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Concept Version 6
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Multivariate Testing

Hotelling's $T$-square statistic allows for the testing of hypotheses on multiple (often correlated) measures within the same sample.

Learning Objective

  • Summarize Hotelling's $T$-squared statistics for one- and two-sample multivariate tests


Key Points

    • Hotelling's $T$-squared distribution is important because it arises as the distribution of a set of statistics which are natural generalizations of the statistics underlying Student's $t$-distribution.
    • In particular, 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.
    • For a one-sample multivariate test, the hypothesis is that the mean vector ($\mu$) is equal to a given vector (${ \mu }_{ 0 }$).
    • For a two-sample multivariate test, the hypothesis is that the mean vectors (${ \mu }_{ 1 }$ and ${ \mu }_{ 2 }$) of two samples are equal.

Terms

  • Hotelling's T-square statistic

    A generalization of Student's $t$-statistic that is used in multivariate hypothesis testing.

  • Type I error

    An error occurring when the null hypothesis ($H_0$) is true, but is rejected.


Full Text

A generalization of Student's $t$-statistic, called Hotelling's $T$-square statistic, allows for the testing of hypotheses on multiple (often correlated) measures within the same sample. For instance, a researcher might submit a number of subjects to a personality test consisting of multiple personality scales (e.g., the Minnesota Multiphasic Personality Inventory). Because measures of this type are usually highly correlated, it is not advisable to conduct separate univariate $t$-tests to test hypotheses, as these would neglect the covariance among measures and inflate the chance of falsely rejecting at least one hypothesis (type I error). In this case a single multivariate test is preferable for hypothesis testing. Hotelling's $T^2$ statistic follows a $T^2$ distribution.

Hotelling's $T$-squared distribution is important because it arises as the distribution of a set of statistics which are natural generalizations of the statistics underlying Student's $t$-distribution. In particular, 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. It is proportional to the $F$-distribution.

One-sample $T^2$ Test

For a one-sample multivariate test, the hypothesis is that the mean vector ($\mu$) is equal to a given vector (${ \mu }_{ 0 }$). The test statistic is defined as follows:

$T^2 = n (\bar{\mathbf{x}}-\mu_0)' \mathbf{S}^{-1} (\bar{\mathbf{x}}-\mu_0)$

where $n$ is the sample size, $\bar { x }$ is the vector of column means and $S$ is a $m \times m$ sample covariance matrix.

Two-Sample T2 Test

For a two-sample multivariate test, the hypothesis is that the mean vectors (${ \mu }_{ 1 },{ \mu }_{ 2 }$) of two samples are equal. The test statistic is defined as:

$T^2 = \dfrac{n_1n_2}{n_1 + n_2}(\bar{\mathbf{x}}_1 - \bar{\mathbf{x}}_2)' {\mathbf{S}_{\text{pooled}}}^{-1} (\bar{\mathbf{x}}_1 - \bar{\mathbf{x}}_2)$

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