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12
STATISTICAL PROCESS CONTROL USING
MICROBIOLOGICAL DATA
Microbiological data are often filed without much consideration as to their further application. When one considers the costs of providing, staffing and running laboratories, of
obtaining and examining samples, etc. it is at best short-sighted not to make effective use
of data generated. Procedures for statistical process control (SPC) have been around for
more than 80 years and have been the subject of many books and papers (see for instance,
Shewhart, 1931; Anon., 1956; Juran, 1974; Duncan, 1986; Beauregard et al., 1992;
Montgomery, 2000;) although few have referred to microbiological applications. The benefits of, and approaches to, SPC for trend analysis of microbiological data was included
in the report of a US FDA-funded programme undertaken by an AOAC Presidential Task
Force programme on ‘Best Practices in Microbiological Methods’. The report (Anon.,
2006a,b) is freely available on the FDA website.
WHAT IS SPC?
SPC is a practical approach to continuous control of quality (and safety) that uses various tools to monitor, analyse, evaluate and control a process and is based on the philosophical concept that it is essential to use the best available data in order to optimise
process efficiency. In this context, the term ‘process’ is any activity that is subject to variability and includes not just production, storage and distribution within an industry, such
as food processing, but also other operations where it is necessary to ensure that day-to-day
operations are in control. It is therefore relevant also to laboratory monitoring and testing
activities, where ‘process’ describes the production of analytical data and to other forms of
quantifiable activity.
The concept was introduced in the late 1920s by Dr William Shewhart whose work helped
to improve efficiency in the United States manufacturing industry by moving from traditional
quality control to one based on quality assurance. Hence, SPC is based on the concept of
Statistical Aspects of the Microbiological Examination of Foods
Copyright © 2008 by Academic Press. All rights of reproduction in any form reserved.
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STATISTICAL ASPECTS OF THE MICROBIOLOGICAL EXAMINATION OF FOODS
using data obtained from process line measurements and/or on samples of intermediate and/
or finished products, rather than reliance on 100% final product inspection. In this context,
quality assurance implies that appropriate checks are ‘built in’ to ensure that a production
process is in control (a concept not dissimilar to that of the HACCP approach to food safety
management). Assurance of production control comes from the regular monitoring of process
parameters and on tests to evaluate product, or other, samples taken during a process. The
target of SPC is to achieve a stable process, the operational performance of which is predictable, through monitoring, analysis and evaluation. It is therefore one of the tools of Total
Quality Management, a subject largely outside the scope of this book, but considered in the
context of the definition and implementation of Food Safety Objectives in Chapter 14.
At its simplest, SPC in manufacturing production is done by operative recording of simple physical (time, temperature, pressure, etc.), chemical (pH values, acidity titrations, etc.)
or other parameters critical to effective operation of a process plant. Such values are plotted
on appropriately designed record forms or charts. An inherent problem in any manufacturing operation is that both management and operatives have a tendency to ‘twiddle’ knobs
and adjust settings, probably with the best of intentions. If it is suspected that a process is
not quite running properly it is a natural human instinct to try to put matters right – but by
so doing matters may get worse. Provided that control systems are operating properly, there
is no necessity for human intervention – but if there is clear evidence of an ‘out of control’
situation developing, then action needs to be taken to identify and correct the cause. SPC
provides an objective approach to aid decision-making for process control.
Data obtained by analysis of critical product compositional parameters can also be charted.
SPC provides a means to assess trends in the data over time in order to ensure that the products
of a process conform to previously approved criteria. Evidence of significant trends away from
the ‘norm’ requires remedial action, for example, to halt production and/or to return the process operation to a controlled status. In a similar way, SPC of internal laboratory quality performance monitoring data can be used to provide assurance that analyses are being undertaken
properly and to identify when potential problems have arisen.
TREND ANALYSIS
At its simplest, trend analysis involves plotting data values, for which a target value can be
established, against values for time, production lot number or some other identifiable parameter. Note that a target value is not a product specification; rather it is a value derived by
analysis of product produced under conditions when the process is known to be ‘in control’.
Figure 12.1 provides a simple schematic of trend lines that lack any form of defined limits.
Although the direction of the trend is obvious, in the absence of control limits, there is no
guidance on whether or not a process has ‘gone out of control’.
One of the philosophical aspects of SPC is that once control has been achieved, the
approach can be converted to Statistical Process Improvement whereby process and, subsequently, product improvement is based on the principles of Total Quality Management
(Beauregard et al., 1992). It is noteworthy that the European legislation on microbiological
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STATISTICAL PROCESS CONTROL USING MICROBIOLOGICAL DATA
227
Product temperature (°C)
8
7
d
6
a
5
b
4
3
c
2
8
9
10
11
12
13
14
Time (h)
FIGURE 12.1 Hypothetical illustration of trends in product temperature: (a) Perfect control (never seen in real
life situations); (b) Regular cycling within limits; (c) Continuous reduction in temperature; (d) Continuous increase
in temperature.
criteria for foodstuffs recommends that manufacturers should use trend analysis to record
and examine the results of routine testing (Anon., 2005).
TOOLS FOR SPC
Shewhart’s original (1931) concept of a process is depicted in Fig. 12.2 as a ‘cause and effect’
(fishtail) diagram comprising materials, methods, manpower, equipment and measurement
systems that together make up the process environment. The extent of variability (i.e. lack of
control) associated with each of these factors impacts on the subsequent stages of the process
and the overall variability in a process can be determined from the individual factors, viz:
Voutput K
∑ VK
i 1
Vmaterials Vequipment Vmethod Vpeople ... VK
where VK variance associated with an individual factor K. Evidence of a high variance in
some measurable quality of the finished product will reflect in the sum of the variances of
the manufacturing stages, the causes of the variances being either random or inherent.
Consequently, the basic statistical tools for SPC are the estimates of the mean (or median)
value, the variance and the distribution characteristics of key parameters associated with a
final, or intermediate stage, product and include the variance associated with the monitoring
and recording of critical control point parameters, such as time, temperature and pressure
(cf. HACCP). From many perspectives, this is similar to the estimation of the measurement
uncertainty of a method (Chapters 10 and 11). A microbiological procedure to assess the
quality or safety of a product can be used to verify that a production process is ‘in control’
but only if the microbiological test procedure is itself ‘in control’.
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STATISTICAL ASPECTS OF THE MICROBIOLOGICAL EXAMINATION OF FOODS
228
Methods
People
Environment
Output
Measurement
system
Machine
FIGURE 12.2
Materials
Schematic ‘Cause and Effect’ description of a process – after Shewhart.
The principle of SPC charts relies on the properties of the normal distribution curve
(Chapter 3), which has a symmetrical distribution around the mean value, such that 95%
of the population falls within the range of mean 2 SDs and 99.7% falls within the range
of mean 3 SDs. However, as we have observed previously, the distribution of microbiological data is inherently asymmetrical and does not usually conform to the normal distribution. It is therefore necessary to use an appropriate transformation of the data, depending
on its actual distribution characteristics. As explained earlier (Chapter 4) the appropriate
transformation for microbiological data could be yi loge xi or yi log10 xi for data that
are log-normal distributed (e.g. colony counts), or yi 冪xi for data conforming to Poisson
(e.g. for low cell or colony counts), or the inverse hyperbolic sine transformation for data
conforming to a negative binomial distribution.
Before any system of SPC can be used it is essential to define the process, the measurement system and the expected variability of the process characteristics to be measured. From
a microbiological perspective, therefore, it is necessary to understand how a product is processed, the microbial associations of the ingredients and the product, and the procedures to be
used to determine the microbial composition of the product.
SPC Performance Standards
It is necessary to introduce ‘performance standards’ for implementation of SPC for microbiological data (Anon., 2006a). Such ‘performance standards’ are not meant to prescribe
procedures that should be used for evaluating processes, but to provide guidance on the
methodology to be used:
1. Charting of data is necessary to gain full benefit from SPC. Charts of output data provide a visual aid to detecting and identifying sources of unexpected variation leading to
their elimination.
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STATISTICAL PROCESS CONTROL USING MICROBIOLOGICAL DATA
229
2. Results to be plotted in a control chart, when the process is known to be ‘in control’
should be normally, or nearly normally, distributed. Where this is not the case then the data
must be ‘normalised’ by an appropriate mathematical transformation.
