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# DILUTION procedure

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

### Options

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

### Parameters

`POSITIVE` = variates 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 `RMETHOD` 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)

### 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.

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
```
Updated on March 8, 2019