Hotelling's T-squared distribution

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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]

Hotelling's T2 distribution
Probability density function
Cumulative distribution function
Parameters p - dimension of the random variables
m - related to the sample size
Support if
otherwise.

Motivation

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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

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If the vector   is Gaussian multivariate-distributed with zero mean and unit covariance matrix   and   is a   random matrix with a Wishart distribution   with unit scale matrix and m degrees of freedom, and d and M are independent of each other, then the quadratic form   has a Hotelling distribution (with parameters   and  ):[3]

 

It can be shown that 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.

Hotelling t-squared statistic

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Let   be 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.[4] The sample covariance matrix of the mean reads  .[5]

The Hotelling's t-squared statistic is then defined as:[6]

 

which is proportional to the Mahalanobis distance between the sample mean and  . Because of this, one should expect the statistic to assume low values if  , and high values if they are different.

From the distribution,

 

where   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   equivalently implies that

 

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

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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.[7]

Proof
Proof

Every positive-semidefinite symmetric matrix   has a positive-semidefinite symmetric square root  , and if it is nonsingular, then its inverse has a positive-definite square root  .

Since  , we have   Consequently   and this is simply the sum of squares of   independent standard normal random variables. Thus its distribution is  

Alternatively, one can argue using density functions and characteristic functions, as follows.

Proof

To show this use the fact that   and derive the characteristic function of the random variable  . As usual, let   denote the determinant of the argument, as in  .

By definition of characteristic function, we have:[8]

 

There are two exponentials inside the integral, so by multiplying the exponentials we add the exponents together, obtaining:

 

Now take the term   off the integral, and multiply everything by an identity  , bringing one of them inside the integral:

 

But the term inside the integral is precisely the probability density function of a multivariate normal distribution with covariance matrix   and mean  , so when integrating over all  , it must yield   per the probability axioms.[clarification needed] We thus end up with:

 

where   is an identity matrix of dimension  . Finally, calculating the determinant, we obtain:

 

which is the characteristic function for a chi-square distribution with   degrees of freedom.  

Two-sample statistic

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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

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

 

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.[9][10]

See also

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References

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  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. ^ Johnson, R.A.; Wichern, D.W. (2002). Applied multivariate statistical analysis. Vol. 5. Prentice hall.
  3. ^ Eric W. Weisstein, MathWorld
  4. ^ a b Mardia, K. V.; Kent, J. T.; Bibby, J. M. (1979). Multivariate Analysis. Academic Press. ISBN 978-0-12-471250-8.
  5. ^ Fogelmark, Karl; Lomholt, Michael; Irbäck, Anders; Ambjörnsson, Tobias (3 May 2018). "Fitting a function to time-dependent ensemble averaged data". Scientific Reports. 8 (1): 6984. doi:10.1038/s41598-018-24983-y. PMC 5934400. Retrieved 19 August 2024.
  6. ^ "6.5.4.3. Hotelling's T squared".
  7. ^ End of chapter 4.2 of Johnson, R.A. & Wichern, D.W. (2002)
  8. ^ Billingsley, P. (1995). "26. Characteristic Functions". Probability and measure (3rd ed.). Wiley. ISBN 978-0-471-00710-4.
  9. ^ 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.
  10. ^ 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.
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