If, on the other hand, the original variable was not associated with the response, it is either not included in the tree (and its importance for this tree is zero by definition), or it is included in the tree by chance. In the latter case, permuting the variable results only in a small random decrease in prediction accuracy, or the permutation of an irrelevant variable can even lead to a small increase in the prediction accuracy (if, by chance, the permutated variable happens to be slightly better suited for splitting than the original one). Thus the permutation importance can even show (small) negative values for irrelevant predictor variables, as illustrated for the irrelevant predictor variables age and lied_to_parents in Figure 8.
