Model-Based Inverse Regression and Its Applications

One fundamental concept in regression is dimension reduction, the basic idea being to reduce the dimension of the predictor space without loss of information on the regression. To avoid the curse of dimensionality, many methods in this field restrict attention to inverse reduction in the framework of inverse regression. This review focuses on model-based inverse regression. First, we consider sufficient reduction for multivariate count data in different contexts, on the basis of the multinomial distribution and its generalizations. Second, we take a different perspective on model-based inverse reduction. Sufficient reduction is achieved in the dual sample-based space, rather than in the primal predictor-based space. The results extend the known duality between principal component analysis and principal coordinate analysis. Finally, we consider an application of inverse modeling to testing the independence between the microbiome composition and a continuous outcome. An adaptive test is presented based on a dynamic slicing technique.