Editing Student's t-test (section)

===Multivariate testing===
{{main|Hotelling's T-squared distribution}}
A generalization of Student's ''t'' statistic, called [[Hotelling's t-squared statistic|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#Limitations of independent assumption|Fisher's Method]] for combining multiple tests with ''[[Type I and type II errors#Type I error|alpha]]'' reduced for positive correlation among tests is one. Another is Hotelling's ''T''{{isup|2}} statistic follows a ''T''{{isup|2}} distribution. However, in practice the distribution is rarely used, since tabulated values for ''T''{{isup|2}} are hard to find. Usually, ''T''{{isup|2}} is converted instead to an ''F'' statistic.

For a one-sample multivariate test, the hypothesis is that the mean vector ({{math|'''μ'''}}) is equal to a given vector ({{math|'''μ'''<sub>0</sub>}}). The test statistic is [[Hotelling's t-squared statistic|Hotelling's ''t''{{isup|2}}]]:

:<math>
t^2=n(\bar{\mathbf x}-{\boldsymbol\mu_0})'{\mathbf S}^{-1}(\bar{\mathbf x}-{\boldsymbol\mu_0})
</math>

where {{math|''n''}} is the sample size, {{math|{{overline|'''x'''}}}} is the vector of column means and {{math|'''S'''}} is an {{math|''m'' × ''m''}} [[sample covariance matrix]].

For a two-sample multivariate test, the hypothesis is that the mean vectors ({{math|'''μ'''<sub>1</sub>, '''μ'''<sub>2</sub>}}) of two samples are equal. The test statistic is [[Hotelling's two-sample t-squared statistic|Hotelling's two-sample ''t''{{isup|2}}]]:

:<math>t^2 = \frac{n_1 n_2}{n_1+n_2}\left(\bar{\mathbf x}_1-\bar{\mathbf x}_2\right)'{\mathbf S_\text{pooled}}^{-1}\left(\bar{\mathbf x}_1-\bar{\mathbf x}_2\right).</math>