Formulas for Cumulative Sums
Positive Shifts
If the shift to be detected is positive, the cusum computed
for the t^{ th} subgroup is

S_{t} = max(0,S_{t1}+(z_{t}k))
for t=1, 2, . . . , n, where S_{0}=0, z_{t} is defined as for
twosided schemes, and the parameter k, termed the reference
value, is positive. The cusum S_{t} is referred to as an
upper cumulative sum. Since S_{t} can be written as
the sequence S_{t} cumulates deviations in the
subgroup means greater than k standard errors from
. If S_{t} exceeds a positive value
h (referred to as the decision interval), a shift or
outofcontrol condition is signaled. This formulation follows that
of Lucas (1976), Lucas and Crosier (1982), and Montgomery (1996).
Negative Shifts
If the shift to be detected is negative, the cusum
computed for the t^{ th} subgroup is

S_{t} = max(0,S_{t1}(z_{t}+k))
for t=1, 2, . . . , n, where S_{0}=0, z_{t} is defined as for
twosided cusum schemes, and the parameter k, termed the
reference value, is positive. The cusum S_{t} is referred
to as a lower cumulative sum. Since S_{t} can be written as
the sequence S_{t} cumulates the absolute value of deviations in
the subgroup means less than k standard errors from .If S_{t} exceeds a positive value h (referred to as the
decision interval), a shift or outofcontrol condition is signaled.
This formulation follows that of Lucas (1976), Lucas and Crosier
(1982), and Montgomery (1996). Note that S_{t} is always positive
and h is always positive, regardless of whether is positive
or negative. For schemes designed to detect a negative shift, some
authors, including van Dobben de Bruyn (1968) and Wadsworth
and others
(1986), define a reflected version of S_{t} for which a shift is
signaled when S_{t} is less than a negative limit.
Headstart Values
Lucas and Crosier (1982) describe the properties of a fast initial
response (FIR) feature for cusum schemes in which the initial cusum
S_{0} is set to a "headstart" value. Average run length
calculations given by Lucas and Crosier (1982) show that the FIR
feature has little effect when the process is in control and that it
leads to a faster response to an initial outofcontrol condition
than a standard cusum scheme. You can provide headstart value S_{0}
with the HEADSTART= option or the variable _HSTART_ in a LIMITS=
data set.
Constant Sample Sizes
When the subgroup sample sizes are constant
(=n), it may be preferable to compute cusums that are scaled in the
same units as the data.
Refer to Montgomery (1996) and Wadsworth and others (1986).
To request this, specify the DATAUNITS option.
Cusums are then computed as
for >0 and the equation
for . In either case, a
shift is signaled if
S_{t} exceeds .Wadsworth and others (1986) use the symbol H for h'.
If the subgroup sample sizes are not constant, you can
specify a constant nominal sample size n with the LIMITN= option
or the variable _LIMITN_ in a LIMITS=
data set.
In this case, only those subgroups with sample size n are
analyzed unless you also specify the option ALLN.
You can further specify the option NMARKERS to request
special symbol markers for points corresponding to sample sizes
not equal to n.
If the cusum scheme is twosided, the cumulative sum S_{t} plotted
for the t^{ th} subgroup is

S_{t}=S_{t1}+z_{t}
for t=1, 2, . . . , n. Here S_{0}=0, and
the term z_{t} is calculated as
where is the t^{ th} subgroup average, and n_{t} is
the t^{ th} subgroup sample size. If the subgroup samples consist
of individual measurements x_{t}, the term z_{t} simplifies to
Since the first equation can be rewritten as
the sequence S_{t} cumulates standardized
deviations of the subgroup averages from the target mean
.In many applications, the subgroup sample sizes n_{i} are constant
(n_{i}=n), and the equation for
S_{t} can be simplified.
In some applications, it may be preferable to compute S_{t} as
which is scaled in the same units as the data.
Refer to Montgomery (1996), Wadsworth and others (1986), and
ASQC Glossary and Tables for Statistical Quality Control.
If the subgroup sample sizes are constant (= n) and if you specify
the DATAUNITS option in the XCHART statement, the CUSUM procedure
computes cusums using the final equation above.
In this case, the procedure rescales the Vmask parameters h
and k to and ,respectively. Wadsworth and others (1986) use the symbols F for
k' and H for h'.
If the subgroup sample sizes are not constant, you can
specify a constant nominal sample size n with the LIMITN= option
or with the variable _LIMITN_ in a LIMITS= data set.
In this case, only those subgroups with sample size n are
analyzed unless you also specify the option ALLN.
You can further specify the option NMARKERS to request
special symbol markers for points corresponding to sample sizes
not equal to n.
If the process is in control and the mean is at or near the
target , the points will not exhibit a trend since
positive and negative displacements from tend to
cancel each other. If shifts in the positive direction, the
points exhibit an upward trend, and if shifts in the negative
direction, the points exhibit a downward trend.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.