Poisson-Gamma and Poisson-Lognormal models for the characterisation of within- batch and between-batch variability in microbial counts in foods

In modelling risk management strategies (i.e., within-lot batch testing, statistical process control), two basic assumptions have been normally made: that the true concentration of microorganisms are log-normally distributed within a batch, and that the variance of the samples is the same for a little or highly contaminated lot. These assumptions were evaluated by characterising the between-batch and within-batch variability in microbial counts within a heterogeneous Poisson theoretical interpretation; this is a gamma or a lognormal true or population distribution for the unknown microbial concentration and a Poisson measurement distribution that produces a discrete Poisson-gamma or a Poisson-lognormal distribution of observed plate counts. Two variants of random-effects regressions based on the Poisson-gamma (m,1/k) and the Poissonlognormal (μ,σ) models were fitted to two microbial data sets of TVC (high counts) and Escherichia coli (low counts) on pre-chill beef carcasses sampled from different production batches of Irish abattoirs. For the high counts data set, the Poisson-lognormal regression with random-effects for within-batch means (μ) provided a better model for the estimation of the within-batch and between-batch standard deviations; whereas for the low counts data set, the Poisson-gamma regressions were by far superior for the characterisation of within-batch and between-batch variability. The additional advantage of the Poissongamma models is that the central tendency estimate is the arithmetic average, which is more appropriate than the mean logs (originated from the PLN models) because it reflects the infrequent high concentrations that largely determine the overall risk. Proposed ways of graphically illustrating the between-batch variability were by means of cumulative functions and forest plots of the fitted parameters of the true within-batch distributions.