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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.

The distribution

If the vector _pd₁ is Gaussian multivariate-distributed with zero mean and unit covariance matrix N(_p0₁,_pI_p) and _mM_p is a p x p matrix with a Wishart distribution with unit scale matrix and m degrees of freedom W(_pI_p,m) then m(₁d' _pM⁻¹_pd₁) has a Hotelling T² distribution with dimensionality parameter p and m degrees of freedom.

If the notation $T_{p,m}^2$ is used to denote a random variable having Hotelling's T-squared distribution with parameters p and m then, if a random variable X has Hotelling's T-squared distribution,

$X\sim T_{p,m}^2$

then

$\frac{m-p+1}{pm}X\sim F_{p,m-p+1}$

where $F_{p,m-p+1}$ is the F-distribution with parameters p and m−p+1.

Hotelling's T-squared statistic

Hotelling's T-squared statistic is a generalization of Student's t statistic that is used in multivariate hypothesis testing, and is defined as follows.

Let ${\mathcal{𝒩}}_p(\mathit{\mu },\mathrm{\Sigma })$ denote a p-variate normal distribution with location $\mathit{\mu }$ and covariance $\mathrm{\Sigma }$. Let

${\mathbf{x}}_1,\dots ,{\mathbf{x}}_n\sim {\mathcal{𝒩}}_p(\mathit{\mu },\mathrm{\Sigma })$

be n independent random variables, which may be represented as $p\times 1$ column vectors of real numbers. Define

$\bar{\mathbf{x}}=\frac{{\mathbf{x}}_1+\cdots +{\mathbf{x}}_n}{n}$

to be the sample mean. It can be shown that

$n(\bar{\mathbf{x}}-\mathit{\mu }{)}^{\prime }{\mathrm{\Sigma }}^{-1}(\bar{\mathbf{x}}-\mathit{\mu })\sim {\chi }_p^2,$

where ${\chi }_p^2$ is the chi-squared distribution with p degrees of freedom. To show this use the fact that $\bar{\mathbf{x}}\sim {\mathcal{𝒩}}_p(\mathit{\mu },\mathrm{\Sigma }/n)$ and then derive the characteristic function of the random variable $\mathbf{y}=n(\bar{\mathbf{x}}-\mathit{\mu }{)}^{\prime }{\mathrm{\Sigma }}^{-1}(\bar{\mathbf{x}}-\mathit{\mu })$. This is done below,

${\varphi }_{\mathbf{y}}(\theta )=\mathrm{E}{e}^{i\theta \mathbf{y}},$

$=\mathrm{E}{e}^{i\theta n(\bar{\mathbf{x}}-\mathit{\mu }{)}^{\prime }{\mathrm{\Sigma }}^{-1}(\bar{\mathbf{x}}-\mathit{\mu })}$

$=\int e^{i\theta n(\bar{\mathbf{x}}-\mathit{\mu }{)}^{\prime }{\mathrm{\Sigma }}^{-1}(\bar{\mathbf{x}}-\mathit{\mu })}(2\pi {)}^{-\frac{p}{2}}|\mathrm{\Sigma }/n{|}^{-\frac{1}{2}}{e}^{-\frac{1}{2}n(\bar{\mathbf{x}}-\mathit{\mu }{)}^{\prime }{\mathrm{\Sigma }}^{-1}(\bar{\mathbf{x}}-\mathit{\mu })}d{x}_1...d{x}_p$

$=\int (2\pi {)}^{-\frac{p}{2}}|\mathrm{\Sigma }/n{|}^{-\frac{1}{2}}{e}^{-\frac{1}{2}n(\bar{\mathbf{x}}-\mathit{\mu }{)}^{\prime }({\mathrm{\Sigma }}^{-1}-2i\theta {\mathrm{\Sigma }}^{-1})(\bar{\mathbf{x}}-\mathit{\mu })}d{x}_1...d{x}_p,$

