In statistics, particularly in hypothesis testing, the Hotelling's T-squared distribution (T<sup>2</sup>), proposed by Harold Hotelling,
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.
Definition
If the vector <math>d</math> is Gaussian multivariate-distributed with zero mean and unit covariance matrix <math>N(\mathbf{0}_{p}, \mathbf{I}_{p, p})</math> and <math>M</math> is a <math>p \times p</math> random matrix with a Wishart distribution <math>W(\mathbf{I}_{p, p}, m)</math> with unit scale matrix and m degrees of freedom, and d and M are independent of each other, then the quadratic form <math>X</math> has a Hotelling distribution (with parameters <math>p</math> and <math>m</math>):
<math display="block">X = m d^T M^{-1} d \sim T^2(p, m).</math>
It can be shown<!-- But someone should put a proof of this fact somewhere in this article. --> that if a random variable X has Hotelling's T-squared distribution, <math>X \sim T^2_{p,m}</math>, then:
The sample covariance matrix of the mean reads <math>\hat{\mathbf \Sigma}_\overline{\mathbf x}=\hat{\mathbf \Sigma}/n</math>.
The Hotelling's t-squared statistic is then defined as:
<math display="block">
t^2=(\overline{\mathbf x}-\boldsymbol{\mu})'\hat{\mathbf \Sigma}_\overline{\mathbf x}^{-1} (\overline{\mathbf x}-\boldsymbol{\mathbf\mu})=n(\overline{\mathbf x}-\boldsymbol{\mu})'\hat{\mathbf \Sigma}^{-1} (\overline{\mathbf x}-\boldsymbol{\mathbf\mu}),
</math>
which is proportional to the Mahalanobis distance between the sample mean and <math>\boldsymbol{\mu}</math>. Because of this, one should expect the statistic to assume low values if <math>\overline{\mathbf x} \approx \boldsymbol{\mu}</math>, and high values if they are different.
From the distribution,
<math display="block">t^2 \sim T^2_{p,n-1}=\frac{p(n-1)}{n-p} F_{p,n-p} ,</math>
where <math>F_{p,n-p}</math> 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 <math>t^2</math> equivalently implies that
<math display="block"> \frac{n-p} {p(n-1)} t^2 \sim F_{p,n-p} .</math>
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 <math>\mathcal{N}_p(\boldsymbol{\mu},{\mathbf \Sigma})</math> denote a p-variate normal distribution with location <math>\boldsymbol{\mu}</math> and known covariance <math>{\mathbf \Sigma}</math>. Let
<math display="block">{\mathbf x}_1,\dots,{\mathbf x}_n\sim \mathcal{N}_p(\boldsymbol{\mu},{\mathbf \Sigma})</math>
be n independent identically distributed (iid) random variables, which may be represented as <math>p\times1</math> column vectors of real numbers. Define
<math display="block">\overline{\mathbf x}=\frac{\mathbf{x}_1+\cdots+\mathbf{x}_n}{n}</math>
to be the sample mean with covariance <math>{\mathbf \Sigma}_\overline{\mathbf x}={\mathbf \Sigma}/ n</math>. It can be shown that
<math display="block">(\overline{\mathbf x}-\boldsymbol{\mu})'{\mathbf \Sigma}_\overline{\mathbf x}^{-1}(\overline{\mathbf x}-\boldsymbol{\mathbf\mu})\sim\chi^2_p ,</math>
where <math>\chi^2_p</math> is the chi-squared distribution with p degrees of freedom.
Alternatively, one can argue using density functions and characteristic functions, as follows.
\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-2 i \theta \mathbf I_p|^{-1/2}
\end{align}
</math>
where <math>I_p</math> is an identity matrix of dimension <math>p</math>. Finally, calculating the determinant, we obtain:
<math display="block">
\begin{align}
& = (1-2 i \theta)^{-p/2}
\end{align}
</math>
which is the characteristic function for a chi-square distribution with <math>p</math> degrees of freedom. <math>\;\;\;\blacksquare</math>
Two-sample statistic
If <math>{\mathbf x}_1,\dots,{\mathbf x}_{n_x}\sim N_p(\boldsymbol{\mu},{\mathbf \Sigma})</math> and <math>{\mathbf y}_1,\dots,{\mathbf y}_{n_y}\sim N_p(\boldsymbol{\mu},{\mathbf \Sigma})</math>, with the samples independently drawn from two independent multivariate normal distributions with the same mean and covariance, and we define
<math display="block">\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</math>
as the sample means, and
<math display="block">\begin{align}
\hat{\mathbf \Sigma}_{\mathbf x} &= \frac{1}{n_x-1}\sum_{i=1}^{n_x} \left(\mathbf{x}_i-\overline{\mathbf x}\right) \left(\mathbf{x}_i-\overline{\mathbf x}\right)' \\
\hat{\mathbf \Sigma}_{\mathbf y}&= \frac{1}{n_y-1}\sum_{i=1}^{n_{y \left(\mathbf{y}_i-\overline{\mathbf y}\right) \left(\mathbf{y}_i-\overline{\mathbf y}\right)'
\end{align}</math>
as the respective sample covariance matrices. Then
<math display="block">\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}</math>
is the unbiased pooled covariance matrix estimate (an extension of pooled variance).
Finally, the Hotelling's two-sample t-squared statistic is
<math display="block">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)</math>
Related concepts
It can be related to the F-distribution by
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's lambda distribution (in multivariate statistics, Wilks's Λ is to Hotelling's T<sup>2</sup> as Snedecor's F is to Student's t in univariate statistics)