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DILUTION procedure

Calculates Most Probable Numbers from dilution series data (M.S. Ridout & S.J. Welham).


PRINT = string tokens Output required (estimates, fitted); default esti, fitt
%LIMITS = scalar Percentage points for confidence limits; default 95
RMETHOD = string token Which type of residuals to form (deviance, Pearson); default devi
MAXCYCLE = scalar Maximum number of iterations allowed for the Newton-Raphson algorithm to converge; default 10
TOLERANCE = scalar Defines the convergence criterion; default 0.0005


POSITIVE = variates Number of positive subsamples at each dilution
NSAMPLE = variates Total number of subsamples tested at each dilution
VOLUME = variates Volume of original sample present in each dilution
FITTED = variates To store the fitted values
RESIDUAL = variates To store the residuals, as specified by option RMETHOD
MPN = scalars To store the maximum likelihood estimate of Most Probable Number
UPPER = scalars To store the upper confidence limit for MPN
LOWER = scalars To store the lower confidence limit for MPN
DEVIANCE = scalars To store the residual deviance
PEARSONCHISQUARE = scalars To store Pearson’s chi-square statistic
DF = scalars To store the degrees of freedom for goodness-of-fit tests (zero if no test is available)


A dilution series experiment seeks to estimate the number of organisms in a sample. This is done by preparing successive dilutions of the original sample (usually with a constant dilution factor at each stage), and then testing for the presence/absence of organisms in several subsamples at each dilution. Under certain assumptions, discussed, for example, by Cochran (1950), it is then possible to estimate, by maximum likelihood, the number of organisms in the original sample. In the context of dilution series data, the maximum likelihood estimator is usually known as the Most Probable Number (MPN) of organisms.

DILUTION calculates the MPN estimator, together with likelihood-based confidence limits for the number of organisms.

The number of positive subsamples at each dilution (i.e. the number of subsamples which show the presence of organisms) must be specified in a variate using the parameter POSITIVE. The total number of subsamples used at each dilution, and the volume of the original sample used at each dilution, must be supplied in variates using parameters NSAMPLE and VOLUME.

Output is controlled by the PRINT option. The estimate setting produces the MPN estimate and associated confidence limits, together with the deviance and Pearson’s chi-square statistic. The fitted setting gives observed and fitted values with residuals. All this information is produced by default. The range of the confidence limits can be set by option %LIMIT, the default being 95% limits, and the type of residuals produced (deviance or Pearson) is controlled by the RMETHOD option.

Both the MPN estimator and the confidence limits are calculated iteratively. Option MAXCYCLE sets the maximum number of iterations allowed in each case, the default being 10. Option TOLERANCE specifies the convergence criterion for the MPN estimator; the estimation process is considered to have converged when the absolute value of the derivative of the log-likelihood is less than TOLERANCE. The default value of TOLERANCE is 0.0005. The iterative calculation of the confidence limits is considered to have converged when the log-likelihood takes the correct value to 2 decimal places.

All the information generated can be saved using parameters of the procedure: MPN saves the estimate; UPPER and LOWER save the upper and lower confidence limits; DEVIANCE, PEARSONCHISQUARE and DF save the goodness of fit statistics and the degrees of freedom; and the fitted values and residuals are saved by FITTED and RESIDUAL.




The Newton-Raphson algorithm is used to find both the MPN and the appropriate confidence limits.

Action with RESTRICT

If any of POSITIVE, NSAMPLE or VOLUME are restricted (these restrictions must be compatible), then only the restricted set of units will be used.


Cochran, W.G. (1950). Estimation of bacterial densities by means of the ‘most probable number’. Biometrics, 6, 105-116.

See also


Commands for: Regression analysis.


CAPTION   'DILUTION example',!t('The original sample consists of several',\
          'volumes of 50 ml. These are diluted 5 times at each stage, and 9',\
          'or 10 subsamples are tested at each dilution.'); STYLE=meta,plain
VARIATE   [NVALUES=6] Positive,Total; !(4,2,0,0,0,0),!(10,10,10,9,10,10)
CALCULATE Volume = 5**!(-1,-2,-3,-4,-5,-6)
 &        Volume = 50 * Volume
DILUTION  Positive; Total; Volume
 &        [PRINT=fitted; RMETHOD=pearson] Positive; Total; Volume
DILUTION  [PRINT=*; %LIMIT=90] Positive; Total; Volume; MPN=Mpn;\ 
          LOWER=Lower; UPPER=Upper; FITTED=Fitted; RESIDUAL=Resid
PRINT     Lower,Mpn,Upper
GRAPH     [NROWS=16; NCOL=40] Resid; Fitted
Updated on March 8, 2019

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