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Table 7 Multivariate Bi-test and Implementations

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

Type Description
  Test bi-hypotheses and twin p-values
test H 0 whether the case-control populations are different, by an inference I nf in the space of multivariate statistics s based on samples from the two populations. H 0 is rejected if \(\phantom {\dot {i}\!}{pp}_{{X}_{1||0}}\!\le \! \alpha \), where the false alarm probability \(\phantom {\dot {i}\!}{pp}_{{X}_{1||0}}\,=\,pp^{o}_{{X}_{1||0}} {rp}_{{X}_{1||0}}\) is given by Equation (93) and α is a prespecified level.
test I 0 whether the dimension m of s is appropriate such that I nf is reliable, with the p-value given by \( p(\neg I_{0} |\neg H_{0}, H_{0}) = p({pp}_{X^{\pi }_{1|0}}\le \alpha |{pp}_{{X}_{1||0}}< \alpha, H_0), \) which is not smaller than \(pp^{o}_{{X}_{1||0}}\) that reflects the relative discriminative information among \( {pp}_{{X}_{1||0}}\phantom {\dot {i}\!}\) while ignoring \({rp}_{{X}_{1||0}}\phantom {\dot {i}\!}\) that reflects the strength of discriminative information.
  Bi-text Implementations
Stochastic way (a) Make the components of s decorrelated by Equation (69). (b) Get \(p({\mathbf {s}}\in \Gamma |\ I_{\textit {nf}}, X_{1||0}^{\pi }, H_0)=p({\mathbf {s}}\in \Gamma (\tilde {\bf {s}}) |H_0)\) by Equation (68) with \(\Gamma (\tilde {\bf s})\) taking one of three choices in Equation (70), and then getting P Π by Equation (92). (c) Get \( {pp}_{{X}_{1||0}}, pp^{o}_{{X}_{1||0}}, {rp}_{{X}_{1||0}}\phantom {\dot {i}\!}\) by Equation (93) and then getting pI 0H 0,H 0) as above. (d) Using \(pp^{o}_{{X}_{1||0}}\) or pI 0H 0,H 0) as J(m) to infer an appropriate \(m^{*}_{o}\) and select the \(m^{*}_{o}\) best components of s.
Nonstochastic way (a) Make the components of s decorrelated by Equation (69). (b) Get {p i } with each p-value p i obatined by an univariate test. (c) Get \( pp^{o}_{{X}_{1||0}}\) by Equation (99) and \({rp}_{{X}_{1||0}}\phantom {\dot {i}\!}\) by Equation (97) with \(p_{X_{1|0}}= \prod _{i} p_{i}\), as well as getting pI 0H 0,H 0) as above. (d) The same as the above (2)(d).