  Model development often takes data structure, subject matter considerations,
model assumptions, and goodness of fit into consideration. To diagnose issues
with any of these factors, it can be helpful to understand regression model
estimates at a more granular level. We propose a new method for decomposing
point estimates from a regression model via weights placed on data clusters.
The weights are informed only by the model specification and data availability
and thus can be used to explicitly link the effects of data imbalance and model
assumptions to actual model estimates. The weight matrix has been understood in
linear models as the hat matrix in the existing literature. We extend it to
Bayesian hierarchical regression models that incorporate prior information and
complicated dependence structures through the covariance among random effects.
We show that the model weights, which we call borrowing factors, generalize
shrinkage and information borrowing to all regression models. In contrast, the
focus of the hat matrix has been mainly on the diagonal elements indicating the
amount of leverage. We also provide metrics that summarize the borrowing
factors and are practically useful. We present the theoretical properties of
the borrowing factors and associated metrics and demonstrate their usage in two
examples. By explicitly quantifying borrowing and shrinkage, researchers can
better incorporate domain knowledge and evaluate model performance and the
impacts of data properties such as data imbalance or influential points.
