A Multivariate Control Chart is used to monitor more than process factor at a time on a single control chart. When the process is stable, it has a stable set of Principal Components. Each Principal Component (PC) is a linear combination of all the process variables. Unlike the process variables, which may be correlated, the PCs are constructed to be orthogonal (independent) of one-another. The PCs may be used to estimate the data and thereby provide a basis for an estimate of the prediction error. The number of PCs may never exceed the number of process variables and is often constrained to be fewer.
1. Begin by constructing the MVA model of the process. Ideally, this model will use data from a period (a range of initial subgroups) of known process stability. The T2 control chart will then detect when the process covariance has shifted relative to this model (or Reference) data. Generally, all Principal Components (or an arbitrary realistic maximum of 10 PC) are used for a first pass, which usually does not provide a realistic model, but allows calculation of a suggested number of PC for a realistic model.
2. Review the Principal Component Statistics table , which indicates (with an asterisk) the suggested number of PC, using either the Cumulative Percentage or the Leverage Correction criteria. Re-generate the model with the suggested number of PC.
3. Evaluate the SPE (Squared Prediction Error) chart. Groups out of control indicate when there is significant error between the data (at that group) and the fitted model. If a few groups are out of control, consider removing them from the model (Reference data). If many groups are out of control, then reconsider the number of PC in the model, the range of groups in the model, or both (i.e. repeat steps 1 and 2, above).
4. If the T2 control chart is in control, then the process has a stable set of PC, and is said to be in control. Consider using this model for evaluating the future status of the process.
5. If the T2 control chart is out of control, consider removing the out of control points from the Model (Reference) data range. This provides a model that excludes known deviations from the remaining model set. In order to determine which PC has the strongest effect on the T2 statistic, consider the following:
a) In the Principal Component t-Scores table, compare the scores from each Principal Component for the out of control group. PC with a higher absolute score influence the group more than other groups.
b) Another approach which is sometimes useful is to copy the Principal Component t-Scores table into a blank Data Editor (.qdb file). Construct an Individual-X chart for each of the PC. The PC are assumed to be independent and Normally distributed. If any of the PC are out of control for the group in question, it is likely to be influencing the T2 control chart.
6. For any PC which seems to be influencing the out of control group on the T2 control chart, determine which underlying process variables have the strongest effect on the PC.
a) Use the Loading Chart to determine which Process Variables have higher influence on the PC in question. (A PC is a linear combination of the Process Variables, and the Loading is the weight applied to each Process Variable in the stated PC).
b) Process Variables having approximately equal Loading are likely to be highly correlated to each other. This can be verified by review of the Process Variable Correlation Matrix . When correlation is high between variables (close to one), it is recommended to remove the extraneous variables from the T2 control chart (i.e. all but one of the highly correlated variables).
c) Use the Score Contribution Chart to determine which Process Variables have the highest influence on the PC in question at the selected group.
Learn more about the SPC principles and tools for process improvement in Statistical Process Control Demystified (2011, McGraw-Hill) by Paul Keller, in his online SPC Concepts short course, or his online SPC certification course.
Interpretation & Calculations