Reconsideration of the derivation of Most Probable Numbers, their standard deviations, confidence bounds and rarity values

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1 Journal of Applied Microbiology ISSN ORIGINAL ARTICLE Reconsideration of the derivation of Most Probable Numbers, their standard deviations, confidence bounds and rarity values B. Jarvis 1, C. Wilrich 2 and P.-T. Wilrich 3 1 Ross Biosciences Ltd., Upton Bishop, Ross-on-Wye, UK 2 Bundesanstalt für Materialforschung und prüfung, Berlin, Germany 3 Institut für Statistik und Ökonometrie, Freie Universität Berlin, Berlin, Germany Keywords confidence bounds, microbiological analyses, most probable number, MPN, rarity value. Correspondence Basil Jarvis, Ross Biosciences Ltd, Upton Bishop, Ross-on-Wye HR9 7UR, UK. basil.jarvis@btconnect.com : received 31 January 2010, revised 3 June 2010 and accepted 4 June 2010 doi: /j x Abstract Aims: The purpose of this work was to derive a simple Excel spreadsheet and a set of standard tables of most probable number (MPN) values that can be applied by users of International Standard Methods to obtain the same output values for MPN, SD of the MPN, 95% confidence limits and test validity. With respect to the latter, it is considered that the Blodgett concept of rarity is more valuable than the frequently used approach of improbability (vide de Man). Methods and Results: The paper describes the statistical procedures used in the work and the reasons for introducing a new set of conceptual and practical approaches to the determination of MPNs and their parameters. Examples of MPNs derived using these procedures are provided. The Excel spreadsheet can be downloaded from wilrich/index.html. Conclusions: The application of the revised approach to the determination of MPN parameters permits those who wish to use tabulated values, and those who require access to a simple spreadsheet to determine values for nonstandard test protocols, to obtain the same output values for any specific set of multiple test results. The concept of rarity is a more easily understood parameter to describe test result combinations that are not statistically valid. Provision of the SD of the log MPN value permits derivation of uncertainty parameters that have not previously been possible. Significance and Impact of the Study: A consistent approach for the derivation of MPNs and their parameters is essential for coherence between International Standard Methods. It is intended that future microbiology standard methods will be based on the procedures described in this paper. Introduction The most probable number (MPN) procedure is used widely to estimate microbial densities in many matrices including foods and water. The procedure, derived from the original work of McCrady (1915), consists of adding a volume of each of several serial dilutions of a sample to a number of replicate tubes of culture medium and, following incubation under appropriate conditions, recording the number of tubes showing growth at each level of inoculum. The estimate of density is based on the application of the theory of probability based on certain assumptions. The first primary assumption is that the inoculum contains a random distribution of microbial cells. This implies that each dilution is thoroughly mixed and clumps or aggregates of cells do not occur or, if they do, that they are not disrupted during further dilution stages an assumption that may not always be correct Journal of Applied Microbiology 109, ª 2010 The Society for Applied Microbiology

