= Hotelling's t-squared Test =

'''Hotelling's t-squared test''' is a multivariate [[Statistics/StudentsTTest|t test]].

<<TableOfContents>>

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== Description ==

Hotelling's t-squared test follows from his [[Analysis/HotellingsTSquaredDistribution|T-squared distribution]].

Note that the distribution is notated as ''T^2^'', while the derived [[Statistics/TestStatistic|test statistics]] are notated as ''t^2^''.

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== Usage ==



=== Two Sample Test ===

To test the null hypothesis that two samples' multivariate means are equal, use the t-squared test.

Given:
 * samples of size ''n,,1,,'' and ''n,,2,,''
 * ''p'' variables
 * sample means for all ''p'' variables as ''X̅,,1,,'' and ''X̅,,2,,''
 * a [[Statistics/PooledVariance|pooled covariance matrix]] as ''Σ''.

Then the [[Statistics/TestStatistic|test statistic]] is calculated as:

{{attachment:tsquared.svg}}

Note similarities to the [[Statistics/MahalanobisDistance|Mahalanobis distance]].

The calculated test statistic is then compared to the critical level of the [[Analysis/FDistribution|F distribution]] with ''p'' numerator degrees of freedom and ''n,,1,,+n,,2,,-p-1'' denominator degrees of freedom. The null hypothesis is rejected if the test statistic is greater.



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CategoryRicottone