  It is widely believed that the prediction accuracy of decision tree models is
invariant under any strictly monotone transformation of the individual
predictor variables. However, this statement may be false when predicting new
observations with values that were not seen in the training-set and are close
to the location of the split point of a tree rule. The sensitivity of the
prediction error to the split point interpolation is high when the split point
of the tree is estimated based on very few observations, reaching 9%
misclassification error when only 10 observations are used for constructing a
split, and shrinking to 1% when relying on 100 observations. This study
compares the performance of alternative methods for split point interpolation
and concludes that the best choice is taking the mid-point between the two
closest points to the split point of the tree. Furthermore, if the (continuous)
distribution of the predictor variable is known, then using its probability
integral for transforming the variable ("quantile transformation") will reduce
the model's interpolation error by up to about a half on average. Accordingly,
this study provides guidelines for both developers and users of decision tree
models (including bagging and random forest).
