Hotelling's T-squared distribution


In statistics Hotelling's T-squared distribution is a multivariate 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.
Hotelling's t-squared statistic is a generalization of Student's t-statistic that is used in multivariate hypothesis testing.

Distribution

Motivation

The distribution arises in multivariate statistics in undertaking tests of the differences between the 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.

Definition

If the vector pd1 is Gaussian multivariate-distributed with zero mean and unit covariance matrix N and pMp is a p x p matrix with unit scale matrix and m degrees of freedom with a Wishart distribution W, then the Quadratic form m has a Hotelling T2 distribution with dimensionality parameter p and m degrees of freedom.
If a random variable X has Hotelling's T-squared distribution,, then:
where is the F-distribution with parameters p and m−p+1.

Statistic

The definition of this multivariate sample statistic follows after it is motivated using a simpler problem.

Motivation

Let denote a p-variate normal distribution with location and known covariance. Let
be n independent identically distributed random variables, which may be represented as column vectors of real numbers. Define
to be the sample mean with covariance. It can be shown that
where is the chi-squared distribution with p degrees of freedom.

Definition

The covariance matrix used above is often unknown.
Here we use instead the sample covariance:
where we denote transpose by an apostrophe. It can be shown that is a positive definite matrix and follows a p-variate Wishart distribution with n−1 degrees of freedom.
The sample covariance matrix of the mean reads.
The Hotelling's t-squared statistic is then defined as:
Also, from the distribution,
where is the F-distribution with parameters p and n − p.
In order to calculate a p-value, note that the distribution of equivalently implies that
Then, use the quantity on the left hand side to evaluate the p-value corresponding to the sample, which comes from the F-distribution.

Two-sample statistic

If and, with the samples independently drawn from two independent multivariate normal distributions with the same mean and covariance, and we define
as the sample means, and
as the respective sample covariance matrices. Then
is the unbiased pooled covariance matrix estimate.
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
with
where is the difference vector between the population means.
In the two-variable case, the formula simplifies nicely allowing appreciation of how the correlation,,
between the variables affects. If we define
and
then
Thus, if the differences in the two rows of the vector are of the same sign, in general, becomes smaller as becomes more positive. If the differences are of opposite sign becomes larger as becomes more positive.
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.