的最坏情况时间复杂度；换言之，从实际应用视角来看，更优质的匹配方案或许能以更短的时间完成构建。该技术借鉴了另一问题——瓶颈指派问题（bottleneck assignment problem）的相关思路，后者的唯一目标是最小化最大组内配对差异；而本文则将该目标转化为最小费用流问题的约束条件。该方法具备可推广性：除约束最大距离外，还可对顺序统计量施加约束。此外，该方法可先最小化倾向得分（propensity scores）的最大差异，再在此前提下优化稳健马氏距离（robust Mahalanobis distance）的最大值。本文借助一项劳动经济学（labor economics）案例对所提方法进行演示，文章的补充材料可在线获取。