Oliver Re icha r t Department of Microbiology and Biotechnology, Unioersity of Horticulture and Food lndustO,,
Budapest, Hungary
(Received 20 February 1990; accepted 28 January. 1991)
The estimation of cell number by using the MPN method was studied mathematically for three parallel inoculations from three decimal dilutions. As a result of computation it can be established that these estimations at a range of microbial numbers over 10 are biased. Confidential estimation could be obtained at cell numbers betw~n 0 and 10 accepting only a few combinations of positive and negative test tubes. These are 1 0 0, 2 0 0, 3 0 0, 3 1 0 and 3 2 0.
Key words: MPN method; Biased estimation; Probability; Relative frequency
Introduction
Nowadays the microbiological standards for foods are becoming increasingly severe, so the MPN method will probably often replace plate counting.
MPN values and confidence intervals can be calculated for any combination of positive and negative test tubes, but the probability of many combinations is so low that they will practically never be obtained.
Leaving out the improbable values, de Man (1975) restricted the MPN tables to results having realistic probability. He suggested two categories. Category 1, normal results, obtained in 95% of cases; Category 2, less likelyresults, obtained only in 4% of cases. Results that are even less likely than those of category 2 are always unacceptable. This results are often due to contamination of sterile tubes.
MPN theory and tables with confidence limits calculated by Man (1983) are critically reviewed by Jarvis (1989). It seems, that the problem of the probability of Most Probable Number has been solved. However, there is an other, neglected problem: estimation of microbial numbers by using MPN method is biased in some cases. The basic question is the following:
If the real microbial number is N, what is the most probable combination of the positive and negative tubes? So the problem of the most probable numbers is reduced to that of the most probable combination of test tube results.
Correspondence address: O. Reichart, Department of Microbiology and Biotechnology, University of Horticulture and Food Industry, Sornloi ut 14-16, H-I l l8 Budapest, Hungary.
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Present work offers a simple solution of this problem calculating the mean values of the estimated MPNs as a function of real cell numbers.
C a l c u l a t i o n m e t h o d
The interval of consistent estimation of microbial numbers was investigated in the most frequently used case, i.e. three parallel inoculations from three decimal dilutions.
Calculations were made by a P C / A T BIOS computer. Calculation method can be described as follows.
Probabi l i t y d is tr ibut ion The probability of a negative (sterile) test tube in the ith dilution:
n! ) . - , , P' = s~i( n - s , ) ! "PSi" (1 - P i (1)
where: Pi, probability of s i sterile tubes; n: number of inoculated tubes; s i, number of sterile tubes; p~, probability of negative (sterile) tube; Pi = e-X~, where Ai is the number of microbes per inoculum. In the case of three decimal dilutions:
h I = N P l = e-N
A2 = 0.1 N p2 = e - °a ~
A 3 = 0.01 N P3 = e-° 'm N
Inoculating three parallel test tubes from every dilution, the probability of sterile tubes can be calculated as follows.
3~ el = -p~,. (I _ pI)3-,, s~!(3 - s l ) !
3! r~ = . p ? . (1 - p~)~- ' :
s2! (3 ~ $2 ~ ~
3~ P3 = .pj3. (1 --p3) 3-s3 $3!(3 S3)!
The probability of the combination of s~, s 2, s 3 sterile tubes
P = P, . P2. P3 (2)
Calculating the P probability at any number of microbes the probability distribu- tion curves can be obtained at any sl, s 2, s 3 combinations (except 3, 3, 3 and 0, 0, 0).
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TABLE I
MPN table for three decimal dilutions with three parallel inoculations (P,,u~ is m a x i m u m value of probabilities)
Number of Pm~ MPN Number of Pmax MPN positive positive results results
* Category 1: obtained in 95$ of cases. * * Category 2: obtained only in 4% of cases.
A c c o r d i n g t o t h e c o n v e n t i o n i n M P N t a b l e s t h e n u m b e r s o f p o s i t i v e t u b e s
( k = n - s ) a r e r e l a t e d t o t h e ce l l n u m b e r s c o r r e s p o n d i n g t o t h e m a x i m u m v a l u e o f
t h e p r o b a b i l i t y c u r v e . B e c a u s e t h e m a x i m u m v a l u e s o f p r o b a b i l i t i e s a r e n o t a v a i l a b l e
i n l i t e r a t u r e t h e s e v a l u e s a r e s u m m a r i z e d i n T a b l e I.
T h e p r o b a b i l i t y c u r v e s b e l o n g i n g t o t h e m o s t p r o b a b l e c o m b i n a t i o n s o f p o s i t i v e
t u b e s c a n b e s e e n i n F i g . 1. T h e f i g u r e s h o w s p r o b a b i l i t y h i s t o g r a m s h a v i n g t h e i r
m a x i m u m o v e r 0 . 0 5 .
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P
05. 3,~ 33,
0 4,, ~o ~3o c . . . ~ ~
I Y / \ ~ -
10 20 30 40 N
Fig. 1. Probability curves (Probability ( P ) versus cell numbers (N) ) corresponding to the combinat ions of positive test tubes (three decimal dilutions with three parallel inoculations).
On the base of probabili ty curves the problem of most probable combinat ion of positive results can be interpreted.