$=|({\mathrm{\Sigma }}^{-1}-2i\theta {\mathrm{\Sigma }}^{-1}{)}^{-1}/n{|}^{\frac{1}{2}}|\mathrm{\Sigma }/n{|}^{-\frac{1}{2}}\int (2\pi {)}^{-\frac{p}{2}}|({\mathrm{\Sigma }}^{-1}-2i\theta {\mathrm{\Sigma }}^{-1}{)}^{-1}/n{|}^{-\frac{1}{2}}{e}^{-\frac{1}{2}n(\bar{\mathbf{x}}-\mathit{\mu }{)}^{\prime }({\mathrm{\Sigma }}^{-1}-2i\theta {\mathrm{\Sigma }}^{-1})(\bar{\mathbf{x}}-\mathit{\mu })}d{x}_1...d{x}_p,$

$=|({\mathbf{I}}_p-2i\theta {\mathbf{I}}_p){|}^{-\frac{1}{2}},$

$=(1-2i\theta {)}^{-\frac{p}{2}}.◼$

However, $\mathrm{\Sigma }$ is often unknown and we wish to do hypothesis testing on the location $\mathit{\mu }$.

Sum of p squared t's

Define

$\mathbf{W}=\frac{1}{n-1}\sum _{i=1}^n({\mathbf{x}}_i-\bar{\mathbf{x}})({\mathbf{x}}_i-\bar{\mathbf{x}}{)}^{\prime }$

to be the sample covariance. Here we denote transpose by an apostrophe. It can be shown that $\mathbf{W}$ is positive-definite and $(n-1)\mathbf{W}$ follows a p-variate Wishart distribution with n−1 degrees of freedom. Hotelling's T-squared statistic is then defined to be

$t^2=n(\bar{\mathbf{x}}-\mathit{\mu }{)}^{\prime }{\mathbf{W}}^{-1}(\bar{\mathbf{x}}-\mathit{\mu })$

and, also from above,

$t^2\sim T_{p,n-1}^2$

i.e.

$\frac{n-p}{p(n-1)}{t}^2\sim F_{p,n-p},$

where $F_{p,n-p}$ is the F-distribution with parameters p and n−p. In order to calculate a p value, multiply the t² statistic by the above constant and use the F-distribution.

Hotelling's two-sample T-squared statistic

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

$\bar{\mathbf{x}}=\frac{1}{n_x}\sum _{i=1}^{n_x}{\mathbf{x}}_i\bar{\mathbf{y}}=\frac{1}{n_y}\sum _{i=1}^{n_y}{\mathbf{y}}_i$

as the sample means, and

$\mathbf{W}=\frac{\sum _{i=1}^{n_x}({\mathbf{x}}_i-\bar{\mathbf{x}})({\mathbf{x}}_i-\bar{\mathbf{x}}{)}^{\prime }+\sum _{i=1}^{n_y}({\mathbf{y}}_i-\bar{\mathbf{y}})({\mathbf{y}}_i-\bar{\mathbf{y}}{)}^{\prime }}{n_x+n_y-2}$

as the unbiased pooled covariance matrix estimate, then Hotelling's two-sample T-squared statistic is

$t^2=\frac{n_x{n}_y}{n_x+n_y}(\bar{\mathbf{x}}-\bar{\mathbf{y}}{)}^{\prime }{\mathbf{W}}^{-1}(\bar{\mathbf{x}}-\bar{\mathbf{y}})\sim T^2(p,n_x+n_y-2)$

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-squared random variable and an independent central Chi-squared 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

$\delta =\frac{n_x{n}_y}{n_x+n_y}{\mathit{\nu }}^{\prime }{\mathbf{V}}^{-1}\mathit{\nu },$

where $\mathit{\nu }$ is the difference vector between the population means.

See also

• Student's t-test in univariate statistics
• Student's t-distribution in univariate probability theory
• Multivariate Student distribution.
• F-distribution (commonly tabulated or available in software libraries, and hence used for testing the T-squared statistic using the relationship given above)
• Wilks' lambda distribution (in multivariate statistics Wilks's Λ is to Hotelling's T² as Snedecor's F is to Student's t in univariate statistics).

References

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