3. During the initial setting up of the charts it is essential that the process be operated in a
stable and controlled manner, so that the data values used to establish the operational parameters of the SPC procedure are normally distributed. The ‘rule of thumb’ requires at least 20
sets of individual results to compute the mean, standard deviation and other summary statistics in order to estimate the distribution of the results and construct control limits and target
value.
4. Rules for evaluating process control require the assessment of both Type I and Type II
errors: a Type I error defines the process to be ‘out of control’ when it is not and a Type
II error defines a process to be ‘in control’ when it is not. The probabilities to be used for
this purpose are based on the and probabilities, respectively, of the normal distribution
curve. In addition, the average run length (ARL), which is the expected number of results
before an ‘out of control’ signal is seen, is also an important parameter.
5. When a process is believed to be ‘in control’, the limits for assessing individual results
are set at some distance from the mean or process target value. In general, the default distance is 3SDs since the (Type I error) should be kept low (i.e. 1%).
6. In using the SPC approach it is necessary to establish rules to assess whether a
change in the process average has occurred. Such rules include the use of moving
averages, and the number of consecutive values moving in a single direction.
7. Process control limits used in SPC are not product criteria limits, set by customer or regulation; rather they are process-related limits. Product specifications should not be shown
on control charts.
A fuller explanation of performance standards for SPC can be found in the BPMM
report that is freely available on the Internet (Anon., 2006a,b).
SETTING CONTROL LIMITS
Assume, for a defined non-sterile food product, processed using a specific manufacturing
system, that the microbial load is considered to be acceptable at the end of processing if
the average aerobic colony count at 30°C is, say, 104 cfu/g (4.0 log10 cfu/g). Assume also
that the log10-transformed counts conform reasonably to a normal distribution and that the
standard uncertainty of the measurement method used to establish the levels of organisms
from the product is 0.25 log10 cfu/g. Since we are interested only in high counts, a lower
limit is not necessary. Then, for an one-sided process control limit, if the mean colony count
level is 104 cfu/g (4.0 log10 cfu/g) then, for 0.01, the upper 99% confidence interval will
be (4.0 2.326 0.25) 4.58 log10 cfu/g. It would then be expected that the probability
of a data value exceeding 4.6 log10 cfu/g would occur on average only on 1 occasion in 100
tests. It is useful also to set a process control warning limit, e.g. for 0.05 the warning
limit would be (4.0 1.645 0.25) 4.41 log10 cfu/g. These upper values, together with
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STATISTICAL ASPECTS OF THE MICROBIOLOGICAL EXAMINATION OF FOODS
the mean value of 4.0 log10 cfu/g, constitute the limit values drawn on the SPC control
chart for average values.
Data from tests on a large number of samples, taken during processing of a single batch
of products, should conform (after transformation if necessary) to a normal distribution
(Fig. 12.3a) and should lie within the upper and lower control limits. However, a data distribution that is skewed (Fig. 12.3b) shows the process to have been out of control even
though all the data values lie within the limits. Data conforming to a normal distribution
with an increased mean value (Fig. 12.3c) will include results that exceed the upper limit
thus showing that some aspect of the process was not properly controlled (for instance, poor
quality ingredients). Other variations could include data with larger variances, such that the
distribution curve is spread more widely, data with smaller variances leading to a reduced
spread of results, negatively skewed data where a majority of the values are below the control target, and multimodal distributions. A bimodal distribution (Fig. 12.3d) suggests that
the process either used two different sources of a major ingredient or that a major change in
process conditions occurred during manufacture.
SHEWHART’S CONTROL CHARTS FOR VARIABLES DATA
Control charts present process data over a period of time and compare these to limits devised
using statistically based inferential techniques. The charts demonstrate when a process is in a
state of control and quickly draw attention to even relatively small, but consistent, changes
away from the ‘steady state’ condition, leading to a requirement to a stop or adjust a process
before defects occur. Most control charts are based on the central limit theory of normal distribution using sample average and variance estimates. Of course, in the case of microbiological data, levels of colony forming units need to be normalised by an appropriate mathematical
transformation (e.g. log transformation).
There are four major types of control chart for variables data:
1.
2.
3.
4.
The x chart, based on the average result;
The R chart, based on the range of the results;
The s chart, based on the sample standard distribution; and
The x chart, based on individual results.
The R and s charts monitor variations in the process whereas the x and x charts monitor the location of results relative to predetermined limits. Often charts are combined, for
example, into an x and R chart or an x and s chart; in addition, special versions of control charts may be used for particular purposes such as batch operations.
x and R Charts
These use the average and range of values determined on a set of samples to monitor the
process location and process variation. Use of a typical x and R chart is illustrated in
Example 12.1. The primary data for setting up a chart is derived by analysis, using the
procedure to be used in process verification, of at least 5 sample units from at least 20 sets
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STATISTICAL PROCESS CONTROL USING MICROBIOLOGICAL DATA
231
(a)
(b)
(c)
(d)
Target
value
Limit
value
FIGURE 12.3 In control or in specification? (a) In control and within limit; (b) In control but not within limit; (c)
Within limit but not in control due to skewness; (d) Bimodal distribution – some, but not all, data conform to the
target and limit values.
of product from the specific process. After normalisation of the data, the average ( x ) and
range (R) of values is derived for each set of samples and the data are plotted as a histogram to confirm that the pattern of distribution conforms reasonably to a normal distribution. Assuming that the distribution is acceptable, the process average (target value x )
and the process average range ( R ) are calculated from the individual sample set averages
and ranges, using the equations:
Target value x ∑x;
n
Target value (R) R ∑R
n
where x sample average, R sample range and n number of sample sets.
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STATISTICAL ASPECTS OF THE MICROBIOLOGICAL EXAMINATION OF FOODS
232
Control limits need to be established for both sample average and sample range data
sets; if both upper and lower confidence limits (CLs) are required, these are located symmetrically about the target (centre) line; in most cases, microbiological tests of quality/safety, require only upper confidence limits (UCLs). Usually these will be set at
UCL x x (A 2 R) and UCLR D4 R for the 99% UCLs, where A2 and D4 are factors
that are dependent on the number of sample units tested (see Table 12.2). The values for the
Target Line and UCLs are drawn on the charts and are used to test the original sets of sample units for conformity (Example 12.1). If all data sets conform then the chart can be used
for routine testing, but if there is evidence of non-conformance, the cause must be investigated and corrected, the process resampled and new targets and UCLs established. The
values for the control lines should be reviewed regularly; evidence that the limits are wider
than previously determined is a cause for concern and the reasons must be investigated. If
process improvements occur over time, the limits can be tightened accordingly. Procedures
for assessing whether or not a process is going out of control are described below.
EXAMPLE 12.1 DATA HANDLING FOR PREPARATION OF
SHEWHART CONTROL CHARTS FOR MEAN,
STANDARD DEVIATION AND RANGE VALUES ( x , s
AND R CHARTS)
Suppose that 24 sets of aerobic colony counts were done on a product prepared in one production plant. Each dataset consists of 5 replicate samples, each of which was examined
in duplicate. How do we prepare an SPC chart?
The counts are log10-transformed and tested for conformance with normality, skewness
and kurtosis – although not perfect, the fit is adequate for purpose. The mean, variance,
standard deviation and range of log counts for each sample set are given in Table 12.1.
The overall mean values were then determined: average mean colony count ( x ) 4.98;
average standard deviation ( s ) 0.18 ; and average range ( R ) 0.43 (all as log10 cfu/g).
Table 12.1.
TABLE 12.1
The Mean, Standard Deviation and Range of Aerobic Colony
Counts for 5 Samples of Each of 24 Sets of a Fresh Food Product
Aerobic colony count (log10 cfu/g)
Occasion
1
2
3
4
5
6
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Mean ( x )
5.05
5.06
4.81
4.76
4.84
4.72
SD (s)
0.343
0.359
0.197
0.140
0.163
0.107
Range (R)
0.72
0.81
0.47
0.34
0.43
0.29
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STATISTICAL PROCESS CONTROL USING MICROBIOLOGICAL DATA
233
Aerobic colony count (log10 cfu/g)
Occasion
Mean ( x )
SD (s)
Range (R)
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
4.95
5.11
5.03
5.06
5.11
5.13
4.74
4.80
4.91
5.13
5.05
4.97
4.95
5.05
5.09
5.04
5.13
5.01
0.192
0.149
0.123
0.235
0.148
0.198
0.199
0.142
0.247
0.144
0.225
0.093
0.066
0.140
0.132
0.145
0.182
0.255
0.42
0.40
0.30
0.59
0.31
0.48
0.44
0.35
0.59
0.35
0.54
0.25
0.17
0.35
0.34
0.37
0.47
0.54
Overall mean
4.98
0.180
0.43
Setting Control Limits (CL)
Control limits are established for each chart at 3 SD above and below the target (mean
value) line; additional control lines can also be inserted at 1 and 2 SDs.
