The Trouble With Hotelling's <i>T</i> ²

ABSTRACT The relationship between the Mahalanobis distance and the χ 2 and Hotelling T 2 distributions is described. The methods are illustrated by two case studies, a 40 × 2 simulation and a 54 × 34 dataset consisting of the NMR of metabolic extracts from maize harvested at 8.5°C. It is shown that if the number of samples is close to but still greater than the number of variables, p values calculated using T 2 as usually defined are higher than should be normally expected. This is interpreted as a problem caused by the difficulty of determining the number of degrees of freedom in multivariate matrices. It is recommended to use the χ 2 distribution in preference when calculating multivariate p values for probabilities that a sample belongs to a predefined distribution. When the sample size is large relative to the number of variables, there is little point in using T 2 , and when small, T 2 can predict misleading p values due to problems determining the number of degrees of freedom.