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

${\mathbf {x} }_{1},\dots ,{\mathbf {x} }_{n}\sim {\mathcal {N}}_{p}({\boldsymbol {\mu }},{\mathbf {\Sigma } })$

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

${\overline {\mathbf {x} }}={\frac {\mathbf {x} _{1}+\cdots +\mathbf {x} _{n}}{n}}$

to be the sample mean. It can be shown that

$n({\overline {\mathbf {x} }}-{\boldsymbol {\mu }})'{\mathbf {\Sigma } }^{-1}({\overline {\mathbf {x} }}-{\boldsymbol {\mathbf {\mu } }})\sim \chi _{p}^{2},$

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

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

$=\operatorname {E} e^{i\theta n({\overline {\mathbf {x} }}-{\boldsymbol {\mu }})'{\mathbf {\Sigma } }^{-1}({\overline {\mathbf {x} }}-{\boldsymbol {\mathbf {\mu } }})}$

$=\int e^{i\theta n({\overline {\mathbf {x} }}-{\boldsymbol {\mu }})'{\mathbf {\Sigma } }^{-1}({\overline {\mathbf {x} }}-{\boldsymbol {\mathbf {\mu } }})}(2\pi )^{-{\frac {p}{2}}}|{\boldsymbol {\Sigma }}/n|^{-{\frac {1}{2}}}\,e^{-{\frac {1}{2}}n({\overline {\mathbf {x} }}-{\boldsymbol {\mu }})'{\boldsymbol {\Sigma }}^{-1}({\overline {\mathbf {x} }}-{\boldsymbol {\mu }})}\,dx_{1}...dx_{p}$

$=\int (2\pi )^{-{\frac {p}{2}}}|{\boldsymbol {\Sigma }}/n|^{-{\frac {1}{2}}}\,e^{-{\frac {1}{2}}n({\overline {\mathbf {x} }}-{\boldsymbol {\mu }})'({\boldsymbol {\Sigma }}^{-1}-2i\theta {\boldsymbol {\Sigma }}^{-1})({\overline {\mathbf {x} }}-{\boldsymbol {\mu }})}\,dx_{1}...dx_{p},$

$=|({\boldsymbol {\Sigma }}^{-1}-2i\theta {\boldsymbol {\Sigma }}^{-1})^{-1}/n|^{\frac {1}{2}}|{\boldsymbol {\Sigma }}/n|^{-{\frac {1}{2}}}\int (2\pi )^{-{\frac {p}{2}}}|({\boldsymbol {\Sigma }}^{-1}-2i\theta {\boldsymbol {\Sigma }}^{-1})^{-1}/n|^{-{\frac {1}{2}}}\,e^{-{\frac {1}{2}}n({\overline {\mathbf {x} }}-{\boldsymbol {\mu }})'({\boldsymbol {\Sigma }}^{-1}-2i\theta {\boldsymbol {\Sigma }}^{-1})({\overline {\mathbf {x} }}-{\boldsymbol {\mu }})}\,dx_{1}...dx_{p},$

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

$=(1-2i\theta )^{-{\frac {p}{2}}}.~~\blacksquare$

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

Sum of p squared t's

Define

${\mathbf {W} }={\frac {1}{n-1}}\sum _{i=1}^{n}(\mathbf {x} _{i}-{\overline {\mathbf {x} }})(\mathbf {x} _{i}-{\overline {\mathbf {x} }})'$

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

$t^{2}=n({\overline {\mathbf {x} }}-{\boldsymbol {\mu }})'{\mathbf {W} }^{-1}({\overline {\mathbf {x} }}-{\boldsymbol {\mathbf {\mu } }})$

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 ${\mathbf {x} }_{1},\dots ,{\mathbf {x} }_{n_{x}}\sim N_{p}({\boldsymbol {\mu }},{\mathbf {V} })$ and ${\mathbf {y} }_{1},\dots ,{\mathbf {y} }_{n_{y}}\sim N_{p}({\boldsymbol {\mu }},{\mathbf {V} })$, with the samples independently drawn from two independent multivariate normal distributions with the same mean and covariance, and we define

${\overline {\mathbf {x} }}={\frac {1}{n_{x}}}\sum _{i=1}^{n_{x}}\mathbf {x} _{i}\qquad {\overline {\mathbf {y} }}={\frac {1}{n_{y}}}\sum _{i=1}^{n_{y}}\mathbf {y} _{i}$

as the sample means, and

${\mathbf {W} }={\frac {\sum _{i=1}^{n_{x}}(\mathbf {x} _{i}-{\overline {\mathbf {x} }})(\mathbf {x} _{i}-{\overline {\mathbf {x} }})'+\sum _{i=1}^{n_{y}}(\mathbf {y} _{i}-{\overline {\mathbf {y} }})(\mathbf {y} _{i}-{\overline {\mathbf {y} }})'}{n_{x}+n_{y}-2}}$

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

$\delta ={\frac {n_{x}n_{y}}{n_{x}+n_{y}}}{\boldsymbol {\nu }}'\mathbf {V} ^{-1}{\boldsymbol {\nu }},$

where ${\boldsymbol {\nu }}$ 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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