A ‘Mean value’ ( x ) chart
The centre line is set at the overall mean value (i.e. for our data this is 4.98 log10 cfu/g).
The upper and lower control limits are determined as:
UCL x x A 2 R
and the LCL x x A 2 R
where x is the overall mean value, A2 is a constant (Table 12.2) and R is the mean
range. An alternative calculation uses the average standard deviation ( s ):
UCL x x A 3 s
and the LCL x x A 3 s
x 4 . 98 and R 0 . 43 , UCL x 4 . 98 0 . 577 0 . 43 5 . 23 ; the
For our data, with alternative procedure using s 0 . 180 , gives UCL x 4 . 98 1 . 427 0 . 180 5 . 24 .We
can determine the LCL similarly: LCL x 4 . 98 0 . 577 0 . 43 4 . 98 0 . 26 4 . 74 .
These control limits are drawn on the graph and the data plotted (Fig. 12.4). All data
values lie within the range 2 SD to 3 SD. Since in most microbiological situations
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STATISTICAL ASPECTS OF THE MICROBIOLOGICAL EXAMINATION OF FOODS
234
lower counts are generally perceived as being beneficial, the LCL is rarely used. This
chart will be used further in the Example 12.2.
A Range (R) chart
The mean value for the range ( R ) chart 0.43; the UCLR and LCLR values are derived
from the equations:
UCL R D4 R
and the LCL R D3 R
where, D3 and D4 are constants (Table 12.2). For our data:
UCL R 2 . 114 0 . 43 0 . 909 ⬇ 0 . 91 and
LCL R 0 . 076 0 . 43 0 . 033 ⬇ 0 . 03
As before, these values are entered as control limits on a graph and the range data
values plotted (Fig. 12.5).
A Standard Deviation (s) Chart
The mean value of the standard deviations ( s ) 0.180. The upper and lower control
limits are determined as: UCL s B4 s and LCL s B3 s , where B3 and B4 are constants
from Table 12.2. For our data:
UCL s 2 . 089 0 . 180 0 . 376 ⬇ 0 . 38 and
LCL s 0 . 030 0 . 180 0 . 0054 ⬇ 0 . 0 1
TABLE 12.2
Some Constants for Control Chart Formulae
x & R Charts
Number of
replicates
2
3
4
5
6
7
8
10
15
20
a
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x -Chart
limits
A2
1.880
1.023
0.729
0.577
0.483
0.419
0.373
0.308
0.223
0.180
x & s Charts
R-Chart
limitsa
D3
–
–
–
–
–
0.076
0.136
0.223
0.347
0.180
x -Chart
limits
D4
A3
3.267
2.574
2.282
2.114
2.004
1.924
1.824
1.777
1.653
1.585
2.659
1.954
1.628
1.427
1.287
1.182
1.099
0.975
0.789
0.680
s-Chart
limitsa
B3
–
–
–
–
0.030
0.118
0.185
0.284
0.428
0.510
B4
3.267
2.568
2.266
2.089
1.970
1.882
1.815
1.716
1.572
1.490
Number of
replicates
2
3
4
5
6
7
8
10
15
20
If a value is not shown, use the next higher value.
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STATISTICAL PROCESS CONTROL USING MICROBIOLOGICAL DATA
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These values are entered onto the chart and the standard deviation values plotted
(Fig. 12.6).
Interpretation of the Standard Reference Charts
The data used to derive the control limits were obtained on product samples taken when
the process was believed to be ‘in control’. In order to confirm that both the process and the
measurements were ‘in control’ it is essential to chart these data before using the control charts for routine purposes. Figures 12.5 and 12.6 show that the data values for
the range and standard deviation charts are randomly distributed around the mid
line with no sequential runs of data on either side and they all lie between the upper
and lower CLs. However, the x chart (Fig. 12.4) shows that several values lie outside
the 2 SD LCL and one value is on that line. In a two-sided test this would indicate
that the process was not absolutely ‘in control’ for these particular batches – but in this
instance, for bacterial counts, we are interested only in the UCL so these low values can
be ignored and the data above the centre line is ‘in control’. These charts will be used in
Example 12.2.
x chart
5.3
UCL 5.24
5.2
x (log cfu/g)
5.1
5.0
Avg 4.98
4.9
4.8
LCL 4.72
4.7
4.6
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Sample number
FIGURE 12.4 Statistical control chart for mean ( x ) colony count (as log10 cfu/g) on 24 sets of replicate examinations of a fresh food product produced under controlled conditions. The heavy dotted
lines (– – – – –) show the upper (UCL) and lower (LCL) 99~~% CL. The lighter dotted lines (……) are
the upper and lower limits for 1 SD and 2 SDs around the centre line.
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R chart
1.00
(s 0.19)
UCL 0.91
0.90
Range (log cfu/g)
0.80
0.70
0.60
0.50
Avg 0.43
0.40
0.30
0.20
0.10
0.00
1 2
3 4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Sample number
FIGURE 12.5 Statistical control chart for range (R ) of log10 cfu/g on 24 sets of replicate examinations of a fresh food product produced under controlled conditions.
s chart
Sample standard deviation
0.4
(s 0.192)
UCL = 0.38
0.35
0.3
0.25
0.2
Avg = 0.18
0.15
0.1
0.05
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Sample number
FIGURE 12.6 Statistical control chart for standard deviation (s) of log10 cfu/g on 24 sets of replicate examinations of a fresh food product produced under controlled conditions.
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STATISTICAL PROCESS CONTROL USING MICROBIOLOGICAL DATA
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x and s Charts
For sample unit sizes greater than 12, the standard deviation (s) should be used to monitor variation rather than the range (R) because it is more sensitive to changes in the process.
However, it should be noted that the distribution of s tends to be symmetrical only when
n 25, otherwise it is positively skewed. Thus the s chart is less sensitive in identifying nonnormal conditions that cause only a single value in a sample to be unusual.
The Target lines for the x and s charts are x and s , respectively. The calculation for x
was shown previously; s is determined by averaging the sample standard deviations of the
individual sample sets:
k
Centre line s ∑ si
i 1
k
s1 s2 ... sk
k
where k number of sample sets and si standard deviation for sample sets i 1, 2, 3, … k.
Note, however, that this is a biased average, since the true average standard deviation is
determined from:
k
s ∑ si2
i 1
k
s12 s22 ... sk2
k
.
The UCLs are calculated from:
UCL (x) x A3 s
UCL(s) B 4 s
where A3 and B4 are standard values (Table 12.2). The charts are prepared and interpreted
in a similar manner to x and R charts.
Interpretation of x , s and R Charts
If a process is ‘in control’ the data from critical parameter tests should conform to the rules
for normal distribution, that is, symmetry about the centre distribution line and randomness; 95% of data should lie within 2 s and 99.7% should lie within 3 s of the centre
line. A process complying with these rules is generally stable. If this is not the case, the
process may be out of control and the cause(s) must be found and rectified.
The key question is ‘How do you recognise out of control situations from the chart?’ At
least 11 ‘out of control’ patterns have been identified (Anon., 1956; Wheeler and Chambers,
1984; Beauregard et al., 1992). As a general rule, 6 patterns cover the majority of situations
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STATISTICAL ASPECTS OF THE MICROBIOLOGICAL EXAMINATION OF FOODS
likely to be encountered (Beauregard et al., 1992). For a single-sided test, such as is used for
microbiological data, these patterns can be summarised as:
1. A single point above the UCL that is set at 3 s above the central line. The probability
for any point to occur above UCL is less than 0.14% (about 1 in 700).
2. Seven consecutive points above the centre line (probability 0.78%; about 1 in 125).
3. Two consecutive points close to the UCL – the probability of 2 consecutive values
occurring close to a limit in a normal distribution is about 0.05% (1 in 2000).
4. Ten out of 11 consecutive points above the centre line – probability 0.54% (about 1 in
200). If reversed (i.e. going downwards) this pattern may also indicate a process change
that warrants investigation.
5. A trend of seven consecutive points in an upward (or downwards) line indicates a lack
of randomness.