2 B. Jarvis et al. Derivation of MPNs and their parameters Secondly, that each volume of inoculum containing at least one viable organism will exhibit growth when incubated in the culture medium. If these assumptions are not met, then the MPN procedure may underestimate true microbial cell density. Over time, different approaches have been made to derive MPN values. Seminal papers include those by Halvorson and Ziegler (1933), Barkworth and Irwin (1938), Haldane (1939), Finney (1947), Cochran (1950) and de Man (1975, 1977, 1983). Although much early work was concerned with testing replicate tubes at a single dilution level, the concept of repeating the tests with replicate tubes at multiple dilution levels soon became normal practice. Haldane (1939) and Cochran (1950) describe the statistical theory behind single and multiple dilution tests and explain some of the problems associated with the calculations of MPN values for multiple tube tests. Cochran (1950) also stressed the need to ensure that the level of inoculum should lie within certain limits in order properly to cover the expected level of organisms in the original sample and provide a means of assessing the standard deviation of the log 10 of the cell density for different combinations of dilution ratios and replicate tests at each level. Since that time, several authors have published tables of MPN values for various combinations of replicate tubes and dilution levels. Pre-eminent are those of de Man (1983) that provide confidence bounds for the MPN estimates together with a measure of the improbability of the estimate. In addition, various workers have developed computer software to enable the calculation of MPN estimates these include Hurley and Roscoe (1983), Klee (1993), Curiale (2000), Garthright and Blodgett (2003), US FDA (2006) and La Budde (2008). Garthright and Blodgett (2003) describe the FDA s preferred MPN methods for standard tests (e.g. five tubes at three dilution levels) and also for large and unusual combinations of tests. Such test systems may use 25 or more replicate tests at each dilution level (Moruzzi et al. 2000) or may use well trays with automatic pipetting for serial dilution assays (Irwin et al. 2000; Walser 2000). One of the benefits of the US FDA (2006) spreadsheet system is that it can accommodate situations where one or more tube in a series has been lost or where overgrowth makes a tube reading unreliable. However, use of the US FDA (2006) spreadsheet system results in determination of confidence bounds that differ from those cited in de Man s and other standard tables, including the FDA tables, thus risking confusion amongst practitioners. The reason for these differences is that different approximations have been used to derive the confidence bounds. Blodgett (personal communication) says that the FDA spreadsheet approach uses the normal approximation described originally by Haldane (1939), whereas the tables are based on a modification of the method devised by de Man (1983). For a more detailed consideration of the different approaches that have been made to determination of MPN confidence intervals, see Garthright and Blodgett (2003). Another important aspect of deriving MPN estimates is that of improbable outcomes. Such outcomes may occur, for instance, as a consequence of laboratory errors. Suppose three sets of results based on five tubes at each of three successive tenfold serial dilutions give outcomes of The calculated estimate of this MPN is 9Æ05 per ml if the inocula are 0Æ1, 0Æ01 and 0Æ001 ml of original sample. But the statistical likelihood for the outcome is only 0Æ In de Man s (1983) MPN tables, and also in those of the US FDA (2006), an improbability index provides guidance to practitioners as to the acceptability or otherwise of a set of results. However, Blodgett (2002) suggested an alternative approach to the assessment of improbability that he describes as the rarity value (Blodgett 2008). Improbability is the sum of the probabilities of all possible outcomes as likely, or less likely, than the actual outcome. In contrast, the rarity value measures the probability of the actual outcome divided by the probability of the most likely outcome. In this work, we have developed a rarity score that is used to indicate whether a particular outcome is likely or not. With the plethora of publications on MPN methods, it is pertinent to ask why we have considered it necessary to reinvent the wheel with yet another publication. In 2008, a number of mistakes were identified in part of a revised International Standard (Anon 2007) dealing inter alia with MPN techniques. The statistics