Adaptively incorporating historical information into analyses of current data can improve the precision of inference without requiring additional new observation. Unfortunately, not all borrowing methods are suitable when limited historical studies are available. When a single historical study is available, the power priors control the amount of information to borrow via specification of a weight parameter that discounts the contribution of the historical data in a likelihood combined with current data. We develop a new type of conditional power prior called the historical-bias power prior using an empirical Bayes approach. It relaxes the assumption of the traditional power priors to allow for historical bias. Moreover, our new weight function controls the amount of borrowing and only borrows when historical data satisfy the borrowing criteria. This is achieved by embedding the Frequentist test-then-pool approach in the weight function. Hence, the historical-bias power prior builds a bridge between the Frequentist test-then-pool and the Bayesian power prior. In the simulation, we examine the impact of historical bias on the operating characteristics for borrowing approaches, which has not been discussed in previous literature. The results show that the historical-bias power prior yields accurate estimation and robustly powerful tests for the experimental treatment effect with good type I error control, especially when historical bias exists.
Keywords: Bayesian adaptive design; empirical Bayes; historical control; power prior.