6. Regular cycling around the centre line.
Beauregard et al. (1992) suggested some possible engineering causes for these patterns but
others may be equally plausible when examining microbiological data. When using microbiological colony counts or other quantitative measures it is essential to ensure that the test is itself
in control. Changes to the culture media, incubation conditions and other factors discussed
previously (Chapter 10) must be taken into account when examining SPC and other forms of
microbiological trend analysis to ensure that it is the process rather than the method of analysis
that is ‘out of control’! Examples 12.1 and 12.2 illustrate some ‘out of control’ patterns.
EXAMPLE 12.2 USE OF THE x AND R SPC CHARTS
Having prepared ‘standard’ SPC charts, how do we now use them to examine data from
routine production and what should we look for to indicate that a process is not ‘in
control’?
We now extend the database used in Example 12.1 by adding data for a further 57
samples, obtained when the plant was running normally. We plot the mean and range of
the log-transformed counts for each sample (Table 12.3) on these charts using the centre
line and UCL values derived for the mean log counts in Example 12.1 and using both the
UCL and LCL for the range of counts. What do the results tell us?
The x chart (Fig. 12.7) provides a ‘mountain range’ pattern with numerous peaks and
troughs. Note firstly that the mean colony counts on samples 26–28, 32, 50–52, 64–69, 75
and 79–81 all exceed the UCL, in some instances by a considerable amount. These process
batches are therefore ‘out of control’ and it would be necessary to investigate the causes.
Are the colony count data recorded correctly or was there some technical reason why high
counts might have been obtained in these instances? Was one or more of the ingredients different? Had a large quantity of ‘reworked’ material been blended in to the batch? And so on.
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STATISTICAL PROCESS CONTROL USING MICROBIOLOGICAL DATA
239
TABLE 12.3
Aerobic Colony Count Data on 81 Sets of Samples of a Fresh Food Product
Replicate colony counts (log10 cfu/g)
Sample
1
2
3
4
5
Mean
Range
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
4.73
4.81
4.93
4.83
4.77
4.72
5.16
5.07
5.02
4.93
4.93
5.22
4.48
4.71
5.20
5.38
5.26
5.06
4.98
5.02
5.00
5.12
5.29
5.06
5.37
5.23
4.47
4.61
4.80
4.84
5.12
5.36
4.91
5.34
4.96
5.20
4.87
4.95
4.78
5.09
4.72
4.82
4.90
4.94
5.29
5.08
5.21
4.78
4.65
4.82
4.90
4.79
4.92
4.75
4.93
5.07
4.92
4.75
5.16
4.78
4.57
4.89
4.61
5.04
5.00
4.97
5.04
4.96
5.13
5.17
5.13
4.72
5.17
4.84
4.94
4.95
4.65
4.55
4.74
5.09
5.07
5.06
5.23
5.26
4.92
4.60
5.13
5.04
5.26
4.96
4.87
5.02
4.94
5.06
5.19
5.25
5.35
5.62
4.80
4.64
5.08
4.76
4.78
4.95
5.21
5.23
5.24
5.18
4.85
4.84
4.85
5.11
5.01
5.01
4.95
5.29
5.10
4.80
4.82
5.26
5.05
5.06
4.81
4.76
4.84
4.72
4.95
5.11
5.03
5.06
5.11
5.13
4.74
4.80
4.91
5.13
5.05
4.97
4.95
5.05
5.09
5.04
5.13
5.01
0.72
0.81
0.47
0.34
0.43
0.29
0.42
0.40
0.30
0.59
0.31
0.48
0.44
0.35
0.59
0.35
0.54
0.25
0.17
0.35
0.34
0.37
0.47
0.54
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
5.66
5.15
4.78
5.48
4.70
4.86
5.45
5.00
5.08
4.80
4.68
4.78
5.01
4.28
4.73
4.85
5.00
5.08
6.17
5.26
4.66
4.99
5.70
5.60
4.90
4.30
4.60
4.97
4.80
4.50
4.82
4.80
4.80
5.78
5.20
5.00
4.58
4.65
6.03
5.20
5.41
4.60
4.63
4.62
4.85
4.74
4.91
5.10
5.01
5.15
5.80
5.20
4.80
4.95
5.97
4.78
4.90
5.08
4.86
4.80
4.85
4.70
5.10
4.95
5.20
5.60
5.97
5.26
5.02
5.02
6.28
5.48
5.58
4.90
4.97
4.85
4.90
4.80
5.05
5.35
5.13
5.35
5.58
5.24
4.75
4.89
5.89
5.21
5.18
4.74
4.75
4.80
4.88
4.60
4.92
5.01
0.86
0.70
1.39
0.48
0.44
0.37
0.83
0.82
0.68
0.78
0.37
0.35
0.21
0.52
0.37
0.55
(continued)
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240
TABLE 12.3 (Continued)
Replicate colony counts (log10 cfu/g)
Sample
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
CH012-N53039.indd 240
1
4.60
4.45
4.55
5.02
4.74
4.80
5.33
5.00
5.32
5.63
5.29
5.15
4.60
4.71
4.80
5.10
5.03
4.90
4.45
4.98
5.57
5.49
5.03
5.10
5.28
5.09
5.31
5.53
4.93
5.18
4.90
4.60
5.27
5.08
4.76
5.12
5.28
5.25
5.28
5.20
5.85
2
3
4
5
4.60
4.60
4.46
4.88
4.81
4.90
4.99
5.35
5.18
5.51
5.33
5.45
4.49
4.84
4.68
5.35
4.70
4.78
4.80
4.71
5.10
5.36
5.95
5.87
5.10
5.42
5.63
5.32
5.18
4.50
5.04
4.30
4.81
5.60
5.31
5.09
5.49
4.90
5.44
5.24
6.09
4.50
4.69
4.50
4.72
4.76
5.30
5.10
5.10
5.45
5.35
5.31
4.78
4.90
5.01
5.15
5.10
4.94
4.88
4.85
5.15
4.87
5.16
5.12
4.98
5.36
5.39
5.84
5.20
5.04
5.08
4.95
4.30
4.79
5.70
4.65
4.80
5.38
5.08
4.98
5.42
5.44
4.30
4.45
4.32
4.24
4.96
5.10
5.25
5.08
5.30
5.10
5.88
5.30
4.89
5.10
5.20
4.80
4.88
4.95
5.10
5.17
5.26
4.89
5.09
5.80
5.90
5.37
5.27
5.04
5.13
4.30
5.08
5.18
4.70
5.20
4.88
5.05
5.18
5.25
5.20
5.47
5.36
4.62
5.10
4.62
4.54
4.81
4.80
5.33
5.50
5.40
5.40
5.40
5.26
5.10
4.82
4.90
4.85
4.98
5.30
5.04
5.28
4.76
5.00
5.01
4.99
5.20
5.49
5.65
5.17
5.12
4.30
5.26
5.08
5.09
5.51
5.20
5.10
4.70
5.40
5.40
5.38
5.63
Mean
Range
4.53
4.66
4.49
4.68
4.82
4.98
5.20
5.21
5.33
5.40
5.44
5.19
4.80
4.90
4.95
5.04
4.91
4.96
4.85
5.06
5.11
5.18
5.24
5.35
5.37
5.35
5.54
5.25
5.08
4.67
5.05
4.69
4.93
5.42
4.96
5.03
5.21
5.18
5.26
5.34
5.67
0.32
0.65
0.30
0.78
0.22
0.50
0.34
0.50
0.27
0.53
0.59
0.67
0.61
0.39
0.52
0.55
0.33
0.52
0.65
0.57
0.81
0.60
0.94
0.89
0.80
0.40
0.57
0.49
0.25
0.88
0.36
0.88
0.57
0.62
0.66
0.32
0.79
0.50
0.46
0.27
0.73
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STATISTICAL PROCESS CONTROL USING MICROBIOLOGICAL DATA
241
x chart
(Avg 4.98, UCL 5.23, for Subgroups 1–24)
6.0
5.8
Mean log cfu/g
5.6
5.4
UCL
5.2
5.0 Average
4.8
4.6
4.4
1
6
11
16
21
27
32
42
47
52
57
62
67
72
77
82
Sample number
FIGURE 12.7 Statistical control chart for mean ( x ) colony count (as log10 cfu/g) on 82 sets of
replicate examinations of a fresh food product produced under controlled conditions. The control lines
were derived from the original 24 samples (Fig. 12.4).