working group (SWG) of ISO TC34 SC9 was asked to recommend amendments to this standard. In so doing, we reviewed all international standards on food, dairy and water microbiology and determined that of 15 published standards, only five standards (Anon 2003, 2004, 2005, 2006a,b) cross-refer to Anon (2007), or its predecessor, whilst others use different sets of MPN tables and or refer the user to one of several different software systems for estimation of MPN values. The SWG recommended that all relevant ISO microbiological standards should be revised to include reference to a revised edition of Anon (2007), which would include both tables of MPNs for standard combinations of tests and reference to a specific, generally available software that could be used for any combination of test systems. We evaluated the free software available and concluded that new approaches would be desirable. We recommended that for an international standard, it was essential that both the tabulated parameters and those determined by use of a spreadsheet should give the same results. We concluded, also, that it would Journal of Applied Microbiology 109, ª 2010 The Society for Applied Microbiology 1661

3 Derivation of MPNs and their parameters B. Jarvis et al. be sensible to replace the concept of improbability with the rarity approach of Blodgett (2002, 2008) and to provide also an estimate of standard deviation that can be used to derive the measurement uncertainty of MPN values. This paper describes the statistical approach used and provides examples of the outputs. Methods MPN estimations are essentially presence or absence tests performed using serial dilutions to estimate the density l [as colony forming units (CFU) per g or ml] of a target micro-organism in a test matrix. Samples drawn from the matrix are diluted to several different levels, and at each dilution level several tubes of culture medium are inoculated with a portion (typically 1 ml) of that dilution. After incubation under defined conditions, the number of tubes that show the presence of the micro-organism at each separate inoculum level is counted. These counts are the basis of the calculation of the MPN as an estimate of the concentration l. We assume that the target micro-organisms are randomly distributed within the matrix, that they are separate, not clustered together, and that they do not repel each other. The numbers of micro-organisms in each quantum of inoculum are independent. We assume also that every tube whose inoculum contains at least one viable target micro-organism will show the presence of the micro-organism after incubation. Under these assumptions, it is reasonable to assume a Poisson distribution of the number of micro-organisms in the tubes. Let k be the number of dilutions, d i the relative dilution level per tube (i.e. for a tenfold serial dilution from 10 )1 to 10 )3, d i =0Æ1, 0Æ01 and 0Æ001, respectively), w i the volume (or weight) of the inoculum at dilution level i, n i the number of tubes and the number of positive tubes (tubes that show the presence of the micro-organism) at dilution i; i ¼ 1; 2; :::; k. Hence, a serial dilution test with k dilutions and its results can be described by the k quadruples (d i, w i,- n i, ), as illustrated in Fig. 1. Figure 1 Screenshot of Excel spreadsheet for determination of Most Probable Number estimates and their parameters Journal of Applied Microbiology 109, ª 2010 The Society for Applied Microbiology

4 B. Jarvis et al. Derivation of MPNs and their parameters Determination of the MPN At each dilution, i the number Y of micro-organisms in the tubes follows the Poisson distribution with expectation d i w i l. The probability of y ¼ 0; 1; 2; ::: microorganisms in a tube is given by PðY ¼ yþ ¼ ðd iw i lþ y expð d i w i lþ; y ¼ 0; 1; 2; ::: y! Particularly, the probability of no micro-organism in a tube is PðY ¼ 0Þ ¼expð d i w i lþ and the probability of at least one micro-organism in a tube is p i ¼ PðY>0Þ ¼1 PðY ¼ 0Þ ¼1 expð d i w i lþ: The number X i of positive tubes at dilution step i follows the binomial distribution