VCBART: Bayesian trees for varying coefficients

Abstract

The linear varying coefficient (VC) model generalizes the conventional linear model by allowing the additive effect of each covariate on the outcome to vary as a function of additional effect modifiers. While there are many existing procedures for VC modeling with a single scalar effect modifier (often assumed to be time), there has, until recently, been comparatively less development for settings with multivariate modifiers. Unfortunately, existing state-of-the-art procedures that can accommodate multivariate modifiers typically make restrictive structural assumptions about the covariate effect functions or require intensive problem-specific hand-tuning that scales poorly to large datasets. In response, we propose VC-BART, which estimates the covariate effect functions in a VC model using Bayesian Additive Regression Trees (BART).On several synthetic and real-world data sets, we demonstrate that, with simple default hyperparameter settings, VC-BART displays covariate effect recovery performance superior to state-of-the-art VC modeling techniques and predictive performance on par with more flexible but less interpretable nonparametric regression procedures. We further demonstrate the theoretical near-optimality of VC-BART by synthesizing recent theoretical results about the VC model and BART to derive posterior concentration rates in settings with independent and correlated errors. An R package implementing VC-BART is available at https://github.com/skdeshpande91/VCBART

Accounting for shared covariates in semi-parametric Bayesian additive regression trees

“…Deshpande, Bai, Balocchi, Starling, and Weiss (2020) introduce Varying Coefficient BART (VCBART), which combines the idea of varying coefficient models (T. Hastie & Tibshirani 1993) with BART and extends the work of Hahn et al (2020) to a framework with multiple covariates. In VCBART, the response is modelled via a linear predictor where the effect of each covariate is approximated by a BART model based on a set of modifiers (i.e.…”

“…In this section, we compare our new SP-BART with GAM, semi-parametric BART of Zeldow et al (2019) (hereafter referred to as separated semi-BART), and VCBART in terms of bias using two sets of synthetic data. The results were generated using R (R Core Team 2020) version 3.6.3 and the R packages mgcv (Wood 2017), semibart (Zeldow et al 2019), and VCBART (Deshpande et al 2020). For both, our SP-BART and separated semi-BART, we use 50 trees, 2,000 MCMC iterations as burn-in, and 2,000 as post-burn-in.…”

“…, p, i ∼ N(0, σ 2 ). This equation is used for benchmarking tree-based methods using synthetic data, and has been used in many other papers (Chipman et al 2010;Deshpande et al 2020;Linero 2018). In this experiment, we set n = 1000, p = (10, 50), σ 2 = (1, 10), totalling four scenarios.…”

Heterogeneous Distributed Lag Models to Estimate Personalized Effects of Maternal Exposures to Air Pollution

“…For example, Starling et al (2020) introduced BART with targeted smoothing (tsBART), which allows a smooth risk function of a single univariate predictor to vary across a population. Deshpande et al (2020) proposed a varying coefficient BART model that uses a separate ensemble of regression trees to modify each regression coefficient in the model. In a non-Bayesian approach to estimating effect heterogeneity, Odden et al (2020) applied a random forest algorithm to identify heterogeneous exposure associations.…”

Bayesian Semiparametric Hidden Markov Tensor Partition Models for Longitudinal Data with Local Variable Selection

“…VC models have been adapted to longitudinal data by considering time as the only modifier (Hoover et al, 1998). More recently BART priors (Deshpande et al, 2020) and variable selection techniques (Koslovsky et al, 2020) have also been adapted to VC settings. While VC models allow for an easy assessment of the predictors' importance, they are restricted in their ability to accommodate interactions between predictors.…”