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Author: David Campbell
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Food Control 27 (2012) 73e80

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Food Control journal homepage: www.elsevier.com/locate/foodcont

Methods for fitting the Poisson-lognormal distribution to microbial testing data Michael S. Williams*, Eric D. Ebel Risk Assessment Division, Office of Public Health Science, Food Safety and Inspection Service, USDA, 2150 Centre Ave, Fort Collins, CO 80526, USA

a r t i c l e i n f o

a b s t r a c t

Article history: Received 22 August 2011 Received in revised form 5 March 2012 Accepted 8 March 2012

The Poisson distribution can be used to describe the number of microorganisms in a serving of food, but in most food-safety applications, the variability of the Poisson distribution is insufficient to describe the heterogeneity in microbial contamination across the population of all servings. To model how contamination varies across the population of all possible servings, the lognormal distribution can be paired with the Poisson distribution to create an over-dispersed distribution, which is referred to as the Poissonlognormal distribution. An advantage of this distribution is that random draws from the distribution are integer-valued. This is beneficial for some food-safety risk assessments because a modeled serving is either contaminated or not. This distribution is also appropriate when the results of a laboratory test are integer-valued, such as when direct plating is used to enumerate samples. While some surveys perform only absence/presence screenings tests, surveys that are more thorough can pair a screening test with enumeration of screen-test positive samples via direct plating. For this application, statistical methods that accommodate censored data are required to fit the data to a Poisson-lognormal distribution. This study compares a Bayesian hierarchical model to a maximum likelihood estimation approach for fitting data to the Poisson-lognormal distribution. Across a range of datasets, the Bayesian method demonstrates superior performance. OpenBUGS and R code are provided to implement both methods. Published by Elsevier Ltd.

Keywords: Markov-chain Monte Carlo Direct plating Screening test Enumeration

1. Introduction The accurate characterization of the distribution of microbial contamination within a food commodity is important for foodsafety risk assessment. Contamination of a food commodity is often the result of processes that cause clustering of contaminants somewhere in the farm-to-fork continuum (e.g., rupture of the digestive track during evisceration). A key component of many food-safety exposure assessments is the estimation of the distribution of contamination from samples collected at some point in the farm-to-table continuum. For example, the Food Safety and Inspection Service in the United States conducts large-scale surveys of microbial contamination on meat and poultry products with samples collected during the slaughter or production process (FSIS, 2009). Statistical distributions that adequately model these clustered processes are often related to the lognormal distribution. When sampling data are non-integer-valued, as is the case when

* Corresponding author. Tel.: þ1 970 492 7189. E-mail address: [email protected] (M.S. Williams). 0956-7135/$ e see front matter Published by Elsevier Ltd. doi:10.1016/j.foodcont.2012.03.007

concentration estimates are derived using the Most Probable Number technique (Haas, 1989), the lognormal distribution is appropriate for modeling these continuous data. When sampling data consists of integer counts, as is the case when plate counts are used, the Poisson-lognormal distribution is recommended, though a number of factors make this distribution difficult to apply in practice (Bassett, Jackson, Jewell, Jongenburger, & Zwietering, 2010). The Poisson component of this distribution accommodates integer inputs (or outputs) to describe the actual number of organisms associated with a single unit or sample. The lognormal component of the distribution describes the over-dispersion in the Poisson rate parameter due to clustering of contaminants and describes how the average concentration of the contaminant varies across the population. One of the complicating factors for the application of the Poisson-lognormal is that the result derived from testing a sample for microbial contamination is often affected by the detection limit (DL) for the assay (Busschaert, Geeraerd, Uyttendaele, & Van Impe, 2010; Helsel, 2005; Pouillot & Delignette-Muller, 2010; Shorten, Pleasants, & Soboleva, 2006). A single sample may also be subjected to multiple tests. For example, a qualitative screening test, with a DL of Lqual , is commonly used to determine if the microorganism

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M.S. Williams, E.D. Ebel / Food Control 27 (2012) 73e80

of interest is present in the sample. Samples that are positive on the screening test undergo additional testing to enumerate the sample. The test used to quantify screen-test positive samples is also subject to a lower bound, which is defined as the quantitation limit (QL) and whose numeric value is denoted Lquan (with Lqual < Lquan ). Measurements whose values are known only above or below specified limits are referred to as censored data in the statistical literature (Helsel, 2005). In this application, the DL is a theoretic censoring threshold that refers to the concentration of organisms that must be present in a sample in order for the sample to test positive. It is not a threshold for the lognormal distribution describing the average concentration across the population. The difference in the DL and QL values is predominantly a function of the volume of the sample material tested. It will be assumed that the test can detect a single viable organism in the sample, so if v is the mass (or volume) of the sample, the limit of either test is the inverse of the sample mass (i.e., 1=v). Enumeration of a sample via a plate count often involves incubation of vplate ¼ 1 g or ml so Lquan ¼ 1. A screening test typically involves incubation of a much larger sample mass, so Lqual ¼ 1=vscreen

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