Microbiological data are often fi led without much consideration as to their further appli-cation. When one considers the costs of providing, staffi ng 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 ben-efi ts 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 vari-ous tools to monitor, analyse, evaluate and control a process and is based on the philo-sophical concept that it is essential to use the best available data in order to optimise process effi ciency. In this context, the term ‘ process ’ is any activity that is subject to vari-ability 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 quantifi able activity.
The concept was introduced in the late 1920s by Dr William Shewhart whose work helped to improve effi ciency 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 PROCESS CONTROL USING
MICROBIOLOGICAL DATA
12
using data obtained from process line measurements and/or on samples of intermediate and/ or fi nished products, rather than reliance on 100% fi nal 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 predict-able, 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 defi nition and implementation of Food Safety Objectives in Chapter 14.
At its simplest, SPC in manufacturing production is done by operative recording of sim-ple 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 manufactur-ing 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 signifi cant trends away from the ‘ norm ’ requires remedial action, for example, to halt production and/or to return the proc-ess operation to a controlled status. In a similar way, SPC of internal laboratory quality per-formance 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 identifi able param-eter. Note that a target value is not a product specifi cation; 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 defi ned 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, sub-sequently, product improvement is based on the principles of Total Quality Management ( Beauregard et al., 1992 ). It is noteworthy that the European legislation on microbiological
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 ’ (fi shtail) 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 VK Vmaterials Vequipment Vmethod Vpeople VK i ... 1 K K
∑
where V K variance associated with an individual factor K. Evidence of a high variance in some measurable quality of the fi nished product will refl ect 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 fi nal, 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 ’ .
2 3 4 5 6 7 8 8 9 10 11 12 13 14 Time (h) Product temper ature ( °C) d c b a
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.
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 microbio-logical data is inherently asymmetrical and does not usually conform to the normal distri-bution. 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 y i log e x i or y i log 10 x i for data that
are log-normal distributed (e.g. colony counts), or y i 冪 x i 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 defi ne the process, the measure-ment 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 proc-essed, 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 micro-biological 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 benefi t from SPC. Charts of output data pro-vide a visual aid to detecting and identifying sources of unexpected variation leading to their elimination. Methods People Environment Measurement system Machine Materials Output
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 param-eters 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 statis-tics 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 defi nes the process to be ‘ out of control ’ when it is not and a Type II error defi nes 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 dis-tance 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 reg-ulation; rather they are process-related limits. Product specifi cations 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 defi ned non-sterile food product, processed using a specifi c 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, 10 4 cfu/g (4.0 log
10 cfu/g). Assume also that the log 10 -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 log 10 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 10 4 cfu/g (4.0 log
10 cfu/g) then, for 0.01, the upper 99% confi dence interval will be (4.0 2.326 0.25) 4.58 log 10 cfu/g. It would then be expected that the probability of a data value exceeding 4.6 log 10 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 log 10 cfu/g. These upper values, together with
the mean value of 4.0 log 10 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 dis-tribution 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 con-trol 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 dis-tribution using sample average and variance estimates. Of course, in the case of microbiologi-cal data, levels of colony forming units need to be normalised by an appropriate mathematimicrobiologi-cal transformation (e.g. log transformation).
There are four major types of control chart for variables data:
1. The x chart, based on the average result; 2. The R chart, based on the range of the results;
3. The s chart, based on the sample standard distribution; and 4. The x chart, based on individual results.
The R and s charts monitor variations in the process whereas the x and x charts moni-tor 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 con-trol 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
of product from the specifi c 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 histo-gram to confi rm that the pattern of distribution conforms reasonably to a normal distribu-tion. 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 Target value ( )
n R R R n
∑
;∑
(a) (b) (c) (d) Target value Limit valueFIGURE 12.3 In control or in specifi cation? (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.
Control limits need to be established for both sample average and sample range data sets; if both upper and lower confi dence limits (CLs) are required, these are located sym-metrically about the target (centre) line; in most cases, microbiological tests of qual-ity/safety, require only upper confi dence limits (UCLs). Usually these will be set at
UCLx (x A2R) and UCLR D4R for the 99% UCLs, where A 2 and D 4 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 sam-ple units for conformity (Examsam-ple 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 inves-tigated 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
pro-duction 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 log 10 -transformed and tested for conformance with normality, skewness
and kurtosis – although not perfect, the fi t 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 log 10 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
Occasion
Aerobic colony count (log 10 cfu/g)
Mean ( x ) SD ( s ) Range ( R ) 1 5.05 0.343 0.72 2 5.06 0.359 0.81 3 4.81 0.197 0.47 4 4.76 0.140 0.34 5 4.84 0.163 0.43 6 4.72 0.107 0.29
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 log 10 cfu/g).
The upper and lower control limits are determined as:
UCLx x A2R and the LCLx x A2R
where x is the overall mean value, A 2 is a constant ( Table 12.2 ) and R is the mean
range. An alternative calculation uses the average standard deviation ( s ):
UCLx x A3s and the LCLx x A3s
For our data, with x 4 98. and R 0 43. , UCLx 4 98. 0 577. 0 43. 5 23. ; the
alternative procedure using s 0 180. , gives UCLx 4 98. 1 427. 0 180. 5 24. .We
can determine the LCL similarly: LCLx 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
Occasion
Aerobic colony count (log 10 cfu/g)
Mean ( x ) SD ( s ) Range ( R ) 7 4.95 0.192 0.42 8 5.11 0.149 0.40 9 5.03 0.123 0.30 10 5.06 0.235 0.59 11 5.11 0.148 0.31 12 5.13 0.198 0.48 13 4.74 0.199 0.44 14 4.80 0.142 0.35 15 4.91 0.247 0.59 16 5.13 0.144 0.35 17 5.05 0.225 0.54 18 4.97 0.093 0.25 19 4.95 0.066 0.17 20 5.05 0.140 0.35 21 5.09 0.132 0.34 22 5.04 0.145 0.37 23 5.13 0.182 0.47 24 5.01 0.255 0.54 Overall mean 4.98 0.180 0.43
TABLE 12.2
Some Constants for Control Chart Formulae
Number of replicates
x & R Charts x & s Charts
Number of replicates x -Chart limits R -Chart limits a x -Chart limits s -Chart limits a A 2 D 3 D 4 A 3 B 3 B 4 2 1.880 – 3.267 2.659 – 3.267 2 3 1.023 – 2.574 1.954 – 2.568 3 4 0.729 – 2.282 1.628 – 2.266 4 5 0.577 – 2.114 1.427 – 2.089 5 6 0.483 – 2.004 1.287 0.030 1.970 6 7 0.419 0.076 1.924 1.182 0.118 1.882 7 8 0.373 0.136 1.824 1.099 0.185 1.815 8 10 0.308 0.223 1.777 0.975 0.284 1.716 10 15 0.223 0.347 1.653 0.789 0.428 1.572 15 20 0.180 0.180 1.585 0.680 0.510 1.490 20
a If a value is not shown, use the next higher value.
lower counts are generally perceived as being benefi cial, 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 UCL R and LCL R values are derived
from the equations:
UCLR D4R and the LCLR D3R
where, D 3 and D 4 are constants ( Table 12.2 ). For our data:
UCL and LCL R R 2 114 0 43 0 909 0 91 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: UCLsB4s and LCLsB3s, where B 3 and B 4 are constants
from Table 12.2 . For our data:
UCL and LCL s s 2 089 0 180 0 376 0 38 0 030 0 180 0 0054 0 . . . . . . . . ⬇ ⬇ 001
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 confi rm that both the process and the measurements were ‘ in control ’ it is essential to chart these data before using the con-trol 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.
FIGURE 12.4 Statistical control chart for mean ( x ) colony count (as log 10 cfu/g) on 24 sets of rep-licate 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.
