Hotelling's T-squared distribution

Definition

If the vector ${\displaystyle d}$ is Gaussian multivariate-distributed with zero mean and unit covariance matrix ${\displaystyle N({\mathbf{0}}_p,{\mathbf{I}}_{p,p})}$ and ${\displaystyle M}$ is a ${\displaystyle p\times p}$ random matrix with a Wishart distribution ${\displaystyle W({\mathbf{I}}_{p,p},m)}$ with unit scale matrix and m degrees of freedom, and d and M are independent of each other, then the quadratic form ${\displaystyle X}$ has a Hotelling distribution (with parameters ${\displaystyle p}$ and ${\displaystyle m}$):

${\displaystyle X=m{d}^T{M}^{-1}d\sim T^2(p,m).}$

It can be shown that if a random variable X has Hotelling's T-squared distribution, ${\displaystyle X\sim T_{p,m}^2}$, then:

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

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

Hotelling t-squared statistic

Let ${\displaystyle \hat{\mathbf{\Sigma }}}$ be the sample covariance:

${\displaystyle \hat{\mathbf{\Sigma }}=\frac{1}{n-1}\sum _{i=1}^n({\mathbf{x}}_i-\overline{\mathbf{x}})({\mathbf{x}}_i-\overline{\mathbf{x}}{)}^{\prime }}$

where we denote transpose by an apostrophe. It can be shown that ${\displaystyle \hat{\mathbf{\Sigma }}}$ is a positive (semi) definite matrix and ${\displaystyle (n-1)\hat{\mathbf{\Sigma }}}$ follows a p-variate Wishart distribution with n − 1 degrees of freedom. The sample covariance matrix of the mean reads ${\displaystyle {\hat{\mathbf{\Sigma }}}_{\overline{\mathbf{x}}}=\hat{\mathbf{\Sigma }}/n}$.

The Hotelling's t-squared statistic is then defined as:

${\displaystyle t^2=(\overline{\mathbf{x}}-\mathit{\mu }{)}^{\prime }{\hat{\mathbf{\Sigma }}}_{\overline{\mathbf{x}}}^{-1}(\overline{\mathbf{x}}-\mathit{\mu })=n(\overline{\mathbf{x}}-\mathit{\mu }{)}^{\prime }{\hat{\mathbf{\Sigma }}}^{-1}(\overline{\mathbf{x}}-\mathit{\mu }),}$

which is proportional to the Mahalanobis distance between the sample mean and ${\displaystyle \mathit{\mu }}$. Because of this, one should expect the statistic to assume low values if ${\displaystyle \overline{\mathbf{x}}\approx \mathit{\mu }}$, and high values if they are different.

From the distribution,

${\displaystyle t^2\sim T_{p,n-1}^2=\frac{p(n-1)}{n-p}{F}_{p,n-p},}$

where ${\displaystyle F_{p,n-p}}$ 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 ${\displaystyle t^2}$ equivalently implies that

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

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.

Motivation

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

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

be n independent identically distributed (iid) random variables, which may be represented as ${\displaystyle p\times 1}$ column vectors of real numbers. Define

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

to be the sample mean with covariance ${\displaystyle {\mathbf{\Sigma }}_{\overline{\mathbf{x}}}=\mathbf{\Sigma }/n}$. It can be shown that

${\displaystyle (\overline{\mathbf{x}}-\mathit{\mu }{)}^{\prime }{\mathbf{\Sigma }}_{\overline{\mathbf{x}}}^{-1}(\overline{\mathbf{x}}-\mathit{\mu })\sim {\chi }_p^2,}$

where ${\displaystyle {\chi }_p^2}$ is the chi-squared distribution with p degrees of freedom.

Two-sample statistic

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

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

as the sample means, and

${\displaystyle {\hat{\mathbf{\Sigma }}}_{\mathbf{x}}=\frac{1}{n_x-1}\sum _{i=1}^{n_x}({\mathbf{x}}_i-\overline{\mathbf{x}})({\mathbf{x}}_i-\overline{\mathbf{x}}{)}^{\prime }}$

${\displaystyle {\hat{\mathbf{\Sigma }}}_{\mathbf{y}}=\frac{1}{n_y-1}\sum _{i=1}^{n_y}({\mathbf{y}}_i-\overline{\mathbf{y}})({\mathbf{y}}_i-\overline{\mathbf{y}}{)}^{\prime }}$

as the respective sample covariance matrices. Then

${\displaystyle \hat{\mathbf{\Sigma }}=\frac{(n_x-1){\hat{\mathbf{\Sigma }}}_{\mathbf{x}}+(n_y-1){\hat{\mathbf{\Sigma }}}_{\mathbf{y}}}{n_x+n_y-2}}$

is the unbiased pooled covariance matrix estimate (an extension of pooled variance).

Finally, the Hotelling's two-sample t-squared statistic is

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

Related concepts

It can be related to the F-distribution by

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

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)

${\displaystyle \frac{n_x+n_y-p-1}{(n_x+n_y-2)p}{t}^2\sim F(p,n_x+n_y-1-p;\delta ),}$

with

${\displaystyle \delta =\frac{n_x{n}_y}{n_x+n_y}{\mathit{d}}^{\prime }{\mathbf{\Sigma }}^{-1}\mathit{d},}$

where ${\displaystyle \mathit{d}=\overline{\mathbf{x}}\mathbf{-}\overline{\mathbf{y}}}$ is the difference vector between the population means.

In the two-variable case, the formula simplifies nicely allowing appreciation of how the correlation, ${\displaystyle \rho }$, between the variables affects ${\displaystyle t^2}$. If we define

${\displaystyle d_1={\overline{x}}_1-{\overline{y}}_1,d_2={\overline{x}}_2-{\overline{y}}_2}$

and

${\displaystyle s_1=\sqrt{{\mathrm{\Sigma }}_{11}}{s}_2=\sqrt{{\mathrm{\Sigma }}_{22}}\rho ={\mathrm{\Sigma }}_{12}/(s_1{s}_2)={\mathrm{\Sigma }}_{21}/(s_1{s}_2)}$

then

${\displaystyle t^2=\frac{n_x{n}_y}{(n_x+n_y)(1-{\rho }^2)}\left[{\left(\frac{d_1}{s_1}\right)}^2+{\left(\frac{d_2}{s_2}\right)}^2-2\rho \left(\frac{d_1}{s_1}\right)\left(\frac{d_2}{s_2}\right)\right]}$

Thus, if the differences in the two rows of the vector ${\displaystyle \mathbf{d}=\overline{\mathbf{x}}-\overline{\mathbf{y}}}$ are of the same sign, in general, ${\displaystyle t^2}$ becomes smaller as ${\displaystyle \rho }$ becomes more positive. If the differences are of opposite sign ${\displaystyle t^2}$ becomes larger as ${\displaystyle \rho }$ 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.