This, together with its applicability to problems with many predictor values, also distinguishes the random forest variable importance from the otherwise appealing approach of Azen, Budescu, and Reiser (2001) and advanced in Azen and Budescu (2003) for assessing the criticality of a predictor variable, termed “dominance analysis”: The authors suggest employing bootstrap sampling and select the best regression model from all possible models for each bootstrap sample in order to estimate the empirical probability distribution of all possible models. From this empirical distribution for each variable the unweighted or weighted sum of probabilities associated with all models containing the predictor is computed and suggested as an intuitive measure of variable importance. This approach, where for p predictor variables 2p − 1 models are fitted in each bootstrap iteration, has the great advantage that it provides sound statistical inference. However, it is computationally prohibitive for problems with many predictor variables of interest, because all possible models have to be fitted on all bootstrap samples.
