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| Hotelling's t-squared test follows from his [[Statistics/HotellingsTSquaredDistribution|T-squared distribution]]. | Hotelling's t-squared test follows from his [[Analysis/HotellingsTSquaredDistribution|T-squared distribution]]. |
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| The calculated test statistic is then compared to the critical level of the [[Statistics/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. | 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. |
Hotelling's t-squared Test
Hotelling's t-squared test is a multivariate t test.
Description
Hotelling's t-squared test follows from his T-squared distribution.
Note that the distribution is notated as T2, while the derived test statistics are notated as t2.
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 n1 and n2
p variables
sample means for all p variables as X̅1 and X̅2
a pooled covariance matrix as Σ.
Then the test statistic is calculated as:
Note similarities to the Mahalanobis distance.
The calculated test statistic is then compared to the critical level of the F distribution with p numerator degrees of freedom and n1+n2-p-1 denominator degrees of freedom. The null hypothesis is rejected if the test statistic is greater.