A comparison between the discrete Poisson-gamma and Poisson-lognormal distributions to characterise microbial counts in foods

The choice of statistical distributions characterising microbial counts is essential in risk assessment and risk management. While the lognormal distribution has been long used to directly model the microbial data obtained from food samples, it does not allow for complete absence of microorganisms in a sample. Within a heterogeneous Poisson theoretical interpretation, a gamma or a lognormal population distribution for the unknown microbial concentration and a Poisson measurement distribution produces a discrete Poisson-gamma (λ, 1/k) or a Poisson-lognormal (μ,σ) distribution of observed plate counts. The capability of both distributions to deal with clustering was compared using six data sets of variable proportion of zero counts: total viable counts, coliforms and Escherichia coli on pre-chill and post-chill beef carcasses. Whereas the Poisson-lognormal distribution fitted better to the high counts data sets, the Poisson-gamma distribution represented the low counts data sets (13–81% zero counts) by far better than the Poisson-lognormal – which invariably tended to have a longer tail, an overestimated mean log and a lower predicted probability of zero counts. The inverse close relationship between the observed proportion of zero counts in the data set and the fitted dispersion factor 1/k suggested the possibility of obtaining a first approximation of 1/k by this means. Finally, in absence of zero counts, it was demonstrated that fitting a Poisson-lognormal to the observed plate count data can be closely approximated by the common practice of fitting a simple normal distribution to the back-calculated ‘unobserved’ mean concentrations in log CFU/g.