最小协变量不平衡问题的网络流方法
Network Flow Methods for the Minimum Covariates Imbalance Problem
摘要：
平衡协变量的问题出现在观察性研究中，其中一组是对照样本，另一组是与对照组不相交的治疗样本。任一组中的每个样本都有几个观察到的标称协变量。每个协变量的值或类别将治疗和对照样本划分为多个子集，称为\textit{levels}，其中每个水平的样本共享相同的协变量值。我们在这里解决的问题，选择一个子集的控制组，以便平衡，尽可能最好的程度，治疗组和控制组的选定子集之间的水平的大小，最小不平衡问题。本文证明了用网络流技术可以有效地解决两个协变量的最小不平衡问题。我们提出了一个整数规划制定的约束矩阵是完全么模的问题，这意味着线性规划松弛的问题有所有的基本解决方案，特别是最佳解决方案，积分。该整数规划公式被链接到最小成本网络流问题，该问题可以在$O(n\cdot (n' + n\log n))$步骤中求解，对于$n$，处理组的大小，以及$n'$，控制组的大小。一个更有效的算法，进一步设计的基础上的替代，最大流量，制定的两个协变量最小不平衡问题，运行在$O(n'^{3/2}\log^2n)$步骤。
英文摘要：
The problem of balancing covariates arises in observational studies where one is given a group of control samples and another group, disjoint from the control group, of treatment samples. Each sample, in either group, has several observed nominal covariates. The values, or categories, of each covariate partition the treatment and control samples to a number of subsets referred to as \textit{levels} where the samples at every level share the same covariate value. We address here a problem of selecting a subset of the control group so as to balance, to the best extent possible, the sizes of the levels between the treatment group and the selected subset of control group, the min-imbalance problem. It is proved here that the min-imbalance problem, on two covariates, is solved efficiently with network flow techniques. We present an integer programming formulation of the problem where the constraint matrix is totally unimodular, implying that the linear programming relaxation to the problem has all basic solutions, and in particular the optimal solution, integral. This integer programming formulation is linked to a minimum cost network flow problem which is solvable in $O(n\cdot (n' + n\log n))$ steps, for $n$ the size of the treatment group and $n'$ the size of the control group. A more efficient algorithm is further devised based on an alternative, maximum flow, formulation of the two-covariate min-imbalance problem, that runs in $O(n'^{3/2}\log^2n)$ steps.
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