In statistics, particularly in hypothesis testing, the Hotelling's T-squared distribution (T2), proposed by Harold Hotelling, 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.[1]
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.[2]
If the vector
d
N(0p,Ip,)
M
p x p
W(Ip,,m)
X
p
m
X=mdTM-1d\simT2(p,m).
It can be shown that if a random variable X has Hotelling's T-squared distribution,
X\sim
2 | |
T | |
p,m |
m-p+1 | |
pm |
X\simFp,m-p+1
Fp,m-p+1
Let
\hat{\Sigma}
\hat{\Sigma}=
1 | |
n-1 |
n | |
\sum | |
i=1 |
(xi-\overline{x
where we denote transpose by an apostrophe. It can be shown that
\hat{\Sigma}
(n-1)\hat{\Sigma}
\hat{\Sigma}\overline{x}=\hat{\Sigma}/n
The Hotelling's t-squared statistic is then defined as:[6]
-1 | |
t | |
\overline{x} |
(\overline{x}-\boldsymbol{\mu})=n(\overline{x}-\boldsymbol{\mu})'\hat{\Sigma}-1(\overline{x}-\boldsymbol{\mu}),
which is proportional to the Mahalanobis distance between the sample mean and
\boldsymbol{\mu}
\overline{x} ≈ \boldsymbol{\mu}
From the distribution,
t2\sim
2 | ||
T | = | |
p,n-1 |
p(n-1) | |
n-p |
Fp,n-p,
where
Fp,n-p
In order to calculate a p-value (unrelated to p variable here), note that the distribution of
t2
n-p | |
p(n-1) |
t2\simFp,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.
Let
l{N}p(\boldsymbol{\mu},{\Sigma})
\boldsymbol{\mu}
{\Sigma}
{x}1,...,{x}n\siml{N}p(\boldsymbol{\mu},{\Sigma})
be n independent identically distributed (iid) random variables, which may be represented as
p x 1
\overline{
|
to be the sample mean with covariance
{\Sigma}\overline{x}={\Sigma}/n
-1 | |
(\overline{x}-\boldsymbol{\mu})'{\Sigma} | |
\overline{x} |
2 | |
(\overline{x}-\boldsymbol{\mu})\sim\chi | |
p |
,
where
2 | |
\chi | |
p |
Alternatively, one can argue using density functions and characteristic functions, as follows.
If
{x}1,...,{x}
nx |
\simNp(\boldsymbol{\mu},{\Sigma})
{y}1,...,{y}
ny |
\simNp(\boldsymbol{\mu},{\Sigma})
nx | ||||
\overline{
| ||||
i=1 |
xi
ny | ||||
\overline{
| ||||
i=1 |
yi
as the sample means, and
\hat{\Sigma}x=
1 | |
nx-1 |
nx | |
\sum | |
i=1 |
(xi-\overline{x})(xi-\overline{x})'
\hat{\Sigma}y=
1 | |
ny-1 |
ny | |
\sum | |
i=1 |
(yi-\overline{y})(yi-\overline{y})'
as the respective sample covariance matrices. Then
\hat{\Sigma}= | (nx-1)\hat{\Sigma |
x |
+(ny-1)\hat{\Sigma}y
is the unbiased pooled covariance matrix estimate (an extension of pooled variance).
Finally, the Hotelling's two-sample t-squared statistic is
t2=
nxny | |
nx+ny |
(\overline{x}-\overline{y})'\hat{\Sigma}-1(\overline{x}-\overline{y}) \simT2(p,nx+ny-2)
It can be related to the F-distribution by[4]
nx+ny-p-1 | |
(nx+ny-2)p |
t2\simF(p,nx+ny-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)
nx+ny-p-1 | |
(nx+ny-2)p |
t2\simF(p,nx+ny-1-p;\delta),
\delta=
nxny | |
nx+ny |
\boldsymbol{d}'\Sigma-1\boldsymbol{d},
\boldsymbol{d}=\overline{x-\overline{y}}
In the two-variable case, the formula simplifies nicely allowing appreciation of how the correlation,
\rho
t2
d1=\overline{x}1-\overline{y}1, d2=\overline{x}2-\overline{y}2
s1=\sqrt{\Sigma11
t2=
nxny | ||||||||||||
|
\left[\left(
d1 | |
s1 |
\right)2+\left(
d2 | |
s2 |
\right)2-2\rho\left(
d1 | |
s1 |
\right)\left(
d2 | |
s2 |
\right)\right]
d=\overline{x}-\overline{y}
t2
\rho
t2
\rho
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]