In statistics, particularly in hypothesis testing, the Hotelling's T-squared distribution (T²), proposed by Harold Hotelling,^[1] is a multivariate probability distribution that is tightly related to the F-distribution and is most notable for arising as the distribution of a set of sample statistics that are natural generalizations of the statistics underlying the Student's t-distribution. The Hotelling's t-squared statistic (t²) is a generalization of Student's t-statistic that is used in multivariate hypothesis testing.^[2]

Quick Facts Parameters, Support ...

Hotelling's T² distribution

Probability density function
Cumulative distribution function
Parameters	p - dimension of the random variables m - related to the sample size
Support	${\displaystyle x\in (0,+\infty )\;}$ if ${\displaystyle p=1}$ ${\displaystyle x\in [0,+\infty )\;}$ otherwise.

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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.^[1]

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 ${\displaystyle p\times p}$ matrix with unit scale matrix and m degrees of freedom with a Wishart distribution ${\displaystyle W(\mathbf {I} _{p,p},m)}$, then the quadratic form ${\displaystyle X}$ has a Hotelling distribution (with parameters ${\displaystyle p}$ and ${\displaystyle m}$):^[3]

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

Furthermore, if a random variable X has Hotelling's T-squared distribution, ${\displaystyle X\sim T_{p,m}^{2}}$, then:^[1]

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

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

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

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

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.^[4] 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:^[5]

${\displaystyle t^{2}=({\overline {\mathbf {x} }}-{\boldsymbol {\mu }})'{\hat {\mathbf {\Sigma } }}_{\overline {\mathbf {x} }}^{-1}({\overline {\mathbf {x} }}-{\boldsymbol {\mathbf {\mu } }}),}$

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

Further information: Multivariate normal distribution § Interval

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

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

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

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

where ${\displaystyle \chi _{p}^{2}}$ is the chi-squared distribution with p degrees of freedom.^[6]

More information To show this use the fact that ...

