  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 practitioner's 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.
