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XCHART Statement |
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 and others (1986).
For one-sided charts, average run lengths are tabulated as a function of h, k, and in Table 12.18. No headstart is assumed in this table. For two-sided charts, average run lengths are tabulated as a function of h, k, and in Table 12.19, which is formatted similarly to Table 2 given by Lucas (1976).
The ARLs in Table 12.18 and Table 12.19 were calculated with the DATA step function CUSUMARL. 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 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 , referred to as the central reference value.
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 and with the ALPHA= and BETA= options or with the variables _ALPHA_ and _BETA_ in a LIMITS= data set. It is not necessary to specify , and the interpretation of 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).
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 12.18: Average Run Lengths for One-Sided V-Mask Cusum Charts as a Function of h, k, and .Parameters | (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 |
Parameters | (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 |
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