XCHART Statement: CUSUM Procedure

Designing a Cusum Scheme

There are three main methods for designing a cusum scheme: the average run length (ARL) approach, the error probability approach, and the economic design approach.

Average Run Length (ARL) Approach

With the ARL approach, the parameters h and k are chosen to yield desired average run lengths when the process is operating at the target mean and when a shift of magnitude $\delta $ has occurred. The average run length is the expected number of samples taken before an out-of-control condition is signaled. Ideally, the ARL should be long when $\mu =\mu _{0}$ and short when $\mu $ shifts away from $\mu _{0}$.

The ARL method typically involves the use of a table or nomogram. Refer to Kemp (1961), van Dobben de Bruyn (1968), Goel and Wu (1971), Duncan (1974), Lucas (1976), Montgomery (1996), and Wadsworth, Stephens, and Godfrey (1986).

For one-sided charts, average run lengths are tabulated as a function of h, k, and $\delta $ in Table 6.3. No headstart is assumed in this table. For two-sided charts, average run lengths are tabulated as a function of h, k, and $\delta $ in Table 6.4, which is formatted similarly to Table 2 given by Lucas (1976).

The ARLs in Table 6.3 and Table 6.4 were calculated with the DATA step function CUSUMARL (see the section CUSUMARL Function). This function uses the method of Goel and Wu (1971). You can use this function to generate more detailed, interpolated versions of the tables or to compute ARLs with headstart values.

It can be shown that the two-sided (V-mask) cusum scheme parameterized by h and k is equivalent to two simultaneously operating one-sided cusum schemes, one that computes an upper cusum and one that computes a lower cusum. Both one-sided schemes use the same parameters h and k.

You can specify h, k, and $\delta $ with the options H=, K=, and DELTA= or with the variables _H_, _K_, and _DELTA_ in a LIMITS= data set. The reference value k is optional, and its default value is $k=|\delta |/2$, referred to as the central reference value.

Error Probability Approach

This approach is available only for two-sided cusum schemes. Values of $\alpha $ (the probability of incorrectly signaling the occurrence of a shift) and $\beta $ (the probability of failing to detect a shift) are specified, and h and k are computed from $\alpha $ and $\beta $ as described in Defining the V-Mask for a Two-Sided Cusum Scheme. The error probability approach interprets the cusum as a sequence of reversed sequential probability ratio tests. Refer to Johnson (1961), Johnson and Leone (1962, 1974), van Dobben de Bruyn (1968), Montgomery (1996), and Wadsworth, Stephens, and Godfrey (1986).

Although the error probability method is intuitively appealing, the actual error probabilities achieved may not be close to those specified since the V-mask does not provide for an acceptance region. This has been pointed out by various authors, including Johnson (1961) and van Dobben de Bruyn (1968). If you follow this approach, it is recommended that you examine the average run lengths for the cusum scheme (these are tabulated by the TABLESUMMARY option and are saved in OUTLIMITS= data sets).

You can specify $\alpha $ and $\beta $ with the ALPHA= and BETA= options or with the variables _ALPHA_ and _BETA_ in a LIMITS= data set. It is not necessary to specify $\beta $, and the interpretation of $\beta $ is somewhat questionable. The SIGMAS= option is an alternative to the ALPHA= option, and the variable _SIGMAS_ is an alternative to the variable _ALPHA_ (if you specify the READSIGMAS option).

Economic Design

The parameters n, h, and k are chosen so that the long-run average cost of the cusum scheme is minimized. Refer to Chiu (1974), Montgomery (1980), Svoboda (1991), and Ho and Case (1994) for reviews of the literature on economic design. This approach typically requires numerical optimization techniques, which are available in SAS/IML software and in the NLP procedure in SAS/OR software.

You can pass the optimal parameters to the CUSUM procedure as values of the variables _LIMITN_, _H_, and _K_ in a LIMITS= data set.

Table 6.3: Average Run Lengths for One-Sided V-Mask Cusum Charts as a Function of h, k, and $\delta $.

   

Parameters

$\delta $ (shift in mean)

                         

