Formal Hypothesis Tests for Additive Structure in Random Forests

ABSTRACT

While statistical learning methods have proved powerful tools for predictive modeling, the black-box nature of the models they produce can severely limit their interpretability and the ability to conduct formal inference. However, the natural structure of ensemble learners like bagged trees and random forests has been shown to admit desirable asymptotic properties when base learners are built with proper subsamples. In this work, we demonstrate that by defining an appropriate grid structure on the covariate space, we may carry out formal hypothesis tests for both variable importance and underlying additive model structure. To our knowledge, these tests represent the first statistical tools for investigating the underlying regression structure in a context such as random forests. We develop notions of total and partial additivity and further demonstrate that testing can be carried out at no additional computational cost by estimating the variance within the process of constructing the ensemble. Furthermore, we propose a novel extension of these testing procedures using random projections to allow for computationally efficient testing procedures that retain high power even when the grid size is much larger than that of the training set.

Keywords:

• Bootstrap/resampling
• Random forests
• Random projections
• U-statistics
• Additive models

Acknowledgments

The authors thank Cornell University's Lab of Ornithology for providing interesting data. Giles Hooker was partially supported by NSF grants DMS-1053252 and DEB-1353039 and NIH grant R03DA036683.