Avg 4.98 UCL 5.24 LCL 4.72 4.6 4.7 4.8 4.9 5.0 5.1 5.2 5.3 1 Sample number x (log cfu/g) 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 x chart
FIGURE 12.5 Statistical control chart for range ( R ) of log 10 cfu/g on 24 sets of replicate examina-tions of a fresh food product produced under controlled condiexamina-tions.
R chart (s 0.19) Avg 0.43 UCL 0.91 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 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
Range (log cfu/g)
FIGURE 12.6 Statistical control chart for standard deviation ( s ) of log 10 cfu/g on 24 sets of repli-cate examinations of a fresh food product produced under controlled conditions.
Avg = 0.18 UCL = 0.38 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 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Sample number Sample standard de viation s chart (s 0.192)
x and s Charts
For sample unit sizes greater than 12, the standard deviation ( s ) should be used to moni-tor 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 non-normal 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:
Centre line s ... s k s s s k i i k k
∑
1 1 2where k number of sample sets and s i 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: ... s s k s s s k i i k k 2 1 12 22 2
∑
. The UCLs are calculated from:UCL A( )x x 3s UCL( )s B4s
where A 3 and B 4 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 random-ness; 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 rectifi ed.
The key question is ‘ How do you recognise out of control situations from the chart? ’ At least 11 ‘ out of control ’ patterns have been identifi ed ( Anon., 1956 ; Wheeler and Chambers, 1984 ; Beauregard et al., 1992 ). As a general rule, 6 patterns cover the majority of situations
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 UC L that is set at 3 s above the central line. The probability for any point to occur above UC L 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 UC L – 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 microbio-logical 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 fi rstly 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 dif-ferent? Had a large quantity of ‘ reworked ’ material been blended in to the batch? And so on.
TABLE 12.3
Aerobic Colony Count Data on 81 Sets of Samples of a Fresh Food Product
Sample
Replicate colony counts (log 10 cfu/g)
1 2 3 4 5 Mean Range 1 4.73 5.37 4.65 5.17 5.35 5.05 0.72 2 4.81 5.23 4.82 4.84 5.62 5.06 0.81 3 4.93 4.47 4.90 4.94 4.80 4.81 0.47 4 4.83 4.61 4.79 4.95 4.64 4.76 0.34 5 4.77 4.80 4.92 4.65 5.08 4.84 0.43 6 4.72 4.84 4.75 4.55 4.76 4.72 0.29 7 5.16 5.12 4.93 4.74 4.78 4.95 0.42 8 5.07 5.36 5.07 5.09 4.95 5.11 0.40 9 5.02 4.91 4.92 5.07 5.21 5.03 0.30 10 4.93 5.34 4.75 5.06 5.23 5.06 0.59 11 4.93 4.96 5.16 5.23 5.24 5.11 0.31 12 5.22 5.20 4.78 5.26 5.18 5.13 0.48 13 4.48 4.87 4.57 4.92 4.85 4.74 0.44 14 4.71 4.95 4.89 4.60 4.84 4.80 0.35 15 5.20 4.78 4.61 5.13 4.85 4.91 0.59 16 5.38 5.09 5.04 5.04 5.11 5.13 0.35 17 5.26 