«Microbiological sampling plans – Statistical aspects* Susanne Dahms, Freie Universität Berlin, Department of Veterinary Medicine, Institute of ...»
Microbiological sampling plans –
Susanne Dahms, Freie Universität Berlin, Department of Veterinary Medicine,
Institute of Biometric and Data Processing, Berlin, Germany
It is nearly 30 years ago that the International Commission on Microbiological
Specifications for Foods (ICMSF) provided urgently needed guidance on the use of
sampling plans and microbiological criteria for foods in international trade. With
publications like “Microorganisms in Foods 2: Sampling for Microbiological Analysis: Principles and Specific Applications” (1) and now “Microorganisms in Foods 7: Microbiological Testing in Food Safety Management” (2) ICMSF introduced concepts of probability and sampling into microbiological criteria and developed a scheme for selection of cases and attributes plans in order to establish criteria for food lot acceptance. Dependent on the conditions in which food is expected to be handled and consumed in the usual course of events and on the degree of concern relative to food utility and health hazard, 15 cases have been distinguished by ICMSF that require increasing stringency of acceptance sampling.
Two general types of sampling plans, attributes sampling plans and variables sampling plans, are used in microbiological testing to make decisions concerning the safety or quality of foods. Attributes plans are used to evaluate qualitative data (presence-absence) or quantitative data that have been grouped (e.g., 10 cfu, 10 to 100 cfu, 100 cfu), whereas variables plans evaluate non-grouped quantitative data.
However, despite their wide use and adoption, microbiological criteria and sampling plans are not fully understood, especially with regard to their statistical background, and in relation to other risk management approaches such as HACCP or Food Safety Objectives. This paper gives an overview on the design of sampling plans forming part of microbiological criteria for foods and on characteristics that determine their reliability and performance.
*Presented at the 36th Symposium of the Swiss Society of Food Hygiene, Zurich, 8 October 2003 32 Mitt. Lebensm. Hyg. 95, 32–44 (2004) Attributes sampling plans Two-class attributes sampling plans A simple way to decide whether to accept or reject a food lot may be based on some microbiological test performed on several sample units. For pathogens this will usually be a test for the presence (positive result) or absence (negative result) of the organism. Concentrations of microorganisms can be assigned to a particular attribute class by determining whether they are above (positive) or below (negative) some preset concentration.
The decision making process of a two-class plan is essentially defined by two numbers. The first, denoted as n, determines the number of sample units that are to be drawn independently and randomly from the lot. The second number, denoted as c, is the maximum allowable number of sample units yielding unsatisfactory test results, for example, the presence of the organism. In case of a two-class plan applied to grouped quantitative data there is one microbiological limit, denoted by m, which separates good quality from non-acceptable or defective quality. In this case the maximum allowable number of sample units exceeding this limit is given by c, which is usually set to zero for pathogens.
To visualize and study the performance of a sampling plan a graphical representation of its Operating Characteristic (OC) curve or function is useful. For a two-class plan this curve has two scales, the horizontal scale showing a measure of lot quality like the fraction or percentage of positive (“defective”) units in the lot being tested, the vertical scale giving the probability of acceptance. The OC curve shown in figure 1, for example, depicts acceptance probabilities for lots in relation to the fraction of defective units when a two-class plan is applied specifying that a number of n=5 sample units are to be drawn and none of them (c =0) are allowed to be positive.
If evaluated for lots containing proportions defective that are regarded as not acceptable, for instance in a risk analysis context, acceptance probabilities characterize the risk that non-conforming lots will be falsely accepted. On the other hand, rejection probabilities, or one minus the according acceptance probabilities, that are derived for actually conforming lots describe the so-called producer’s risk.
When sampling plans are compared and their stringency in making decisions is considered, different aspects of their performance can be addressed. In an idealized situation the OC curve would fall down from a 100% probability of acceptance to a 100% probability of rejection just at the limit proportion defective that distinguishes between conforming and non-conforming lot quality. In practice, no sampling plan can achieve this ideal, but the steeper the curve, the closer the sampling plan comes to approaching the ideal. In general steeper curves can only be achieved by increasing the number of sample units n to be drawn from a lot. This should be distinguished from a shift of the OC curve that is achieved by decreasing the acceptance number c.
A lower value for c will result in a general reduction of consumer’s risk by stating a different limit of acceptability, whereas the producer’s risk will be increased.
Mitt. Lebensm. Hyg. 95 (2004) 33 There are two additional vertical lines in figure 1 highlighting lot qualities that may be referred to as characterizing sampling plan performance. From a consumer’s or regulator’s point of view the prospects to ensure food safety by applying a sampling plan can be evaluated by examining which lot quality would be rejected with high probability, for example 95% (or accepted with low probability). Food producers, however, will be more interested in examining which lot quality would be accepted with high probability, say 95%, to adjust their production processes accordingly.
Figure 1 OC-curve for a two-class sampling plan in relation to proportion defective Thus, performance of the sampling plan in general, lot qualities that are actually rejected or accepted with high probability, and steepness of the OC curve depend on sampling plan specifications n and c. Figure 2 gives an impression how much these characteristics change in case the number of sample units is increased to n=10 or n=20 resulting in more steeply falling curves and lower acceptance probabilities, and thus in better assurance that lots with high proportions defective will be rejected.
Three-class attributes sampling plans In situations where decisions are not based on results of presence-absence tests but on quantitative analytical results, three-class plans can be applied as an alternative to two-class plans working with data grouped according to a single microbiological limit m.
