  The heterogeneity of treatment effect lies at the heart of precision
medicine. Randomized controlled trials are gold-standard for treatment effect
estimation but are typically underpowered for heterogeneous effects. While
large observational studies have high predictive power but are often confounded
due to lack of randomization of treatment. We show that the observational
study, even subject to hidden confounding, may empower trials in estimating the
heterogeneity of treatment effect using the notion of confounding function. The
confounding function summarizes the impact of unmeasured confounders on the
difference in the potential outcomes between the treated and untreated groups
accounting for the observed covariates, which is unidentifiable based only on
the observational study. Coupling the randomized trial and observational study,
we show that the heterogeneity of treatment effect and confounding function are
nonparametrically identifiable. We then derive the semiparametric efficient
scores and the rate-doubly robust integrative estimators of the heterogeneity
of treatment effect and confounding function under parametric structural
models. We clarify the conditions under which the integrative estimator of the
heterogeneity of treatment effect is strictly more efficient than the trial
estimator. We illustrate the integrative estimators via simulation and an
application.
