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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. CH012-N53039.indd 225 225 5/26/2008 8:47:09 PM 226 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 CH012-N53039.indd 226 5/26/2008 8:47:10 PM 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’. CH012-N53039.indd 227 5/26/2008 8:47:10 PM 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. CH012-N53039.indd 228 5/26/2008 8:47:10 PM 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 CH012-N53039.indd 229 5/26/2008 8:47:11 PM 230 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 CH012-N53039.indd 230 5/26/2008 8:47:11 PM 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. CH012-N53039.indd 231 5/26/2008 8:47:11 PM 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 CH012-N53039.indd 232 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 5/26/2008 8:47:11 PM 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 CH012-N53039.indd 233 5/26/2008 8:47:12 PM 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 CH012-N53039.indd 234 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. 5/26/2008 8:47:12 PM STATISTICAL PROCESS CONTROL USING MICROBIOLOGICAL DATA 235 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. CH012-N53039.indd 235 5/26/2008 8:47:13 PM STATISTICAL ASPECTS OF THE MICROBIOLOGICAL EXAMINATION OF FOODS 236 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. CH012-N53039.indd 236 5/26/2008 8:47:13 PM STATISTICAL PROCESS CONTROL USING MICROBIOLOGICAL DATA 237 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 CH012-N53039.indd 237 5/26/2008 8:47:14 PM 238 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. CH012-N53039.indd 238 5/26/2008 8:47:14 PM 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) CH012-N53039.indd 239 5/26/2008 8:47:14 PM STATISTICAL ASPECTS OF THE MICROBIOLOGICAL EXAMINATION OF FOODS 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 5/26/2008 8:47:14 PM 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 CH012-N53039.indd 241 5/26/2008 8:47:15 PM 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. CH012-N53039.indd 242 5/26/2008 8:47:16 PM STATISTICAL PROCESS CONTROL USING MICROBIOLOGICAL DATA 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. CH012-N53039.indd 243 5/26/2008 8:47:16 PM STATISTICAL ASPECTS OF THE MICROBIOLOGICAL EXAMINATION OF FOODS 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 5/26/2008 8:47:16 PM 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. CH012-N53039.indd 245 5/26/2008 8:47:17 PM STATISTICAL ASPECTS OF THE MICROBIOLOGICAL EXAMINATION OF FOODS 246 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 5/26/2008 8:47:17 PM 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. CH012-N53039.indd 247 5/26/2008 8:47:17 PM STATISTICAL ASPECTS OF THE MICROBIOLOGICAL EXAMINATION OF FOODS 248 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 CH012-N53039.indd 248 5/26/2008 8:47:18 PM 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). CH012-N53039.indd 249 5/26/2008 8:47:18 PM STATISTICAL ASPECTS OF THE MICROBIOLOGICAL EXAMINATION OF FOODS 250 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. CH012-N53039.indd 250 5/26/2008 8:47:18 PM STATISTICAL PROCESS CONTROL USING MICROBIOLOGICAL DATA 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 CH012-N53039.indd 251 5/26/2008 8:47:19 PM STATISTICAL ASPECTS OF THE MICROBIOLOGICAL EXAMINATION OF FOODS 252 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). CH012-N53039.indd 252 5/26/2008 8:47:20 PM STATISTICAL PROCESS CONTROL USING MICROBIOLOGICAL DATA 253 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 CH012-N53039.indd 253 5/26/2008 8:47:20 PM STATISTICAL ASPECTS OF THE MICROBIOLOGICAL EXAMINATION OF FOODS 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. CH012-N53039.indd 254 5/26/2008 8:47:20 PM STATISTICAL PROCESS CONTROL USING MICROBIOLOGICAL DATA 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. CH012-N53039.indd 255 5/26/2008 8:47:20 PM STATISTICAL ASPECTS OF THE MICROBIOLOGICAL EXAMINATION OF FOODS 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. CH012-N53039.indd 256 5/26/2008 8:47:21 PM STATISTICAL PROCESS CONTROL USING MICROBIOLOGICAL DATA 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. CH012-N53039.indd 257 5/26/2008 8:47:21 PM 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 Anon. (1956) Statistical Quality Control Handbook, 2nd edition. Delco Remy, Anderson, IN. 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 Control – 1. Appendices for Statistical Process Control. USFDA. http://www.cfsan.fda. gov/~acrobat/bpmm-f1.pdf Beauregard, MR, Mikulak, RJ, and Olson, BA (1992) A Practical Guide to Statistical Quality Improvement. Van Nostrand Reinhold, New York. Duncan, AJ (1986) Quality Control and Industrial Statistics, 5th edition. Richard D Irwin, 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. Inc., New York. Wheeler, DJ and Chambers, D (1984) Understanding Statistical Process Control, 2nd edition. Statistical Process Control Inc., Knoxville, TN. CH012-N53039.indd 258 5/26/2008 8:47:22 PM