6.
Process or Product Monitoring and Control
6.5. Tutorials 6.5.4. Elements of Multivariate Analysis 6.5.4.3. Hotelling's T squared


Estimate \(\mu\) with \(\bar{\bar{x}}\)  Since \(\mu\) is generally unknown, it is necessary to estimate \(\mu\) analogous to the way that \(\mu\) is estimated when an \(\bar{X}\) chart is used. Specifically, when there are rational subgroups, \(\mu\) is estimated by \(\bar{\bar{x}}\), with $$ \bar{{\bf x}} = \left[ \begin{array}{c} \bar{\bar{x}}_1 \\ \bar{\bar{x}}_2 \\ \vdots \\ \\ \bar{\bar{x}}_p \end{array} \right] \, . $$  
Obtaining the \(\bar{\bar{x}}_i\)  Each \(\bar{\bar{x}}_i, \, i=1, \, 2, \, \ldots, \, p\), is obtained the same way as with an \(\bar{X}\) chart, namely, by taking \(k\) subgroups of size \(n\) and computing $$ \bar{\bar{x}}_i = \frac{1}{k} \sum_{l=1}^k \bar{x}_{il} \, . $$ Here \(\bar{x}_{il}\) is used to denote the average for the \(l\)th subgroup of the \(i\)th variable. That is, $$ \bar{x}_{il} = \frac{1}{n}\sum_{r=1}^n x_{ilr} \, , $$ with \(x_{ilr}\) denoting the \(r\)th observation (out of \(n\)) for the \(i\)th variable in the \(l\)th subgroup.  
Estimating the variances and covariances  The variances and covariances are similarly averaged over the subgroups. Specifically, the \(s_{ij}\) elements of the variancecovariance matrix \({\bf S}\) are obtained as $$ s_{ij} = \frac{1}{k} \sum_{l=1}^k s_{ijl} \, , $$ with \(s_{ijl}\) for \(i \ne j\) denoting the sample covariance between variables \(X_i\) and \(X_j\) for the \(l\)th subgroup, and \(s_{ij}\) for \(i = j\) denotes the sample variance of \(X_i\). The variances \(s_{ijl}^2 (=s_{iil})\) for subgroup \(l\) and for variables \(i = 1, \, 2, \, \ldots, \, p\) are computed as $$ \frac{1}{n1} \sum_{r=1}^n (x_{ilr}  \bar{x}_{il})^2 \, . $$ Similarly, the covariances \(s_{ijl}\) between variables \(X_i\) and \(X_j\) for subgroup \(l\) are computed as $$ \frac{1}{n1} \sum_{r=1}^n (x_{ilr}  \bar{x}_{il})(x_{jlr}  \bar{x}_{jl}) \, . $$  
Compare \(T^2\) against control values  As with an \(\bar{X}\) chart (or any other chart), the \(k\) subgroups would be tested for control by computing \(k\) values of \(T^2\) and comparing each against the UCL. If any value falls above the UCL (there is no lower control limit), the corresponding subgroup would be investigated.  
Formula for plotted \(T^2\) values  Thus, one would plot $$ T_{j}^2 = n (\bar{{\bf x}}^{(j)}  \bar{\bar{{\bf x}}})' {\bf S}_p^{1} (\bar{{\bf x}}^{(j)}  \bar{\bar{{\bf x}}}) $$ for the \(j\)th subgroup (\(j = 1, \, 2, \, \ldots, \, k\)), with \(\bar{{\bf x}}\) denoting a vector with \(p\) elements that contains the subgroup averages for each of the \(p\) characteristics for the \(j\)th subgroup. (\({\bf S}_p^{1}\) is the inverse matrix of the "pooled" variancecovariance matrix, \({\bf S}_p\), which is obtained by averaging the subgroup variancecovariance matrices over the \(k\) subgroups.)  
Formula for the upper control limit  Each of the \(k\) values of \(T_j^2\) given in the equation above would be compared with $$ UCL = \left( \frac{knp  kp  np + p}{kn  k  p + 1} \right) F_{\alpha, \, (p, \, knkp+1)} \, . $$  
Lower control limits  A lower control limit is generally not used in multivariate control chart applications, although some control chart methods do utilize a LCL. Although a small value for \(T_j^2\) might seem desirable, a value that is very small would likely indicate a problem of some type as we would not expect every element of \(\bar{{\bf x}}^{(j)}\) to be virtually equal to every element in \(\bar{\bar{x}}\).  
Delete outofcontrol points once cause discovered and corrected  As with any Phase I control chart procedure, if there are any points that plot above the UCL and can be identified as corresponding to outofcontrol conditions that have been corrected, the point(s) should be deleted and the UCL recomputed. The remaining points would then be compared with the new UCL and the process continued as long as necessary, remembering that points should be deleted only if their correspondence with outofcontrol conditions can be identified and the cause(s) of the condition(s) were removed. 