Bi-linear matrix-variate analyses, integrative hypothesis tests, and case-control studies

We pursue a threefold purpose in this paper. First, we suggest a Kullback-Leibler formulation for developing a statistics and making discriminative projection for case-control studies, based on which existing typical methods are revisited and then further extended to matrix-variate counterparts. Second, we propose a bi-linear matrix form, based on which multivariate discriminative analysis and logistic, Cox, and linear mixed regression are extended into their matrix-variate counterparts. Third, we systematically address the necessity, feasibility, and methodology of integrative hypothesis tests (IHT) from the complementarity of model-based test and boundary-based test (BBT) in the data (D)-space, statistics (S)-space, and probability (P)-space. We elaborate four IHT components (modelling, comparison, classification, and assurance) and summarise four IHT types in the D-space. Then, we extend the existing efforts on multivariate tests to BBTs in the S-space. Particularly, we extend the classic univariate one-tail z-test to the multivariate ones, which is then applied to a multivariate sample-pairing delta (SPD) test for detecting a collective inclining dominance. Also, we propose a SPD discriminative analysis that extends this SPD test. Moreover, we propose a multivariate bi-test that tests the classic null and also a null about the inference reliability due to test space complexity, including a further development of Fisher combination. Finally, we suggest possible applications for gene expression biomarkers and exome-sequencing-based joint single-nucleotide variant (SNV) detection.