Figure 2 OC-curves for two-class sampling plans in relation to proportion defective with varying number of sample units Three-class plans were devised for situations where the quality of food lots can be divided into three attribute classes. As in two-class plans based on quantitative analytical results, sample results above a concentration m, which in a three-class plan separates good quality from marginally acceptable quality, are not desirable, but a certain number, denoted as c, can be accepted. However, sample results above a second microbiological limit M are unacceptable (or defective), and usually a lot is rejected if any analytical result for the n sample units drawn from the lot being tested exceeds M.
For three-class plans acceptance probabilities for lots being tested depend on two fractions describing lot quality, the percentage of marginally acceptable units with microbiological concentrations between m and M, and the percentage of unacceptable units with concentrations exceeding M. Therefore, depiction of OC functions for three-class plans results in three-dimensional graphs, which are difficult to compare with two-dimensional OC curves visualizing performance of two-class plans as described earlier.
Attributes sampling plans for assessment of mean microbiological concentrations Only when the result of a microbiological analysis is given in a quantitative manner, for instance as a count, there is a choice between types of sampling plans like two- or three-class plans, and thus need for some way to compare their perMitt. Lebensm. Hyg. 95 (2004) 35 formance. The decision for a suitable sampling plan depends on the given purpose and on available prior information on production processes. When dealing with quantitative analytical results for sample units in a lot, questions arise concerning the frequency distributions of sample results and whether there is any previous information on shape, location, and spread of these distributions.
The following considerations are restricted to a situation where the production process is known and well documented showing some evidence that log-transformed sampling results from food lots follow a normal distribution. Based on this assumption sampling plans can be compared by means of their OC function, which is again calculated and plotted for various lot qualities, but now lot quality is described by the mean concentration of microbes for all units in the lot and their standard deviation. To relate the performance of attributes sampling plans to concentration the frequency distribution of analytical results in sample units is used to establish the proportion of defective samples in the lot, as proposed by Hildebrandt et al. (3). Assuming a normal distribution for log-concentrations of microbes, the area under the normal density function above m, as shown in figure 3, is used to define the value for the proportion defective for a two-class sampling plan. For a three-class sampling plan the area between m and M defines the value for the proportion marginally acceptable, as shown in figure 4, and the area above M defines the value for the proportion defective. With given frequency distribution for the lot, these proportions can be derived, making it possible to calculate acceptance probabilities for both types of attributes plans and to relate them to mean concentration in the lot being sampled.
Figure 4 Frequency distribution (log-normal) describing lot quality and proportions marginally acceptable and defective for a three-class sampling plan OC curves related to mean concentrations then can be developed by
• increasing the mean of a normal distribution with fixed standard deviation through a range of values,
• deriving the corresponding proportions defective (and marginally acceptable) for each distribution,
• calculating acceptance probabilities according to prescriptions for n and c,
• and plotting them against the normal distribution means.
A spreadsheet to facilitate these calculations has been developed by Legan et al.
(4) and can be downloaded from the ICMSF homepage (see references).
As an example, figure 5 shows the resulting OC curve for a two-class plan with n=5, c=0, and a microbiological limit set at m=100 cfu/g (or m=2 in log-units), assuming log concentrations in sample units are normally distributed with a standard deviation of 0.8 log-units.
Mean concentrations characterizing lot qualities that will be rejected or accepted with high probability, say 95%, can be highlighted for this type of plot as well (figure 5). Likewise, the effects of changes in sampling plan prescriptions can be studied. Figure 6, for example, visualizes shifts of the OC curve gained by increasing the number of sample units to n=10 and n=20.
However, when dealing with quantitative analytical results, performance of the sampling plan does not only depend on n and c, the number of sample units and the maximum allowable number of non-acceptable units, but on the microbiological limits m and M as well. Furthermore, calculation of acceptance probabilities
Figure 6 OC-curves for two-class sampling plans in relation to mean microbial concentration with varying number of sample units requires assumptions be made regarding the shape and spread of the frequency distribution of sample results. Thus, the effect of using an attributes plan is also dependent on the validity of the underlying assumptions for the frequency distribuMitt. Lebensm. Hyg. 95 (2004) tion, especially with regard to its standard deviation. However, with prior experience substantiating the assumptions made, even attributes plans can be used to assess mean microbiological concentrations in lots of food.
For two-class plans suggested by ICMSF for situations classified as cases 10 to 15 (serious to severe concern) mean microbial concentrations that are rejected with 95% probability are given in table 1, assuming a lot standard deviation of 0.8 logunits and a microbiological limit of m=0 cfu/25 g (i.e., 25 g samples are drawn that should be negative with regard to the target microorganism). Likewise, table 2 is listing mean microbial concentrations that are accepted with 95 % probability. With respect to a scale expressed in cfu/g these means should be interpreted as geometric means, as they are derived by taking arithmetic means on the log-scale to base 10.
Variables sampling plans When the underlying distribution of microbial concentrations within lots is known, or can be assumed, an alternative option is to use variables sampling plans.
As such plans make full use of the microbial counts, rather than ascribing them to categories or classes, variables plans can be more useful under some conditions than attributes plans. The following is an example of the way in which a variables plan Mitt. Lebensm. Hyg. 95 (2004) 39 may be designed. In this case, the decision rule is based on the assumption that the underlying distribution of microbial concentrations in the lot is log-normal, as assumed for attributes plans.