# 7.1.3 - Hotelling’s T-Square

7.1.3 - Hotelling’s T-Square

A more preferable test statistic is Hotelling’s $$T^2$$ and we will focus on this test.

To motivate Hotelling's $$T^2$$, consider the square of the t-statistic for testing a hypothesis regarding a univariate mean. Recall that under the null hypothesis t has a distribution with n-1 degrees of freedom. Now consider squaring this test statistic as shown below:

$t^2 = \frac{(\bar{x}-\mu_0)^2}{s^2/n} = n(\bar{x}-\mu_0)\left(\frac{1}{s^2}\right)(\bar{x}-\mu_0) \sim F_{1, n-1}$

When you square a t-distributed random variable with n-1 degrees of freedom, the result is an F-distributed random variable with 1 and n-1 degrees of freedom. We reject $$H_{0}$$ at level $$α$$ if $$t^2$$ is greater than the critical value from the F-table with 1 and n-1 degrees of freedom, evaluated at level $$α$$.

$$t^2 > F_{1, n-1,\alpha}$$

Hotelling's T-Square

Consider the last term in the above expression for $$t^2$$. In the expression for Hotelling's $$T^2$$, the difference between the sample mean and $$\mu_{0}$$ is replaced with the difference between the sample mean vector and the hypothesized mean vector $$\boldsymbol{\mu _{0}}$$. The inverse of the sample variance is replaced by the inverse of the sample variance-covariance matrix S, yielding the expression below:

$$T^2 = n\mathbf{(\overline{X}-\mu_0)'S^{-1}(\overline{X}-\mu_0)}$$

## Notes on $$\mathbf{T^2}$$

For large n, $$T^2$$ is approximately chi-square distributed with p degrees of freedom.

If we replace the sample variance-covariance matrix, S, by the population variance-covariance matrix, $$Σ$$

$$n\mathbf{(\overline{X}-\mu_0)'\Sigma^{-1}(\overline{X}-\mu_0)},$$

then the resulting test is exactly chi-square distributed with p degrees of freedom when the data are normally distributed.

For small samples, the chi-square approximation for $$T^2$$ does not take into account variation due to estimating $$Σ$$ with the sample variance-covariance matrix S.

Better results can be obtained from the transformation of the Hotelling $$T^2$$ statistic as below:

$F = \frac{n-p}{p(n-1)}T^2 \sim F_{p,n-p}$

Under null hypothesis, $$H_{0}\colon \boldsymbol{\mu} = \boldsymbol{\mu_{0}}$$, this will have a F distribution with p and n-p degrees of freedom. We reject the null hypothesis, $$H_{0}$$, at level $$α$$ if the test statistic F is greater than the critical value from the F-table with p and n-p degrees of freedom, evaluated at level $$α$$.

$$F > F_{p, n-p, \alpha}$$

To illustrate the Hotelling's $$T^2$$ test we will return to the USDA Women’s Health Survey data.

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