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# Hotelling's T-squared statistic

Hotelling’s T-squared statistic can be used to test averages in multivariate data.

## Calculating Hotelling's T-squared statistic

Neither SAS nor Minitab have built-in functions for calculating Hotelling's T-squared statistic. However, it can be derived from various other parameters. If you have a hypothetical mean and the variance-covariance matrix , then it is possible to calculate,

This requires a few steps and realistically must be done using software. The equation reads as follows,

• Begin with subtracting expected mean values from observed mean values, to make the vector

• Take that vector, and make the transpose,

• Multiply that transpose with the variance-covariance matrix

• Multiply the resulting vector by , resulting in a single number

• Multiply that number with

### Calculating the F-statistic from the Hotelling's T-squared value

Calculating the F-statistic will allow us to test the following,

• Null hypothesis: . Our hypothetical mean may be equivalent to the actual mean, at a certain significance level.

• Hypothesis: . Our hypothetical mean is not equivalent to the actual mean, at a certain significance level.

Thankfully, it is easy to use the Hotelling's T-squared value to calculate the F-statistic used to test this hypothesis. There are two common use cases:

• In one case, we have two subpopulations and we set one one mean vector as and another mean vector as , and then test for their equality. For example, if we have patients' data from two hospitals, we may test their mean vectors for equality.

• In another case, we want to test the observations against a set of standards of recommendations. For example, we may have blood sample test data, and want to test the data's mean vector against ideal values blood test results to see if patients are generally healthy on those measurements.

We compare this to the F critical value we extract from statistical or spreadsheet software,

If our F-statistic is greater than the T-squared value, then we reject our null hypothesis.

## Minitab result for MANOVA

Below is the layout of a conventional Minitab table for MANOVA Tests for type,

Criterion

Test Statistic

F

DF Num

DF Denom

P

Wilks'

Lawley-Hotelling

Pillai's

Roy's

s = ___, m = ___, n = ___

When conducting a MANOVA test, then,

• The usual T-squared test statistic can be calculated from Minitab's output using the relationship T-squared = (N - 2)U

• N is the total number of observations in the data set and U is the Lawley-Hotelling trace.