The SHEWHART Procedure

Multivariate Control Charts

Subsections:

Note: See Creating Multivariate Control Charts in the SAS/QC Sample Library.

In many industrial applications, the output of a process characterized by p variables that are measured simultaneously. Independent variables can be charted individually, but if the variables are correlated, a multivariate chart is needed to determine whether the process is in control.

Many types of multivariate control charts have been proposed; refer to Alt (1985) for an overview. Denote the ith measurement on the jth variable as $X_{ij}$ for $i=1,2,\ldots ,n$, where n is the number of measurements, and $j=1,2,\ldots ,p$. Standard practice is to construct a chart for a statistic $T^2_ i$ of the form

\[ T^2_ i = (\mb{X}_ i - \bar{\mb{X}}_ n)^{\prime } \mb{S}^{-1}_ n (\mb{X}_ i - \bar{\mb{X}}_ n) \]

where

\[ \begin{array}{lcr} \bar{X}_ j = \frac{1}{n} \sum _{i=1}^ n X_{ij}~ ~ , & \mb{X}_ i = \left[ \begin{array}{c} X_{i1} \\ X_{i2} \\ \vdots \\ X_{ip} \end{array} \right] , & \bar{\mb{X}}_ n = \left[ \begin{array}{c} \bar{X}_{1} \\ \bar{X}_{2} \\ \vdots \\ \bar{X}_{p} \end{array} \right] \end{array} \]

and

\[ \mb{S}_ n = \frac{1}{n-1} \sum _{i=1}^ n (\mb{X}_ i - \bar{\mb{X}}_ n) (\mb{X}_ i - \bar{\mb{X}}_ n)^{\prime } \]

It is assumed that $\mb{X}_ i$ has a p-dimensional multivariate normal distribution with mean vector $\mb{\mu } = (\mu _1 \mu _2 \cdots \mu _ p)^\prime $ and covariance matrix $\bSigma $ for $i=1,2,\ldots ,n$. Depending on the assumptions made about the parameters, a $\chi ^2$, Hotelling $T^2$, or beta distribution is used for $T^2_ i$, and the percentiles of this distribution yield the control limits for the multivariate chart.

In this example, a multivariate control chart is constructed using a beta distribution for $T^2_ i$. The beta distribution is appropriate when the data are individual measurements (rather than subgrouped measurements) and when $\mb{\mu }$ and $\bSigma $ are estimated from the data being charted. In other words, this example illustrates a start-up phase chart where the control limits are determined from the data being charted.