Optimizing Balance and Sample Size in Matching Methods for Causal Inference

We propose a greatly simplified approach to matching for causal inference that simultaneously optimizes both balance (between the treated and control groups) and matched sample size. This procedure resolves two widespread (bias-variance trade off-related) tensions in the use of this powerful and popular methodology. First, current practice is to run a matching method that maximizes one balance metric (such as a propensity score or average Mahalanobis distance), but then to check whether it succeeds with respect to a different balance metric for which it was not designed (such as differences in means or L1). Second, current matching methods either fix the sample size and maximize balance (e.g., Mahalanobis or propensity score matching), fix balance and maximize the sample size (such as coarsened exact matching), or are arbitrary compromises between the two (such as calipers with ad hoc thresholds applied to other methods). These tensions lead researchers to either try to optimize manually, by iteratively tweaking their matching method and rechecking balance, or settle for suboptimal solutions. We address these tensions by first defining the frontier in the balance-sample size trade-off as the set of matching solutions with maximum balance for each possible sample size. Researchers can then choose one, several, or all matching solutions from the frontier for analysis. Computation is fast and requires no iteration or manual tweaking. We offer easy-to-use software that implements these ideas and present a range of empirical analyses that demonstrate their value. ∗Our thanks to Carter Coberley, Stefano Iacus, Giuseppe Porro, and Aaron Wells. †Albert J. Weatherhead III University Professor, Institute for Quantitative Social Science, 1737 Cambridge Street, Harvard University, Cambridge MA 02138; http://GKing.harvard.edu, [email protected], (617) 500-7570. ‡Ph.D. Candidate, Institute for Quantitative Social Science, 1737 Cambridge Street, Harvard University, Cambridge MA 02138. §Assistant Professor, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge MA 02139; http://www.mit.edu/∼rnielsen, [email protected], (857) 998-8039.