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Scientists in all disciplines attempt to identify and document causal relationships. Those not fortunate enough to be able to design and implement randomized control trials must resort to observational studies. To make causal inferences outside the experimental realm, researchers attempt to control for bias sources by postprocessing observational data. Finding the subset of data most conducive to unbiased or least biased treatment effect estimation is a challenging, complex problem. However, the rise in computational power and algorithmic sophistication leads to an operations research solution that circumvents many of the challenges presented by methods employed over the past 30 years.

Observational data are prevalent in many fields of research, and it is desirable to use this data to make causal inferences. Because this data is nonrandom, additional assumptions are needed in order to construct unbiased estimators for causal effects. The standard assumption is strong ignorability, though it is often impossible to achieve the level of covariate balance that it requires. As such, researchers often settle for lesser balance levels within their datasets. However, these balance levels are generally insufficient to guarantee an unbiased estimate of the treatment effect without further assumptions. This article presents several extensions to the strong ignorability assumption that address this issue. Under these additional assumptions, specific levels of covariate balance are both necessary and sufficient for the treatment effect estimate to be unbiased. There is a trade‐off, however: as balance decreases, stronger assumptions are required to guarantee estimator unbiasedness. These results unify parametric and nonparametric adjustment methods for causal inference and are actualized by the Balance Optimization Subset Selection framework, which identifies the best level of balance that can be achieved within a dataset. © 2017 Wiley Periodicals, Inc. Naval Research Logistics 64: 323–344, 2017

Matching is widely used in the estimation of treatment effects in observational studies. However, the matching paradigm may be too restrictive in many cases because exact matches often do not exist in the available data. One mechanism for overcoming this issue is to relax the requirement of exact matching on some or all of the covariates (attributes that may affect the response to treatment) to a requirement of balance on the covariate distributions for the treatment and control groups. The balance optimization subset selection (BOSS) model can be used to identify a control group featuring optimal covariate balance. This paper explores the relationship between the matching and BOSS models and shows how BOSS subsumes matching. Complexity and approximation results are presented for the resulting models. Computational results demonstrate some of the important trade-offs between matching and BOSS. Data, as supplemental material, are available at http://dx.doi.org/10.1287/ijoc.2013.0583 . There is a video associated with this paper. Click here to view the Video Overview . To save the file, right click and choose “Save Link As” from the menu.

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