  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.
