9.6
Counted Data Cusum Control Charts
The following information is supplemental to the text.
• For moderate or low count events (such as nonconformities or defects), it is common to assume
the distribution of the counts follows a Poisson distribution.
• In this case, we are assuming that the probability that a defect occurs in a very small volume
of product is proportional to the volume of the product, and defects occur independently of
each other.
• This assumption is usually reasonable for production processes when the process is in a state
of statistical control. When the process goes out of control, the counts will increase and their
distribution may depart from being Poisson. For example, this would occur if defects cluster
spatially when the process is out of control.
• Properties of counted data from a production process can be monitored by a counted data
cusum. Counted data cusums should be used in moderate and low count in-control situations
(although it can often be practical to use in high count situations).
• When defects or other undesirable events are being monitored with a counted data cusum,
the ideal case would be a count of zero. The average count, of course, will always be greater
than zero. Thus, there is no direct analogy to the aim value or the variance in a counted data
cusum.
• Although counted data cusums can, in practice, be designed to detect either increases or
decreases in the number of defects in a sampling interval, the primary application is to detect
increases. Therefore, the following discussion will be restricted to one-sided schemes to detect
an increase in the counts per sampling interval.
Design of a Counted Data Cusum Scheme
• Let Yi be the observed defect count in the ith sample. Thus, Yi is a non-negative integer.
• The mean acceptable (sample) count µa and a high (sample) count µd are set by the
process expert and will be used to design the counted data cusum. The high count level is a
level which the cusum is designed to detect quickly.
• The cusum should be designed so that the ARL will (i) be large if the current mean count
per sample remains at the mean acceptable count and (ii) be small if the current mean count
per sample increases to an unacceptable level.
• The cusum formula applied to a counted data property (e.g., defects) is
Si = max(0, Yi − K + Si−1 )
• The parameter K is the boundary count level below which the counted data cusum is not
designed to react. Note that the cusum only increases if Yi > K.
• The value of K should be chosen to be between the mean acceptable count and the high count
level. It is recommended that the value of K be selected as the integer closest to
µd − µa
(High count level) − (Acceptable count level)
=
ln(High count level) − ln(Acceptable count level)
lnµd − lnµa
194
• For a given ARL at the acceptable count level, this value of K will give the shortest ARL at
the high count level. The value of K need not be an integer although ARL Tables 28.01 and
28.02 do not show fractional values of K.
• Like the continuous data cusums, we need to select a decision interval H. That is, if the
cusum reaches or exceeds H, then an out-of-control signal occurs.
• The value of H can be selected from Table 28.01 (no FIR) or Table 28.02 (FIR) to give an
appropriately large ARL when the process is running at or below the acceptable count level.
It should also be chosen to give an appropriately small ARL value when the process is running
at or greater than the high level that is to be detected quickly.
• The FIR feature should be used to give quicker detection in case of potential process start-up
problems. For a FIR, we will use S0 = H/2.
Example
• In the following example suppose
µa = the acceptable count level = 4
µd = the high count level = 7
The counted data cusum is designed with a K value close to
7−4
µd − µa
=
= 5.36.
lnµd − lnµa
ln(7) - ln(4)
After rounding to the nearest integer, we use K = 5.
• In the Lucas article “Counted Data CUSUMs”, the notation kb is the reciprocal of this ratio.
For this example, kb = 1/5.35 ≈ .187. This would be the value of k to enter in SAS.
• To use ARL Table 28.01 or 28.02, the acceptable count level µa and the high count level µd
must first be expressed in normalized units by dividing by K. Thus, the normalize values are
µ∗a =
µa
4
= = .8
K
5
and
µ∗d =
µd
7
= = 1.4
K
5
These values correspond to the values in the “Mean, as a multiple of K” columns.
• Next, go to the K = 5 rows and select H that has a large ARL in the µ∗a = 8 column and a
small value in the µ∗d = 1.4 column.
For example: in Table 28.01, when K = 5 and H = 10, the ARLs are 422 and 5.59 for
normalized values of µ∗a = 0.8 and µ∗d = 1.4.
