Table 7 Multivariate Bi-test and Implementations

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 p(¬I ₀|¬H ₀,H ₀) as above. (d) Using \(pp^{o}_{{X}_{1||0}}\) or p(¬I ₀|¬H 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 p(¬I ₀|¬H ₀,H ₀) as above. (d) The same as the above (2)(d).