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

6.5.4.3.2.

T2 Chart for Subgroup Averages -- Phase II

Phase II requires recomputing S_p and and different control limits

Determining the UCL that is to be subsequently applied to future subgroups entails recomputing, if necessary, S_p and , 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 S_p and . (The same thing happens with 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 subgroups had been discarded (with possibly a = 0) so that k - a subgroups are used in obtaining and . We shall let these two values be represented by and to distinguish them from the original values, and , before any subgroups are deleted. Future values to be plotted on the multivariate chart would then be obtained from

with 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

with a denoting the number of the original subgroups that are deleted before computing and . 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 and .