  Effect modification occurs when the effect of the treatment on an outcome
varies according to the level of other covariates and often has important
implications in decision making. When there are tens or hundreds of covariates,
it becomes necessary to use the observed data to select a simpler model for
effect modification and then make valid statistical inference. We propose a two
stage procedure to solve this problem. First, we use Robinson's transformation
to decouple the nuisance parameters from the treatment effect of interest and
use machine learning algorithms to estimate the nuisance parameters. Next,
after plugging in the estimates of the nuisance parameters, we use the Lasso to
choose a low-complexity model for effect modification. Compared to a full model
consisting of all the covariates, the selected model is much more
interpretable. Compared to the univariate subgroup analyses, the selected model
greatly reduces the number of false discoveries. We show that the conditional
selective inference for the selected model is asymptotically valid given the
rate assumptions in classical semiparametric regression. Extensive simulation
studies are conducted to verify the asymptotic results and an epidemiological
application is used to demonstrate the method.