195
196
197
198
199
200
Upper One-Sided Cusum for Count Data
The CUSUM Procedure
Cumulative Sum Chart Summary for accident
Individual
date Value
Cusum
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
2.0
1.0
2.0
0.0
3.0
2.0
6.0
2.0
2.0
3.0
5.0
0.0
1.0
2.0
1.0
0.0
4.0
1.0
1.0
3.0
3.0
0.0
2.0
1.0
1.0
1.0
0.0
2.0
2.0
4.0
3.0
2.0
1.0
1.0
4.0
4.0
1.0
3.0
1.0
1.0
1.0
4.0
1.0
1.0
3.0
2.0
1.0
7.0
3.0
1.0
2.0
0.0
2.0
1.0
0.0
1.0
2.0
4.0
4.0
2.0
0.00000
0.00000
0.00000
0.00000
0.42108
0.00000
3.42108
2.84216
2.26324
2.68432
5.10540
2.52648
0.94756
0.36864
0.00000
0.00000
1.42108
0.00000
0.00000
0.42108
0.84216
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
1.42108
1.84216
1.26324
0.00000
0.00000
1.42108
2.84216
1.26324
1.68432
0.10540
0.00000
0.00000
1.42108
0.00000
0.00000
0.42108
0.00000
0.00000
4.42108
4.84216
3.26324
2.68432
0.10540
0.00000
0.00000
0.00000
0.00000
0.00000
1.42108
2.84216
2.26324
Decision
Interval
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
Individual
date Value
Cusum
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
1.0
1.0
1.0
2.0
2.0
0.0
0.0
2.0
0.0
0.0
0.0
0.0
0.0
1.0
0.0
2.0
1.0
0.0
1.0
0.0
1.0
1.0
0.0
1.0
2.0
1.0
4.0
1.0
3.0
3.0
0.0
1.0
0.0
2.0
1.0
1.0
0.0
1.0
0.0
1.0
0.0
1.0
0.0
1.0
1.0
0.0
0.0
1.0
1.0
1.0
2.0
1.0
0.0
3.0
2.0
2.0
0.0
1.0
2.0
1.0
0.68432
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
1.42108
0.00000
0.42108
0.84216
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.42108
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
201
Decision
Interval
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
Lower One-Sided Cusum for Count Data
The CUSUM Procedure
Cumulative Sum Chart Summary for accident
Individual
date Value
Cusum
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
2.0
1.0
2.0
0.0
3.0
2.0
6.0
2.0
2.0
3.0
5.0
0.0
1.0
2.0
1.0
0.0
4.0
1.0
1.0
3.0
3.0
0.0
2.0
1.0
1.0
1.0
0.0
2.0
2.0
4.0
3.0
2.0
1.0
1.0
4.0
4.0
1.0
3.0
1.0
1.0
1.0
4.0
1.0
1.0
3.0
2.0
1.0
7.0
3.0
1.0
2.0
0.0
2.0
1.0
0.0
1.0
2.0
4.0
4.0
2.0
0.00000
0.42108
0.00000
1.42108
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
1.42108
1.84216
1.26324
1.68432
3.10540
0.52648
0.94756
1.36864
0.00000
0.00000
1.42108
0.84216
1.26324
1.68432
2.10540
3.52648
2.94756
2.36864
0.00000
0.00000
0.00000
0.42108
0.84216
0.00000
0.00000
0.42108
0.00000
0.42108
0.84216
1.26324
0.00000
0.42108
0.84216
0.00000
0.00000
0.42108
0.00000
0.00000
0.42108
0.00000
1.42108
0.84216
1.26324
2.68432
3.10540
2.52648
0.00000
0.00000
0.00000
Decision
Interval
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
