7.1.3  Hotelling’s TSquare
7.1.3  Hotelling’s TSquareA 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 tstatistic for testing a hypothesis regarding a univariate mean. Recall that under the null hypothesis t has a distribution with n1 degrees of freedom. Now consider squaring this test statistic as shown below:
\[t^2 = \frac{(\bar{x}\mu_0)^2}{s^2/n} = n(\bar{x}\mu_0)\left(\frac{1}{s^2}\right)(\bar{x}\mu_0) \sim F_{1, n1}\]
When you square a tdistributed random variable with n1 degrees of freedom, the result is an Fdistributed random variable with 1 and n1 degrees of freedom. We reject \(H_{0}\) at level \(α\) if \(t^2\) is greater than the critical value from the Ftable with 1 and n1 degrees of freedom, evaluated at level \(α\).
\(t^2 > F_{1, n1,\alpha}\)
 Hotelling's TSquare

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 variancecovariance 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 chisquare distributed with p degrees of freedom.
If we replace the sample variancecovariance matrix, S, by the population variancecovariance matrix, \(Σ\)
\(n\mathbf{(\overline{X}\mu_0)'\Sigma^{1}(\overline{X}\mu_0)},\)
then the resulting test is exactly chisquare distributed with p degrees of freedom when the data are normally distributed.
For small samples, the chisquare approximation for \(T^2\) does not take into account variation due to estimating \(Σ\) with the sample variancecovariance matrix S.
Better results can be obtained from the transformation of the Hotelling \(T^2\) statistic as below:
\[F = \frac{np}{p(n1)}T^2 \sim F_{p,np}\]
Under null hypothesis, \(H_{0}\colon \boldsymbol{\mu} = \boldsymbol{\mu_{0}}\), this will have a F distribution with p and np 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 Ftable with p and np degrees of freedom, evaluated at level \(α\).
\(F > F_{p, np, \alpha}\)
To illustrate the Hotelling's \(T^2\) test we will return to the USDA Women’s Health Survey data.