Hotelling's T-squared distribution

Type of probability distribution

Hotelling's T² distribution
Probability density function
Cumulative distribution function
Parameters	p - dimension of the random variables m - related to the sample size
Support	$x\in (0,+\infty )\;$ if $p=1$ $x\in [0,+\infty )\;$ otherwise.

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]

Motivation

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]

Definition

If the vector $d$ is Gaussian multivariate-distributed with zero mean and unit covariance matrix $N(\mathbf {0} _{p},\mathbf {I} _{p,p})$ and $M$ is a $p\times p$ matrix with unit scale matrix and m degrees of freedom with a Wishart distribution $W(\mathbf {I} _{p,p},m)$ , then the quadratic form $X$ has a Hotelling distribution (with parameters $p$ and $m$ ):^[3]

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

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

{\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 t-squared statistic

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

{\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 ${\hat {\mathbf {\Sigma } }}$ is a positive (semi) definite matrix and $(n-1){\hat {\mathbf {\Sigma } }}$ follows a p-variate Wishart distribution with n − 1 degrees of freedom.^[4] The sample covariance matrix of the mean reads ${\hat {\mathbf {\Sigma } }}_{\overline {\mathbf {x} }}={\hat {\mathbf {\Sigma } }}/n$ .

The Hotelling's t-squared statistic is then defined as:^[5]

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 ${\boldsymbol {\mu }}$ . Because of this, one should expect the statistic to assume low values if ${\overline {\mathbf {x} }}\approx {\boldsymbol {\mu }}$ , and high values if they are different.

From the distribution,

t^{2}\sim T_{p,n-1}^{2}={\frac {p(n-1)}{n-p}}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 (unrelated to p variable here), note that the distribution of $t^{2}$ equivalently implies that

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

{\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 $p\times 1$ column vectors of real numbers. Define

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

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

({\overline {\mathbf {x} }}-{\boldsymbol {\mu }})'{\mathbf {\Sigma } }_{\overline {\mathbf {x} }}^{-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.^[6]

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

Proof

Proof

To show this use the fact that ${\overline {\mathbf {x} }}\sim {\mathcal {N}}_{p}({\boldsymbol {\mu }},{\mathbf {\Sigma } }/n)$ and derive the characteristic function of the random variable $\mathbf {y} =({\bar {\mathbf {x} }}-{\boldsymbol {\mu }})'{\mathbf {\Sigma } }_{\bar {\mathbf {x} }}^{-1}({\bar {\mathbf {x} }}-{\boldsymbol {\mathbf {\mu } }})=({\bar {\mathbf {x} }}-{\boldsymbol {\mu }})'({\mathbf {\Sigma } }/n)^{-1}({\bar {\mathbf {x} }}-{\boldsymbol {\mathbf {\mu } }})$ . As usual, let $|\cdot |$ denote the determinant of the argument, as in $|{\boldsymbol {\Sigma }}|$ .

By definition of characteristic function, we have:^[7]

{\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:

{\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 $|{\boldsymbol {\Sigma }}/n|^{-1/2}$ off the integral, and multiply everything by an identity $I=|({\boldsymbol {\Sigma }}^{-1}-2i\theta {\boldsymbol {\Sigma }}^{-1})^{-1}/n|^{1/2}\;\cdot \;|({\boldsymbol {\Sigma }}^{-1}-2i\theta {\boldsymbol {\Sigma }}^{-1})^{-1}/n|^{-1/2}$ , bringing one of them inside the integral:

{\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 $({\boldsymbol {\Sigma }}^{-1}-2i\theta {\boldsymbol {\Sigma }}^{-1})^{-1}/n=\left[n({\boldsymbol {\Sigma }}^{-1}-2i\theta {\boldsymbol {\Sigma }}^{-1})\right]^{-1}$ and mean $\mu$ , so when integrating over all $x_{1},\dots ,x_{p}$ , it must yield $1$ per the probability axioms.^{[clarification needed]} We thus end up with:

{\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 $I_{p}$ is an identity matrix of dimension $p$ . Finally, calculating the determinant, we obtain:

{\begin{aligned}&=(1-2i\theta )^{-p/2}\end{aligned}}

which is the characteristic function for a chi-square distribution with $p$ degrees of freedom. $\;\;\;\blacksquare$

Two-sample statistic

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

{\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

{\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} }})'

{\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

{\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

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)

Related concepts

It can be related to the F-distribution by^[4]

{\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 {d}}'\mathbf {\Sigma } ^{-1}{\boldsymbol {d}},

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

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

d_{1}={\overline {x}}_{1}-{\overline {y}}_{1},\qquad d_{2}={\overline {x}}_{2}-{\overline {y}}_{2}

and

s_{1}={\sqrt {\Sigma _{11}}}\qquad s_{2}={\sqrt {\Sigma _{22}}}\qquad \rho =\Sigma _{12}/(s_{1}s_{2})=\Sigma _{21}/(s_{1}s_{2})

then

t^{2}={\frac {n_{x}n_{y}}{(n_{x}+n_{y})(1-r^{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 $\mathbf {d} ={\overline {\mathbf {x} }}-{\overline {\mathbf {y} }}$ are of the same sign, in general, $t^{2}$ becomes smaller as $\rho$ becomes more positive. If the differences are of opposite sign $t^{2}$ becomes larger as $\rho$ 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.^[8]^[9]

References

^ ^a ^b ^c Hotelling, H. (1931). "The generalization of Student's ratio". Annals of Mathematical Statistics. 2 (3): 360–378. doi:10.1214/aoms/1177732979.
^ Johnson, R.A.; Wichern, D.W. (2002). Applied multivariate statistical analysis. Vol. 5. Prentice hall.
^ Eric W. Weisstein, MathWorld
^ ^a ^b Mardia, K. V.; Kent, J. T.; Bibby, J. M. (1979). Multivariate Analysis. Academic Press. ISBN 978-0-12-471250-8.
^ "6.5.4.3. Hotelling's T squared".
^ End of chapter 4.2 of Johnson, R.A. & Wichern, D.W. (2002)
^ Billingsley, P. (1995). "26. Characteristic Functions". Probability and measure (3rd ed.). Wiley. ISBN 978-0-471-00710-4.
^ Marozzi, M. (2016). "Multivariate tests based on interpoint distances with application to magnetic resonance imaging". Statistical Methods in Medical Research. 25 (6): 2593–2610. doi:10.1177/0962280214529104. PMID 24740998.
^ Marozzi, M. (2015). "Multivariate multidistance tests for high-dimensional low sample size case-control studies". Statistics in Medicine. 34 (9): 1511–1526. doi:10.1002/sim.6418. PMID 25630579.

External links

Prokhorov, A.V. (2001) [1994], T²-distribution "Hotelling T²-distribution", Encyclopedia of Mathematics, EMS Press

Probability distributions (list)

Discrete
univariate

with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric negative Poisson binomial Rademacher soliton discrete uniform Zipf Zipf–Mandelbrot
with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Flory–Schulz Gauss–Kuzmin geometric logarithmic mixed Poisson negative binomial Panjer parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous
univariate

supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular continuous Bernoulli Irwin–Hall Kumaraswamy logit-normal noncentral beta PERT raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi chi-squared noncentral inverse scaled Dagum Davis Erlang hyper exponential hyperexponential hypoexponential logarithmic F noncentral folded normal Fréchet gamma generalized inverse gamma/Gompertz Gompertz shifted half-logistic half-normal Hotelling's T-squared inverse Gaussian generalized Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal log-t Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami Pareto phase-type Poly-Weibull Rayleigh relativistic Breit–Wigner Rice truncated normal type-2 Gumbel Weibull discrete Wilks's lambda
supported on the whole real line	Cauchy exponential power Fisher's z Kaniadakis κ-Gaussian Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t Tracy–Widom variance-gamma Voigt
with support whose type varies	generalized chi-squared generalized extreme value generalized Pareto Marchenko–Pastur Kaniadakis κ-exponential Kaniadakis κ-Gamma Kaniadakis κ-Weibull Kaniadakis κ-Logistic Kaniadakis κ-Erlang q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda