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

Abstract

We pursue a threefold purpose in this paper. First, we suggest a Kullback-Leibler formulation for developing a statistics and making discriminative projection for casecontrol 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.

Joint Covariate Detection on Expression Profiles for Selecting Prognostic miRNAs in Glioblastoma

“…Moreover, the variables for stratification of patients using the risk score defined by (4) correspond to small p values as seen in (8). In fact, this conception derives from Integrative Hypothesis Testing (IHT) proposed by Xu [19]. The obtained covariates may indicate not only a close correlation with survival time but also distinct stratification of patients.…”

A new multivariate test formulation: theory, implementation, and applications to genome-scale sequencing and expression

“…As addressed in Ref. Xu (2015a), many tasks of big data analyses demand extending vector-based sampling units to sampling units in matrix format. Illustrated in Fig.…”

“…Xu 2015a) and matrix-variate discriminative analysis (see the details from Eqs (33) to (43) in Ref. Xu 2015a). For the former, test is made on regression coefficients in two vectors, which is still a multivariate test.…”

“…Xu (2015a) as one example of the boundary-based two-sample test or BBT in short Learning LDA score We may also perform either a directional test or a non-directional test as above, but with → β obtained by (a) Support vector machine (SVM) (Suykens 1999;Suykens et al 2002), as suggested in Fig. 7 A case-control study that considers matrix-variate samples.…”

“…Xu (2015a), considering both T and N jointly in a 2D vector by a two-variate PT test paves a new road for reconsidering the task. As shown in Fig.…”