Combining matching and regression for causal inference provides double-robustness in removing treatment effect estimation bias due to confounding variables. In most real-world applications, however, treatment and control populations are not large enough for matching to achieve perfect or near-perfect balance on all confounding variables and their nonlinear/interaction functions, leading to trade-offs. [this fact is independent of regression, so a bit disjointed from first sentence.] Furthermore, variance is as important of a contributor as bias towards total error in small samples, and must therefore be factored into the methodological decisions. In this paper, we develop a mathematical framework for quantifying the combined impact of matching and linear regression on bias and variance of treatment effect estimation. The framework includes expressions for bias and variance in a misspecified linear regression, theorems regarding impact of matching on bias and variance, and a constrained bias estimation approach for quantifying misspecification bias and combining it with variance to arrive at total error. Methodological decisions can thus be based on minimization of this total error, given the practitioners assumption/belief about an intuitive parameter, which we call `omitted R-squared. The proposed methodology excludes the outcome variable from analysis, thereby avoiding overfit creep and making it suitable for observational study designs. All core functions for bias and variance calculation, as well as diagnostic tools for bias-variance trade-off analysis, matching calibration, and power analysis are made available to researchers and practitioners through an open-source R library, MatchLinReg.
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