Uncertainty in most probable number calculations for microbiological assays

Microbiological assays commonly use incubations of multiple tubes in a dilution series, and microorganism concentration is read as a most probable number (MPN) in standard tables for the observed pattern of positive tubes. Published MPN tables differ, sometimes substantially, because of use of approximate MPN calculation procedures, different rounding conventions in the results, and different methods of calculating confidence or credible intervals. We conclude that the first 2 issues can now be resolved by using recently developed exact MPN calculation methods and by reporting rounding conventions in standard tables. The third issue is not amenable to complete resolution, especially if credible interval (as opposed to confidence interval) limits are desired--as we think they most often are. In that case, Bayesian statistics are called for and the analyst must provide a distribution of concentration that was presumed to be true before the assay was performed. This is mathematically combined with the assay data, resulting in a posterior concentration distribution. These distributions may then be used to quantify the uncertainty in the MPN estimate, and the best approach is to use the highest posterior density regions of these distributions. If based on diffuse prior information (positing that, prior to an assay being performed, all positive concentrations are equally likely), then established procedures might be used to calculate the limits and publish them in standard tables. In the event that this prior assumption is held to be not satisfactory, we show results for an empirical Bayes procedure, with a Poisson prior distribution, giving credible interval widths much narrower than in the other cases examined.