Computing Control Limits :: SAS/QC(R) 13.1 User's Guide

Computing $T^2$ Control Limits

The control limits for the $T^2$ chart are the same for all the $T^2$ statistics in the chart. The control limits are computed using the method of Tracy, Young, and Mason (1992), in which the distribution of the $T^2$ statistic is shown to follow a scaled beta distribution:

$T^2_ i \sim \frac{(n-1)^2}{n} B \left( \frac{j}{2}, \frac{n-j-1}{2} \right) \qquad j \geq 2,\, n \geq j+1$

where i is the observation, j is the number of principal components in the model, and n is the number of observations used to build the principal component model.

The upper control limit is computed as the $(1-\frac{\alpha }{2})$ quantile of this distribution, and the lower control limit is computed by the $\frac{\alpha }{2}$ quantile. You can use the ALPHA= option in the TSQUARECHART statement to specify $\alpha$ .

See the section Computing the $T^2$ and SPE Statistics for details of computing the $T^2$ statistic based on a principal component model.

The MVPMONITOR Procedure

Computing Control Limits

Computing $T^2$ Control Limits