Cusum Basics Mel Alexander, ASQ Fellow, CQE. Tutorial: ASQ - Baltimore Section January 13, 2004 Phone: 410-712-7426/work E-mail: email@example.com or firstname.lastname@example.org . Purpose.
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Tutorial: ASQ - Baltimore Section
January 13, 2004
E-mail: email@example.com or
Wald’s SPRT used a 3-way sequential sampling where samples of n 1 were taken at stages to:
(1) declare that a process is in control;
(2) find the process out-of-control;
(3) take additional observations
The advantage SPRT had over fixed-sized sampling was decisions regarding the two risks associated with shift detection and the size of the shift were determined in advance.
The two risks regarding shift detection are:
- finding false alarms (finding process shifts that did not occur)
- missing shifts that did occur (fail to detect process shifts that occurred)
With fixed-sized sampling, is usually specified first, while is computed for different process level values
Wald and G.A. Barnard (in 1945, 1946) showed that the acceptance and reject limit numbers (Ca and Cr, respectively) must satisfy the relationships: and
so that a decision interval boundary could be formed Ca < SPRT < Cr
Cusums (Si) plot the cumulative sums of deviations of sample values (Xis) from a target value or aim (T) over time
Si = where
Xi = process output value of the i-th item or sample (sometimes = i-th mean may be used),
T = Target value or aim, T may be estimated with the in- control mean ,
n= number of samples collected, tested, or baselined
Uses deviations above (below) the target T that is calculated as:
Upper Cusum: (Shi,i) = max[0, Shi,i-1 + Xi – (T - k)]
Lower Cusum: (Slo,i) = max[0, Slo,i-1 + (T- k) - Xi]
where starting values Shi,0 = Slo,0 = 0,
Next, we find parameter values K and H
K = reference value (a.k.a. allowance or slack) equal to some constant (multiple, coefficient) times - sigma,
i.e, standard deviation estimated from values, subgroup ranges, or average moving range.
Usually, K= 0.5 x Delta = where
Delta =the amount of shift from the target (T) we seek to detect. Usually, Delta equals sigma (= )
Xout = out-of-control value of the mean (= T + K )
The parameter H serves as a decision point (like a control limit) that works as follows:
H=4 or 5 indicates an out-of-control signal
Shi,i > H or Slo,i > H for sample item (or subgroup) i
Parameters H and K are designed to yield large Average Run Lengths (number of samples before an signaling an out-of-control condition) when process is on target, denoted as ARL(0).
As the process shifts by the size of Delta, the Average Run Lengths should be small, denoted as ARL(Delta).
Tables exist that show the relationship of ARLs to H, K, and Delta
Cumulative Sum (St )
- semi angle
Subgroup Index (t)
Lead distance d =
H = d tan( )
Statistical significance tests for relative changes are performed on adjacent local means that help identify new problems
Change-Point Analysis Example: Plot of US Trade Deficit Data Showing Changes in Background
For more information, visit Wayne Taylor’s web site: http://www.variation.com/cpa/tech/changepoint.html