Randomized controlled trials (RCTs) provide reliable evidence for approval of new treatments, informing clinical practice, and coverage decisions. The participants in RCTs are often not a representative sample of the larger at-risk population. Hence it is argued that the average treatment effect from the trial is not generalizable to the larger at-risk population. An essential premise of this argument is that there is significant heterogeneity in the treatment effect (HTE). We present a new method to extrapolate the treatment effect from a trial to a target group that is inadequately represented in the trial, when HTE is present. Our method integrates trial and observational data (cross-design synthesis). The target group is assumed to be well-represented in the observational database. An essential component of the methodology is the estimation of calibration adjustments for unmeasured confounding in the observational sample. The estimate of treatment effect, adjusted for unmeasured confounding, is projected onto the target sample using a weighted G-computation approach. We present simulation studies to demonstrate the methodology for estimating the marginal treatment effect in a target sample that differs from the trial sample to varying degrees. In a companion paper, we demonstrate and validate the methodology in a clinical application
