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ANALYSIS OF PROPORTIONS (ANOP)
In many situations, you need to examine differences among several proportions. The ANOP
provides a “confidence interval type of approach” that allows you to determine which, if any, of
the c groups has a proportion significantly different from the overall average of all the group
proportions combined.
The ANOP is really an extension of the ANOM procedure developed in Section 11.1. In
that section, you focused on evaluating several groups of numerical data. In this section, you
will learn how to evaluate several groups of categorical data. Instead of looking at the lower
and upper limits of a confidence interval, you will be studying which of the c group proportions are not contained in an interval formed between a lower decision line and upper decision
line. Any individual group proportion not contained in this interval is deemed significantly
larger than the overall average of all group proportions if it lies above the upper decision line.
Similarly, any group proportion that falls below the lower decision line is declared significantly
smaller than the overall average of all group proportions.
As with confidence interval estimation, to compute the upper and lower decision lines
(UDL and LDL) for ANOP, you must add and subtract a measure of sampling error around the
statistic of interest. That is,
FORMAT FOR ALL CONFIDENCE INTERVALS
statistic ; sampling error
statistic ; (critical value) (standard error of the statistic)
For ANOP this is demonstrated in the following equations:
OBTAINING THE UDL AND LDL
p ; hc, q
p(1 - p) (c - 1)
B
n
B c
where
c = number of groups in the study
j = representation for a particular group; j = 1, 2, . .., c
nj = sample size for group j
n = total number of observations where each of the nj sample sizes are equal;
n = n 1 + n2 + . . . + n c
n = average sample size over all c groups; n = n>c
p = pooled proportion; the overall average of all c sample proportions;
X1 + X2 + Á + Xc
X
=
p =
n
n1 + n2 + Á + nc
hc, q = critical value of Nelson’s h statistic with c groups and “large” sample sizes, nj,
per group obtained using the “infinity” row in the table
so that
UPPER DECISION LINE FOR ANOP
UDL = p + hc, q
p(1 - p) (c - 1)
B
n
B c
1
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Analysis of Proportions (ANOP)
and
LOWER DECISION LINE FOR ANOP
LDL = p - hc, q
B
p(1 - p) (c - 1)
n
B c
Note that Nelson’s h statistic is found from the combination of c groups with the “infinity”
row for common group sample sizes. This is because the ANOP procedure expects the sample
sizes to be large. The major assumption for applying the ANOP procedure is that the c group
sample sizes be sufficiently large to enable the expected number of items of interest in each
group to be at least 5. That is, nj pj Ú 5 and also nj(1 - pj) Ú 5.
To demonstrate how the ANOP procedure is used, suppose that the Ramsey & Ramsey
food processing plant funnels its baked cookies to sealing machines used for product packaging. In monitoring the process, Philip Pat, the product manager, finds that the quality of the
seals, as defined by the proportion of defective seals, is at an unacceptably high level. Because
Philip Pat feels that the temperature setting on the sealing machine may affect the quality of the
seals, he designs an experiment in which five different temperature settings are evaluated. At
each temperature setting, 500 boxes are sealed and examined for their seal quality.
The following table presents the number of boxes with defective and non-defective package seals at each of the five temperature settings.
Cross-Classification of
Observed Frequencies
from the Sealing
Machine TemperatureSetting Experiment
Sealing Machine Temperature Setting
Packaging Result
Defective seals
Non-defective seals
Totals
No. 1
No. 2
No. 3
No. 4
No. 5
Totals
22
478
500
20
480
500
34
466
500
23
477
500
41
459
500
140
2,360
2,500
From the data you observe that with five temperature settings used to evaluate the sealing
process for samples of 500 different boxes per temperature setting, there is no problem with the
ANOP procedure assumption regarding expected numbers of defective and non-defective
seals. To apply the ANOP procedure, you first need to compute the key statistics from the experiment as well as the critical value of Nelson’s h statistic. From the data displayed in the table
above, you compute the following:
p =
=
X1 + X2 + . . . + Xc
X
=
n
n1 + n 2 + . . . + n c
(22 + 20 + 34 + 23 + 41)
140
=
= 0.056
(500 + 500 + 500 + 500 + 500)
2,500
Because the estimated overall proportion of defective seals is 0.056, its complement, (1 - p),
or 0.944, is the estimated proportion of non-defective or conforming seals.
The critical values of Nelson’s h statistic for obtaining the 95% UDL and LDL are found in
the following table.
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Analysis of Proportions (ANOP)
Selected Critical Values
of Nelson’s h Statistic for
Obtaining 95% Upper
and Lower Decision Lines
(UDL and LDL)
Number of Groups, c
Sample Size per Group, nj
4
5
6
7
8
10
12
15
20
“infinity”
3
4
5
6
7
2.79
2.67
2.60
2.55
2.52
2.49
2.46
2.43
2.40
2.34
2.85
2.74
2.68
2.65
2.62
2.59
2.56
2.55
2.53
2.47
2.88
2.79
2.74
2.71
2.69
2.66
2.64
2.62
2.61
2.56
2.91
2.83
2.79
2.77
2.74
2.72
2.70
2.68
2.66
2.62
2.94
2.87
2.83
2.80
2.79
2.76
2.75
2.73
2.71
2.68
Note: Values in italics were obtained through linear interpolation.
Source: Table 2 of L. S. Nelson, “Exact Critical Values for Use with the Analysis of Means,” Journal of Quality
Technology, 15(1), 1983. Reprinted with permission of the American Society for Quality.
Given that you have five temperature settings (c = 5) and samples of 500 boxes per setting
(nj = 500), the critical value of Nelson’s h statistic for the sealing machine temperature-setting
experiment is h5, q = 2.56.
You now compute the UDL and LDL as follows:
UDL = p + hc, q
p(1 - p) (c - 1)
B
n
B c
= 0.056 + (2.56)
(0.056)(0.944) 4
500
B5
B
= 0.056 + 0.0235
= 0.0795
and
LDL = p - hc, q
B
p(1 - p) (c - 1)
n
B c
= 0.056 - (2.56)
(0.056)(0.944) 4
500
B5
B
= 0.056 - 0.0235
= 0.0325
From the cross-classification table, the proportions of defective seals out of 500 boxes
sampled for each of the five sealing machine temperature settings are
p1 = 0.044
p2 = 0.040
p3 = 0.068
The following is a graphical display for the ANOP.
p4 = 0.046
p5 = 0.082
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Analysis of Proportions (ANOP)
The proportions of defective box seals for each of the five sealing machine temperature
settings are plotted on the vertical axis. From top to bottom, the three horizontal lines represent
the UDL, p (the pooled proportion or overall average of all five proportions of defective box
seals combined), and the LDL.
From this figure, you observe that the proportion of defective boxes produced by temperature setting 5 is significantly higher than the average proportion, based on all five temperature
settings combined. You would suggest to the product manager Philip Pat that temperature setting 5 should not be used.