# Random forest splitting rule and importance in package ranger

The package ranger implements random forests in R. Among other things, the function used to fit a random forest allows to choose among several splitting rules, and several ways to compute the importance of the features. The documentation says:

For importance:

Variable importance mode, one of 'none', 'impurity', 'impurity_corrected', 'permutation'. The 'impurity' measure is the Gini index for classification, the variance of the responses for regression and the sum of test statistics (see splitrule) for survival.

What is impurity_corrected? I understand that impurity in regression is a measure based on the variance reduction for each split where the considered variable is used, but how is it corrected?

For splitting rules:

Splitting rule. For classification and probability estimation "gini", "extratrees" or "hellinger" with default "gini". For regression "variance", "extratrees", "maxstat" or "beta" with default "variance". For survival "logrank", "extratrees", "C" or "maxstat" with default "logrank".

What are maxstat and beta splitting rules in a regression setting?