Most Probable Number calculation



Most Probable Number Equations & References

Assumption

As explained in [1: Download the Article Here], there are two assumptions.

1. The liquid is completely mixed so that the organisms in the liquid are randomly distributed.

2. “Each sample from the liquid, when incubated in the culture medium, is certain to exhibit growth whenever the sample contains one or more organisms.” Otherwise, the MPN underestimates the true density of organisms.

Calculation

Consider the single organism. The probability that the organism is present in the sample is simply given by the ratio of the volume of the sample [pic] to the total volume [pic] and the probability that the organism is not in the sample is given by

[pic]

Due to the completely mixed assumption, this probability is irrelevant to the locations of the other organisms. Assume that there are [pic] organisms in the liquid. The probability that the sample does not contain any of the organisms, [pic], is given by

[pic]

If [pic] is small, this probability is described by

[pic]

Since [pic] is the density [pic] of organisms, the probability is given by

[pic]

Therefore, the probability of obtaining a fertile sample, [pic], is

[pic]

If [pic] samples of volume [pic] are taken, the probability observing [pic] fertile samples is given by the binomial distribution as

[pic]

What we want to know is the “most probable number (MPN)”, the density which maximizes the probability of obtaining the observed result. Such a density is obtained by finding root of the derivative of log likelihood function.

[pic]

Therefore, the MPN is given by the root of the following equation

[pic]

In general, multiple dilution sets are used to obtain the MPN. The joint probability of observing [pic] fertile samples in one of m dilution sets with volume [pic] and [pic] samples is given by the following likelihood function.

[pic]

Like a single dilution case, the MPN is given by the root of the following equation

[pic]

The MPN obtained by the above method can be approximated by the log normal distribution with mean equal to the logarithm of the MPN and standard error as shown in the following equation [2: Download the Article Here].

[pic]

This standard error is obtained by the following equations. The variance of the maximum likelihood estimator is obtained by the following equation.

[pic]

where [pic], and [pic] is expected information calculated by

[pic]

The expected information of the logarithm of the likelihood function obtained above is described as the following equation.

[pic]

Since [pic], the variance of [pic] is described by

[pic]

Therefore, the standard error of [pic] is described as the following equation

[pic]

References [3,4,5] also provide further explanation on the derivation of these equations

By using [pic], 95% confidence bound is obtained by the following equation.

[pic]

IDEXX QuantiTray 2000

IDEXX QuantiTray 2000 consists of 49 large wells of volume 1.86ml and 48 small wells of volume 0.186ml. Therefore the above equation is described as

[pic]

where S1 and S2 is the number of positive large and small wells respectively.

References

[1] William G. Cochran, Estimation of bacterial densities by means of the ‘Most Probable Number’, Biometrics 6 (1950), no. 2, 105–116.

[2] M. A. Hurley and M. E. Roscoe, Automated statistical-analysis of microbial enumeration by dilution series, Journal of Applied Bacteriology 55 (1983), no. 1, 159–164.

[3] Fisher, R.A. On the mathematical foundations of theoretical statistics. Philosophical Transactions of the Royal Society of London (1921), A 222, 309-368.

[4] Loyer, M.W. and M.A. Hamilton. Interval Estimation of the Density of Organisms Using a Serial-Dilution Experiment, Biometrics, Vol. 40, No. 4 (Dec., 1984), pp. 907-916

Other References

[5] Finney, D.J. 1964 Statistical Methods in Biological Assay, 2nd Ed. High Wycombe: Charles Griffin & Co.

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