  Investigators often use multi-source data (e.g., multi-center trials,
meta-analyses of randomized trials, pooled analyses of observational cohorts)
to learn about the effects of interventions in subgroups of some well-defined
target population. Such a target population can correspond to one of the data
sources of the multi-source data or an external population in which the
treatment and outcome information may not be available. We develop and evaluate
methods for using multi-source data to estimate subgroup potential outcome
means and treatment effects in a target population. We consider identifiability
conditions and propose doubly robust estimators that, under mild conditions,
are non-parametrically efficient and allow for nuisance functions to be
estimated using flexible data-adaptive methods (e.g., machine learning
techniques). We also show how to construct confidence intervals and
simultaneous confidence bands for the estimated subgroup treatment effects. We
examine the properties of the proposed estimators in simulation studies and
compare performance against alternative estimators. We also conclude that our
methods work well when the sample size of the target population is much larger
than the sample size of the multi-source data. We illustrate the proposed
methods in a meta-analysis of randomized trials for schizophrenia.
