  In clinical settings, we often face the challenge of building prediction
models based on small observational data sets. For example, such a data set
might be from a medical center in a multi-center study. Differences between
centers might be large, thus requiring specific models based on the data set
from the target center. Still, we want to borrow information from the external
centers, to deal with small sample sizes. There are approaches that either
assign weights to each external data set or each external observation. To
incorporate information on differences between data sets and observations, we
propose an approach that combines both into weights that can be incorporated
into a likelihood for fitting regression models. Specifically, we suggest
weights at the data set level that incorporate information on how well the
models that provide the observation weights distinguish between data sets.
Technically, this takes the form of inverse probability weighting. We explore
different scenarios where covariates and outcomes differ among data sets,
informing our simulation design for method evaluation. The concept of effective
sample size is used for understanding the effectiveness of our subgroup
modeling approach. We demonstrate our approach through a clinical application,
predicting applied radiotherapy doses for cancer patients. Generally, the
proposed approach provides improved prediction performance when external data
sets are similar. We thus provide a method for quantifying similarity of
external data sets to the target data set and use this similarity to include
external observations for improving performance in a target data set prediction
modeling task with small data.