Note secondly that the values for batches 32–33, 50–51, and 78–79 all lie within
the bounds of 2 SD to 3 SD (UCL) – another prime indicator of an ‘out of control’
situation that requires investigation. In addition, several of these batches are linked to
others where the colony counts exceed the UCL.
Then note that several consecutive batches of product have results lying in the area
above the average control line (e.g. batches 20–28, 51–55 and 62–70); in each case there
is a sequential run of 7 or more points above the line. However, it should be noted that
batches 20–24 were part of the ‘in control’ batches used to set up the control chart. If
we include also some values below the average line (e.g. 47–49) we have an even longer
sequential run of increasing mean colony count values.
The R chart (Fig. 12.8) is less dramatic though it still provides valuable information. It
shows successive runs of range values above the average line for batches 29–35 and 61–68.
In addition, the ranges on samples from batches 32 and 65 exceed the UCL for ranges,
indicating that one or more of the colony counts was considerably greater or smaller
than the other data values in the set. If you look at the data in Table 12.3, you will
note that for batch 27 the recorded colony counts were 4.78, 6.17, 5.20, 5.80 and 5.97 –
the analyst should have been suspicious of these wildly disparate results – were the
results calculated correctly or was this a heterogeneous batch of product? Similarly, for
batch 63 we have counts of 5.03, 5.95, 5.12, 5.09 and 5.01 log cfu/g. The second count is
significantly higher than the others – should it have been 4.95 not 5.95? Had a simple
calculation or transcription error resulted in a set of data that indicates an out of control
situation?
These are just examples of how SPC charts can aid both microbiological laboratory
control and also process control. Further rules for describing out of control situations
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STATISTICAL ASPECTS OF THE MICROBIOLOGICAL EXAMINATION OF FOODS
242
R Chart
(Avg 0.43, UCL 0.91, LCL None)
1.60
1.40
Range (log cfu/g)
1.20
1.00
UCL
0.80
0.60
Average
0.40
0.20
0.00
1
6
16
21
27
32
37
42
47
52
57
62
67
72
77
82
Sample number
FIGURE 12.8 Statistical control chart for range (R) of colony counts (as log10 cfu/g) on 82 sets
of replicate examinations of a fresh food product produced under controlled conditions. The control
lines were derived from the original 24 samples (Fig. 12.6).
within SPC charts are given in the text of this chapter. Finally, a word of warning: if you
use a computer program to generate control charts it is essential, once the control limits
have been set, to ensure that they are not updated automatically whenever the chart is
updated with new data!
CUSUM Charts
CUmulative SUM charts (CUSUM) provide a particularly useful means of process control.
They are more sensitive than the x and R charts for detection of small changes (0.5 –
1.5 SD) in process output but are less sensitive when large (2 SD) process shifts occur.
Nonetheless, because of their sensitivity to smaller shifts they signal changes in the process
mean much faster than do conventional charts.
CUSUM charts are constructed differently. The sum of the deviation of a normalised
value from a target statistic is plotted (see Example 12.3) and they do not have control
limits as such. Instead, they use decision criteria to determine when a process shift has
occurred. For standard CUSUM charts a ‘mask’ is used to cover the most recent data point;
if the mask covers any of the previously plotted points it indicates that a shift in process
conditions occurred at that point. Construction of the mask is complex and it is difficult to
use (Montgomery, 1990). Beauregard et al. (1992) recommend that it is not used and that a
CUSUM signal chart is used instead.
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243
EXAMPLE 12.3 USE OF A CUSUM CHART TO MONITOR A CRITICAL
CONTROL POINT
ATP bioluminescence is used to provide a rapid and simple method to monitor the
hygienic status of a process plant Critical Control Point (CCP). The data, in the form
of Relative Light Units (RLU), measure the amount of microbial or other ATP at the CCP
and the hygienic status of the process plant at that point is assessed by comparison with
previously determined maximum RLU levels on the basis of a Pass/Fail status. If the
test indicates that the plant has not been properly cleaned then cleaning and testing are
repeated until satisfactory results are obtained, before starting process operations. Can we
use a CUSUM chart to examine performance in monitoring this CCP?
Trend analysis of RLU data provides a simple means to monitor the efficiency of
cleaning operations over a period of time. Hayes et al. (1997) used the CUSUM approach
to analyse data from a dairy company. They argued that since the data set consists of
only single measurements (n 1) other forms of SPC were generally not suitable and the
raw data do not conform to a normal distribution. This is not strictly true since alternative procedures do exist. However, their data provides a useful set for CUSUM analysis.
The raw data and several alternative transformations are shown in Table 12.4.
Although a square root transformation (as used by Hayes et al., 1997) improves the normality of the distribution, there is still considerable skewness and kurtosis; both forms
of logarithmic transformation effectively normalise the data and remove both skewness and kurtosis (Table 12.5). Figure 12.9 shows the distributions of the raw and the
Ln-transformed data, ‘box and whisker’ plots (that show outliers values in the raw data)
and ‘normality plots’ and demonstrates the importance of choosing the correct transformation procedure to normalise data (Table 12.5).
The CUSUM plot is set up as follows using a spreadsheet (Table 12.6):
In column A, we list the data reference value (e.g. day numbers) and in column B the
actual data values (for this example these are the actual RLUs measured on each day). In
column C we transform the data values (column B) into the natural logarithms, that is,
the Ln-RLU values and in column D we show the process ‘target’ value (Ln 110 4.70).
We subtract the target value from the transformed data value in column E and in column
F we sequentially add the values from column E to generate the CUSUM values. These
values (column F) are then plotted against the day reference value (column A).
An ‘individuals’ plot of the ATP measurements (as RLUs) for each day of testing (Fig.
12.10) shows a day-to-day lack of consistency amongst the readings with values in excess
of the arbitrary control limit value of 110 RLUs, set prior to the routine use of the system, on days 46–47, 58, 69, 70, 71, 74 and 87. The standard plot shows the individual
values above the limit but fails to indicate any significant adverse trends. A similar situation is seen in a plot of the Ln-transformed data (Fig. 12.11). By comparison, the CUSUM
plot (Fig. 12.12) shows several adverse trends including a major adverse trend from day
65 onwards, thereby indicating that the process is going out of control, i.e. development
of a potential problem, before it actually occurred. This is where the CUSUM plot scores
over other forms of trend analysis – it is predictive and indicates that a process is going
‘out of control’ before it actually happens. However, to use the CUSUM system effectively
it is necessary to develop a CUSUM Mask, which is both difficult and very time consuming. An alternate procedure is shown in Example 12.4.
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244
TABLE 12.4
ATP ‘Hygiene Test’ Measurements on a CCP in a Dairy Plant, Before and After Transformation Using
the Square Root Function, the Natural Logarithm (Ln) and the Logarithm to Base 10 (log10)
Transformed RLU
Day
RLU
(x)
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
23
46
39
62
33
21
20
56
69
26
27
24
22
69
42
97
56
33
33
69
24
34
29
29
19
57
81
99
43
80
68
112
25
52
24
35
15
47
104
53
81
34
17
62
23
123
冪x
4.80
6.78
6.24
7.87
5.74
4.58
4.47
7.48
8.31
5.10
5.20
4.90
4.69
8.31
6.48
9.85
7.48
5.74
5.74
8.31
4.90
5.83
5.39
5.39
4.36
7.55
9.00
9.95
6.56
8.94
8.25
10.58
5.00
7.21
4.90
5.92
3.87
6.86
10.20
7.28
9.00
5.83
4.12
7.87
4.80
11.09
Transformed RLU
Ln (x)
log10 (x)
Day
RLU (x)
3.14
3.83
3.66
4.13
3.50
3.04
3.00
4.03
4.23
3.26
3.30
3.18
3.09
4.23
3.74
4.57
4.03
3.50
3.50
4.23
3.18
3.53
3.37
3.37
2.94
4.04
4.39
4.60
3.76
4.38
4.22
4.72
3.22
3.95
3.18
3.56
2.71
3.85
4.64
3.97
4.39
3.53
2.83
4.13
3.14
4.81
1.36
1.66
1.59
1.79
1.52
1.32
1.30
1.75
1.84
1.41
1.43
1.38
1.34
1.84
1.62
1.99
1.75
1.52
1.52
1.84
1.38
1.53
1.46
1.46
1.28
1.76
1.91
2.00
1.63
1.90
1.83
2.05
1.40
1.72
1.38
1.54
1.18
1.67
2.02
1.72
1.91
1.53
1.23
1.79
1.36
2.09
47
48
49
50
51
52
53
54
55
56
57
58
59
60
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
147
13
12
15
14
76
28
29
44
15
16
123
5
41
56
9
82
22
22
63
85
34
158
164
155
79
94
319
38
19
64
50
16
25
28
35
52
28
35
52
139
89
62
55
12
67
冪x
12.12
3.61
3.46
3.87
3.74
8.72
5.29
5.39
6.63
3.87
4.00
11.09
2.24
6.40
7.48
3.00
9.06
4.69
4.69
7.94
9.22
5.83
12.57
12.81
12.45
8.89
9.70
17.86
6.16
4.36
8.00
7.07
4.00
5.00
5.29
5.92
7.21
5.29
5.92
7.21
11.79
9.43
7.87
7.42
3.46
8.19
Ln (x)
4.99
2.56
2.48
2.71
2.64
4.33
3.33
3.37
3.78
2.71
2.77
4.81
1.61
3.71
4.03
2.20
4.41
3.09
3.09
4.14
4.44
3.53
5.06
5.10
5.04
4.37
4.54
5.77
3.64
2.94
4.16
3.91
2.77
3.22
3.33
3.56
3.95
3.33
3.56
3.95
4.93
4.49
4.13
4.01
2.48
4.20
log10 (x)
2.17
1.11
1.08
1.18
1.15
1.88
1.45
1.46
1.64
1.18
1.20
2.09
0.70
1.61
1.75
0.95
1.91
1.34
1.34
1.80
1.93
1.53
2.20
2.21
2.19
1.90
1.97
2.50
1.58
1.28
1.81
1.70
1.20
1.40
1.45
1.54
1.72
1.45
1.54
1.72
2.14
1.95
1.79
1.74
1.08
1.83
Source: Based on the data of Hayes et al. (1997).
