In statistics, particularly in hypothesis testing, the Hotelling's T-squared distribution (T2), proposed by Harold Hotelling,[1] is a multivariate probability distribution that is tightly related to the F-distribution and is most notable for arising as the distribution of a set of sample statistics that are natural generalizations of the statistics underlying the Student's t-distribution.
The Hotelling's t-squared statistic (t2) is a generalization of Student's t-statistic that is used in multivariate hypothesis testing.
[2] 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.
is Gaussian multivariate-distributed with zero mean and unit covariance matrix
random matrix with a Wishart distribution
with unit scale matrix and m degrees of freedom, and d and M are independent of each other, then the quadratic form
):[3] It can be shown that if a random variable X has Hotelling's T-squared distribution,
be the sample covariance: where we denote transpose by an apostrophe.
follows a p-variate Wishart distribution with n − 1 degrees of freedom.
[4] The sample covariance matrix of the mean reads
[5] The Hotelling's t-squared statistic is then defined as:[6] which is proportional to the Mahalanobis distance between the sample mean and
Because of this, one should expect the statistic to assume low values if
is the F-distribution with parameters p and n − p. In order to calculate a p-value (unrelated to p variable here), 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.
A confidence region may also be determined using similar logic.
denote a p-variate normal distribution with location
Let be n independent identically distributed (iid) random variables, which may be represented as
is the chi-squared distribution with p degrees of freedom.
, and if it is nonsingular, then its inverse has a positive-definite square root
independent standard normal random variables.
and derive the characteristic function of the random variable
By definition of characteristic function, we have:[8] There are two exponentials inside the integral, so by multiplying the exponentials we add the exponents together, obtaining: Now take the term
, bringing one of them inside the integral: But the term inside the integral is precisely the probability density function of a multivariate normal distribution with covariance matrix
Finally, calculating the determinant, we obtain: which is the characteristic function for a chi-square distribution with
, 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.
Finally, the Hotelling's two-sample t-squared statistic is It can be related to the F-distribution by[4] 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,
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.