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 XCHART Statement

## Creating a One-Sided Cusum Chart with a Decision Interval

 See CUSONES1 in the SAS/QC Sample Library

An alternative to the V-mask cusum chart is the one-sided cusum chart with a decision interval, which is sometimes referred to as the "computational form of the cusum chart." This example illustrates how you can create a one-sided cusum chart for individual measurements.

A can of oil is selected every hour for fifteen hours. The cans are weighed, and their weights are saved in a SAS data set named CANS:*

   data cans;
length comment $16; label hour = 'Hour'; input hour weight comment$16. ;
datalines;
1  8.024
2  7.971
3  8.125
4  8.123
5  8.068
8  8.072
9  8.066
10  8.089
11  8.058
12  8.147
13  8.141
14  8.047
15  8.125
;


Suppose the problem is to detect a positive shift in the process mean of one standard deviation () from the target of 8.100 ounces. Furthermore, suppose that

• a known value is available for the process standard deviation
• an in-control average run length (ARL) of approximately 100 is required
• an ARL of approximately five is appropriate for detecting the shift

Table 12.18 indicates that these ARLs can be achieved with the decision interval h=3 and the reference value k=0.5. The following statements use these parameters to create the chart and tabulate the cusum scheme:

   title "One-Sided Cusum Analysis";
proc cusum data=cans;
xchart weight*hour /
mu0      = 8.100     /* target mean for process     */
sigma0   = 0.050     /* known standard deviation    */
delta    = 1         /* shift to be detected        */
h        = 3         /* cusum parameter h           */
k        = 0.5       /* cusum parameter k           */
scheme   = onesided  /* one-sided decision interval */
tableall             /* table                       */
cinfill  = ywh
cframe   = bigb
cout     = salmon
cconnect = salmon
climits  = black
coutfill = bilg;
label weight = 'Cusum of Weight';
run;


The chart is shown in Figure 12.5.

Figure 12.5: One-Sided Cusum Chart with Decision Interval

The cusum plotted at HOUR=t is

St = max(0,St-1+(zt-k))
where S0=0, and zt is the standardized deviation of the t th measurement from the target.
The cusum St is referred to as an upper cumulative sum. A shift is signaled at the seventh hour since S7 exceeds h. For further details, see "One-Sided Cusum Schemes" .

The option TABLEALL requests the tables shown in Figure 12.6, Figure 12.7, and Figure 12.8. The table in Figure 12.6 summarizes the cusum scheme, and it confirms that an in-control ARL of 117.6 and an ARL of 6.4 at are achieved with the specified h and k.

 One-Sided Cusum Analysis

 The CUSUM Procedure

 Cusum Parameters Process Variable weight (Cusum of Weight) Subgroup Variable hour (Hour) Scheme One-Sided Target Mean (Mu0) 8.1 Sigma0 0.05 Delta 1 Nominal Sample Size 1 h 3 k 0.5 Average Run Length (Delta) 6.40390895 Average Run Length (0) 117.595692
Figure 12.6: Summary Table

The table in Figure 12.7 tabulates the information displayed in Figure 12.5.

 The CUSUM Procedure

 Cumulative Sum Chart Summary for weight hour SubgroupSampleSize IndividualValue Cusum DecisionInterval DecisionIntervalExceeded 1 1 8.0240000 0.0000000 3.0000 2 1 7.9710000 0.0000000 3.0000 3 1 8.1250000 0.0000000 3.0000 4 1 8.1230000 0.0000000 3.0000 5 1 8.0680000 0.0000000 3.0000 6 1 8.1770000 1.0400000 3.0000 7 1 8.2290000 3.1200000 3.0000 Upper 8 1 8.0720000 2.0600000 3.0000 9 1 8.0660000 0.8800000 3.0000 10 1 8.0890000 0.1600000 3.0000 11 1 8.0580000 0.0000000 3.0000 12 1 8.1470000 0.4400000 3.0000 13 1 8.1410000 0.7600000 3.0000 14 1 8.0470000 0.0000000 3.0000 15 1 8.1250000 0.0000000 3.0000
Figure 12.7: Tabulation of One-Sided Chart

The table in Figure 12.8 presents the computational form of the cusum scheme described by Lucas (1976).

 The CUSUM Procedure

 Computational Cumulative Sum for weight hour SubgroupSampleSize IndividualValue UpperCusum Number ofConsecutiveUpper Sums > 0 1 1 8.0240000 0.0000000 0 2 1 7.9710000 0.0000000 0 3 1 8.1250000 0.0000000 0 4 1 8.1230000 0.0000000 0 5 1 8.0680000 0.0000000 0 6 1 8.1770000 1.0400000 1 7 1 8.2290000 3.1200000 2 8 1 8.0720000 2.0600000 3 9 1 8.0660000 0.8800000 4 10 1 8.0890000 0.1600000 5 11 1 8.0580000 0.0000000 0 12 1 8.1470000 0.4400000 1 13 1 8.1410000 0.7600000 2 14 1 8.0470000 0.0000000 0 15 1 8.1250000 0.0000000 0
Figure 12.8: Computational Form of Cusum Scheme

Following the method of Lucas (1976), the process average at the out-of-control point (HOUR=7) can be estimated as

where S7 =3.12 is the upper sum at HOUR=7, and N7 =2 is the number of successive positive upper sums at HOUR=7.

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