Multivariate Testing

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:

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