structural:reformatting
Multivariate testing

A generalization of Student's t statistic, called Hotelling's t -squared 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 positively 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. Fisher's Method for combining multiple tests with alpha reduced for positive correlation among tests is one. Another is Hotelling's T statistic follows a T distribution. However, in practice the distribution is rarely used, since tabulated values for T are hard to find. Usually, T is converted instead to an F statistic.

For a one-sample multivariate test, the hypothesis is that the mean vector (μ) is equal to a given vector (μ). The test statistic is Hotelling's t:

where n is the sample size, x is the vector of column means and S is an m × m sample covariance matrix.

For a two-sample multivariate test, the hypothesis is that the mean vectors (μ, μ) of two samples are equal. The test statistic is Hotelling's two-sample t: