Double soft-thresholded model for multi-group scalar on vector-valued image regression

In this paper, we develop a novel spatial variable selection method for scalar on vector-valued image regression in a multi-group setting. Here, 'vector-valued image' refers to the imaging datasets that contain vector-valued information at each pixel/voxel location, such as in RGB color images, multimodal medical images, DTI imaging, etc. The focus of this work is to identify the spatial locations in the image having an important effect on the scalar outcome measure. Specifically, the overall effect of each voxel is of interest. We thus develop a novel shrinkage prior by soft-thresholding the \ell_2 norm of a latent multivariate Gaussian process. It will allow us to estimate sparse and piecewise-smooth spatially varying vector-valued regression coefficient functions. Furthermore, motivated by our clinical application, the regression effect is decomposed into shared and group-specific parts, where a double soft-thresholding based prior using ST2N is introduced. For posterior inference, an efficient MCMC algorithm is developed. We establish the posterior contraction rate for parameter estimation and consistency for variable selection of the proposed Bayesian model, assuming that the true regression coefficients are Holder smooth. Finally, we demonstrate the better performance in simulations and illustrate in an ADNI dataset for modeling MMSE scores based on DTI-based vector-valued imaging markers.