Effectiveness of sampling plans by attributes based on mixture distributions characterising microbial clustering in food Arianna Mussida a, * , Ursula Gonzales-Barron a, b , Francis Butler a a UCD School of Biosystems Engineering, University College of Dublin, Belfield, Dublin 4, Ireland b CIMO Mountain Research Centre, School of Agriculture, Polytechnic Institute of Braganza, Portugal article info Article history: Received 24 November 2012 Received in revised form 26 March 2013 Accepted 2 April 2013 Keywords: Clusters Negative binomial Poisson-logarithmic Zero-inflated Poisson Zero-inflated negative binomial Operating characteristic curves abstract As in many cases, pathogenic microorganisms contaminate the food material as clusters or group of individual cells; the effectiveness of sampling plans based on mixture distributions representing bac- terial agglomeration was assessed. In general, sampling plans that do not take into account such consideration lead to higher probabilities of accepting defective lots. Since quite often no scientific data are available in order to determine the degree of over-dispersion or clustering of the target microor- ganisms, in this theoretical study we compare the variance-to-mean ratio and the reciprocal of the exponent k of the negative binomial distribution (NB) as measures of dispersion. The mixture Poisson- logarithmic (Plog) model is proposed as a special case of the NB distribution, where the bacterial clus- ters are Poisson distributed while the individuals in each cluster follow a logarithmic distribution. In order to describe microbial data characterised by an excess of zero counts (1p), we assess the zero- inflated Poisson (ZIP) and zero-inflated negative binomial (ZINB) distributions as alternative statistical models. The Operating Characteristic (OC) curves generated on the basis of the zero-inflated distributions were compared for fixed values of the variance-to-mean ratio and the parameter p at any mean level of contamination and sample weight adopted. The results show that assuming fixed 1/k and p for the NB and ZIP distributions, respectively, both models converge to a Poisson distribution at the producer’s quality level. In contrast, the consumer’s quality level is highly affected by assuming fixed values of 1/k and p since it increases. The OC curves generated for the NB and ZIP distributions assuming fixed values of the variance-to-mean ratio at any mean level of contamination and sample weight adopted, reveal that both the consumer’s and producer’s quality level are affected, as they both increase. Within the ZINB distribution, a separate investigation is conducted to determine which parameters are mostly responsible for describing microbial over-dispersion. As a general conclusion, for the design of sampling plans based on any statistical distribution, OC curves that reflect microbial agglomeration should be constructed considering that variance is not constant but dependant on the level of microbial concentration of the lot. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction In order to protect consumers from pathogenic microorganisms that may be present in foods, it is important to implement an appropriate sampling plan, defined by the number of samples (n) and size of the samples (w) to be taken. Therefore, European regulators (European Commission, 2005) have established single two-class attribute sampling plans (presence/absence) as the monitoring system for several pathogens, such as Cronobacter spp. and Salmonella, in food material. In food safety, the acceptance number is often set to zero, since it would appear that the use of any greater acceptance number implies the acceptance of defective or contaminated lots. The performance of a sampling plan is given by its Operating Characteristic (OC) curve, which, in the context of microorganisms in food, plots the conditional probability of acceptance of a lot against possible values of the mean level of contamination of the lot (Van Schothorst, Zwietering, Ross, Buchanan, & Cole, 2009). In attributes sampling plans, the OC curves are influenced by the number of samples (n), the sample weight (w) and the statistical distribution assumed to characterise the dispersion of the pathogen in the food matrix. Based on the OC curve, the producer’s risk is defined as the probability of rejecting a lot having a low (acceptable) mean level of contamination; and the consumer’s risk, as the probability of accepting an unsatisfactory (highly contaminated) lot. The mean levels of contamination at the producer’s and consumer’s risk are defined as the producer’s (PQL) * Corresponding author. Tel.: þ353 1 7167473. E-mail address: arianna.mussida@ucd.ie (A. Mussida). Contents lists available at SciVerse ScienceDirect Food Control journal homepage: www.elsevier.com/locate/foodcont 0956-7135/$ e see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.foodcont.2013.04.001 Food Control 34 (2013) 50e60