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Abstract:
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Standard propensity score methods (e.g., propensity-score-based weighting or matching) for estimation of an average causal effect of treatment A on outcome Y assume all covariates are perfectly measured; we refer to such covariates as Z. However, if among the covariates there is one measured with error (unobserved true X, observed measurement W), the use of W (as a representation of X) fails to obtain balance on X, leading to residual bias in the estimated causal effect. We show that X_E ?= E[X | W,Z,A] (as a representation of X) helps obtain mean-balance on X. Specifically, balancing the distribution of (Z,X_E) between treatment conditions results in balancing the distribution of Z and the mean of X. This result is consistent with the known importance of imputation-analysis model compatibility in missing data theory (Meng, 1994). It is also in dialogue with recent results about valid weighting/matching functions (McCaffrey et al., Statistical Science, 2013; Lockwood & McCaffrey, JASA, 2016) -- on how to get as close as we can to the valid function when X is not observed. To consider potential applications of this idea we discuss situations where one can estimate E[X | W,Z,A].
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