Multi forests: Variable importance for multi-class outcomes

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.

Sequential Permutation Testing of Random Forest Variable Importance Measures

Hypothesis testing of random forest (RF) variable importance measures (VIMP) remains the subject of ongoing research. Among recent developments, heuristic approaches to parametric testing have been proposed whose distributional assumptions are based on empirical evidence. Other formal tests under regularity conditions were derived analytically. However, these approaches can be computationally expensive or even practically infeasible. This problem also occurs with non-parametric permutation tests, which are, however, distribution-free and can generically be applied to any type of RF and VIMP. Embracing this advantage, it is proposed here to use sequential permutation tests and sequential p-value estimation to reduce the high computational costs associated with conventional permutation tests. The popular and widely used permutation VIMP serves as a practical and relevant application example. The results of simulation studies confirm that the theoretical properties of the sequential tests apply, that is, the type-I error probability is controlled at a nominal level and a high power is maintained with considerably fewer permutations needed in comparison to conventional permutation testing. The numerical stability of the methods is investigated in two additional application studies. In summary, theoretically sound sequential permutation testing of VIMP is possible at greatly reduced computational costs. Recommendations for application are given. A respective implementation is provided through the accompanying R package $rfvimptest$. The approach can also be easily applied to any kind of prediction model.