Multi forests: Variable importance for multi-class outcomes

TL;DR

This paper introduces multi forests (MuFs), a random-forest variant that combines multi-way and binary splits to address multi-class outcomes, with a dedicated multi-class VIM to identify covariates specifically associated with individual classes and a discriminatory VIM to capture general discriminative power. The multi-class VIM relies on out-of-bag data and a split-criterion that rewards covariate regions highly dominated by single classes, enabling interpretable identification of class-associated covariates, while the discriminatory VIM mirrors conventional VIM behavior for broader influence. Through extensive simulations, the authors show that the multi-class VIM consistently prioritizes class-associated covariates (especially with the wsquared variants), whereas conventional VIMs often rank non-class-specific covariates highly. Real-data analyses on 121 datasets reveal that MuFs provide only modestly lower predictive performance compared with conventional random forests, suggesting MuFs' primary value lies in interpretability and covariate discovery rather than superior prediction. The work offers practical guidance, including recommending wsquared_wgini as the default MuF variant, and discusses limitations, particularly as the number of classes grows, where MuFs may be better suited for VIM computation than for prediction.

Abstract

In prediction tasks with multi-class outcomes, identifying covariates specifically associated with one or more outcome classes can be important. Conventional variable importance measures (VIMs) from random forests (RFs), like permutation and Gini importance, focus on overall predictive performance or node purity, without differentiating between the classes. Therefore, they can be expected to fail to distinguish class-associated covariates from covariates that only distinguish between groups of classes. We introduce a VIM called multi-class VIM, tailored for identifying exclusively class-associated covariates, via a novel RF variant called multi forests (MuFs). The trees in MuFs use both multi-way and binary splitting. The multi-way splits generate child nodes for each class, using a split criterion that evaluates how well these nodes represent their respective classes. This setup forms the basis of the multi-class VIM, which measures the discriminatory ability of the splits performed in the respective covariates with regard to this split criterion. Alongside the multi-class VIM, we introduce a second VIM, the discriminatory VIM. This measure, based on the binary splits, assesses the strength of the general influence of the covariates, irrespective of their class-associatedness. Simulation studies demonstrate that the multi-class VIM specifically ranks class-associated covariates highly, unlike conventional VIMs which also rank other types of covariates highly. Analyses of 121 datasets reveal that MuFs often have slightly lower predictive performance compared to conventional RFs. This is, however, not a limiting factor given the algorithm's primary purpose of calculating the multi-class VIM.

Figures (6)

• Figure 1: VIM values obtained for wsquared_wgini and the permutation VIM (perm) obtained for all simulated datasets with $n=500$. For visual clarity, the VIM values of only five of the 50 noise covariates are shown.
• Figure 2: Mean AUC values per considered sample size and method for $C=4$. The line types distinguish the different VIM types, where conventional corresponds to conventional VIMs, multi-class to multi-class VIMs, and multi-class diff. to the differences between the multi-class VIM values and the corresponding discriminatory VIM values.
• Figure 3: Mean AUC values per considered sample size and method for $C=6$. The line types distinguish the different VIM types, where conventional corresponds to conventional VIMs, multi-class to multi-class VIMs, and multi-class diff. to the differences between the multi-class VIM values and the corresponding discriminatory VIM values.
• Figure 4: Mean AUC values per considered sample size and method for $C=10$. The line types distinguish the different VIM types, where conventional corresponds to conventional VIMs, multi-class to multi-class VIMs and multi-class diff. to the differences between the multi-class VIM values and the corresponding discriminatory VIM values.
• Figure 5: Ranks of the MuF versions and RF with respect to the different performance metrics. Each stacked bar represents the number of datasets for which the respective method achieved the indicated ranks among all other methods.