h

k

0.00

0.25

0.50

0.75

1.00

1.50

2.00

2.50

3.00

4.00

5.00

2.50

0.25

27.27

13.43

7.96

5.42

4.06

2.71

2.06

1.68

1.42

1.11

1.01

4.00

0.25

77.08

26.68

13.29

8.38

6.06

3.91

2.93

2.38

2.05

1.61

1.23

6.00

0.25

350.80

51.34

20.90

12.37

8.73

5.51

4.07

3.26

2.74

2.13

1.90

8.00

0.25

736.78

84.00

28.76

16.37

11.39

7.11

5.21

4.15

3.48

2.67

2.14

10.00

0.25

2071.51

124.66

36.71

20.37

14.06

8.71

6.36

5.04

4.20

3.20

2.65

2.00

0.50

38.55

18.19

10.00

6.32

4.45

2.74

1.99

1.58

1.32

1.07

1.01

3.00

0.50

117.60

39.47

17.35

9.68

6.40

3.75

2.68

2.12

1.77

1.31

1.07

4.00

0.50

335.37

77.08

26.68

13.29

8.38

4.75

3.34

2.62

2.19

1.71

1.31

5.00

0.50

930.89

141.69

38.01

17.05

10.38

5.75

4.01

3.11

2.57

2.01

1.69

6.00

0.50

2553.11

250.80

51.34

20.90

12.37

6.75

4.68

3.62

2.98

2.24

1.95

1.50

0.75

42.57

21.09

11.59

7.09

4.78

2.73

1.90

1.48

1.24

1.04

1.00

2.25

0.75

139.71

51.46

22.38

11.66

7.13

3.73

2.51

1.91

1.56

1.16

1.02

3.00

0.75

442.80

117.60

39.47

17.35

9.68

4.73

3.12

2.36

1.93

1.41

1.11

3.75

0.75

1375.71

258.96

65.65

24.16

12.37

5.73

3.71

2.79

2.27

1.72

1.31

4.50

0.75

4251.69

559.95

105.12

32.09

15.15

6.73

4.31

3.21

2.59

1.97

1.60

1.00

1.00

35.29

19.22

11.21

7.03

4.75

2.63

1.78

1.38

1.17

1.02

1.00

1.50

1.00

93.85

42.57

21.09

11.59

7.09

3.50

2.24

1.66

1.34

1.07

1.01

2.00

1.00

258.67

94.34

38.55

18.19

10.00

4.45

2.74

1.99

1.58

1.16

1.02

2.50

1.00

716.00

205.97

68.19

27.27

13.43

5.42

3.25

2.34

1.85

1.31

1.07

3.00

1.00

1962.79

442.80

117.60

39.47

17.35

6.40

3.75

2.68

2.12

1.52

1.16

3.50

1.00

5341.40

943.73

199.57

55.69

21.76

7.39

4.25

3.01

2.37

1.73

1.31

0.70

1.50

67.72

36.03

20.26

12.07

7.63

3.66

2.18

1.55

1.25

1.04

1.00

1.10

1.50

184.28

86.36

42.72

22.50

12.74

5.17

2.80

1.86

1.43

1.08

1.01

1.50

1.50

549.69

221.49

93.85

42.57

21.09

7.09

3.50

2.24

1.66

1.16

1.02

1.90

1.50

1762.09

595.61

210.95

80.54

34.26

9.38

4.26

2.64

1.92

1.29

1.05

2.30

1.50

5897.30

1638.15

476.90

151.04

54.47

12.00

5.03

3.04

2.20

1.45

1.12

 


Table 6.4: Average Run Lengths for Two-Sided V-Mask Cusum Charts as a Function of h, k, and $\delta $.

   

Parameters

$\delta $ (shift in mean)

                         

h

k

0.00

0.25

0.50

0.75

1.00

1.50

2.00

2.50

3.00

4.00

5.00

2.50

0.25

13.64

11.22

7.67

5.38

4.06

2.71

2.06

1.68

1.42

1.11

1.01

4.00

0.25

38.54

24.71

13.20

8.38

6.06

3.91

2.93

2.38

2.05

1.61

1.23

6.00

0.25

125.40

50.33

20.89

12.37

8.73

5.51

4.07

3.26

2.74

2.13

1.90

8.00

0.25

368.39

83.63

28.76

16.37

11.39

7.11

5.21

4.15

3.48

2.67

2.14

10.00

0.25

1035.75

124.55

36.71

20.37

14.06

8.71

6.36

5.04

4.20

3.20

2.65

2.00

0.50

19.27

15.25

9.63

6.27

4.44

2.74

1.99

1.58

1.32

1.07

1.01

3.00

0.50

58.80

36.24

17.20

9.67

6.40

3.75

2.68

2.12

1.77

1.31

1.07

4.00

0.50

167.68

74.22

26.63

13.29

8.38

4.75

3.34

2.62

2.19

1.71

1.31

5.00

0.50

465.44

139.49

38.00

17.05

10.38

5.75

4.01

3.11

2.57

2.01

1.69

6.00

0.50

1276.55

249.26

51.34

20.90

12.37

6.75

4.68

3.62

2.98

2.24

1.95

1.50

0.75

21.28

17.22

11.01

7.00

4.77

2.73

1.90

1.48

1.24

1.04

1.00

2.25

0.75

69.85

45.97

22.04

11.63

7.13

3.73

2.51

1.91

1.56

1.16

1.02

3.00

0.75

221.40

110.95

39.31

17.34

9.68

4.73

3.12

2.36

1.93

1.41

1.11

3.75

0.75

687.85

251.56

65.58

24.16

12.37

5.73

3.71

2.79

2.27

1.72

1.31

4.50

0.75

2125.85

552.11

105.09

32.09

15.15

6.73

4.31

3.21

2.59

1.97

1.60

1.00

1.00

17.65

15.03

10.39

6.88

4.72

2.63

1.78

1.38

1.17

1.02

1.00

1.50

1.00

46.92

35.70

20.31

11.49

7.07

3.50

2.24

1.66

1.34

1.07

1.01

2.00

1.00

129.34

84.00

37.93

18.14

10.00

4.45

2.74

1.99

1.58

1.16

1.02

2.50

1.00

358.00

191.48

67.76

27.25

13.43

5.42

3.25

2.34

1.85

1.31

1.07

3.00

1.00

981.39

423.29

117.32

39.47

17.35

6.40

3.75

2.68

2.12

1.52

1.16

3.50

1.00

2670.70

917.89

199.40

55.69

21.76

7.39

4.25

3.01

2.37

1.73

1.31

0.70

1.50

33.86

28.41

18.90

11.84

7.59

3.66

2.18

1.55

1.25

1.04

1.00

1.10

1.50

92.14

71.41

40.91

22.29

12.71

5.17

2.80

1.86

1.43

1.08

1.01

1.50

1.50

274.84

191.58

91.58

42.39

21.07

7.09

3.50

2.24

1.66

1.16

1.02

1.90

1.50

881.05

536.07

208.31

80.41

34.25

9.38

4.26

2.64

1.92

1.29

1.05

2.30

1.50

2948.65

1523.15

474.09

150.96

54.47

12.00

5.03

3.04

2.20

1.45

1.12