6. Process or Product Monitoring and Control
6.5. Tutorials
6.5.4. Elements of Multivariate Analysis

## 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 $$\begin{eqnarray} t^2 & = & \frac{(\bar{x} - \mu_0)^2}{s^2 / n} \\ & & \\ & = & n (\bar{x} - \mu_0)(s^2)^{-1} (\bar{x} - \mu_0) \, . \end{eqnarray}$$
Generalize to $$p$$ variables When $$T^2$$ is generalized to $$p$$ variables it becomes $$T^2 = n (\bar{{\bf x}} - {\bf \mu}_0) {\bf S}^{-1} (\bar{{\bf x}} - {\bf \mu}_0) \, ,$$ with $$\bar{{\bf x}} = \left[ \begin{array}{c} \bar{x}_1 \\ \bar{x}_2 \\ \vdots \\ \bar{x}_p \end{array} \right] \,\,\,\,\,\,\,\,\,\,\,\, {\bf \mu}_0 = \left[ \begin{array}{c} \mu_1^0 \\ \mu_2^0 \\ \vdots \\ \mu_p^0 \end{array} \right] \, .$$ $${\bf S}^{-1}$$ is the inverse of the sample variance-covariance matrix, $${\bf S}$$, and $$n$$ is the sample size upon which each $$\bar{x}_i, \, i=1, \, 2, \, \ldots, \, p$$, 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 $$\mu = \mu_0$$ $$T^2 \sim \frac{p(n-1)}{n-p} F_{(p, \, n-p)} \, ,$$ with $$F_{(p, \, n-p)}$$ 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 $$\frac{p(n-1)}{n-p} F_{\alpha \, (p, \, n-p)}$$ for a suitably chosen $$\alpha$$.
Result does not apply directly to multivariate Shewhart-type charts Although this result applies to hypothesis testing, it does not apply directly to multivariate Shewhart-type charts (for which there is no $$\mu_0$$, although the result might be used as an approximation when a large sample is used and data are in subgroups, with the upper control limit (UCL) of a chart based on the approximation.
Three-sigma limits from univariate control chart When a univariate control chart is used for Phase I (analysis of historical data), and subsequently for Phase II (real-time process monitoring), the general form of the control limits is the same for each phase, although this need not be the case. Specifically, three-sigma limits are used in the univariate case, which skirts the relevant distribution theory for each Phase.
Selection of different control limit forms for each Phase Three-sigma units are generally not used with multivariate charts, however, which makes the selection of different control limit forms for each Phase (based on the relevant distribution theory), a natural choice.