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Hotelling's T-squared distribution

In statistics Hotelling's T-squared distribution (T2) is a multivariate 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. Hotelling's t-squared statistic (t2) is a generalization of Student's t-statistic that is used in multivariate hypothesis testing.[1]

Contents

DistributionEdit

MotivationEdit

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]

DefinitionEdit

If the vector pd1 is Gaussian multivariate-distributed with zero mean and unit covariance matrix N(p01,pIp) and pMp is a p x p matrix with unit scale matrix and m degrees of freedom with a Wishart distribution W(pIp,m), then the Quadratic form m(1dT p M−1pd1) has a Hotelling T2(p,m) distribution with dimensionality parameter p and m degrees of freedom.[2]

If a random variable X has Hotelling's T-squared distribution,  , then:[1]

 

where   is the F-distribution with parameters p and m−p+1.

StatisticEdit

The definition follows after it is motivated using a simpler problem.

MotivationEdit

Let   denote a p-variate normal distribution with location   and known covariance  . Let

 

be n independent identically distributed (iid) random variables, which may be represented as   column vectors of real numbers. Define

 

to be the sample mean with covariance  . It can be shown that

 

where   is the chi-squared distribution with p degrees of freedom.

DefinitionEdit

The covariance matrix   used above is often unknown. Here we use instead the sample covariance:

 

where we denote transpose by an apostrophe. It can be shown that   is a positive (semi) definite matrix and   follows a p-variate Wishart distribution with n−1 degrees of freedom.[3] The sample covariance matrix of the mean reads  .

Hotelling's t-squared statistic is then defined as:[4]

 

Also, from the distribution,

 

where   is the F-distribution with parameters p and n − p. In order to calculate a p-value (unrelated to the p variable here), divide the t2 statistic by the above fraction and use the F-distribution.

Two-sample statisticEdit

If   and  , with the samples independently drawn from two independent multivariate normal distributions with the same mean and covariance, and we define

 

as the sample means, and

 
 

as the respective sample covariance matrices. Then

 

is the unbiased pooled covariance matrix estimate (an extension of pooled variance).

Finally, the Hotelling's two-sample t-squared statistic is

 

Related conceptsEdit

It can be related to the F-distribution by[3]

 

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)

 

with

 

where   is the difference vector between the population means.

In the two-variable case, the formula simplifies nicely allowing appreciation of how the correlation,  , between the variables affects  . If we define

 

and

 

then

 

Thus, if the differences in the two rows of the vector   are of the same sign, in general,   becomes smaller as   becomes more positive. If the differences are of opposite sign   becomes larger as   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.[5][6]

See alsoEdit

ReferencesEdit

  1. ^ 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.
  2. ^ Eric W. Weisstein, MathWorld
  3. ^ a b Mardia, K. V.; Kent, J. T.; Bibby, J. M. (1979). Multivariate Analysis. Academic Press. ISBN 978-0-12-471250-8.
  4. ^ "6.5.4.3. Hotelling's T squared".
  5. ^ 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.
  6. ^ 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 linksEdit