Characterisation of within-batch and between-batch variability in microbial counts in foods using Poisson-gamma and Poisson-lognormal regression models

In modelling risk management strategies (i.e., acceptance sampling plans, 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. Within a heterogeneous Poisson theoretical framework, these two assumptions have been evaluated by characterising the between-batch and within-batch variability in microbial counts. To this effect, three variants of regressions (random effects for within-batch means only, correlated and uncorrelated random effects for within-batch means and spread measures) based on the Poisson-gamma (m,1/k) and the Poisson-lognormal (μ,σ) models were fitted to six microbial data sets of TVC, coliforms and Escherichia coli on pre-chill and post-chill beef carcasses sampled from different production batches. For the high counts data sets, 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 deviation; whereas for the low counts data sets, the Poisson-gamma regressions were superior for the characterisation of within-batch and between-batch variability. However, the selection of a complex Poisson-gamma model with correlated (m,1/k) random effects against a simple Poisson-gamma with variable means (m) depended on the extent of between-batch heterogeneity in the dispersion factor 1/k. The need to introduce the between-batch variability notion in risk management was further highlighted by assessing the real effectiveness of a hypothetical sampling plan operating under the best-fit correlated random effects Poisson-gamma approach, whereby the within-batch dispersion factor was variable and conditional on the within-batch mean.