See SHWT2 in the SAS/QC Sample LibraryIn 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 
for 
, where n is the number of measurements, and 
. Standard practice is to construct a chart for a statistic 
of the form
![\[ T^2_ i = (\mb {X}_ i - \bar{\mb {X}}_ n)^{\prime } \mb {S}^{-1}_ n (\mb {X}_ i - \bar{\mb {X}}_ n) \]](images/qcug_shewhart0432.png){kind=small}
where
![\[ \begin{array}{lcr} \bar{X}_ j = \frac{1}{n} \sum _{i=1}^ n X_{ij}~ ~ , & \mb {X}_ i = \left[ \begin{array}{c} X_{i1} \\ X_{i2} \\ \vdots \\ X_{ip} \end{array} \right] , & \bar{\mb {X}}_ n = \left[ \begin{array}{c} \bar{X}_{1} \\ \bar{X}_{2} \\ \vdots \\ \bar{X}_{p} \end{array} \right] \end{array} \]](images/qcug_shewhart0433.png)
and
![\[ \mb {S}_ n = \frac{1}{n-1} \sum _{i=1}^ n (\mb {X}_ i - \bar{\mb {X}}_ n) (\mb {X}_ i - \bar{\mb {X}}_ n)^{\prime } \]](images/qcug_shewhart0434.png){kind=small}
It is assumed that 
has a p-dimensional multivariate normal distribution with mean vector 
and covariance matrix 
for . Depending on the assumptions made about the parameters, a 
, Hotelling 
, or beta distribution is used for , 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 . The beta distribution is appropriate when the data are individual measurements (rather than subgrouped measurements) and
when 
and 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.