In statistics Hotelling's T-squared distribution is a univariate distribution proportional to the F-distribution and arises importantly as the distribution of a set of statistics which are natural generalizations of the statistics underlying Student's t-distribution. In particular, the distribution arises in multivariate statistics in undertaking tests of the differences between the (multivariate) means of different populations, where tests for univariate problems would make use of a t-test. The distribution is named for Harold Hotelling, who developed it as a generalization of Student's t-distribution.
Contents
Distribution
If the vector pd1 is Gaussian multivariate-distributed with zero mean and unit covariance matrix N(p01,pIp) and pMp is a p x p matrix with unit scale matrix and m degrees of freedom with a Wishart distribution W(pIp,m), then the Quadratic form m(1dT p M−1pd1) has a Hotelling T2(p,m) distribution with dimensionality parameter p and m degrees of freedom.
If a random variable X has Hotelling's T-squared distribution,
where
Statistic
Hotelling's t-squared statistic is a generalization of Student's t statistic that is used in multivariate hypothesis testing. The definition follows after it is motivated using a simpler problem.
Motivation
Let
be n independent random variables, which may be represented as
to be the sample mean with covariance
where
Definition
The covariance matrix
where we denote transpose by an apostrophe. It can be shown that
Hotelling's t-squared statistic is then defined as:
Also, from the #Distribution,
where
Two-sample statistic
If
as the sample means, and
as the respective sample covariance matrices. Then
is the unbiased pooled covariance matrix estimate (an extension of pooled variance).
Finally, the Hotelling's two-sample t-squared statistic is
Related concepts
It can be related to the F-distribution by
The non-null distribution of this statistic is the noncentral F-distribution (the ratio of a non-central Chi-squared random variable and an independent central Chi-squared random variable)
with
where
In the two-variable case, the formula simplifies nicely allowing appreciation of how the correlation,
and
then
Thus, if the differences in the two rows of the vector
A univariate special case can be found in Welch's t-test.
More robust and powerful tests than Hotelling's two-sample test have been proposed in the literature, see for example the interpoint distance based tests which can be applied also when the number of variables is comparable with, or even larger than, the number of subjects.