CH012-N53039.indd 244
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245
60
25
50
20
40
Frequency
Frequency
STATISTICAL PROCESS CONTROL USING MICROBIOLOGICAL DATA
30
20
15
10
5
10
0
0
5
3
4
2
Normal quantile
Normal quantile
6
3
2
1
0
1
1
0
1
2
2
3
3
0
50 100 150 200 250 300 350
RLU
1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
Ln RLU
FIGURE 12.9 Data distributions, ‘box and whisker’ plots and ‘normality’ plots for untransformed
and Ln-transformed data (as relative light units, RLU) determined by ATP analysis of the mandrel
of a dairy plant bottling machine.
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TABLE 12.5
Descriptive Statistics of Data from Table 12.4. Note that the Tests for Normality, Skewness and
Kurtosis are Identical for the Two Alternative Logarithmic Transformations Both of Which
Demonstrate Conformance with Normality; Note Also that Neither the Original Data nor the Square
Root Transformed Data Comply with a Normal Distribution and Show Marked Skewness and Kurtosis
Transformed RLU
Parameter
Number of
Tests (n)
Mean
Median
Variance
Standard
deviation
Normality
test a
Skewness
Kurtosis
RLU (x)
冪x
Ln (x)
log10 (x)
92
92
92
92
54.6
41.5
2109.8
45.93
6.90
6.44
7.044
2.654
3.726
3.726
0.558
0.747
1.618
1.618
0.105
0.324
4.69 (P 0.0001)
1.33 (P 0.0019)
0.205 (P 0.8723)
0.205 (P 0.8723)
2.67 (P 0.0001)
11.45 (P 0.0001)
1.13 (P 0.0001)
2.18 (P 0.0055)
0.004 (P 0.9864)
0.025 (P 0.8904)
0.004 (P 0.9864)
0.025 (P 0.8904)
Source: Based on the data of Hayes et al. (1997).
a
Anderson-Darling Test.
TABLE 12.6
Spreadsheet Layout for Data Used to Derive a Cumulative Sum (CUSUM) Chart
A
B
C
D
E
F
1
2
Day
RLU
Ln-RLU
Target (Ln-RLU)
Difference (C – D)
CUSUM
3
0
4
1
23
3.135
3.912
0.777
0.777
5
2
46
3.829
3.912
0.083
0.860
1.108
6
3
39
3.664
3.912
0.248
7
4
62
4.127
3.912
0.215
0.893
8
5
33
3.497
3.912
0.416
1.309
9
6
21
3.045
3.912
0.868
2.176
10
7
20
2.996
3.912
0.916
3.093
11
8
56
4.025
3.912
0.113
2.979
12
9
69
4.234
3.912
0.322
2.657
13
10
26
3.258
3.912
0.654
3.311
…
…
…
…
…
…
…
…
…
…
…
…
90
55
4.007
3.912
0.095
12.090
92
CH012-N53039.indd 246
93
91
12
2.485
3.912
1.427
13.517
94
92
67
4.205
3.912
0.293
13.224
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STATISTICAL PROCESS CONTROL USING MICROBIOLOGICAL DATA
350
247
Individuals chart
(Target 50, UCL 110, LCL 0, s 33.12, for Subgroups 1–92)
300
250
RLU
200
150
UCL
100
Tartget
50
0
1
10
19
28
37
46
55
64
73
82
91
Sample number
FIGURE 12.10 Individuals plot of the ATP data from hygiene tests on a dairy bottling mandrel.
Individuals chart
(Target 3.90, UCL 4.70)
6.0
5.5
5.0
UCL
Ln RLU
4.5
Target
4.0
3.5
3.0
2.5
2.0
1.5
1.0
1
11
21
31
41
51
61
71
81
91
Sample number
FIGURE 12.11 Individuals plot of the ATP data (as Ln-RLU) from hygiene tests on a dairy bottling mandrel.
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0.0
Cumulative sum of Ln RLU
5.0
10.0
15.0
20.0
25.0
0
10
20
30
40
50
60
70
80
90
Day number
FIGURE 12.12 CUSUM plot of the ATP data (as Ln-RLU) from hygiene tests on a dairy bottling
mandrel.
The CUSUM Signal Chart
This consists of two parallel graphs of modified cumulative sum values.
The upper graph is a plot of cumulative sum upper signal values in relation to an upper
signal alarm limit (USAL). The lower graph records the cumulative sum lower signal values
against a lower signal alarm limit (LSAL). The cumulative signal values are calculated by
adjusting the nominal mean by a signal factor (SF), which is a derived value for the level of
shift to be detected (Example 12.4):
Cumulative USAL ∑ (x v
SF)
Cumulative LSAL ∑ (x v
SF)
where, x v is the nominalised mean ( x – a target value). Limits may be applied to both
the upper and lower signal values. The upper signal value can never be negative so that
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STATISTICAL PROCESS CONTROL USING MICROBIOLOGICAL DATA
249
if the calculation indicates a negative value then it is put equal to zero and is not plotted.
Similarly, the lower signal value can never be positive so a positive value is put as zero and
is not plotted. Example 12.4 illustrates the derivation of a CUSUM signal plot.
EXAMPLE 12.4 CUSUM SIGNAL PLOT
Can we modify the CUSUM chart to provide more definitive information on ‘out of control’ situations?
The signal chart consists of two parallel graphs of modified CUSUM values. The upper
graph plots a ‘cumulative upper signal value’ that is monitored against an upper signal
alarm limit (USAL). The lower graph monitors a ‘cumulative lower signal value’ against
a LSAL. Each data value is ‘nominalised’ by subtraction of a target value, the nominalised mean ( xv ) ( x – target value). The upper and lower signal values are calculated
by adjusting the ‘nominalised’ mean value by a ‘signal factor’ (SF), which is a function
of the extent of change (level of shift) to be detected and the standard deviation of the
data and is calculated as: SF (change to be detected) (sample standard deviation/2).
If a change of 1 standard deviation is to be signalled by the CUSUM chart, then the
SF SD/2. The signal values are calculated as:
Cumulative upper signal value ( xv SF)
Cumulative lower signal value ( xv SF)
The USAL and LSAL, against which any change is monitored, are set at 10 SF
and 10 SF, respectively. To use the system, the data are ‘nominalised’, adjusted by the
SF and then summed cumulatively. Table 12.7 shows how the data for analysis is set out
in a spreadsheet; for this example, the data are the individual Ln-transformed values
used in Example 12.3.
Note that in column E and G, respectively, the cumulative US values 0 and the
cumulative LS values 0 are shown as [0]. Zero values are not plotted for either signal
value. In deriving the Table, the SD used (s 0.75) is that of the Ln-transformed group
(see Table 12.5) and the ‘nominal value’ used to ‘nominalise’ the data is the approximate
mean value of the entire group (3.7).
