  Examples of "doubly robust" estimator for missing data include augmented
inverse probability weighting (AIPWT) models (Robins et al., 1994) and
penalized splines of propensity prediction (PSPP) models (Zhang and Little,
2009). Doubly-robust estimators have the property that, if either the response
propensity or the mean is modeled correctly, a consistent estimator of the
population mean is obtained. However, doubly-robust estimators can perform
poorly when modest misspecification is present in both models (Kang and
Schafer, 2007). Here we consider extensions of the AIPWT and PSPP models that
use Bayesian Additive Regression Trees (BART; Chipman et al., 2010) to provide
highly robust propensity and mean model estimation. We term these
"robust-squared" in the sense that the propensity score, the means, or both can
be estimated with minimal model misspecification, and applied to the
doubly-robust estimator. We consider their behavior via simulations where
propensities and/or mean models are misspecified. We apply our proposed method
to impute missing instantaneous velocity (delta-v) values from the 2014
National Automotive Sampling System Crashworthiness Data System dataset and
missing Blood Alcohol Concentration values from the 2015 Fatality Analysis
Reporting System dataset. We found that BART applied to PSPP and AIPWT,
provides a more robust and efficient estimate compared to PSPP and AIPWT, with
the BART-estimated propensity score combined with PSPP providing the most
efficient estimator with close to nominal coverage.