with parameters p i and n i. The probability of ¼ 0; 1; :::; n i positive tubes is given by PðX i ¼ Þ¼ ¼ n i n i p xi i ð1 p iþ ðni xiþ ð1 expð d i w i lþþ xi ðexpð d i w i lþþ ðni xiþ : The probability of observing (x 1, x 2,..., x k ) positive tubes at the k dilutions is PðX 1 ¼ x 1 ; :::; X k ¼ x k Þ ¼ Yk n i ð1 expð d i w i lþþ xi ðexpð d i w i lþþ ðni xiþ : Given the results (x 1, x 2,..., x k ) of the serial dilution test, this is the likelihood function of l, L ¼ Lðl; d i ; w i ; ; i ¼ 1; :::; kþ ¼ Yk n i ð1 expð d i w i lþþ xi ðexpð d i w i lþþ ðni xiþ : The likelihood function L(l) gives, for each possible concentration l, the probability of the result of the serial dilution test that has been observed. As an estimated MPN of the concentration l, we use the value of l that maximizes the likelihood function. As the logarithm 1 1 For convenience we use natural logarithms ln(x) = log e (x). log (f(x)) of a function has its maximum at the same value as the function f(x), we use the Loglikelihood function ln L ¼ Xk ln n i þ lnð1 expð d i w i lþþ ðn i Þd i w i l and calculate the MPN as the value ^l at which the first derivative of the Loglikelihood function with respect to l, is 0: X k ¼ ln ¼ Xk d i w i expð d i w i lþ 1 expð d i w i lþ ðn i Þd i w i d i w i expð d i w i^lþ 1 expð d i w þd i w i ð1 expð d i w i^lþþ n i d i w i i^lþ 1 expð d i w i^lþ d i w i 1 expð d i w n id i w i ¼0: i^lþ An estimate V^arð^lÞ of the variance Varð^lÞ of ^l is obtained from the second derivative of the Loglikelihood function, 2 ln 2 V^arð^lÞ ¼ ¼ Xk 2 ln L 2 l ¼ ^l " # ðd i w i Þ 2 expð d i w i lþ ð1 expð d i w i lþþ 2 ; P k 1 " #: ðd i w i Þ 2 expð d i w i^lþ ð1 expð d i w i^lþ Þ 2 The standard deviation of the estimate ^l is given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Varð^lÞ : If only negative test results have been observed, = 0 for i = 1,..., k, the MPN is ^l ¼ 0; and if only positive test results have been observed, = n i for i = 1,..., k, the MPN is infinity. A confidence interval for the concentration There are various ways to construct a confidence interval for the concentration l. The calculation of an exact 1 ) a = 95% confidence interval [l L, l U ] for the concentration l is tedious. It is described in detail by de Man (1983). Its limits additionally depend on the rule by which the a = 5% improbable concentrations are divided into those below l L and those above l U and hence, Journal of Applied Microbiology 109, ª 2010 The Society for Applied Microbiology 1663

5 Derivation of MPNs and their parameters B. Jarvis et al. different authors end up with different confidence intervals. We propose to use an easy approximation. As a maximum likelihood estimator of ln l, the (natural) logarithm ln ^l of ^l follows an approximately normal distribution with estimated variance ^r 2 ln ^l ¼ V^arðln ^lþ ¼V^arð^lÞ=^l2 : Hence, the interval ln ^l 2^r ln ^l ; ln ^l þ 2^r ln ^l is an approximate 95% confidence interval for ln l and ^l expð 2^r ln ^l Þ; ^l expð2^r ln ^l Þ is an approximate 95% confidence interval for the concentration l. If only negative test results have been observed, = 0 for i = 1,..., k, the lower confidence limit l L is 0, and the upper confidence limit l U is the value l for which the likelihood function Lðl U Þ¼0 025; where 0Æ025 = (1 ) 0Æ95) 2: Y k expð d i w i l U Þ n i ¼ 0 025: positive tubes we find MPN = ^l ¼ 006 CFU ml )1. However, the result x 1 ¼ 0; x 2 ¼ 5; x 3 ¼ 10 strongly contradicts our expectation that, with increasing dilution levels, the numbers of positive results should decrease. Hence, this result violates our assumptions underlying the MPN determination. As the MPN calculation works irrespective of the unlikeliness of the result of the serial dilution test, we need a measure of this unlikeliness to decide whether to trust the result of the test or not. The rarity index (r), introduced by Blodgett (2002, 2008), is the ratio of two likelihood values: r ¼ Lð^lÞ L 0 ð^lþ ; In the numerator, we have the likelihood Lð^lÞ ¼exp Xk ln n i! ðn i Þd i w i^l þ lnð1 expð d i w i^lþþ of the result (x 1, x 2,..., x k ) of the serial dilution test, i.e. the value that we find if we insert ^l into the likelihood function L(l). In the denominator, we have the maximum of the likelihood Lð^lÞwith respect to (x 1,..., x k ), L 0 ð^lþ ¼ max ðx 1 ; :::; x k Þ flð^lþ g ¼ max ( exp Xk ln n!) i þ lnð1 expð d i w i^lþþ ðn i Þd i w i^l ; ðx 1 ; :::; x k Þ This gives ln 0025 l U ¼ P k d i w i n i ¼ P k ln 40 : d i w i n i If only positive test results have been observed, = n i for i = 1,..., k, the upper confidence limit l U is infinity, and the lower confidence limit l L is the value for which the likelihood function L(l L )=0Æ025: Y k ð1 expð d i w i l L ÞÞ n i ¼ 0025: This is a nonlinear equation for l L. The rarity index If we perform a serial dilution test with k = 3 dilutions, dilution factors d 1 ¼ 1; d 2 ¼ 01; d 3 ¼ 001, inocula volumes w 1 = w 2 = w 3 and numbers of tubes n 1 = n 2 = n 3 = 10 and observe x 1 ¼ 0; x 2 ¼ 5; x 3 ¼ 10 i.e. the value of the likelihood if the result of the serial dilution test was most likely under a concentration l equal to the estimate ^l of the concentration; this maximum is achieved if ¼½ðn i þ 1Þð1 expð d i w i^lþš; i ¼ 1; :::; k where [x] denotes the largest integer not larger than x. The rarity indes a value between 0 and 1. It is 1 if the result of the serial dilution test is most likely a concentration equal to the estimated MPN. If it is in the neighbourhood of 0, the result of the serial dilution test is very unlikely for a concentration equal to the estimated MPN. Following the approach of de Man (1975, 1983), we use three categories of rarity: Category 1: The MPN value would be very likely to occur if its rarity value falls within the range 0Æ05 1Æ00 (0Æ05 r 1Æ00). Category 2: The MPN value would be expected to occur only rarely if its rarity value falls within the range 0Æ01 0Æ05 (0Æ01 r <0Æ05) Journal of Applied Microbiology 109, ª 2010 The Society for Applied Microbiology

6 B. Jarvis et al. Derivation of MPNs and their parameters Category 3: The MPN value would be expected to occur extremely rarely if its rarity value falls within the range 0 0Æ01 (0 < r <0Æ01). If only negative results or only positive results have been observed, the rarity indes r = 1 and hence, the category is also 1. Software An Excel spreadsheet for estimation of the MPN and its parameters has been developed that is freely available from iter/wilrich/index.html. Results Table 1 illustrates the MPN parameters for a three-tube assay with tenfold dilutions. The MPN values, derived using the procedures described by Arndt et al. (1981), are essentially identical to those found in the tables of de Man (1983) and on the BAM website (Garthright and Blodgett 2003; US FDA, 2006). The data columns in the table are as follows: Columns 1 3 show the numbers of positive test results for inoculum quanta of 1Æ0, 0Æ1 and 0Æ01 g or ml of sample, respectively. Column 4 shows the derived MPN estimates referenced to the primary level of inoculum (i.e. 1Æ0 g or ml) rounded to two significant figures. Respectively, MPN values for larger or smaller quantities of primary inocula should be divided or multiplied by the additional factor involved, e.g. if the series contains 10, 1 and 0Æ1 g inoculum, the listed MPN value per g should be divided by 10; if the series contains 0Æ01, 0Æ001 and 0Æ0001 g inoculum, the MPN value per g should be multiplied by 100. Columns 5 and 6 show the log MPN and the standard deviation, respectively, of the log MPN. Thus, it is possible to derive an estimate of the expanded microbiological uncertainty for the calculated MPN value and combine it with uncertainties stemming from other sources. Columns 7 and 8 provide approximate bounds of the 95% confidence interval for each MPN value. Column 9 lists the calculated rarity value for the MPN result, based on the procedure of Blodgett (2008) that is used to determine the acceptability category of potential MPN results (Column 10). Table 1 Most Probable Number (MPN) table for a 3 3 design (i.e. three sequential tenfold dilution levels and a reference quantum of 1Æ0 g) for outcomes with rarity