The use of an identity in Anderson for multivariate multiple testing

Consider the model where there are II independent multivariate normal treatment populations with p×1p×1 mean vectors μiμi, i=1,…,Ii=1,…,I, and covariance matrix ΣΣ. Independently the (I+1)(I+1)st population corresponds to a control and it too is multivariate normal with mean vector μI+1μI+1 and covariance matrix ΣΣ. Now consider the following two multiple testing problems.Problem   1: Test Hij:μij-μ(I+1)j=0Hij:μij-μ(I+1)j=0 vs Kij:μij-μ(I+1)j≠0Kij:μij-μ(I+1)j≠0, i=1,…,I;j=1,…,p.Problem   2: Test Hi:μi-μ(I+1)=0Hi:μi-μ(I+1)=0 vs Ki:μi-μ(I+1)≠0Ki:μi-μ(I+1)≠0.For each problem an identity in Anderson [1984. An Introduction to Multivariate Statistical Analysis, second ed., Wiley, New York] is used to derive families of likelihood ratio statistics that can be used to determine step-down multiple testing procedures with desirable properties.