4.72 5.00 5.26 5.01 5.05 0.54 18 5.06 4.82 4.97 4.96 5.01 4.97 0.25 19 4.98 4.90 5.04 4.87 4.95 4.95 0.17 20 5.02 4.94 4.96 5.02 5.29 5.05 0.35 21 5.00 5.29 5.13 4.94 5.10 5.09 0.34 22 5.12 5.08 5.17 5.06 4.80 5.04 0.37 23 5.29 5.21 5.13 5.19 4.82 5.13 0.47 24 5.06 4.78 4.72 5.25 5.26 5.01 0.54 25 5.66 5.00 4.80 5.01 5.20 5.13 0.86 26 5.15 5.08 5.78 5.15 5.60 5.35 0.70 27 4.78 6.17 5.20 5.80 5.97 5.58 1.39 28 5.48 5.26 5.00 5.20 5.26 5.24 0.48 29 4.70 4.66 4.58 4.80 5.02 4.75 0.44 30 4.86 4.99 4.65 4.95 5.02 4.89 0.37 31 5.45 5.70 6.03 5.97 6.28 5.89 0.83 32 5.00 5.60 5.20 4.78 5.48 5.21 0.82 33 5.08 4.90 5.41 4.90 5.58 5.18 0.68 34 4.80 4.30 4.60 5.08 4.90 4.74 0.78 35 4.68 4.60 4.63 4.86 4.97 4.75 0.37 36 4.78 4.97 4.62 4.80 4.85 4.80 0.35 37 5.01 4.80 4.85 4.85 4.90 4.88 0.21 38 4.28 4.50 4.74 4.70 4.80 4.60 0.52 39 4.73 4.82 4.91 5.10 5.05 4.92 0.37 40 4.85 4.80 5.10 4.95 5.35 5.01 0.55 (continued)
TABLE 12.3 (Continued )
Sample
Replicate colony counts (log 10 cfu/g)
1 2 3 4 5 Mean Range 41 4.60 4.60 4.50 4.30 4.62 4.53 0.32 42 4.45 4.60 4.69 4.45 5.10 4.66 0.65 43 4.55 4.46 4.50 4.32 4.62 4.49 0.30 44 5.02 4.88 4.72 4.24 4.54 4.68 0.78 45 4.74 4.81 4.76 4.96 4.81 4.82 0.22 46 4.80 4.90 5.30 5.10 4.80 4.98 0.50 47 5.33 4.99 5.10 5.25 5.33 5.20 0.34 48 5.00 5.35 5.10 5.08 5.50 5.21 0.50 49 5.32 5.18 5.45 5.30 5.40 5.33 0.27 50 5.63 5.51 5.35 5.10 5.40 5.40 0.53 51 5.29 5.33 5.31 5.88 5.40 5.44 0.59 52 5.15 5.45 4.78 5.30 5.26 5.19 0.67 53 4.60 4.49 4.90 4.89 5.10 4.80 0.61 54 4.71 4.84 5.01 5.10 4.82 4.90 0.39 55 4.80 4.68 5.15 5.20 4.90 4.95 0.52 56 5.10 5.35 5.10 4.80 4.85 5.04 0.55 57 5.03 4.70 4.94 4.88 4.98 4.91 0.33 58 4.90 4.78 4.88 4.95 5.30 4.96 0.52 59 4.45 4.80 4.85 5.10 5.04 4.85 0.65 60 4.98 4.71 5.15 5.17 5.28 5.06 0.57 61 5.57 5.10 4.87 5.26 4.76 5.11 0.81 62 5.49 5.36 5.16 4.89 5.00 5.18 0.60 63 5.03 5.95 5.12 5.09 5.01 5.24 0.94 64 5.10 5.87 4.98 5.80 4.99 5.35 0.89 65 5.28 5.10 5.36 5.90 5.20 5.37 0.80 66 5.09 5.42 5.39 5.37 5.49 5.35 0.40 67 5.31 5.63 5.84 5.27 5.65 5.54 0.57 68 5.53 5.32 5.20 5.04 5.17 5.25 0.49 69 4.93 5.18 5.04 5.13 5.12 5.08 0.25 70 5.18 4.50 5.08 4.30 4.30 4.67 0.88 71 4.90 5.04 4.95 5.08 5.26 5.05 0.36 72 4.60 4.30 4.30 5.18 5.08 4.69 0.88 73 5.27 4.81 4.79 4.70 5.09 4.93 0.57 74 5.08 5.60 5.70 5.20 5.51 5.42 0.62 75 4.76 5.31 4.65 4.88 5.20 4.96 0.66 76 5.12 5.09 4.80 5.05 5.10 5.03 0.32 77 5.28 5.49 5.38 5.18 4.70 5.21 0.79 78 5.25 4.90 5.08 5.25 5.40 5.18 0.50 79 5.28 5.44 4.98 5.20 5.40 5.26 0.46 80 5.20 5.24 5.42 5.47 5.38 5.34 0.27 81 5.85 6.09 5.44 5.36 5.63 5.67 0.73
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 signifi cantly 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
Average UCL 4.4 1 6 11 16 21 27 32 42 47 52 57 62 67 72 77 82 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0 Sample number x chart
(Avg 4.98, UCL 5.23, for Subgroups 1–24)
Mean log cfu/g
FIGURE 12.7 Statistical control chart for mean ( x ) colony count (as log 10 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 ).
