Quality and Reliability Methods - SAS

Chapter 6 Cumulative Sum Control Charts 105
Formulas for CUSUM Charts
for t = 1, 2,..., n, where S 0 = 0, z t is defined as for two-sided 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 μ 0 . If S t
exceeds a positive value h (referred to as the decision interval), a shift or out-of-control condition is signaled.
Negative Shifts
X i – ( μ 0
+ kσ )
Xi
max0,
S t – 1
+ ---------------------------------------
σ
Xi
If the shift to be detected is negative, the cusum computed for the t th subgroup is
S t = max(0, S t – 1 – (z t + k))
for t = 1, 2,..., n, where S 0 = 0, z t is defined as for two-sided 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
X i – ( μ 0
– kσ )
Xi
max0,
S t – 1
– --------------------------------------
σ
Xi
the sequence S t cumulates the absolute value of deviations in the subgroup means less than k standard errors
from μ 0 . If S t exceeds a positive value h (referred to as the decision interval), a shift or out-of-control
condition is signaled.
Note that S t is always positive and h is always positive, regardless of whether δ is positive or negative. For
schemes designed to detect a negative shift, some authors define a reflected version of S t for which a shift is
signaled when S t is less than a negative limit.
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 “head start” value. Average run length calculations given
by them 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 out-of-control condition than a standard CUSUM scheme. You can provide head start
values on the dialog or through JSL.
Constant Sample Sizes
When the subgroup sample sizes are constant (= n), it might be preferable to compute cusums that are
scaled in the same units as the data. Cusums are then computed as
S t
= max( 0,
S t – 1
+ ( X i – ( μ 0
+ kσ ⁄ n)
))
for δ > 0 and the equation
S t
= max( 0,
S t – 1
– ( X i – ( μ 0
– kσ ⁄ n)
))
for δ < 0. In either case, a shift is signaled if S t exceeds
h' = hσ ⁄ n
. Some authors use the symbol H for h'.