PPD-CPP: Pointwise predictive density calibrated-power prior in dynamically borrowing historical information

Abstract: Incorporating historical or real-world data into analyses of treatment effects for rare diseases has become increasingly popular. A major challenge, however, lies in determining the appropriate degree of congruence between historical and current data. In this study, we devote ourselves to the capacity of historical data in replicating the current data, and propose a new congruence measure/estimand $p_{CM}$. $p_{CM}$ quantifies the heterogeneity between two datasets following the idea of the marginal posterior predictive $p$-value, and its asymptotic properties were derived. Building upon $p_{CM}$, we develop the pointwise predictive density calibrated-power prior (PPD-CPP) to dynamically leverage historical information. PPD-CPP achieves the borrowing consistency and allows modeling the power parameter either as a fixed scalar or case-specific quantity informed by covariates. Simulation studies were conducted to demonstrate the performance of these methods and the methodology was illustrated using the Mother's Gift study and \textit{Ceriodaphnia dubia} toxicity test.