Sufficient dimension reduction for a novel class of zero-inflated graphical models

Abstract

Graphical models allow modeling of complex dependencies among components of a random vector. In many applicationsof graphical models, however, for example microbiome data, the data have an excess number of zero values. Wepresent new pairwise graphical models with distributions in an exponential family, that accommodate excess numbersof zeros in the random vector components. First we characterise these multivariate distributions in terms of univariateconditional distributions. We then model predictors that arise from such a pairwise graphical model with excess zerosas a function of an outcome, and derive the corresponding first order sufficient dimension reduction (SDR). That is,we find linear combinations of the predictors that contain all the information for the regression of the outcome as afunction of the predictors. We estimate the SDR using pseudo-likelihood with a hierarchical penalty that accounts forthe graphical model structure, for variable selection, by allowing interactions only for variables that are associatedwith outcome also through main effects. This method yields consistent estimators of the reduction and can be appliedto continuous or categorical outcomes. We then illustrate our methods by studying normal, Poisson and truncatedPoisson graphical models with excess zeros in simulations and by analyzing microbiome data from the AmericanGut Project. Our models provided robust variable selection and the Poisson zero-inflation pairwise graphical modelresulted in predictive performance that was equal or better than that obtained from applying other available methodsfor the analysis of microbiome data.