Individual
date Value
Cusum
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
1.0
1.0
1.0
2.0
2.0
0.0
0.0
2.0
0.0
0.0
0.0
0.0
0.0
1.0
0.0
2.0
1.0
0.0
1.0
0.0
1.0
1.0
0.0
1.0
2.0
1.0
4.0
1.0
3.0
3.0
0.0
1.0
0.0
2.0
1.0
1.0
0.0
1.0
0.0
1.0
0.0
1.0
0.0
1.0
1.0
0.0
0.0
1.0
1.0
1.0
2.0
1.0
0.0
3.0
2.0
2.0
0.0
1.0
2.0
1.0
0.42108
0.84216
1.26324
0.68432
0.10540
1.52648
2.94756
2.36864
3.78972
5.21080
6.63188
8.05296
9.47404
9.89512
11.31620
10.73728
11.15836
12.57944
13.00052
14.42160
14.84268
15.26376
16.68484
17.10592
16.52700
16.94808
14.36916
14.79024
13.21132
11.63240
13.05348
13.47456
14.89564
14.31672
14.73780
15.15888
16.57996
17.00104
18.42212
18.84320
20.26428
20.68536
22.10644
22.52752
22.94860
24.36968
25.79076
26.21184
26.63292
27.05400
26.47508
26.89616
28.31724
26.73832
26.15940
25.58048
27.00156
27.42264
26.84372
27.26480
202
Decision
Interval
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
7.9072
1
2
3
4
5
6
7
8
9
10
11
Lower 12
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
Lower
DM ’LOG; CLEAR; OUT; CLEAR;’;
* ODS PRINTER PDF file=’C:\COURSES\ST528\SAS\cusum_ct.pdf’;
ODS LISTING;
OPTIONS NODATE NONUMBER PS=500 LS=76;
DATA lucas;
DO date = 1 to 120;
INPUT accident @@; OUTPUT;
END;
LINES;
2 1 2 0 3 2 6 2 2 3 5 0
1 2 1 0 4 1 1 3
1 1 0 2 2 4 3 2 1 1 4 4
1 3 1 1 1 4 1 1
3 1 2 0 2 1 0 1 2 4 4 2
1 1 1 2 2 0 0 2
0 1 0 2 1 0 1 0 1 1 0 1
2 1 4 1 3 3 0 1
0 1 0 1 0 1 0 1 1 0 0 1
1 1 2 1 0 3 2 2
;
SYMBOL1 V=dot WIDTH=2;
3
3
0
0
0
0
2
0
2
1
2
1
0
1
2
1
7
0
1
1
PROC CUSUM DATA=lucas;
XCHART accident*date=’1’
/ MU0=2 SIGMA0=1.412 NPANELPOS=121
h=5.6 k=0.41 DATAUNITS DELTA=1
HAXIS=0 to 120 by 1 NOMASK;
INSET ARL0 ARLDELTA H K SHIFT / POS = sw;
LABEL accident =’Accidents Cusum’
date
=’Month (beginning January 1970)’;
TITLE ’CUSUM for Number of Accidents per Month’;
TITLE2 ’(Data taken from Lucas 1985)’;
PROC CUSUM DATA=lucas;
XCHART accident*date=’1’
/ MU0=2 SIGMA0=1.412 NPANELPOS=121
h=5.6 k=0.41 DATAUNITS DELTA=1
HAXIS=0 to 120 by 1
SCHEME=onesided TABLESUMMARY TABLEOUT;
INSET ARL0 ARLDELTA H K SHIFT / POS = ne;
LABEL accident =’Accidents Cusum’
date
=’Month (beginning January 1970)’;
TITLE ’Upper One-Sided Cusum for Count Data’;
PROC CUSUM DATA=lucas;
XCHART accident*date=’1’
/ MU0=2 SIGMA0=1.412 NPANELPOS=121
h=5.6 k=0.41 DATAUNITS DELTA=-1
HAXIS=0 to 120 by 1
SCHEME=onesided TABLESUMMARY TABLEOUT;
LABEL accident =’Accidents Cusum’
date
=’Month (beginning January 1970)’;
TITLE ’Lower One-Sided Cusum for Count Data’;
RUN;
203
204
205
206
207
208
209
210
211
212
213
214
215
216