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


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 variancecovariance 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 offdiagonal 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 \(\mu = \mu_0\) $$ T^2 \sim \frac{p(n1)}{np} F_{(p, \, np)} \, , $$ with \(F_{(p, \, np)}\) representing the F distribution with \(p\) degrees of freedom for the numerator and \(np\) 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 $$ \frac{p(n1)}{np} F_{\alpha \, (p, \, np)} $$ for a suitably chosen \(\alpha\).  
Result does not apply directly to multivariate Shewharttype charts  Although this result applies to hypothesis testing, it does not apply directly to multivariate Shewharttype 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.  
Threesigma limits from univariate control chart  When a univariate control chart is used for Phase I (analysis of historical data), and subsequently for Phase II (realtime process monitoring), the general form of the control limits is the same for each phase, although this need not be the case. Specifically, threesigma 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  Threesigma 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. 