Matching and Weighting With Functions of Error-Prone Covariates for Causal Inference

Matching estimators are commonly used to estimate causal effects in nonexperimental settings. Covariate measurement error can be problematic for matching estimators when observational treatment groups differ on latent quantities observed only through error-prone surrogates. We establish necessary and sufficient conditions for matching and weighting with functions of observed covariates to yield unconfounded causal effect estimators, generalizing results from the standard (i.e., no measurement error) case. We establish that in common covariate measurement error settings, including continuous variables with continuous measurement error, discrete variables with misclassification, and factor and item response theory models, no single function of the observed covariates computed for all units in a study is appropriate for matching. However, we demonstrate that in some circumstances, it is possible to create different functions of the observed covariates for treatment and control units to construct a variable appropriate for matching. We also demonstrate the counterintuitive result that in some settings, it is possible to selectively contaminate the covariates with additional measurement error to construct a variable appropriate for matching. We discuss the implications of our results for the choice between matching and weighting estimators with error-prone covariates. Supplementary materials for this article are available online.