Hotelling's T-squared distribution

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^[1] 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.^[2]

If the notation $T^2_{p,m}$ 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^2_{p,m}

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'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.^[1]

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 $p\times1$ 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^2_p ,

where $\chi^2_p$ 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,(this needs a citation or justification for the 4th equality)

\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}-2 i \theta \boldsymbol\Sigma^{-1})(\overline{\mathbf x}-\boldsymbol\mu) }\,dx_{1}...dx_{p},

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

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

= (1-2 i \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 a positive (semi) definite matrix and $(n-1)\mathbf W$ follows a p-variate Wishart distribution with n−1 degrees of freedom.^[3] Hotelling's T-squared statistic is then defined^[4] 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^2_{p,n-1}

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^[3]

\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.

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.^[5]^[6]

In the two variable case, the formula simplifies nicely allowing appreciation of how the correlation, $r$ , 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

SD_1 = \sqrt{W_{11}} \qquad SD_2 = \sqrt{W_{22}}

then

t^2 = \frac{n_x n_y}{(n_x+n_y)(1-r^2)} \left [ \left ( \frac{d_1}{SD_1} \right )^2+\left ( \frac{d_2}{SD_2} \right )^2-2r \left ( \frac{d_1}{SD_1} \right )\left ( \frac{d_2}{SD_2} \right ) \right ]

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

References

1 2 3 Hotelling, H. (1931). "The generalization of Student's ratio". Annals of Mathematical Statistics 2 (3): 360–378. doi:10.1214/aoms/1177732979.
↑ Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics, Second Edition, Chapman & Hall/CRC, 2003, p. 1408
1 2 Mardia, K. V.; Kent, J. T.; Bibby, J. M. (1979). Multivariate Analysis. Academic Press. ISBN 0-12-471250-9.
↑ http://www.itl.nist.gov/div898/handbook/pmc/section5/pmc543.htm
↑ Marozzi, M. (2014). "Multivariate tests based on interpoint distances with application to magnetic resonance imaging". Statistical Methods in Medical Research. doi:10.1177/0962280214529104.
↑ Marozzi, M. (2015). "Multivariate multidistance tests for high-dimensional low sample size case-control studies". Statistics in Medicine 34. doi:10.1002/sim.6418.

External links

Prokhorov, A.V. (2001), "Hotelling T²-distribution", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

Some common univariate probability distributions

Continuous	beta Cauchy chi-squared exponential F gamma Laplace log-normal normal Pareto Student's t uniform Weibull

Discrete	Bernoulli binomial discrete uniform geometric hypergeometric negative binomial Poisson

List of probability distributions

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