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A substantial amount of work exists for tensor regression analysis in a variety of clinical settings, including neuroimaging, genomics, and dental medicine. Our motivation for this paper is from a study of periodontal disease (PD) with a three-dimensional tensor response: multiple biomarkers measured at pre-specified tooth sites within each tooth, for each subject. A careful investigation would reveal considerable skewness in the responses, in addition to response missingness. To mitigate the shortcomings of existing multivariate regression tools (that ignore the inherent tensor structure) and tensor normal based methods (that ignore response skewness), we propose a new Bayesian tensor response regression method that facilitates interpretation of covariate effects on both marginal and joint distributions of the tensor response, accommodating missing responses under a closure property. Furthermore, we present a prudent evaluation of the overall covariate effects, as well as identifying their possible variations on only a sparse subset of the tensor components. Our method promises MCMC tools that are readily implementable. We illustrate substantial advantages of our estimation proposal over existing methods via simulation studies and application to the PD dataset. We propose a general class of skewed elliptical distributions for tensor responses to ensure that any linear combinations of tensor variables still follow tensor elliptical distributions. Additionally, the marginal density has the same form as the conditional density of skewed tensor elliptical distribution. Our class of skewed elliptical distributions has useful properties including the following: the class is closed under marginalization and includes skewed tensor normal and skewed tensor-t distributions as special cases. Exploiting tensor form provides multiple benefits; 1) we can use maximum information of tensor structure that cannot be employed by multivariate methods, 2) tensor covariance structure captures the dependence of each direction of tensor. Practical applications of this new class are provided via Bayesian tensor response regression analysis with two types of prior for tensor regression coefficients: tensor normal (TN) prior, and tensor spike-and-slab lasso (TSSL) prior. We illustrate the practical advantages of TSSL prior to detecting fast decaying teeth types and sites in a periodontal disease study.
