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

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

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

:$\{mathbf\; W\}=sum\_\{i=1\}^n\; (mathbf\{x\}\_i-overline\{mathbf\; x\})(mathbf\{x\}\_i-overline\{mathbf\; x\})\text{'}/(n-1)$

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

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

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:

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

Hotelling's two-sample T-square statistic

If and , with the samples independently drawn from two independent multivariate normal distributions with the same mean and covariance, and we define

:$overline\{mathbf\; x\}=frac\{1\}\{n\_x\}sum\_\{i=1\}^\{n\_x\}\; mathbf\{x\}\_i\; qquad\; overline\{mathbf\; y\}=frac\{1\}\{n\_y\}sum\_\{i=1\}^\{n\_y\}\; mathbf\{y\}\_i$as the sample means, and:$\{mathbf\; W\}=\; frac\{sum\_\{i=1\}^\{n\_x\}(mathbf\{x\}\_i-overline\{mathbf\; x\})(mathbf\{x\}\_i-overline\{mathbf\; x\})\text{'}+sum\_\{i=1\}^\{n\_y\}(mathbf\{y\}\_i-overline\{mathbf\; y\})(mathbf\{y\}\_i-overline\{mathbf\; y\})\text{'}\}\{n\_x+n\_y-2\}$as the unbiased pooled covariance matrix estimate, then Hotelling's two-sample T-square statistic is

:$t^2\; =\; frac\{n\_x\; n\_y\}\{n\_x+n\_y\}(overline\{mathbf\; x\}-overline\{mathbf\; y\})\text{'}\{mathbf\; W\}^\{-1\}(overline\{mathbf\; x\}-overline\{mathbf\; y\})sim\; T^2(p,\; n\_x+n\_y-2)$

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 :where 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