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!
R Chart
(Avg 0.43, UCL 0.91, LCL None)
Average UCL 0.00 1 6 16 21 27 32 37 42 47 52 57 62 67 72 77 82 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 Sample number
Range (log cfu/g)
FIGURE 12.8 Statistical control chart for range ( R ) of colony counts (as log 10 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 ).
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 diffi cult to use ( Montgomery, 1990 ). Beauregard et al. (1992) recommend that it is not used and that a CUSUM signal chart is used instead.
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 effi ciency 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 alterna-tive 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 nor-mality of the distribution, there is still considerable skewness and kurtosis; both forms of logarithmic transformation effectively normalise the data and remove both skew-ness 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 transfor-mation 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 sys-tem, 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 signifi cant adverse trends. A similar situa-tion 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 diffi cult and very time
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 (log 10 )
Day RLU ( x ) Transformed RLU Day RLU ( x ) Transformed RLU 冪x Ln ( x ) log 10 ( x ) 冪x Ln ( x ) log 10 ( x ) 1 23 4.80 3.14 1.36 47 147 12.12 4.99 2.17 2 46 6.78 3.83 1.66 48 13 3.61 2.56 1.11 3 39 6.24 3.66 1.59 49 12 3.46 2.48 1.08 4 62 7.87 4.13 1.79 50 15 3.87 2.71 1.18 5 33 5.74 3.50 1.52 51 14 3.74 2.64 1.15 6 21 4.58 3.04 1.32 52 76 8.72 4.33 1.88 7 20 4.47 3.00 1.30 53 28 5.29 3.33 1.45 8 56 7.48 4.03 1.75 54 29 5.39 3.37 1.46 9 69 8.31 4.23 1.84 55 44 6.63 3.78 1.64 10 26 5.10 3.26 1.41 56 15 3.87 2.71 1.18 11 27 5.20 3.30 1.43 57 16 4.00 2.77 1.20 12 24 4.90 3.18 1.38 58 123 11.09 4.81 2.09 13 22 4.69 3.09 1.34 59 5 2.24 1.61 0.70 14 69 8.31 4.23 1.84 60 41 6.40 3.71 1.61 15 42 6.48 3.74 1.62 61 56 7.48 4.03 1.75 16 97 9.85 4.57 1.99 62 9 3.00 2.20 0.95 17 56 7.48 4.03 1.75 63 82 9.06 4.41 1.91 18 33 5.74 3.50 1.52 64 22 4.69 3.09 1.34 19 33 5.74 3.50 1.52 65 22 4.69 3.09 1.34 20 69 8.31 4.23 1.84 66 63 7.94 4.14 1.80 21 24 4.90 3.18 1.38 67 85 9.22 4.44 1.93 22 34 5.83 3.53 1.53 68 34 5.83 3.53 1.53 23 29 5.39 3.37 1.46 69 158 12.57 5.06 2.20 24 29 5.39 3.37 1.46 70 164 12.81 5.10 2.21 25 19 4.36 2.94 1.28 71 155 12.45 5.04 2.19 26 57 7.55 4.04 1.76 72 79 8.89 4.37 1.90 27 81 