Semiparametric Bayesian Latent Variable Regression for Skewed Multivariate Data

Abstract

For many real‐life studies with skewed multivariate responses, the level of skewness and association structure assumptions are essential for evaluating the covariate effects on the response and its predictive distribution. We present a novel semiparametric multivariate model and associated Bayesian analysis for multivariate skewed responses. Similar to multivariate Gaussian densities, this multivariate model is closed under marginalization, allows a wide class of multivariate associations, and has meaningful physical interpretations of skewness levels and covariate effects on the marginal density. Other desirable properties of our model include the Markov Chain Monte Carlo computation through available statistical software, and the assurance of consistent Bayesian estimates of the parameters and the nonparametric error density under a set of plausible prior assumptions. We illustrate the practical advantages of our methods over existing alternatives via simulation studies and the analysis of a clinical study on periodontal disease.

Bayesian additive regression trees for multivariate skewed responses

“…Additionally, each of the individual components of the MSN distribution of () has a univariate SN marginal of (). Sahu et al 20 and Bhingare et al 5 present physical interpretations of the univariate SN density of (), and the MSN density (), in terms of convolution of a skewing shocks $$$ \boldsymbol{Z} $$$ to a symmetric distribution.…”

“…In our data, we have 60% subjects having poorly controlled T2D (ie, with HbA1c $$$ \ge 7 $$$), while the rest 40% are well‐controlled (not T2D free). Unlike previous approaches based on parametric regression functions 29 and semiparametric formulations with skewed errors, 5 our approach incorporates tree‐based unknown nonparametric regression function to capture both nonlinear and interaction effects of these covariates. Code for implementing these models is available on GitHub at https://github.com/Seungha-Um/skewBART.…”

“…In this paper, we focus on the Bayesian additive regression trees (BART) model introduced by Chipman et al 4 Unlike the parametric regression functions utilized to model skewed multivariate responses, 5 decision tree ensembles efficiently capture both nonlinear and interaction effects of covariates in $$$ \boldsymbol{f}\left(\boldsymbol{x}\right) $$$, automatically. Additionally, compared to other machine learning algorithms, BART models provide a number of appealing advantages.…”

“…For example, in studies with highly skewed responses, the error density $$$ {h}_{\epsilon}\left(\cdotp \right) $$$ being a multivariate Gaussian density may not be appropriate. Bhingare et al, 5 showed that the mean PPD and mean CAL are highly skewed, and methods which adapt to skewed responses tend to outperform methods based on Gaussian errors regarding precision of the estimated parameters and predictions of future outcomes. Second, as illustrated by Linero and Yang, 7 the estimate of $$$ \boldsymbol{f}\left(\cdotp \right) $$$ obtained from the usual BART models lack smoothness, leading to suboptimal performance when the true $$$ \boldsymbol{f}\left(\boldsymbol{x}\right) $$$ varies smoothly with $$$ \boldsymbol{x} $$$.…”

“…For univariate case with k = 1, a popular strategy for modeling f is to use an ensemble of decision trees, such as random forests 2 or boosted decision trees. 3 In this paper, we focus on the Bayesian additive regression trees (BART) model introduced by Chipman et al 4 Unlike the parametric regression functions utilized to model skewed multivariate responses, 5 decision tree ensembles efficiently capture both nonlinear and interaction effects of covariates in f(x), automatically. Additionally, compared to other machine learning algorithms, BART models provide a number of appealing advantages.…”