DILUTION procedure

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

Options

Parameters

Description

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.

Options: PRINT, %LIMITS, RMETHOD, MAXCYCLE, TOLERANCE.

Parameters: POSITIVE, NSAMPLE, VOLUME, FITTED, RESIDUAL, MPN, UPPER, LOWER, DEVIANCE, PEARSONCHISQUARE, DF.

Method

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.

Reference

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

See also

Procedures: PROBITANALYSIS, WADLEY.

Commands for: Regression analysis.

Example

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