9.00 4.39 1.91 73 94 9.70 4.54 1.97 28 99 9.95 4.60 2.00 74 319 17.86 5.77 2.50 29 43 6.56 3.76 1.63 75 38 6.16 3.64 1.58 30 80 8.94 4.38 1.90 76 19 4.36 2.94 1.28 31 68 8.25 4.22 1.83 77 64 8.00 4.16 1.81 32 112 10.58 4.72 2.05 78 50 7.07 3.91 1.70 33 25 5.00 3.22 1.40 79 16 4.00 2.77 1.20 34 52 7.21 3.95 1.72 80 25 5.00 3.22 1.40 35 24 4.90 3.18 1.38 81 28 5.29 3.33 1.45 36 35 5.92 3.56 1.54 82 35 5.92 3.56 1.54 37 15 3.87 2.71 1.18 83 52 7.21 3.95 1.72 38 47 6.86 3.85 1.67 84 28 5.29 3.33 1.45 39 104 10.20 4.64 2.02 85 35 5.92 3.56 1.54 40 53 7.28 3.97 1.72 86 52 7.21 3.95 1.72 41 81 9.00 4.39 1.91 87 139 11.79 4.93 2.14 42 34 5.83 3.53 1.53 88 89 9.43 4.49 1.95 43 17 4.12 2.83 1.23 89 62 7.87 4.13 1.79 44 62 7.87 4.13 1.79 90 55 7.42 4.01 1.74 45 23 4.80 3.14 1.36 91 12 3.46 2.48 1.08 46 123 11.09 4.81 2.09 92 67 8.19 4.20 1.83
0 10 20 30 40 50 60 F requency 0 5 10 15 20 25 F requency
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.
3 2 1 0 0 50 100 150 200 250 300 350 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 1 2 3 4 5 6 RLU Nor mal quantile 3 2 1 0 1 2 3 Ln RLU Nor mal quantile
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
Parameter RLU ( x ) Transformed RLU 冪x Ln ( x ) log 10 ( x ) Number of Tests ( n ) 92 92 92 92 Mean 54.6 6.90 3.726 1.618 Median 41.5 6.44 3.726 1.618 Variance 2109.8 7.044 0.558 0.105 Standard deviation 45.93 2.654 0.747 0.324 Normality test a 4.69 ( P 0.0001) 1.33 ( P 0.0019) 0.205 ( P 0.8723) 0.205 ( P 0.8723) Skewness 2.67 ( P 0.0001) 1.13 ( P 0.0001) 0.004 ( P 0.9864) 0.004 ( P 0.9864) Kurtosis 11.45 ( P 0.0001) 2.18 ( P 0.0055) 0.025 ( P 0.8904) 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
Day RLU Ln-RLU Target (Ln-RLU) Difference (C – D) CUSUM
2 3 0 4 1 23 3.135 3.912 0.777 0.777 5 2 46 3.829 3.912 0.083 0.860 6 3 39 3.664 3.912 0.248 1.108 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 … … … … … … … … 92 90 55 4.007 3.912 0.095 12.090 93 91 12 2.485 3.912 1.427 13.517 94 92 67 4.205 3.912 0.293 13.224
FIGURE 12.10 Individuals plot of the ATP data from hygiene tests on a dairy bottling mandrel.
Individuals chart
(Target 50, UCL 110, LCL 0, s 33.12, for Subgroups 1–92)
Tartget UCL 0 50 100 150 200 250 300 350 1 10 19 28 37 46 55 64 73 82 91 Sample number RLU Individuals chart (Target 3.90, UCL 4.70) Target UCL 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 1 11 21 31 41 51 61 71 81 Sample number Ln RLU 91
FIGURE 12.11 Individuals plot of the ATP data (as Ln-RLU) from hygiene tests on a dairy bot-tling mandrel.