The cumulative Upper Signal (US) values (column E) and the cumulative Lower
Signal (LS) values are plotted against sample number, to give a series of individual
points and lines. Whilst it is simple to plot the data manually, it is difficult to achieve
the effect seen in Fig. 12.13 using normal Excel graphics – these lines were produced
using an add-on programme for SPC. Figure 12.13 shows a continuous rise in US values
from day 69 to day 74, with days 74 to 78 all exceeding the USAL that is set at 10 SF
(i.e. 10(s/2) 3.75).
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TABLE 12.7
Spread sheet Data Layout for Deriving CUSUM Signal Values – Only the First 18 Values are Shown. The Nominal
Value is Taken as 3.7 (Mean Ln Value for All 92 Data Points) and s 0.75 (from Table 12.5)
1
A
B
C
D
E
F
G
Day
Ln RLU
Nominalised
(Nominal 3.7)
Nominalised
SF
Cumulative US
value
Nominalised
+SF
Cumulative LS
Value
xv ( x 3 . 7)
Xv s / 2
∑ ( xv s / 2)
Xv s / 2
Xv s / 2
2
x
3
1
3.135
0.56
0.94
[0]
4
2
3.829
0.13
0.25
[0]
0.50
[0] b
5
3
3.664
0.04
0.41
[0]
0.34
[0]
6
4
4.127
0.43
0.05
7
5
3.497
0.20
0.58
0.19
a
0.05
[0]
0.19
0.80
[0]
0.17
[0]
8
6
3.045
0.66
1.03
[0]
0.28
0.28
9
7
2.996
0.70
1.08
[0]
0.33
0.61
[0]
10
8
4.025
0.33
0.05
11
9
4.234
0.53
0.16
0.16
0.70
[0]
0.91
[0]
12
10
3.258
0.44
0.82
[0]
0.07
0.07
13
11
3.296
0.40
0.78
[0]
0.03
0.10
14
12
3.178
0.52
0.90
[0]
0.15
0.24
15
13
3.091
0.61
0.98
[0]
0.23
0.48
16
14
4.234
0.53
0.16
17
15
3.738
0.04
0.34
0.16
[0]
0.91
[0]
0.41
[0]
18
16
4.575
0.87
0.50
0.50
1.25
[0]
19
17
4.025
0.33
0.05
0.45
0.70
[0]
20
18
3.497
0.20
0.58
0.17
[0]
[0]
Note: The value of Ln-RLU is shown as x since in most CUSUM signal charts this would be the mean value
of two or more values. The cumulative US and cumulative LS values are plotted in Fig. 12.13.
a
values 0 are shown as [0] and not included in the chart.
b
values 0 are shown as [0] and not included in the chart.
CUSUM signal chart
5
Signal value
USAL 3.75
Avg 0.00
LSAL 3.75
-5
1
11
21
31
41
51
61
71
81
91
Sample number
FIGURE 12.13 CUSUM signal plot of the ATP data (as Ln-RLU) from hygiene tests on a dairy
bottling mandrel 䊏 Upper Signal values; 䉬 Lower Signal values.
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251
The Interpretation of CUSUM Signal Charts
Interpretation is relatively straightforward. Any value that exceeds the USAL (or for some
purposes the LSAL) is deemed to be ‘out of control’ since the signal value limits are set to
enable the detection of a predetermined change, usually 10 times the SF. Smaller signals may
be indicative of less extreme changes.
Control Charts for Attribute Data
Attribute control charts can be used for the results of quantal tests (e.g. tests for the presence of salmonellae in a given quantity of sample), but are probably worthwhile only if
many sample units are tested. The charts can be set up for numbers or proportions of
non-conforming sample units. For microbiological tests, a number chart for non-conforming units is probably the most useful but the number of sample units tested must generally
remain constant. Anon. (2006b) provides special measures for examining and plotting data
when different numbers of samples are tested.
For an np chart the centre line is the average number of non-conforming units ( np ),
where p is the average proportion of positive results, (npi) number of non-conforming
samples in sample set i 1 to k and k number of sample sets:
k
np ∑ (np)i
i 1
k
(np)1 (np)2 ... (np)k
k
The upper and lower control limits are calculated from:
UCLnp np 3
np(1 − np)
n
LCLnp np 3
np(1 np)
n
where, n the sample size. The use of this system is shown in Examples 12.5 and 12.6.
EXAMPLE 12.5 SPC OF ATTRIBUTES DATA
How can I set up an SPC for data on the prevalence of organisms from quantal tests?
Special versions of SPC are available for analysis of attributes data. Attributes can
include the occurrence of defective products (Example 12.6), the prevalence of detection
of pathogenic or index organisms in food samples, or some other measurable attribute.
Let us assume that tests were done for campylobacters on swabs of 30 randomlydrawn poultry carcases per flock in a processing plant; let us assume further that several flocks of poultry had been slaughtered and processed sequentially and that the
first eight flocks had previously been designated as being free from campylobacters
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TABLE 12.8
Occurrence of Positive Campylobacter Samples in Swabs of Processed Chickens from
Sequential Flocks Passing Through a Processing Plant
Tests for Campylobacter
Flock number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Total 1–8
Total 9–20
Number of ve
np
Number of ve
Proportion ve
p
1
0
2
1
1
2
0
1
13
20
12
12
15
16
14
23
18
11
19
25
29
30
28
29
29
28
30
29
17
10
18
18
15
14
16
7
12
19
11
5
0.03
0.00
0.07
0.03
0.03
0.07
0.00
0.03
0.43
0.67
0.40
0.40
0.50
0.53
0.47
0.77
0.60
0.37
0.63
0.83
8
198
232
162
0.03
0.55
(‘camp ve’) flocks whilst the campylobacter history of the others flocks were unknown.
The results of the examinations are summarised in Table 12.8.
We will examine the data using an np control chart, where p the proportion of positive samples and n is the total number of replicate tests per flock
(n 30) to compare the number of positive samples for each set of tests (Fig. 12.14).
The chart shows the low prevalence of campylobacter positive results from the
initial eight flocks and, a high but variable prevalence of campylobacters in the remaining flocks. The average number of all positive tests is np 10.3 out of 30 tests (n) for
which the upper (UCL) and lower (LCL) control limits can be determined as UCL 18.1
and the LCL 2.5, using the equations UCL np 3 np(1 (np / n)) and,
LCL np np 3 np(1 (np / n)) .
np
These
values
are
shown
on
the
chart
(Fig. 12.14).
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np control chart
(Avg 10.3, UCL 18.1, LCL 2.5, n 30, for subgroups 120)
30
Number positive
25
20
UCL
15
10
Avg
5
LCL
0
1
2
3
4
5
6
7
8
9
10 11 12
Flock number
13
14
15
16
17
18
19
20
FIGURE 12.14 Attributes np -chart for prevalence of campylobacter in poultry processed
sequentially through a commercial plant. The first eight flocks were believed to be campylobacterfree flocks but the status of the remainder was not known. Note the overall average number of positive tests was 10.3 out of 30. The average (Avg), upper control limit (UCL) and lower control limit
(LCL) are shown by dotted lines.
However, we need to set control limits using the data from the uncontaminated flocks
(1–8) for which the value of np np/k 8/8 1. Since this value of np is 5, we cannot use the standard equations to derive the values for the UCL and LCL which will
be asymmetrical about the mean value ( np ). We must derive these estimates from the
binomial distribution.
The probability with which we can determine the occurrence of defective samples is
based on the equation: Px [n ! ]/[(n i) ! i ! ]pi (1 p)ni , where i number of positive
results, n number of replicate tests 30 and p the average proportion of positive re
sults 1/30 0.03333. The cumulative binomial probability is the sum of the individual
probabilities for i 0…x. The probability for zero positives (i.e. i 0) is given by:
30 !
Pi0 0 . 033330 (1 0 . 03333)(300)
(30 0) ! 0 !
0 . 966730 0 . 3617
Similarly, the probability for 1 positive (x 1) is given by:
30 !
0 . 033331 (1 0 . 03333)(301)
(30 1) ! 1 !
