Doubly Flexible Estimation under Label Shift

In studies ranging from clinical medicine to policy research, complete data are usually available from a population (Formula presented.), but the quantity of interest is often sought for a related but different population (Formula presented.) which only has partial data. We consider the setting when both outcome Y and covariate X are available from (Formula presented.) but only X is available from (Formula presented.), under the label shift assumption; that is, the conditional distribution of X given Y is the same in the two populations. To estimate the parameter of interest in (Formula presented.) by leveraging information from (Formula presented.), three ingredients are essential: (a) the common conditional distribution of X given Y, (b) the regression model of Y given X in (Formula presented.), and (c) the density ratio of the outcome Y between the two populations. We propose an estimation procedure that only needs some standard nonparametric technique to approximate the conditional expectations with respect to (a), while by no means needs an estimate or model for (b) or (c); that is, doubly flexible to the model misspecifications of both (b) and (c). This is conceptually different from the well-known doubly robust estimation in that, double robustness allows at most one model to be misspecified whereas our proposal can allow both (b) and (c) to be misspecified. This is of particular interest in label shift because estimating (c) is difficult, if not impossible, by virtue of the absence of the Y-data from (Formula presented.). While estimating (b) is occasionally off-the-shelf, it may encounter issues related to the curse of dimensionality or computational challenges. We develop the large sample theory for the proposed estimator, and examine its finite-sample performance through simulation studies as well as an application to the MIMIC-III database. Supplementary materials for this article are available online including a standardized description of the materials available for reproducing the work.