On the Rejection of Hotelling's Single Sample T

Abstract

Multivariate hypotheses regarding the centroid of a single population can be tested using Hotelling's T ². Upon rejection of such an hypothesis, there seems to be no available technique by which the relative contribution of each variable to the rejection can be gauged. An examination of the derivation of the T ² statistic, however, shows that while T ² is the only nonzero eigenvalue of the matrix ND ^-1(X - μ₀)(X - μ₀)', the elements of its associated eigenvector define the weighting scheme used in combining the original variables for the purposes of the test. When adjusted to provide dimensionless numbers, these elements reflect the relative contribution of each variable to the rejection of the original hypothesis.