Data-Adaptive Integration with External Summary Data for Outcome Mean Estimation

Abstract

Combining an internal individual-level study with readily available external summary statistics promises major efficiency gains at minimal additional cost, yet heterogeneity between sources can bias estimates for the internal target population. We develop a generalized entropy-balancing integration strategy that calibrates the internal individual-level sample to externally reported moments while retaining the internal population as the target, explicitly permitting a biased external sample. The weighted-regression version of our estimator is doubly robust: it remains consistent when either the outcome-regression model or the entropy-balancing model is correctly specified. When multiple balancing specifications are plausible, we introduce a data-adaptive entropy-family selection rule. For the final borrowing decision, we propose a bootstrap-based criterion comparing stabilized mean squared error (MSE) estimates for the selected entropy-balancing estimator and the internal sample mean. This criterion is selection consistent under fixed alternatives and reverts to the internal estimator when a nonvanishing bias is detected. Separately, under a linear homoscedastic benchmark, the asymptotic efficiency criteria admit geometric interpretations through the Mahalanobis distance and Pearson chi-squared divergence. The entropy-balancing estimators and numerical-experiment routines are implemented in the R package daisy. Simulations show stable MSE reductions for the weighted-regression estimator across calibrated distributional shifts and the predicted reversion toward the internal estimator under fixed simultaneous misspecification as the sample size increases. An application to nationwide public-access defibrillation records in Japan illustrates the resulting MSE-based borrowing decision.