ࡱ> (*'vy bjbjEE 4<''::}}}}}l%5555=%?%?%?%?%?%?%$j' *vc%}c%}}55x%ggg:}5}5=%g=%gg:#,a$5PbK$ )%%0%$R*4*a$a$*}u$" g"6c%c%%*: C: Most Probable Number Equations & References Assumption As explained in [1: HYPERLINK "Cochran_biometrics1950.pdf"Download the Article Here], there are two assumptions. The liquid is completely mixed so that the organisms in the liquid are randomly distributed. 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  EMBED Equation.3  to the total volume  EMBED Equation.3  and the probability that the organism is not in the sample is given by  EMBED Equation.3  Due to the completely mixed assumption, this probability is irrelevant to the locations of the other organisms. Assume that there are  EMBED Equation.3  organisms in the liquid. The probability that the sample does not contain any of the organisms,  EMBED Equation.3 , is given by  EMBED Equation.3  If  EMBED Equation.3  is small, this probability is described by  EMBED Equation.3  Since  EMBED Equation.3  is the density  EMBED Equation.3  of organisms, the probability is given by  EMBED Equation.3  Therefore, the probability of obtaining a fertile sample,  EMBED Equation.3 , is  EMBED Equation.3  If  EMBED Equation.3  samples of volume  EMBED Equation.3  are taken, the probability observing  EMBED Equation.3  fertile samples is given by the binomial distribution as  EMBED Equation.3  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.  EMBED Equation.3  Therefore, the MPN is given by the root of the following equation  EMBED Equation.3  In general, multiple dilution sets are used to obtain the MPN. The joint probability of observing  EMBED Equation.3  fertile samples in one of m dilution sets with volume  EMBED Equation.3  and  EMBED Equation.3  samples is given by the following likelihood function.  EMBED Equation.3  Like a single dilution case, the MPN is given by the root of the following equation  EMBED Equation.3  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: HYPERLINK "Hurley_jab1983.pdf"Download the Article Here].  EMBED Equation.3  This standard error is obtained by the following equations. The variance of the maximum likelihood estimator is obtained by the following equation.  EMBED Equation.3  where  EMBED Equation.3 , and  EMBED Equation.3  is expected information calculated by  EMBED Equation.3  The expected information of the logarithm of the likelihood function obtained above is described as the following equation.  EMBED Equation.3  Since  EMBED Equation.3 , the variance of  EMBED Equation.3  is described by  EMBED Equation.3  Therefore, the standard error of  EMBED Equation.3  is described as the following equation  EMBED Equation.3  References [3,4,5] also provide further explanation on the derivation of these equations By using  EMBED Equation.3 , 95% confidence bound is obtained by the following equation.  EMBED Equation.3  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  EMBED Equation.3  where S1 and S2 is the number of positive large and small wells respectively. References [HYPERLINK "Cochran_biometrics1950.pdf"1] William G. Cochran, Estimation of bacterial densities by means of the Most Probable Number, Biometrics 6 (1950), no. 2, 105116. [HYPERLINK "Hurley_jab1983.pdf"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, 159164. [HYPERLINK "Fisher_pt1921.pdf"3] Fisher, R.A. On the mathematical foundations of theoretical statistics. Philosophical Transactions of the Royal Society of London (1921), A 222, 309-368. [HYPERLINK "Loyer_biometrics1984.pdf"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. 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