Proof

Proof To show this use the fact that ${\displaystyle {\overline {\mathbf {x} }}\sim {\mathcal {N}}_{p}({\boldsymbol {\mu }},{\mathbf {\Sigma } }/n)}$ and derive the characteristic function of the random variable ${\displaystyle \mathbf {y} =({\bar {\mathbf {x} }}-{\boldsymbol {\mu }})'{\mathbf {\Sigma } }_{\bar {\mathbf {x} }}^{-1}({\bar {\mathbf {x} }}-{\boldsymbol {\mathbf {\mu } }})=({\bar {\mathbf {x} }}-{\boldsymbol {\mu }})'({\mathbf {\Sigma } }/n)^{-1}({\bar {\mathbf {x} }}-{\boldsymbol {\mathbf {\mu } }})}$. As usual, let ${\displaystyle |\cdot |}$ denote the determinant of the argument, as in ${\displaystyle |{\boldsymbol {\Sigma }}|}$. By definition of characteristic function, we have:[7] ${\displaystyle {\begin{aligned}\varphi _{\mathbf {y} }(\theta )&=\operatorname {E} e^{i\theta \mathbf {y} },\\[5pt]&=\operatorname {E} e^{i\theta ({\overline {\mathbf {x} }}-{\boldsymbol {\mu }})'({\mathbf {\Sigma } }/n)^{-1}({\overline {\mathbf {x} }}-{\boldsymbol {\mathbf {\mu } }})}\\[5pt]&=\int e^{i\theta ({\overline {\mathbf {x} }}-{\boldsymbol {\mu }})'n{\mathbf {\Sigma } }^{-1}({\overline {\mathbf {x} }}-{\boldsymbol {\mathbf {\mu } }})}(2\pi )^{-p/2}|{\boldsymbol {\Sigma }}/n|^{-1/2}\,e^{-(1/2)({\overline {\mathbf {x} }}-{\boldsymbol {\mu }})'n{\boldsymbol {\Sigma }}^{-1}({\overline {\mathbf {x} }}-{\boldsymbol {\mu }})}\,dx_{1}\cdots dx_{p}\end{aligned}}}$ There are two exponentials inside the integral, so by multiplying the exponentials we add the exponents together, obtaining: ${\displaystyle {\begin{aligned}&=\int (2\pi )^{-p/2}|{\boldsymbol {\Sigma }}/n|^{-1/2}\,e^{-(1/2)({\overline {\mathbf {x} }}-{\boldsymbol {\mu }})'n({\boldsymbol {\Sigma }}^{-1}-2i\theta {\boldsymbol {\Sigma }}^{-1})({\overline {\mathbf {x} }}-{\boldsymbol {\mu }})}\,dx_{1}\cdots dx_{p}\end{aligned}}}$ Now take the term ${\displaystyle |{\boldsymbol {\Sigma }}/n|^{-1/2}}$ off the integral, and multiply everything by an identity ${\displaystyle I=|({\boldsymbol {\Sigma }}^{-1}-2i\theta {\boldsymbol {\Sigma }}^{-1})^{-1}/n|^{1/2}\;\cdot \;|({\boldsymbol {\Sigma }}^{-1}-2i\theta {\boldsymbol {\Sigma }}^{-1})^{-1}/n|^{-1/2}}$, bringing one of them inside the integral: ${\displaystyle {\begin{aligned}&=|({\boldsymbol {\Sigma }}^{-1}-2i\theta {\boldsymbol {\Sigma }}^{-1})^{-1}/n|^{1/2}|{\boldsymbol {\Sigma }}/n|^{-1/2}\int (2\pi )^{-p/2}|({\boldsymbol {\Sigma }}^{-1}-2i\theta {\boldsymbol {\Sigma }}^{-1})^{-1}/n|^{-1/2}\,e^{-(1/2)n({\overline {\mathbf {x} }}-{\boldsymbol {\mu }})'({\boldsymbol {\Sigma }}^{-1}-2i\theta {\boldsymbol {\Sigma }}^{-1})({\overline {\mathbf {x} }}-{\boldsymbol {\mu }})}\,dx_{1}\cdots dx_{p}\end{aligned}}}$ But the term inside the integral is precisely the probability density function of a multivariate normal distribution with covariance matrix ${\displaystyle ({\boldsymbol {\Sigma }}^{-1}-2i\theta {\boldsymbol {\Sigma }}^{-1})^{-1}/n=\left[n({\boldsymbol {\Sigma }}^{-1}-2i\theta {\boldsymbol {\Sigma }}^{-1})\right]^{-1}}$ and mean ${\displaystyle \mu }$, so when integrating over all ${\displaystyle x_{1},\dots ,x_{p}}$, it must yield ${\displaystyle 1}$ per the probability axioms.[clarification needed] We thus end up with: ${\displaystyle {\begin{aligned}&=\left|({\boldsymbol {\Sigma }}^{-1}-2i\theta {\boldsymbol {\Sigma }}^{-1})^{-1}\cdot {\frac {1}{n}}\right|^{1/2}|{\boldsymbol {\Sigma }}/n|^{-1/2}\\&=\left|({\boldsymbol {\Sigma }}^{-1}-2i\theta {\boldsymbol {\Sigma }}^{-1})^{-1}\cdot {\frac {1}{\cancel {n}}}\cdot {\cancel {n}}\cdot {\boldsymbol {\Sigma }}^{-1}\right|^{1/2}\\&=\left|\left[({\cancel {{\boldsymbol {\Sigma }}^{-1}}}-2i\theta {\cancel {{\boldsymbol {\Sigma }}^{-1}}}){\cancel {\boldsymbol {\Sigma }}}\right]^{-1}\right|^{1/2}\\&=|\mathbf {I} _{p}-2i\theta \mathbf {I} _{p}|^{-1/2}\end{aligned}}}$ where ${\displaystyle I_{p}}$ is an identity matrix of dimension ${\displaystyle p}$. Finally, calculating the determinant, we obtain: ${\displaystyle {\begin{aligned}&=(1-2i\theta )^{-p/2}\end{aligned}}}$ which is the characteristic function for a chi-square distribution with ${\displaystyle p}$ degrees of freedom. ${\displaystyle \;\;\;\blacksquare }$

Close

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

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

as the sample means, and

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

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

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

${\displaystyle t^{2}={\frac {n_{x}n_{y}}{n_{x}+n_{y}}}({\overline {\mathbf {x} }}-{\overline {\mathbf {y} }})'{\hat {\mathbf {\Sigma } }}^{-1}({\overline {\mathbf {x} }}-{\overline {\mathbf {y} }})\sim T^{2}(p,n_{x}+n_{y}-2)}$

• 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's 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)