**Hotelling’s T-Squared** (Hotelling, 1931) is the multivariate counterpart of the T-test. “Multivariate” means that you have data for **more than one parameter for each sample**. For example, let’s say you wanted to compare how well two different sets of students performed in school. You could compare univariate data (e.g. mean test scores) with a t-test. Or, you could use Hotelling’s T-squared to compare multivariate data (e.g. the mutivariate mean of test scores, GPA, and class grades).

Hotelling’s T-Squared is based on Hotelling’s T^{2} distribution and forms the basis for various multivariate control charts.

## Test Versions

Two versions of the test exist with the following null hypotheses:

**One sample**: The multivariate vector means for a group equals a hypothetical vector of means.**Two sample**: The multivariate vector of means for two groups are equal.

For more than two samples, one option is to run a MANOVA.

## Two-sample Hotelling’s T-Squared

If you know how to run a two sample t-test, then you know how to run a two-sample Hotelling’s T-squared.** The basic steps are the same,** although you’ll use a different formula to calculate the t-squared value and you’ll use a different table (the F-table) to find the critical value.

Hotelling’s T-squared has **several advantages** over the t-test (Fang, 2017):

- The Type I error rate is well controlled,
- The relationship between multiple variables is taken into account,
- It can generate an overall conclusion even if multiple (single) t-tests are inconsistent. While a t-test will tell you which variable differ between groups, Hotelling’s summarizes the between-group differences.

The** test hypotheses** are:

**Null hypothesis**(H_{0}): the two samples are from populations with the same multivariate mean.**Alternate hypothesis**(H_{1}): the two samples are from populations with different multivariate means.

**Three major assumptions** are that the samples:

- …have underlying normal distributions.
- …are independent.
- …have equal variance-covariance matrices (for the two sample test only). Run Bartlett’s test to check this assumption.

^{2}is first transformed into an F-statistic:

**Where**:

- N
_{1}& N_{2}= sample sizes, - p = number of variables measured,
- N
_{1}+ N_{2}– p – 1 = degrees of freedom.

Reject the null hypothesis (at a chosen significance level) if the calculated value is greater than the F-table critical value. Rejecting the null hypothesis means that at least one of the parameters, or a combination of one or more parameters working together, is significantly different.

**References:**

Fang, J. (2017). Handbook of Medical Statistics.

Hotelling H (1931) The generalization of Student’s ratio. Ann Math Stat. 2(3):360–378.