  Nonparametric regression models such as Bayesian Additive Regression Trees
(BART) can be useful in fitting flexible functions of a set of covariates to a
response, while accounting for nonlinearities and interactions. However, they
are often cumbersome to interpret. Breaking down the function into additive
components, if appropriate, could simplify the interpretation and improve the
utility of the model. On the other hand, establishing nonadditivity can be
useful in determining the need for individualized predictions and treatment
selection. Testing additivity of single covariates in nonparametric regression
models has been extensively studied. However, additivity assessment of
nonparametric functions of disjoint sets of variables has not received as much
attention. We propose a method for detection of nonadditivity of two disjoint
sets of variables by fitting the sum of two BART models, each using its own set
of variables. We then compare the pseudo marginal likelihood (PsML) of this
sum-of- BARTs model vs. a single-BART model with all the variables together, in
a ratio known as Pseudo Bayes Factor (PsBF). A special case of our method
checks additivity between one variable of interest and another set of
variables, where the additive model allows for direct interpretation of the
variable of interest while adjusting for the remaining variables in a flexible,
nonparametric manner. We extended the above approaches to allow a binary
response using a logit link. We also propose a systematic way to design
simulations that are used in additivity assessment. In simulation studies, PsBF
showed better performance compared to out-of-sample prediction error in
choosing the correct model, while avoiding computationally expensive
cross-validation and providing an interpretable criterion for model selection.
We applied our approach to two different examples with a continuous and binary
outcomes.
