Abstract
In this paper, we propose a method to construct tree regression models including linear regression terms. Ordinary tree regression models reliably detect high order interactions compared with multiple regression models, but they tend to make large, deep trees because they explain all covariate effects using only stratification of samples. In particular, if there are many effects common to all samples or some sample group, they estimate a tree with many almost identical subtrees. In order to avoid this redundancy, we propose a tree regression model that explains main effects by linear regression terms in each node, and heterogeneity of the effects by stratification. When we take this strategy, there may be a huge number of candidate models; hence model selection is one of the main tasks. Thus we propose an algorithm to estimate the models using a criterion based on the MDL principle. This criterion can be interpreted as a criterion to select split variables which have the maximum interaction effects. Finally, we demonstrate the efficiency of our method using numerical examples as well as real data on the amount of time caregivers provided individually to elderly persons.
