Calculates Most Probable Numbers from dilution series data (M.S. Ridout & S.J. Welham).
|Output required (
||Percentage points for confidence limits; default 95|
||Which type of residuals to form (deviance, Pearson); default
||Maximum number of iterations allowed for the Newton-Raphson algorithm to converge; default 10|
||Defines the convergence criterion; default 0.0005|
||Number of positive subsamples at each dilution|
||Total number of subsamples tested at each dilution|
||Volume of original sample present in each dilution|
||To store the fitted values|
||To store the residuals, as specified by option
||To store the maximum likelihood estimate of Most Probable Number|
||To store the upper confidence limit for MPN|
||To store the lower confidence limit for MPN|
||To store the residual deviance|
||To store Pearson’s chi-square statistic|
||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
Output is controlled by 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
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;
LOWER save the upper and lower confidence limits;
DF save the goodness of fit statistics and the degrees of freedom; and the fitted values and residuals are saved by
The Newton-Raphson algorithm is used to find both the MPN and the appropriate confidence limits.
If any of
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.
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