Optimal Matching for Observational Studies
Rosenbaum, Paul R., Journal of the American Statistical Association
1.1 Two Literatures on Matching
There are two essentially disjoint literatures on matching. The first is the statistical literature on the construction of matched samples for observational studies. The second is the literature in discrete mathematics, computer science, and operations research on matching in graphs and networks. This article uses ideas from the second literature as they relate to problems in the first.
The article is organized as follows. Section 1.2 reviews certain statistical aspects of matching in observational studies. Section 1.3 discusses a tangible example that illustrates the difference between an optimal matching and a matching constructed by the greedy heuristics that are currently used by statisticians. The key point is that two or more treated units may have the same control as their best match, and conventional heuristics resolve this bottleneck in an arbitrary way, typically yielding a suboptimal match, that is, a matched sample that could be improved with the data at hand. Greedy and optimal matching are compared in Section 1.4. Relevant network flow theory is briefly reviewed in Section 2, with extensive references. Network flow methods are used to solve a series of statistical matching problems in Section 3, including matching with multiple controls, matching with a variable number of controls, and balanced matching. Computational considerations are discussed in Section 4.
1.2 Constructing Matched Samples in Observational Studies: A Short Review
An observational study is an attempt to estimate the effects of a treatment when, for ethical or practical reasons, it is not possible to randomly assign units to treatment or control; see Cochran (1965) for a review of issues that arise in such studies. The central problem in observational studies is that treated and control units may not be comparable prior to treatment, so differences in outcomes in treated and control groups may or may not indicate effects actually caused by the treatment. This problem has two aspects: The treated and control groups may be seen to differ prior to treatment with respect to various recorded measurements, or they may be suspected to differ in ways that have not been recorded. Observed pretreatment differences are controlled by adjustments, for example by matched sampling, the method discussed here. Even after adjustments have been made for recorded pre-treatment differences, there is always a concern that some important differences were not recorded, so no adjustments could be made. See Rosenbaum (1987a, b) and the references given there for discussion of methods for addressing unobserved pretreatment differences.
Pretreatment measurements are available for N treated units, numbered n = 1, ..., N, and a reservoir of M potential controls, numbered m = 1, ..., M. Often M is much larger than N, but this is not essential, and it is assumed only that M [greater than or equal to] N. Each unit has a vector of pretreatment measurements, say [x.sub.n] for the nth treated unit and [w.sub.m] for the mth potential control. A matched pair is an ordered pair (n, m) with 1 [less than or equal to] n [less than or equal to] N and 1 [less than or equal to] m [less than or equal to] M, indicating that the nth treated unit is matched with the mth potential control. A complete matched-pair sample is a set [Imaginary part] of N disjoint matched pairs, that is, N matched pairs in which each treated unit appears once, and each control appears either once or not at all. An incomplete matched-pair sample is a set of < N disjoint matched pairs; however, there are strong reasons for avoiding incomplete matched-pair samples (Rosenbaum and Rubin 1985a), and little attention will be given to them here.
There are two notions of a "good" complete matched-pair sample. The first involves closely matched individual pairs, and the second involves balanced treated and control groups. …