Additivity Assessment in Nonparametric Models Using Ratio of Pseudo Marginal Likelihoods

Abstract

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.