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.
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).