A Bayesian Bootstrap Approach for Dynamic Borrowing for Minimizing Mean Squared Error

Abstract

For dynamic borrowing to leverage external data to augment the control arm of small RCTs, the key step is determining the amount of borrowing based on the similarity of the outcomes in the controls from the trial and the external data sources. A simple approach for this task uses the empirical Bayesian approach, which maximizes the marginal likelihood (maxML) of the amount of borrowing, while a likelihood-independent alternative minimizes the mean squared error (minMSE). We consider two minMSE approaches that differ from each other in the way of estimating the parameters in the minMSE rule. The classical one adjusts for bias due to sample variance, which in some situations is equivalent to the maxML rule. We propose a simplified alternative without the variance adjustment, which has asymptotic properties partially similar to the maxML rule, leading to no borrowing if means of control outcomes from the two data sources are different and may have less bias than that of the maxML rule. In contrast, the maxML rule may lead to full borrowing even when two datasets are moderately different, which may not be a desirable property. For inference, we propose a Bayesian bootstrap (BB) based approach taking the uncertainty of the estimated amount of borrowing and that of pre-adjustment into account. The approach can also be used with a pre-adjustment on the external controls for population difference between the two data sources using, e.g., inverse probability weighting. The proposed approach is computationally efficient and is implemented via a simple algorithm. We conducted a simulation study to examine properties of the proposed approach, including the coverage of 95 CI based on the Bayesian bootstrapped posterior samples, or asymptotic normality. The approach is illustrated by an example of borrowing controls for an AML trial from another study.