Abstract：
What is the best way to split one stratum into two to maximally reduce the within-stratum imbalance in many covariates? We formulate this as an integer program and approximate the solution by randomized rounding of a linear program. A linear program may assign a fraction of a person to each refined stratum. Randomized rounding views fractional people as probabilities, assigning intact people to strata using biased coins. Randomized rounding is a well-studied theoretical technique for approximating the optimal solution of certain insoluble integer programs. When the number of people in a stratum is large relative to the number of covariates, we prove the following new results: (i) randomized rounding to split a stratum does very little randomizing, so it closely resembles the linear programming relaxation without splitting intact people; (ii) the linear relaxation and the randomly rounded solution place lower and upper bounds on the unattainable integer programming solution; and because of (i), these bounds are often close, thereby ratifying the usable randomly rounded solution. We illustrate using an observational study that balanced many covariates by forming matched pairs composed of 2016 patients selected from 5735 using a propensity score. Instead, we form 5 propensity score strata and refine them into 10 strata, obtaining excellent covariate balance while retaining all patients. An R package optrefine at CRAN implements the method. Supplementary materials are available online.
摘要：
将一个层分为两个以最大程度地减少许多协变量中的层内不平衡的最佳方法是什么？我们将其公式化为整数程序，并通过线性程序的随机舍入来近似求解。线性程序可以将人的一部分分配给每个细化的阶层。随机舍入将分数人视为概率，使用有偏见的硬币将完整的人分配到地层中。随机舍入是一种经过充分研究的理论技术，可以近似某些不可解决的整数程序的最佳解决方案。当一个阶层中的人数相对于协变量的数量很大时，我们证明了以下新结果：(I)随机舍入以分割地层几乎没有随机化，所以它非常类似于线性规划松弛而不分裂完整的人；(ii)线性松弛和随机舍入解在不可达的整数规划解上放置下界和上界；并且由于(i)，这些界限通常很接近，从而批准可用的随机舍入解决方案。我们使用一项观察性研究进行了说明，该研究通过形成由使用倾向评分从5735中选出的2016年患者组成的匹配对来平衡许多协变量。相反，我们形成5个倾向评分层，并将它们细化为10个层，获得良好的协变量平衡，同时保留所有患者。CRAN的R包optrefine实现了该方法。补充材料可在线获得。
