Conditional variable importance for random forests

Abstract

Background: Random forests are becoming increasingly popular in many scientific fields because they can cope with "small n large p" problems, complex interactions and even highly correlated predictor variables. Their variable importance measures have recently been suggested as screening tools for, e.g., gene expression studies. However, these variable importance measures show a bias towards correlated predictor variables.

Results: We identify two mechanisms responsible for this finding: (i) A preference for the selection of correlated predictors in the tree building process and (ii) an additional advantage for correlated predictor variables induced by the unconditional permutation scheme that is employed in the computation of the variable importance measure. Based on these considerations we develop a new, conditional permutation scheme for the computation of the variable importance measure.

Conclusion: The resulting conditional variable importance reflects the true impact of each predictor variable more reliably than the original marginal approach.

Figure 1

Selection rates. Relative selection rates for twelve variables in the first splits (left) and in all splits (right) of all trees in random forests built with different values for mtry.

Figure 2

Permutation scheme for the original marginal (left) and for the newly suggested conditional (right) permutation importance.

Figure 3

Permutation importance. Median permutation importance for marginal (dashed) and conditional (solid) permutation scheme along with inter-quartile range. Note that the ordering of variables in the plot is arbitrary.

Figure 4

Example: peptide-binding data. Marginal (top) and conditional (bottom) permutation importance of 104 predictors of peptide-binding.