Let us assume that the number of microbes in the first dilution is 15. In this case the most probable score of positive tubes is 3, 3, 0 instead of 3, 2, 1 although MPN3.2, t = 15. The probable results in the order of their probabil i ty are as follows:
MPN3,3. 0 = 24
MPN3.2, 0 = 9.3
MPN3.3.1 -- 46
MPN3.2.1 = 15
MPN3.1. 0 = 4.3
MPN3,1A = 7.5
The mean of cell numbers estimated by MPN method can be calculated as follows:
= E Pi (MPN)i E P, (3)
where: N, mean of estimated number of microbes; Pi, probabili ty of k t, k2, k 3 combination belonging to the real cell number (N) ; (MPN) i, MPN belonging to the k 1, k 2, k 3 combination. By introducing relative frequency
Pi (4) f i= 2 p i
the (3) equation can be calculated:
= y ' f i (MPN), (5)
TABLE II
Calculation of the mean of estimated MPNs (N =15, Pk = 0.01)
For explanation of symbols see section on calculation method.
Using the fi relative frequency, information can be obtained about the reality of a k~, k 2, k 3 combination. Assuming that the real cell number N - - 1 5 and the probability threshold Pk ffi 0.01 (every kl, k2, k 3 combination having probability less than 0.01 at N = 15 is neglected), the calculated mean of MPNs IV = 22.16. (The results of calculation are given in Table II.)
i; 70
• O x X
xxx x~
5o xXX " x / ,~,N
1¢ xX ~ x P~ : oo~ OP~ =025
;o ~o ~ ~o ~o Fig. 2. Estimated mean of MPNs ( N ) as a function of real cell number (N) . Pk = probability threshold
of obtaining a combination.
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R e s u l t s a n d D i s c u s s i o n
F ig . 2 s h o w s r e s u l t s o b t a i n e d b y c a l c u l a t i n g t h e m e a n o f e s t i m a t e d M P N s a t
d i f f e r e n t p r o b a b i l i t y t h r e s h o l d s . C o m p a r i n g t h e e s t i m a t e d ce l l n u m b e r w i t h t h e
TABLE III
Estimated cell numbers at different probability thresholds (Pk) and real cell numbers ( N )
When the test is performed many times and results of category 2 are accepted, too, the reliability of the estimation decreases (Fig. 4).
Tables IV-VIII summarize results obtained by calculating the estimated mean of MPNs at different probability thresholds and microbial numbers below 10. In the Tables beside the N values and relative frequency of the combinations the standard deviations are given as well. Standard deviation (SD) was calculated as follows:
10 )2
E()9- N S D = 1 9 (6)
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TABLE VI
Relative frequency ( f ) of MPN results as a function of the real cell number (N). Probability threshold: Pk = 0.20. Standard deviation: SD
Relative frequency ( f ) of MPN results as a function of the real cell number (N). Probability threshold: Pk = 0.30. Standard deviation (SD)
N Results MPN Probability f Estimated N ( N )
1 2 0 0 0.91 0.3170 1.0000
2 3 0 0 2.3 0.3341 1.0000
3 3 1 0 4.3 0.3346 0.5121 3 0 0 2.3 0.3188 0.4879
4 3 1 0 4.3 0.3729 1.001~
5 3 1 0 4.3 0.3663 1.0000
6 3 1 0 4.3 0.3380 1.0000
7 3 2 0 9.3 0.3052 0.5034 3 1 0 4.3 0.3011 0.4966
8 3 2 0 9.3 0.3212 1.0000
9 3 2 0 9.3 0.3278 1.0000
10 3 2 0 9.3 0.3266 1.0000
0.91
2.3
3.32
4.3
4.3
4.3
6.82
9.3
9.3
9.3
SD = 0.814
When compar ing SD values corresponding to different probabi l i ty thresholds a m i n i m u m value at Pk = 0.25 can be established. In this case only the following
combina t ion of positive tubes can be obta ined: 2 0 0, 3 0 0, 3 1 0, 3 2 0. Beside these combinat ions , taking into account other probable combina t ions (see Tables I V - V I I I ) the acceptable results a r e : 1 0 0 , 1 1 0 , 2 0 0 , 2 1 0 , 3 0 0 , 3 1 0 , 3 1 1 , 3 2 0 , 3 2 1 * 3 3 1 *. The results marked with * are less likely than others, so their acceptance may result in an erroneous est imation.
From the calculat ions the following conclusions can be drawn. (1) The MP N method results in biased est imat ion of cell n u m b e r over 10. (2) Microbial n u m b e r s below 10 could be est imated by accepting only a few combina t ions of positive and negative test tubes. These are the following; 1 0 0, 2 0 0, 3 0 0, 3 1 0 and 3 2 0. By accepting only these combina t ions the mean difference between the est imated and real microbial n u m b e r does not exceed 1 cell number .
References
De Man. J.C. (1975) The probability of Most Probable Numbers. Eur. J. Appl. Microbiol. 1.67-68. De Man. J.C. (1983) MPN tables, Corrected. Eur. J. Appl. Microbiol. Biotechnol. 17, 301-305. Jarvis, B. (1989) Statistical Aspects of the Microbiological Analysis of Foods, pp. 117-141. Elsevier,
