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

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 http://www.wiwiss.fu‐berlin.de/institute/iso/mitarbeiter/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. 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 log₁₀ 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 0‐0‐5. The calculated estimate of this MPN is 9·05 per ml if the inocula are 0·1, 0·01 and 0·001 ml of original sample. But the statistical likelihood for the outcome is only 0·000025. 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 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 μ [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 μ.

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 = 0·1, 0·01 and 0·001, respectively), w_i the volume (or weight) of the inoculum at dilution level i, n_i the number of tubes and x_i the number of positive tubes (tubes that show the presence of the micro‐organism) at dilution . Hence, a serial dilution test with k dilutions and its results can be described by the k quadruples (d_i, w_i, n_i, x_i), as illustrated in Fig. 1.

Figure 1

Screenshot of Excel spreadsheet for determination of Most Probable Number estimates 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_iw_iμ. The probability of micro‐organisms in a tube is given by

Particularly, the probability of no micro‐organism in a tube is

and the probability of at least one micro‐organism in a tube is

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 positive tubes is given by

The probability of observing (x₁, x₂, ..., x_k) positive tubes at the k dilutions is

Given the results (x₁, x₂, ..., x_k) of the serial dilution test, this is the likelihood function of μ,

The likelihood function L(μ) gives, for each possible concentration μ, the probability of the result of the serial dilution test that has been observed. As an estimated MPN of the concentration μ, we use the value of μ that maximizes the likelihood function. As the logarithm¹1 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

and calculate the MPN as the value at which the first derivative of the Loglikelihood function with respect to μ,

is 0:

An estimate of the variance of is obtained from the second derivative of the Loglikelihood function,

as

The standard deviation of the estimate is given by

If only negative test results have been observed, x_i = 0 for i = 1, ..., k, the MPN is and if only positive test results have been observed, x_i = 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 μ. The calculation of an ‘exact’ 1 − α = 95% confidence interval [μ_L, μ_U] for the concentration μ is tedious. It is described in detail by de Man (1983). Its limits additionally depend on the rule by which the α = 5% improbable concentrations are divided into those below μ_L and those above μ_U and hence, different authors end up with different confidence intervals. We propose to use an easy approximation. As a maximum likelihood estimator of ln μ, the (natural) logarithm of follows an approximately normal distribution with estimated variance

Hence, the interval

is an approximate 95% confidence interval for ln μ and

is an approximate 95% confidence interval for the concentration μ.

If only negative test results have been observed, x_i = 0 for i = 1, ..., k, the lower confidence limit μ_L is 0, and the upper confidence limit is the value μ for which the likelihood function where 0·025 = (1 − 0·95)/2 :

This gives

If only positive test results have been observed, x_i = n_i for i = 1, ..., k, the upper confidence limit μ_U is infinity, and the lower confidence limit μ_L is the value for which the likelihood function L(μ_L) = 0·025:

This is a nonlinear equation for μ_L.

The rarity index

If we perform a serial dilution test with k = 3 dilutions, dilution factors , inocula volumes w₁ = w₂ = w₃ and numbers of tubes n₁ = n₂ = n₃ = 10 and observe positive tubes we find MPN = CFU ml⁻¹. However, the result 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:

In the numerator, we have the likelihood

of the result (x₁, x₂, ..., x_k) of the serial dilution test, i.e. the value that we find if we insert into the likelihood function L(μ).

In the denominator, we have the maximum of the likelihood with respect to (x₁, ..., x_k),

i.e. the value of the likelihood if the result of the serial dilution test was most likely under a concentration μ equal to the estimate of the concentration; this maximum is achieved if

where [x] denotes the largest integer not larger than x.

The rarity index is 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 0·05–1·00 (0·05 ≤ r ≤ 1·00).

Category 2: The MPN value would be expected to occur only rarely if its rarity value falls within the range 0·01–0·05 (0·01 ≤ r < 0·05).

Category 3: The MPN value would be expected to occur extremely rarely if its rarity value falls within the range 0–0·01 (0 < r < 0·01).

