6.5.4.3.

Hotelling's T squared

Hotelling's \(T^2\) distribution A multivariate method that is the multivariate counterpart of Student's \(t\) and which also forms the basis for certain multivariate control charts is based on Hotelling's \(T^2\) distribution, which was introduced by Hotelling (1947).

Univariate \(t\)-test for mean Recall, from Section 1.3.5.2, $$ t = \frac{\bar{x} - \mu}{s/\sqrt{n}} $$ has a \(t\) distribution provided that \(X\) is normally distributed, and can be used as long as \(X\) doesn't differ greatly from a normal distribution. If we wanted to test the hypothesis that \(\mu = \mu_0\), we would then have $$ t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} $$ so that $$ \begin{eqnarray} t^2 & = & \frac{(\bar{x} - \mu_0)^2}{s^2 / n} \\ & & \\ & = & n (\bar{x} - \mu_0)(s^2)^{-1} (\bar{x} - \mu_0) ,円 . \end{eqnarray} $$

Generalize to \(p\) variables When \(T^2\) is generalized to \(p\) variables it becomes $$ T^2 = n (\bar{{\bf x}} - {\bf \mu}_0) {\bf S}^{-1} (\bar{{\bf x}} - {\bf \mu}_0) ,円 , $$ with $$ \bar{{\bf x}} = \left[ \begin{array}{c} \bar{x}_1 \\ \bar{x}_2 \\ \vdots \\ \bar{x}_p \end{array} \right] ,円,円,円,円,円,円,円,円,円,円,円,円 {\bf \mu}_0 = \left[ \begin{array}{c} \mu_1^0 \\ \mu_2^0 \\ \vdots \\ \mu_p^0 \end{array} \right] ,円 . $$ \({\bf S}^{-1}\) is the inverse of the sample variance-covariance matrix, \({\bf S}\), and \(n\) is the sample size upon which each \(\bar{x}_i, ,円 i=1, ,円 2, ,円 \ldots, ,円 p\), is based. (The diagonal elements of \({\bf S}\) are the variances and the off-diagonal elements are the covariances for the \(p\) variables. This is discussed further in Section 6.5.4.3.1.)

Distribution of \(T^2\) It is well known that when \(\mu = \mu_0\) $$ T^2 \sim \frac{p(n-1)}{n-p} F_{(p, ,円 n-p)} ,円 , $$ with \(F_{(p, ,円 n-p)}\) representing the F distribution with \(p\) degrees of freedom for the numerator and \(n-p\) for the denominator. Thus, if \(\mu\) were specified to be \(\mu_0\), this could be tested by taking a single \(p\)-variate sample of size \(n\), then computing \(T^2\) and comparing it with $$ \frac{p(n-1)}{n-p} F_{\alpha ,円 (p, ,円 n-p)} $$ for a suitably chosen \(\alpha\).