  A core step of every algorithm for learning regression trees is the selection
of the best splitting variable from the available covariates and the
corresponding split point. Early tree algorithms (e.g., AID, CART) employed
greedy search strategies, directly comparing all possible split points in all
available covariates. However, subsequent research showed that this is biased
towards selecting covariates with more potential split points. Therefore,
unbiased recursive partitioning algorithms have been suggested (e.g., QUEST,
GUIDE, CTree, MOB) that first select the covariate based on statistical
inference using p-values that are adjusted for the possible split points. In a
second step a split point optimizing some objective function is selected in the
chosen split variable. However, different unbiased tree algorithms obtain these
p-values from different inference frameworks and their relative advantages or
disadvantages are not well understood, yet. Therefore, three different popular
approaches are considered here: classical categorical association tests (as in
GUIDE), conditional inference (as in CTree), and parameter instability tests
(as in MOB). First, these are embedded into a common inference framework
encompassing parametric model trees, in particular linear model trees. Second,
it is assessed how different building blocks from this common framework affect
the power of the algorithms to select the appropriate covariates for splitting:
observation-wise goodness-of-fit measure (residuals vs. model scores),
dichotomization of residuals/scores at zero, and binning of possible split
variables. This shows that specifically the goodness-of-fit measure is crucial
for the power of the procedures, with model scores without dichotomization
performing much better in many scenarios.
