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

T2 Chart for Subgroup Averages -- Phase II

Phase II requires recomputing \({\bf S}_p\) and \(\bar{\bar{\bf x}}\) and different control limits Determining the UCL that is to be subsequently applied to future subgroups entails recomputing, if necessary, \({\bf S}_p\) and \(\bar{\bar{\bf x}}\), and using a constant and an \(F\) value that are different from the form given for the Phase I control limits. The form is different because different distribution theory is involved since future subgroups are assumed to be independent of the "current" set of subgroups that is used in calculating \({\bf S}_p\) and \(\bar{\bar{\bf x}}\). (The same thing happens with \(\bar{X}\) charts; the problem is simply ignored through the use of 3-sigma limits, although a different approach should be used when there is a small number of subgroups -- and the necessary theory has been worked out.)
Illustration To illustrate, assume that \(a\) a subgroups had been discarded (with possibly \(a = 0\)) so that \(k - a\) subgroups are used in obtaining \({\bf S}_p\) and \(\bar{\bar{\bf x}}\). We shall let these two values be represented by \({\bf S}_p^\bullet\) and \(\bar{\bar{\bf x}}^\bullet\) to distinguish them from the original values, \({\bf S}_p\) and \(\bar{\bar{\bf x}}\), before any subgroups are deleted. Future values to be plotted on the multivariate chart would then be obtained from $$ n (\bar{{\bf x}}^{(future)} - \bar{\bar{{\bf x}}}^\bullet)' ({\bf S}_p^\bullet)^{-1} (\bar{{\bf x}}^{(future)} - \bar{\bar{{\bf x}}}^\bullet) \, ,$$ with \(\bar{{\bf x}}^{(future)}\) denoting an arbitrary vector containing the averages for the \(p\) characteristics for a single subgroup obtained in the future. Each of these future values would be plotted on the multivariate chart and compared with
Phase II control limits $$ UCL = \left( \frac{p(k-a+1)(n-1)}{(k-a)n - k + a - p + 1} \right) F_{\alpha, \, (p, \, (k-a)n-k+a-p+1)} \, , $$ with \(a\) denoting the number of the original subgroups that are deleted before computing \({\bf S}_p^\bullet\) and \(\bar{\bar{\bf x}}^\bullet\). Notice that the equation for the control limits for Phase II given here does not reduce to the equation for the control limits for Phase I when \(a=0\), nor should we expect it to since the Phase I UCL is used when testing for control of the entire set of subgroups that is used in computing \({\bf S}_p\) and \(\bar{\bar{\bf x}}\).
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