30 0 . 0333 0 . 966 729 0 . 3741
Pi1 Hence, the cumulative probability of finding up to, and including, 1 positive is given by:
Pi1 Pi0 Pi1 0 . 3617 0 . 3741 0 . 7358
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254
The individual probabilities of occurrence for values of i from 2 to 6 are calculated
similarly and the cumulative probabilities are determined:
Number of positive
tests (i)
0
1
2
3
4
5
6
Probability of
occurrence
0.3617
0.3741
0.1871
0.0602
0.0140
0.0025
0.00036
Cumulative
probability
0.3617
0.7358
0.9229
0.9831
0.9971
0.9996
0.9999
For a single-sided limit, the UCL must be set at an value of 0.00270 such that
1 0.9973. We must therefore use the value of x 5 for the UCL, since this is the
first value for which the cumulative probability (0.9996) exceeds the limit probability
of 0.9973. By definition, the LCL is zero. We can now set the control limits based on the
initial 8 data sets: the average number of positive tests 1 and the UCL 5. However, if
the chart were to be based on two-sided limits, the UCL would be based on 0.9956 in
which case we would use the value for x 4.
The chart shows that all of the succeeding data sets exceed this UCL (Fig. 12.15) and
therefore demonstrate that the prevalence of campylobacters in the ‘camp ve’ flocks is
‘out of control’ by comparison with the ‘camp ve’ flocks.
The control charts used in this example were prepared using an Excel add-in program
(SPC for Excel from Business Process Improvement, Cypress TX, USA) but the charts
could have been derived manually using the procedures described above.
np control chart
(Limits based on subgroups 1–8, n 30)
30
Number positive
25
20
15
10
5
UCL
Avg
0
Flock number (k)
FIGURE 12.15 Attributes np -chart for numbers of positive tests for campylobacter in a poultry
processing plant. The average and UCL (dotted lines) were derived for the first eight flocks of ‘camp
ve’ birds giving an average number of positive tests of 1 and an upper control limit of 5. Note that
the prevalence of contamination in all of the unknown flocks was greater than the UCL.
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255
EXAMPLE 12.6 USE OF ATTRIBUTE DATA CHARTS IN PROBLEM
INVESTIGATION
During a spell of hot weather, a beverage manufacturer received a high number of complaints regarding spoilage in several lots of one particular brand of a flash-pasteurised
beverage. Spoilage was evidenced by cloudiness and gas formation. Production was halted
and the product was recalled from the marketplace. The frequency of overt spoilage in
bottles of product from consecutive lots and the presence of spoilage yeasts in apparently
unspoiled product were determined to try to identify the cause(s) of the problem. The
results are summarised in Table 12.9.
It is clear from the results that the spoilage problem appeared to start suddenly – probably as a consequence of the increased ambient temperatures. The plot of the spoilage
data using an np Attributes chart without control limits (Fig. 12.16) shows a sudden
steep rise in incidence for lots 4037 onwards, followed by a gradual fall-off in lots 4042–
4047. However, even in unspoiled product there was evidence of significant levels of contamination by spoilage yeasts in the majority of the unspoiled products in all lots from
4030 onwards. Examination of cool-stored reference stock from the earlier lots also
showed evidence of yeast contamination at an average level of slightly greater than 1
organism/750 ml bottle (Fig. 12.17).
60
55
Prevalence of defective bottles
50
45
40
35
30
25
20
15
10
5
0
4030 4031 4032 4033 4034 4035 4036 4037 4038 4039 4040 4041 4042 4043 4044 4045 4046 4047
Lot number
FIGURE 12.16 Multiple np -chart showing the prevalence of spoiled products (_____) and of
unspoiled products with evidence of yeast contamination (-- - - - - -). The arrow indicates the onset of
the heat wave.
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256
TABLE 12.9
Incidence of Overt Spoilage and Yeast Contamination in a Bottled Beverage
Number of products a
Lot number
4030
4031
4032
4033
4034
4035
4036
4037
4038
4039
4040
4041
4042
4043
4044
4045
4046
4047
4048
4049
4050
Spoiled
1
3
4
1
0
1
3
13
46
54
52
59
53
48
46
12
8
1
0
0
0
Contaminated
35
32
36
48
39
25
42
10
5
4
8
1
7
12
13
48
50
59
60
60
57
Spoiled or
contaminated (%)
Incidence of low level
contamination b (%)
60
58
67
82
65
43
75
38
85
97
100
100
100
100
98
100
97
100
100
100
95
40
50
70
50
80
60
90
60
100
100
100
100
100
100
100
100
100
100
100
100
100
a
60 bottles tested/lot.
b
10 cool-stored reference products tested/lot.
A cause and effect study was undertaken prior to examination of the process plant;
Fig. 12.18 shows some of the potential causal factors, to illustrate the way in which a
‘cause and effect’ diagram can be used as an aid to problem-solving. After much investigation it was concluded that the actual cause was a major breakdown in pasteurisation efficiency, although the process records (not shown) indicated that the pasteurisation
plant had operated within defined time and temperature limits. A strip-down examination of the pasteurisation plant identified a defective plate seal in the pre-heat section
that had permitted a low level contamination of pasteurised product by unpasteurised
material. The presence of low levels of viable yeast cells in reference samples of earlier
batches of product indicated that previous lots had also been contaminated but that the
organisms had not grown sufficiently to cause overt spoilage at the ambient temperatures
then occurring.
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257
100
90
Per cent defective
80
70
60
50
40
30
4030 4031 4032 4033 4034 4035 4036 4037 4038 4039 4040 4041 4042 4043 4044 4045 4046 4047
Target 0
Lot number
FIGURE 12.17 Comparison of prevalence of actual product spoilage (–––––––) and the prevalence
of low-level yeast contamination in cool-stored reference samples of the product (- - - - - - - - - -).
Measurement
People
Environment
Policies
Process
Management
Storage temperature
Storage time
Temperature
Time
Practices
Distribution system
Quality control
Records
Quality assurance
Records
Process operators
QC/QA staff
Warehouse operatives
Product
spoilage
Production plant
Maintenance
Filling plant
Pasteurisation plant
HACCP Scheme
CCPs
Monitoring procedures/Records
Verification procedures/Records
Microbial load
Ingredients
Ingredient quality
Process batch
Packaging materials
Process operation
Cleaning procedures
Process records
Machines
Methods
Materials
FIGURE 12.18 Example of a simplified ‘Cause and Effect’ (Fishbone) diagram used in the investigation of product spoilage.
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258
STATISTICAL ASPECTS OF THE MICROBIOLOGICAL EXAMINATION OF FOODS
Why had these yeasts not been detected in the regular checks made on products?
Simply, because the level of contamination was only slightly more than 1 viable yeast
cell per 750 ml bottle of product and the methodology used for routine testing relied on
membrane filtration of 100 ml of product from each bottle tested. Hence, the likelihood of
detecting such a low prevalence of yeasts was not high.
In this example the causal effect of contamination was a pasteurisation plant defect,
but a major contributory factor was the elevated ambient temperature. Another major
contributory factor, albeit not a causal factor, was the inadequacy of the microbiological
test system that was not fit for the purpose required since it could not detect the presence of such a low level of product contamination. This example serves also as a warning
that it is not possible to ‘test quality in a production process’!
CONCLUSION
Control charts, as a means of simple trend analysis, provide visual aids to interpretation of
data that are more readily understood than are tables of data values. They identify when a
process went out of control so that appropriate remedial action can be taken before ‘out of
control’ equates to ‘out of specification’ with a subsequent manufacturing crisis.
References
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Anon. (2005) Commission Regulation (EC) No 2073/2005 of 15 November 2005 on microbiological
criteria for foodstuffs. Official J Europ. Union, L138, 1–26. 22 December 2005.
Anon. (2006a) Final Report and Executive Summaries from the AOAC International Presidential
Task Force on Best Practices in Microbiological Methodology: Part II F Statistical Process Control.
USFDA. http://www.cfsan.fda.gov/~acrobat/bpmm-f.pdf
Anon. (2006b) Final Report and Executive Summaries from the AOAC International Presidential
Task Force on Best Practices in Microbiological Methodology: Part II F Statistical Process
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gov/~acrobat/bpmm-f1.pdf
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Homewood, IL..
Hayes, GD, Scallan, AJ, and Wong, JHF (1997) Applying statistical process control to monitor and
evaluate the hazard analysis critical control point hygiene data. Food Cont., 8, 173–176.
Juran, JM (1974) Quality Control Handbook, 3rd edition. McGraw Hill Book Co., New York.
Montgomery, DC (2000) Introduction to Statistical Quality Control, 4th edition. John Wiley & Sons,
New York.
Shewhart, WA (1931) Economic Control of Quality of Manufactured Product. Van Nostrand Co.
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