  When the marginal causal effect comparing the same treatment pair is
available from multiple trials, we wish to transport all results to make
inference on the target population effect. To account for the differences
between populations, statistical analysis is often performed controlling for
relevant variables. However, when transportability assumptions are placed on
conditional causal effects, rather than the distribution of potential outcomes,
we need to carefully choose these effect measures. In particular, we present
identifiability results in two cases: target population average treatment
effect for a continuous outcome and causal mean ratio for a positive outcome.
We characterize the semiparametric efficiency bounds of the causal effects
under the respective transportability assumptions and propose estimators that
are doubly robust against model misspecifications. We highlight an important
discussion on the tension between the non-collapsibility of conditional effects
and the variational independence induced by transportability in the case of
multiple source trials.