25.0 20.0 15.0 10.0 5.0 0.0 Day number Cum ulativ e sum of Ln RLU 0 10 20 30 40 50 60 70 80 90
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 modifi ed 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
∑
(xv SF) Cumulative LSAL∑
(xv SF)where, xv is the nominalised mean ( x – a target value). Limits may be applied to both
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 defi nitive information on ‘ out of
con-trol ’ situations?
The signal chart consists of two parallel graphs of modifi ed 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
nominal-ised 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 (xvSF)
Cumulative lower signal value (xvSF)
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 diffi cult 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
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 )
A B C D E F G
1 Day Ln RLU Nominalised
(Nominal 3.7) Nominalised SF Cumulative US value
Nominalised +SF Cumulative LS Value 2 x xv(x3 7 . ) Xv / 2 s ∑(xvs/ )2 Xv / 2 s Xv / 2 s 3 1 3.135 0.56 0.94 [0] a 0.19 0.19 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 0.05 0.80 [0] 7 5 3.497 0.20 0.58 [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 10 8 4.025 0.33 0.05 [0] 0.70 [0] 11 9 4.234 0.53 0.16 0.16 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 0.16 0.91 [0] 17 15 3.738 0.04 0.34 [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] 0.17 [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.
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.
CUSUM signal chart
Avg 0.00 USAL 3.75 LSAL 3.75 -5 5 1 11 21 31 41 51 61 71 81 Sample number Signal v a lue 91
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 pres-ence 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-conform-ing 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, ( np i ) number of non-conforming samples in sample set i 1 to k and k number of sample sets:
np np k np np np k i i k k ( ) ( ) ( ) ... ( ) 1 1 2
∑
The upper and lower control limits are calculated from:UCLnp np np np n 3 (1− ) LCLnp np np np n 3 (1 )
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 randomly-drawn poultry carcases per fl ock in a processing plant; let us assume further that sev-eral fl ocks of poultry had been slaughtered and processed sequentially and that the fi rst eight fl ocks had previously been designated as being free from campylobacters
( ‘ camp ve ’ ) fl ocks whilst the campylobacter history of the others fl ocks 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
propor-tion of positive samples and n is the total number of replicate tests per fl ock
( 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 fl ocks and, a high but variable prevalence of campylobacters in the
remain-ing fl ocks. 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
npnp3 np(1(np n/ )) and,
LCLnpnp3 np(1(np n/ )) . These values are shown on the chart
( Fig. 12.14 ).
TABLE 12.8
Occurrence of Positive Campylobacter Samples in Swabs of Processed Chickens from Sequential Flocks Passing Through a Processing Plant
Flock number
Tests for Campylobacter Number of ve np Number of ve Proportion ve p 1 1 29 0.03 2 0 30 0.00 3 2 28 0.07 4 1 29 0.03 5 1 29 0.03 6 2 28 0.07 7 0 30 0.00 8 1 29 0.03 9 13 17 0.43 10 20 10 0.67 11 12 18 0.40 12 12 18 0.40 13 15 15 0.50 14 16 14 0.53 15 14 16 0.47 16 23 7 0.77 17 18 12 0.60 18 11 19 0.37 19 19 11 0.63 20 25 5 0.83 Total 1–8 8 232 0.03 Total 9–20 198 162 0.55
However, we need to set control limits using the data from the uncontaminated fl ocks
(1–8) for which the value of np np / k 8/8 1. Since this value of np is 5, we
can-not 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!]/[(ni i) ! !]pi(1p)n i, 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:
Pi 0 30 30 0 0 0 03333 1 0 03333 0 966730 0 3617 0 30 ! ( ) ! ! . ( . ) . . ( 0)
Similarly, the probability for 1 positive ( x 1) is given by:
Pi1 1 30 1 30 30 1 1 0 03333 1 0 03333 30 0 0333 0 966 ! ( ) ! ! . ( . ) . . ( ) 7 729 .0 3741
Hence, the cumulative probability of fi nding up to, and including, 1 positive is given by:
Pi 1Pi0Pi10 3617. 0 3741. 0 7358.
