  One fundamental statistical question for research areas such as precision
medicine and health disparity is about discovering effect modification of
treatment or exposure by observed covariates. We propose a semiparametric
framework for identifying such effect modification. Instead of using the
traditional outcome models, we directly posit semiparametric models on
contrasts, or expected differences of the outcome under different treatment
choices or exposures. Through semiparametric estimation theory, all valid
estimating equations, including the efficient scores, are derived. Besides
doubly robust loss functions, our approach also enables dimension reduction in
presence of many covariates. The asymptotic and non-asymptotic properties of
the proposed methods are explored via a unified statistical and algorithmic
analysis. Comparison with existing methods in both simulation and real data
analysis demonstrates the superiority of our estimators especially for an
efficiency improved version.
