A graphical model for skewed matrix-variate non-randomly missing data

Abstract

Summary Epidemiological studies on periodontal disease (PD) collect relevant bio-markers, such as the clinical attachment level (CAL) and the probed pocket depth (PPD), at pre-specified tooth sites clustered within a subject’s mouth, along with various other demographic and biological risk factors. Routine cross-sectional evaluation are conducted under a linear mixed model (LMM) framework with underlying normality assumptions on the random terms. However, a careful investigation reveals considerable non-normality manifested in those random terms, in the form of skewness and tail behavior. In addition, PD progression is hypothesized to be spatially-referenced, i.e. disease status at proximal tooth-sites may be different from distally located sites, and tooth missingness is non-random (or informative), given that the number and location of missing teeth informs about the periodontal health in that region. To mitigate these complexities, we consider a matrix-variate skew-$t$ formulation of the LMM with a Markov graphical embedding to handle the site-level spatial associations of the bivariate (PPD and CAL) responses. Within the same framework, the non-randomly missing responses are imputed via a latent probit regression of the missingness indicator over the responses. Our hierarchical Bayesian framework powered by relevant Markov chain Monte Carlo steps addresses the aforementioned complexities within an unified paradigm, and estimates model parameters with seamless sharing of information across various stages of the hierarchy. Using both synthetic and real clinical data assessing PD status, we demonstrate a significantly improved fit of our proposition over various other alternative models.

Bayesian Regression Analysis of Skewed Tensor Responses

“…Here, the tensor response is composed of two biomarkers: the periodontal pocket depth (PPD) and clinical attachment level (CAL), measured at prespecified tooth–sites, within each tooth of a study participant. We strive to present a new model and analysis of clinical data derived from the GAAD study over existing analyses of such PD studies that often ignore or contort the original tensor response structure (Bhingare et al., 2019; Zhang & Bandyopadhyay, 2020), leading to the imminent loss of data information and lack of proper evaluation of differences in covariate effects on various tensor components. For example, one of our analysis goals is to evaluate whether the effects of HbA1c (covariate measuring Type‐2 diabetes, T2D) on some tooth–site combinations are more pronounced, than its overall effect on the rest.…”