Defining the V-Mask for a Two-Sided Cusum Scheme
The dimensions of the V-mask can be specified using two
distinct sets of two parameters.
- , defined as half of the angle formed by the V-mask arms,
and d, the distance between the origin and
the vertex, as shown in Figure 12.13.
This parameterization is used by many authors, including
Johnson and Leone (1962, 1974) and Montgomery (1996).
- h, the vertical distance between the origin and the upper
(or lower) V-mask arm, and k, the rise (drop) in the lower
(upper) arm corresponding to an interval of one subgroup unit
on the horizontal axis. You can specify the definition of an
interval with the INTERVAL= option. This parameterization is
used by Lucas (1976) and Wadsworth and others (1986).
Lucas (1976) uses the symbols h* for h and k* for
k, and Wadsworth and others (1986) use the symbol f
in place of k.
The two parameterizations are related by the equations
d = h/k
where the aspect ratio a is the number of units on the
vertical axis corresponding to one unit on the horizontal axis.
The CUSUM procedure uses the h and k parameterization
because it eliminates the need for working with aspect ratios, which
are dependent on the graphics device.
h and k are
also useful for average run length computations and for
parameterizing one-sided cusum schemes.
Figure 12.13: V-Mask Parameters
You can specify the V-mask in two ways:
If you provide and , h and k are computed using
If you provide but not , h and k are computed
using the formulas
In the preceding equations, the error probability is
divided by two because two-sided deviations from the target mean are
Refer to Johnson and Leone (1962, 1974).
Interpreting Two-Sided Cusum Charts
The origin of the V-mask is located at the most
recently plotted point, as illustrated in Figure 12.13.
As additional data are collected and the cumulative sum sequence is
updated, the origin is relocated at the newest point.
A shift or out-of-control condition is signaled at time t if one or
more of the points plotted up to time t cross an arm of the V-mask.
An upward shift is signaled by points crossing the lower arm, and a
downward shift is signaled by points crossing the upper arm.
The time at which the shift occurred corresponds to the time at which
a distinct change is observed in the slope of the plotted points.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.