Hotelling's T-square distribution

In statistics, Hotelling's T-square statistic, [ H. Hotelling (1931) "The generalization of Student's ratio", Ann. Math. Statist., Vol. 2, pp 360–378.] named for Harold Hotelling,is a generalization of Student's t statistic that is used in multivariate hypothesis testing.

Hotelling's T-square statistic is defined as

:$$t^2=n({mathbf x}-{mathbfmu})'{mathbf W}^{-1}({mathbf x}-{mathbfmu})

where "n" is a number of points (see below) $${mathbf x} is a column vector of $$p elements and $${mathbf W} is a $$p imes p sample covariance matrix.

If $$xsim N_p(mu,{mathbf V}) is a random variable with a multivariate Gaussian distribution and $${mathbf W}sim W_p(m,{mathbf V}) (independent of "x") has a Wishart distribution with the same non-singular variance matrix $$mathbf V and with $$m=n-1,then the distributionof $$t^2 is $$T^2(p,m), Hotelling's T-square distribution with parameters "p" and "m".It can be shown that

:$$frac{m-p+1}{pm}T^2sim F_{p,m-p+1}where $$F is the F-distribution.

Now suppose that

:$${mathbf x}_1,dots,{mathbf x}_n

are "p"×1 column vectors whose entries are real numbers. Let

:$$overline{mathbf x}=(mathbf{x}_1+cdots+mathbf{x}_n)/n

be their mean. Let the "p"×"p" positive-definite matrix

:$${mathbf W}=sum_{i=1}^n (mathbf{x}_i-overline{mathbf x})(mathbf{x}_i-overline{mathbf x})'/(n-1)

be their "sample variance" matrix. (The transpose of any matrix "M" is denoted above by "M"′). Let μ be some known "p"×1 column vector (in applications a hypothesized value of a population mean). Then Hotelling's T-square statistic is

:$$t^2=n(overline{mathbf x}-{mathbfmu})'{mathbf W}^{-1}(overline{mathbf x}-{mathbfmu}).

Note that $$t^2 is closely related to the squared Mahalanobis distance.

In particular, it can be shownK.V. Mardia, J.T. Kent, and J.M. Bibby (1979) "Multivariate Analysis", Academic Press.] that if $\{$mathbf x}_1,dots,{mathbf x}_nsim N_p(mu,{mathbf V}), are independent, and $$overline{mathbf x} and $\{$mathbf W} are as defined above then $\{$mathbf W} has a Wishart distribution with "n" − 1 degrees of freedom

:$$mathbf{W} sim W_p(V,n-1)

and is independent of $$overline{mathbf x}, and

:$$overline{mathbf x}sim N_p(mu,V/n).

This implies that:

:$$t^2 = n(overline{mathbf x}-{mathbfmu})'{mathbf W}^{-1}(overline{mathbf x}-{mathbfmu}) sim T^2(p, n-1).

Hotelling's two-sample T-square statistic

If $${mathbf x}_1,dots,{mathbf x}_{n_x}sim N_p(oldsymbol{mu},{mathbf V}) and $${mathbf y}_1,dots,{mathbf y}_{n_y}sim N_p(oldsymbol{mu}_Y,{mathbf V}), with the samples independently drawn from two independent multivariate normal distributions with the same mean and covariance, and we define

:$$overline{mathbf x}=frac{1}{n_x}sum_{i=1}^{n_x} mathbf{x}_i qquad overline{mathbf y}=frac{1}{n_y}sum_{i=1}^{n_y} mathbf{y}_ias the sample means, and:$${mathbf W}= frac{sum_{i=1}^{n_x}(mathbf{x}_i-overline{mathbf x})(mathbf{x}_i-overline{mathbf x})'+sum_{i=1}^{n_y}(mathbf{y}_i-overline{mathbf y})(mathbf{y}_i-overline{mathbf y})'}{n_x+n_y-2}as the unbiased pooled covariance matrix estimate, then Hotelling's two-sample T-square statistic is

:$$t^2 = frac{n_x n_y}{n_x+n_y}(overline{mathbf x}-overline{mathbf y})'{mathbf W}^{-1}(overline{mathbf x}-overline{mathbf y})sim T^2(p, n_x+n_y-2)

and it can be related to the F-distribution by

:$$frac{n_x+n_y-p-1}{(n_x+n_y-2)p}t^2 sim F(p,n_x+n_y-1-p).

The non-null distribution of this statistic is the noncentral F-distribution (the ratio of a non-central Chi-square random variable and an independent central Chi-square random variable) :$$frac{n_x+n_y-p-1}{(n_x+n_y-2)p}t^2 sim F(p,n_x+n_y-1-p;delta),with :$$delta = frac{n_x n_y}{n_x+n_y}oldsymbol{ u}'mathbf{V}^{-1}oldsymbol{ u},where $$oldsymbol{ u} is the difference vector between the population means.

ee also

* Student's t-distribution (the univariate equivalent)
* F-distribution (commonly tabulated or available in software libraries, and hence used for testing the T-square statistic using the relationship given above)
* Wilks' lambda distribution (in multivariate statistics Wilks' $$Lambda is to Hotelling's $$T^2 as Snedecor's $$F is to Student's $$t in univariate statistics).

References