  Measuring average differences in an outcome across racial or ethnic groups is
a crucial first step for equity assessments, but researchers often lack access
to data on individuals' races and ethnicities to calculate them. A common
solution is to impute the missing race or ethnicity labels using proxies, then
use those imputations to estimate the disparity. Conventional standard errors
mischaracterize the resulting estimate's uncertainty because they treat the
imputation model as given and fixed, instead of as an unknown object that must
be estimated with uncertainty. We propose a dual-bootstrap approach that
explicitly accounts for measurement uncertainty and thus enables more accurate
statistical inference, which we demonstrate via simulation. In addition, we
adapt our approach to the commonly used Bayesian Improved Surname Geocoding
(BISG) imputation algorithm, where direct bootstrapping is infeasible because
the underlying Census Bureau data are unavailable. In simulations, we find that
measurement uncertainty is generally insignificant for BISG except in
particular circumstances; bias, not variance, is likely the predominant source
of error. We apply our method to quantify the uncertainty of prevalence
estimates of common health conditions by race using data from the American
Family Cohort.
