  Mean decrease in impurity (MDI) is a popular feature importance measure for
random forests (RFs). We show that the MDI for a feature $X_k$ in each tree in
an RF is equivalent to the unnormalized $R^2$ value in a linear regression of
the response on the collection of decision stumps that split on $X_k$. We use
this interpretation to propose a flexible feature importance framework called
MDI+. Specifically, MDI+ generalizes MDI by allowing the analyst to replace the
linear regression model and $R^2$ metric with regularized generalized linear
models (GLMs) and metrics better suited for the given data structure. Moreover,
MDI+ incorporates additional features to mitigate known biases of decision
trees against additive or smooth models. We further provide guidance on how
practitioners can choose an appropriate GLM and metric based upon the
Predictability, Computability, Stability framework for veridical data science.
Extensive data-inspired simulations show that MDI+ significantly outperforms
popular feature importance measures in identifying signal features. We also
apply MDI+ to two real-world case studies on drug response prediction and
breast cancer subtype classification. We show that MDI+ extracts
well-established predictive genes with significantly greater stability compared
to existing feature importance measures. All code and models are released in a
full-fledged python package on Github.
