  While sample sizes in randomized clinical trials are large enough to estimate
the average treatment effect well, they are often insufficient for estimation
of treatment-covariate interactions critical to studying data-driven precision
medicine. Observational data from real world practice may play an important
role in alleviating this problem. One common approach in trials is to predict
the outcome of interest with separate regression models in each treatment arm,
and estimate the treatment effect based on the contrast of the predictions.
Unfortunately, this simple approach may induce spurious treatment-covariate
interaction in observational studies when the regression model is misspecified.
Motivated by the need of modeling the number of relapses in multiple sclerosis
patients, where the ratio of relapse rates is a natural choice of the treatment
effect, we propose to estimate the conditional average treatment effect (CATE)
as the ratio of expected potential outcomes, and derive a doubly robust
estimator of this CATE in a semiparametric model of treatment-covariate
interactions. We also provide a validation procedure to check the quality of
the estimator on an independent sample. We conduct simulations to demonstrate
the finite sample performance of the proposed methods, and illustrate their
advantages on real data by examining the treatment effect of dimethyl fumarate
compared to teriflunomide in multiple sclerosis patients.