index category 1 and 2 Number of positive results for inoculum volume (ml or g) MPN SD of 95% confidence limits 1Æ00 0Æ10 0Æ01 per ml or per g log 10 MPN log 10 MPN Lower Upper Rarity index Category NA* NA 0 1Æ1 1Æ Æ30 )0Æ52 0Æ43 0Æ041 2Æ3 0Æ Æ36 )0Æ45 0Æ44 0Æ048 2Æ7 1Æ Æ72 )0Æ14 0Æ31 0Æ17 3Æ0 0Æ Æ74 )0Æ13 0Æ31 0Æ18 3Æ1 0Æ Æ1 0Æ056 0Æ26 0Æ35 3Æ7 0Æ Æ92 )0Æ037 0Æ32 0Æ21 4Æ0 1Æ Æ4 0Æ16 0Æ26 0Æ42 4Æ8 0Æ Æ5 0Æ17 0Æ27 0Æ43 5Æ0 0Æ Æ0 0Æ31 0Æ23 0Æ69 6Æ0 0Æ Æ1 0Æ32 0Æ24 0Æ71 6Æ2 0Æ Æ3 0Æ36 0Æ31 0Æ55 9Æ7 1Æ Æ8 0Æ59 0Æ31 0Æ Æ Æ3 0Æ63 0Æ33 0, Æ Æ5 0Æ87 0Æ30 1Æ9 30 0Æ Æ1 0Æ26 3Æ6 37 0Æ Æ3 0Æ97 0Æ32 2Æ2 40 1Æ Æ2 0Æ27 4Æ4 51 0Æ Æ3 0Æ24 7Æ2 64 0Æ Æ4 0Æ32 5Æ Æ Æ7 0Æ34 9Æ Æ Æ0 0Æ Æ NA NA 36 1Æ000 1 *NA, not available. Journal of Applied Microbiology 109, ª 2010 The Society for Applied Microbiology 1665

7 Derivation of MPNs and their parameters B. Jarvis et al. Table 2 Examples of Most Probable Number (MPN) estimates for large and unusual combinations of tests Inocula (ml) Weight of sample* (g) No. tubes No. positives MPN per ml or per g log 10 MPN log 10 SD MPN 95% confidence limits Lower Upper Rarity index Rarity category 1Æ0, 0Æ1, 0Æ01 1Æ0 20, 20, 20 20, 14, Æ1 0Æ11 7Æ6 21 0Æ Æ0, 0Æ1, 0Æ01 1Æ0 50, 50, 50 50, 35, Æ1 0Æ071 9Æ0 17 0Æ Æ0, 0Æ1, 0Æ01 2Æ0 50, 50, 50 50, 35, 7 6Æ3 0Æ80 0Æ071 4Æ5 8Æ7 0Æ Æ0, 0Æ1, 0Æ01 2Æ0 50, 49, 49 50, 34, 7 6Æ2 0Æ79 0Æ071 4Æ5 8Æ6 0Æ Æ0, 1Æ0, 0Æ1, 0Æ01 1Æ0 1, 10, 10, 10 1, 9, 4, 1 3Æ3 0Æ51 0Æ15 1Æ7 6Æ4 0Æ Æ0, 1Æ0, 0Æ1, 0Æ01 1Æ0 1, 10, 10, 10 1, 4, 2, 1 0Æ80 )0Æ096 0Æ17 0Æ37 1Æ7 0Æ Æ0, 1Æ0, 0Æ1, 0Æ01 1Æ0 1, 10, 10, 10 0, 5, 1, 0 0Æ33 )0Æ49 0Æ18 0Æ14 0Æ74 0Æ Æ0, 1Æ0, 0Æ5, 0Æ1, 0Æ05 1Æ0 1, 5, 5, 5, 5 1, 5, 3, 1, 1 2Æ7 0Æ42 0Æ16 1Æ3 5Æ5 0Æ512 1 *Quantity of sample in 1 ml of the initial homogenate (w i ). In Table 1, and also in the output of the computer programme, we show that if results at all test levels are negative, then the MPN is 0; this value is supplemented by the 95% confidence interval (with the lower confidence limit 0). Most published MPN tables give a value <x where s the MPN for the next set of results with the same number of tubes and only one positive result. For instance, if in the design (3 1Æ0, 3 0Æ1, 3 0Æ01 ml) all results are negative, we report MPN = 0 with a 95% confidence interval [0, 1Æ1], whereas for this design some tables (e.g. de Man 1983) and computer programmes (e.g. Curiale 2000) show MPN < 0Æ30, with no confidence limits. We believe that our statement is much more informative: i.e. 0 is the most probable concentration, but a concentration up to 1Æ1 is possible with 95% confidence. Similarly, if all test results are positive the MPN is infinity ( ) with 95% confidence interval [36, ], whereas de Man (1983), Curiale (2000) and others give MPN > 110 with no confidence limits. Table 2 illustrates results for some different combinations of dilution factor, inoculum level and number of replicate tests undertaken. These values were derived using the Excel spreadsheet version of the programme, for which a partial screenshot is presented as Fig. 1. Discussion The procedure described here provides a means of obtaining MPN estimates, and their parameters, for both standard and nonstandard assay combinations using either derived tables of values or a freely available spreadsheet, both of which provide identical outputs for the same inputs. The system is to be included in the revised ISO 7218 Standard to which other international standards will cross-refer. Acknowledgements The authors are grateful to their colleagues on the SWG of ISO TC34 SC9 and the anonymous reviewers for helpful comments and discussion. References Anon (2003) Microbiology of Food and Animal Feeding Stuffs Horizontal Method for the Enumeration of Coagulasepositive Staphylococci (Staphylococcus aureus and Other Species) Part 3: Detection and MPN Technique for Low Numbers. ISO 68883:2003. Geneva: International Organisation for Standardisation. Anon (2004) Microbiology of Food and Animal Feeding Stuffs Horizontal Methods for the Detection and Enumeration of Enterobacteriaceae Part 1: Detection and Enumeration by MPN Technique with Pre-enrichment. ISO :2004. Geneva: International Organisation for Standardisation. Anon (2005) Microbiology of Food and Animal Feeding Stuffs Horizontal Method for the Enumeration of Beta-glucuronidase-positive Escherichia coli Part 3: Most Probable Number Technique Using 5-bromo-4-chloro-3-indolyl-beta-Dglucuronide. ISO :2005. Geneva: International Organisation for Standardisation. Anon (2006a) Microbiology of Food and Animal Feeding Stuffs Horizontal Method for the Detection and Enumeration of Coliforms Most Probable Number Technique. ISO 4831:2006. Geneva: International Organisation for Standardisation. Anon (2006b) Microbiology of Food and Animal Feeding Stuffs Horizontal Method for the Determination of Low Numbers of Presumptive Bacillus cereus Most Probable Number Technique and Detection Method. ISO 21871:2006. Geneva: International Organisation for Standardisation. Anon (2007) Microbiology of Food and Animal Feeding Stuffs General Requirements and Guidance for 1666 Journal of Applied Microbiology 109, ª 2010 The Society for Applied Microbiology

8 B. Jarvis et al. Derivation of MPNs and their parameters Microbiological Examinations. ISO 7218:2007. Geneva: International Organisation for Standardisation. Arndt, G., Weiss, H. and Hampe, F. (1981) On the investigation of some statistical properties of the Most Probable Number (MPN)-procedure for estimating the density of microorganisms by use of computer simulations. In Computational Statistics ed. Büning, H. and Naeve, P. pp Berlin, NY: de Gruyter. Barkworth, H. and Irwin, J.O. (1938) Distribution of coliform organisms in milk and the accuracy of the presumptive coliform test. J Hyg, Camb 38, Blodgett, R.J. (2002) Measuring improbability of outcomes from a serial dilution test, Communications in Statistics. Theory Methods 31, Blodgett, R.J. (2008) Judging a model of an experiment. A Paper Presented to the Joint Statistical Meetings of the American Statistical Association, Denver, 14 August Cochran, W.G. (1950) Estimation of bacterial densities by means of the Most Probable Number. Biometrics 6, Curiale, M. (2000) MPN Calculator for Excel. (accessed 23 June 2010). Finney, D.J. (1947) The principles of biological assay. J Roy Stat Soc, Ser B 9, Garthright, W.E. and Blodgett, R.J. (2003) FDA s preferred MPN methods for standard, large or unusual tests, with a spreadsheet. Food Microbiol 20, Haldane, J.B.S. (1939) Sampling errors in the determination of bacterial or virus density by the dilution method. J Hyg, Camb 39, Halvorson, H.O. and Ziegler, N.R. (1933) Application of statistics to problems in bacteriology. J Bacteriol 25, Hurley, M.A. and Roscoe, M.E. (1983) Automated statistical analysis of microbial enumeration by dilution series. J Appl Bacteriol 55, Irwin, P., Tu, S., Damert, W. and Phillips, J. (2000) A modified Gauss Newton algorithm and ninety-six well micro-technique for calculating MPN using Excel spreadsheets. J Rapid Methods Autom Microbiol 8, Klee, A.J. (1993) A computer program for the determination of most probable number and its confidence limits. J Microbiol Methods 18, La Budde, R. (2008) A simple MPN calculator. (accessed 23 June 2010). de Man, J.C. (1975) The probability of most probable numbers. Eur J Appl Microbiol 1, de Man, J.C. (1977) MPN tables for more than one test. Eur J Appl Microbiol 4, de Man, J.C. (1983) MPN Tables corrected. Eur J Appl Microbiol 17, McCrady, M.H. (1915) The numerical interpretation of fermentation-tube results. J Infect Dis 17, Moruzzi, G., Garthright, W.E. and Floros, J.D. (2000) Aseptic packaging machine pre-sterilization and package sterilization: statistical aspects of microbiological validation. Food Control 11, US FDA (2006) Bacteriological Analytical Manual Online Appendix 2: Most Probable Number from Serial Dilutions. Methods/BacteriologicalAnalyticalManualBAM/acm htm (accessed 23 June 2010). Walser, P.E. (2000) Using conventional microtiter plate technology for the automation of microbiological testing of drinking water. J Rapid Methods Autom Microbiol 8, Journal of Applied Microbiology 109, ª 2010 The Society for Applied Microbiology 1667