Nonparametric Identification and Estimation of Ratios of Multi-Category Means under Preferential Sampling
Multi-category data arise in diverse fields including marketing, chemistry, public policy, genomics, political science, and ecology. We consider the problem of estimating ratios of category-specific means in a fully nonparametric setting, allowing for both observational units and categories to be preferentially sampled. We consider covariate-adjusted and unadjusted estimands that are non-parametrically defined and straightforward to interpret. While identifiability for related models has been established through parametric distributions or restrictions on the conditional mean (e.g., log-linearity), we show that identifiability can be obtained through an independence assumption or a category constraint, such as a reference category or a centering function. We develop an efficient, doubly-robust targeted minimum loss based estimator with excellent finite-sample performance, including in the setting of a large number of infrequently observed categories. We contrast the performance of our method with related approaches via simulation, and apply it to identify bacteria that are differentially abundant in diarrheal cases compared to controls. Our work provides a general framework for studying parameter identifiability in compositional data settings without requiring parametric assumptions on the data distribution.