np control chart
(Avg 10.3, UCL 18.1, LCL 2.5, n 30, for subgroups 120)
Avg UCL LCL 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 5 10 15 20 25 30 Flock number Number positiv e
FIGURE 12.14 Attributes np -chart for prevalence of campylobacter in poultry processed sequentially through a commercial plant. The fi rst eight fl ocks were believed to be campylobacter-free fl ocks but the status of the remainder was not known. Note the overall average number of posi-tive tests was 10.3 out of 30. The average (Avg), upper control limit (UCL) and lower control limit (LCL) are shown by dotted lines.
The individual probabilities of occurrence for values of i from 2 to 6 are calculated similarly and the cumulative probabilities are determined:
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
fi rst value for which the cumulative probability (0.9996) exceeds the limit probability
of 0.9973. By defi nition, 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 ’ fl ocks is
‘ out of control ’ by comparison with the ‘ camp ve ’ fl ocks.
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)
Avg UCL 0 5 10 15 20 25 30 Flock number (k) Number positiv e
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 fi rst eight fl ocks 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 fl ocks was greater than the UCL.
Number of positive tests ( i ) Probability of occurrence Cumulative probability 0 0.3617 0.3617 1 0.3741 0.7358 2 0.1871 0.9229 3 0.0602 0.9831 4 0.0140 0.9971 5 0.0025 0.9996 6 0.00036 0.9999
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 com-plaints regarding spoilage in several lots of one particular brand of a fl ash-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 – prob-ably 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 signifi cant levels of con-tamination 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 ).
0 5 10 15 20 25 30 35 40 45 50 55 60 Lot number Pre v alence of def ectiv e bottles 4030 4031 4032 4033 4034 4035 4036 4037 4038 4039 4040 4041 4042 4043 4044 4045 4046 4047
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.
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 inves-tigation it was concluded that the actual cause was a major breakdown in pasteurisa-tion effi ciency, although the process records (not shown) indicated that the pasteurisapasteurisa-tion plant had operated within defi ned time and temperature limits. A strip-down examina-tion of the pasteurisaexamina-tion plant identifi ed a defective plate seal in the pre-heat secexamina-tion 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 suffi ciently to cause overt spoilage at the ambient temperatures then occurring.
TABLE 12.9
Incidence of Overt Spoilage and Yeast Contamination in a Bottled Beverage
Lot number
Number of products a
Spoiled or contaminated (%)
Incidence of low level contamination b (%) Spoiled Contaminated 4030 1 35 60 40 4031 3 32 58 50 4032 4 36 67 70 4033 1 48 82 50 4034 0 39 65 80 4035 1 25 43 60 4036 3 42 75 90 4037 13 10 38 60 4038 46 5 85 100 4039 54 4 97 100 4040 52 8 100 100 4041 59 1 100 100 4042 53 7 100 100 4043 48 12 100 100 4044 46 13 98 100 4045 12 48 100 100 4046 8 50 97 100 4047 1 59 100 100 4048 0 60 100 100 4049 0 60 100 100 4050 0 57 95 100 a 60 bottles tested/lot.
Target 0 30 4030 4031 4032 4033 4034 4035 4036 4037 4038 4039 4040 4041 4042 4043 4044 4045 4046 4047 40 50 60 70 80 90 100 Lot number P e r cent def ectiv e
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
Machines Methods Materials Process Temperature Time Quality control Quality assurance Maintenance Policies Practices Process operators QC/QA staff Warehouse operatives Storage temperature Storage time Process operation Process records Filling plant Pasteurisation plant Production plant Cleaning procedures HACCP Scheme CCPs Monitoring procedures/Records Verification procedures/Records Ingredient quality Process batch Ingredients Microbial load Packaging materials Records Records Distribution system Product spoilage Management
FIGURE 12.18 Example of a simplifi ed ‘ Cause and Effect ’ (Fishbone) diagram used in the investi-gation of product spoilage.