If only negative results or only positive results have been observed, the rarity index is 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 http://www.wiwiss.fu‐berlin.de/institute/iso/mitarbeiter/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:

Table 1. Most Probable Number (MPN) table for a 3 × 3 design (i.e. three sequential tenfold dilution levels and a reference quantum of 1·0 g) for outcomes with rarity index category 1 and 2

Number of positive results for inoculum volume (ml or g)			MPN per ml or per g	log10 MPN	SD of log10 MPN	95% confidence limits		Rarity index	Category
1·00	0·10	0·01				Lower	Upper
0	0	0	0	NA*	NA	0	1·1	1·000	1
0	1	0	0·30	−0·52	0·43	0·041	2·3	0·087	1
1	0	0	0·36	−0·45	0·44	0·048	2·7	1·000	1
1	0	1	0·72	−0·14	0·31	0·17	3·0	0·021	2
1	1	0	0·74	−0·13	0·31	0·18	3·1	0·211	1
1	2	0	1·1	0·056	0·26	0·35	3·7	0·021	2
2	0	0	0·92	−0·037	0·32	0·21	4·0	1·000	1
2	0	1	1·4	0·16	0·26	0·42	4·8	0·041	2
2	1	0	1·5	0·17	0·27	0·43	5·0	0·426	1
2	1	1	2·0	0·31	0·23	0·69	6·0	0·019	2
2	2	0	2·1	0·32	0·24	0·71	6·2	0·069	1
3	0	0	2·3	0·36	0·31	0·55	9·7	1·000	1
3	0	1	3·8	0·59	0·31	0·93	16	0·084	1
3	1	0	4·3	0·63	0·33	0,95	19	1·000	1
3	1	1	7·5	0·87	0·30	1·9	30	0·209	1
3	1	2	12	1·1	0·26	3·6	37	0·021	2
3	2	0	9·3	0·97	0·32	2·2	40	1·000	1
3	2	1	15	1·2	0·27	4·4	51	0·420	1
3	2	2	21	1·3	0·24	7·2	64	0·068	1
3	3	0	24	1·4	0·32	5·6	100	1·000	1
3	3	1	46	1·7	0·34	9·6	220	1·000	1
3	3	2	110	2·0	0·32	25	480	1·000	1
3	3	3	∞	NA	NA	36	∞	1·000	1

• *NA, not available.

Columns 1–3 show the numbers of positive test results for inoculum quanta of 1·0, 0·1 and 0·01 g or ml of sample, respectively.

Column 4 shows the derived MPN estimates referenced to the primary level of inoculum (i.e. 1·0 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 0·1 g inoculum, the listed MPN value per g should be divided by 10; if the series contains 0·01, 0·001 and 0·0001 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).

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 x is 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 × 1·0, 3 × 0·1, 3 × 0·01 ml) all results are negative, we report MPN = 0 with a 95% confidence interval [0, 1·1], whereas for this design some tables (e.g. de Man 1983) and computer programmes (e.g. Curiale 2000) show MPN < 0·30, 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 1·1 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.

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	log10 MPN	log10 SD MPN	95% confidence limits		Rarity index	Rarity category
							Lower	Upper
1·0, 0·1, 0·01	1·0	20, 20, 20	20, 14, 3	13	1·1	0·11	7·6	21	0·794	1
1·0, 0·1, 0·01	1·0	50, 50, 50	50, 35, 7	13	1·1	0·071	9·0	17	0·806	1
1·0, 0·1, 0·01	2·0	50, 50, 50	50, 35, 7	6·3	0·80	0·071	4·5	8·7	0·806	1
1·0, 0·1, 0·01	2·0	50, 49, 49	50, 34, 7	6·2	0·79	0·071	4·5	8·6	0·746	1
10·0, 1·0, 0·1, 0·01	1·0	1, 10, 10, 10	1, 9, 4, 1	3·3	0·51	0·15	1·7	6·4	0·089	1
10·0, 1·0, 0·1, 0·01	1·0	1, 10, 10, 10	1, 4, 2, 1	0·80	−0·096	0·17	0·37	1·7	0·017	2
10·0, 1·0, 0·1, 0·01	1·0	1, 10, 10, 10	0, 5, 1, 0	0·33	−0·49	0·18	0·14	0·74	0·004	3
5·0, 1·0, 0·5, 0·1, 0·05	1·0	1, 5, 5, 5, 5	1, 5, 3, 1, 1	2·7	0·42	0·16	1·3	5·5	0·512	1

• *Quantity of sample in 1 ml of the initial homogenate (w_i).

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.