Accounting for shared covariates in semiparametric Bayesian additive regression trees

We propose some extensions to semiparametric models based on Bayesian additive regression trees (BART). In the semiparametric BART paradigm, the response variable is approximated by a linear predictor and a BART model, where the linear component is responsible for estimating the main effects and BART accounts for nonspecified interactions and nonlinearities. Previous semiparametric models based on BART have assumed that the set of covariates in the linear predictor and the BART model are mutually exclusive in an attempt to avoid poor coverage properties and reduce bias in the estimates of the parameters in the linear predictor. The main novelty in our approach lies in the way we change the tree-generation moves in BART to deal with this bias and resolve nonidentifiability issues between the parametric and nonparametric components, even when they have covariates in common. This allows us to model complex interactions involving the covariates of primary interest, both among themselves and with those in the BART component. Our novel method is developed with a view to analysing data from an international education assessment, where certain predictors of students’ achievements in mathematics are of particular interpretational interest. Through additional simulation studies and another application to a well-known benchmark dataset, we also show competitive performance when compared to regression models, alternative formulations of semiparametric BART, and other tree-based methods. The implementation of the proposed method is available at https://github.com/ebprado/CSP-BART.