the "Resampling" step draws a weighted sample from the prior
m=round(N/40)
index=sample(seq(1,N),size=m,replace=T,prob=w)
p.posterior=p.prior[index]
```
#### **6. Example**

Previously published examples of the proposed probabilistic framework and Bayesian method (Williams & Ebel, 2012; Williams, Ebel & Vose, 2011a) focus on the prediction of changes in human illness in a farm-to-table model. The following example is a departure from the farm-to-table model and it is presented to highlight similarities and the utility of the proposed framework in applications where the data represent a surveillance system.

The data used in this study spans the time period between January of 2007 and December 2009. The annual number of sampled collected was 9401, 10760, and 10774 for each of the three years. The number of positive samples observed in each year was 26,43 and 35, with the

A Bayesian Approach for Calibrating Risk Assessment Models 307

The recorded data values are the result of two tests. The first being a qualitative test that provides a positive or negative result for each sample. The sample for this qualitative PCR test that consists of a 325 *g* subsample of the original 900 *g* sample that is incubated for 24 hours using an enrichment broth. The size of the sample implies a detection limit of 0.003

The second test is only performed on the qualitatively positive samples. This test provides an estimate of the number of organisms per gram, derived from an additional 33.3 *g* sample of the remaining portion of the original 900 *g*. The estimate is derived using the most probable number (MPN) method (Harrigan, 1998) using three dilutions and three tubes per dilution. The dilutions used for MPN testing were 10, 1, and 0.1 *g*. Thus, the 33.3 *g* sample is divided into 3 10-*g*, 3 1-*g* and 3 0.1-*g* subsamples, incubated in a test tube, and tested for *E. coli*

The two tests have different levels of detection and there are three possible outcomes for any sample (i.e., -/-,+/-, +/(estimated number or organisms per *g*)). Samples that are only positive on the first test will be referred to as qualitatively positive, while samples that were positive

The Food Safety Inspection Service (FSIS, 1998) provides additional descriptions of the testing

Ground beef at the point of production comprise no natural units, so it can be viewed as a homogeneously mixed product that mimics a viscous liquid. As such, the estimation of the levels of pathogens in ground beef follows a similar methodology as used for describing the distribution of contaminates in other fluids, such as water. The grinding process inevitably introduces some microscopic voids in the medium, so the unit of measurement is typically in

The sampling data are censored in the sense that the true number of organisms is not observed for all samples. This occurs because the test has a level of detection (*LOD*) at which the probability of a positive is low even though the original sample contained one or more viable

Contamination generally occurs at very low levels for the vast majority of ground beef production. Nevertheless, there are situations where high contamination levels can occur. The commonly used biologically plausible model assumes that the average concentration of contamination varies according to a Lognormal distribution (i.e.,*<sup>X</sup>* <sup>∼</sup> *Lognormal*(*μ*, *<sup>σ</sup>*2) (Haas

**8.1 Estimating the distribution for the average concentration of** *E.coli* **O157:H7**

change in the enrichment media occurring in January of 2008.

O157:H7 using the same PCR technique.

on both tests will be referred to as quantitatively positive.

methodology since the inception of the program.

*cfu/g*.

**8. Methods**

grams rather than milliliters.

organisms (Helsel, 2005).

et al., 1999).

Surveillance sensitivity, in the context of testing for pathogens in food, is the probability that the pathogen is detected given that it exists in a sampled unit. Surveillance sensitivity is a function of test sensitivity in the sense that not only does a contaminated unit need to be sampled, but the results of testing must properly classify the unit as containing the pathogen of interest. When the sensitivity of a test is *Se*, and the number of units in the population is large in relation to the number of samples collected (*n*), and *p*(*S*+) is the proportion of the units that are contaminated. Then the surveillance system sensitivity is typically given by

$$P(\text{detecting one or more positives}) = 1 - (1 - Se \times p(\text{S}^+))^n. \tag{11}$$

Note that the role of test sensitivity is to modify the true prevalence term *p*(*S*+)) to provide the apparent prevalence.

The concern with the standard approaches used in defining *Se* and equation 11 is that test sensitivity can decrease as the level of the pathogen drops, especially in cases where the average concentration is less than 1 cfu/tested unit (e.g., a test that uses 10 g of a 100 g food unit with only 1 organism will have an average concentration of 0.1 *cfu/g*).

In the testing for pathogens that occur at low levels, test sensitivity can be improved by employing enrichment techniques, increasing incubation time, and increasing the volume of material sampled. All of these methods increase the number of pathogens in the medium to be tested. The bonding and potential encapsulation of a microbe within fatty tissues, insufficient time for cells to leave a quiescent state during incubation, the possibility of cells entering a viable but nonculturable state (Oliver, 2005; Oliver et al., 2005), and the small volume of material tested all can lead to reductions in test sensitivity.

In this study, a situation is examined in which the prevalence of positive samples doubled over a one-year period. During this time, a minor modification was made to the enrichment technique used for testing. While the laboratory had performed testing to determine the equivalence of the new and old methodology, insufficient evidence exists to determine if the observed increase in prevalence is due to the change of the enrichment methodology or whether the change was due to an actual change in pathogen prevalence.

This study is predicated on the assumption that the observed increase in *E.coli* O157:H7 positive samples is the result of a change in enrichment media, rather than an actual increase in contamination. The analyses presented assess what the change in test sensitivity would be if this assumption where true. This initial analysis is used to specify both the required sample and the concentration of *E.coli* O157:H7 used to spike validation samples.

#### **7. Data description**

The Food Safety and Inspection Service of the United States Department of Agriculture has been collecting ground beef samples from all slaughter and grinding facilities producing ground beef products since the beginning of the year 2000. Nevertheless, the enumeration of positive samples was only begun in January 2007. This more limited dataset was used in this analysis. Each facility that produces ground beef for distribution is sampled on multiple occasions every year and the sample unit consists of approximately 900 *g* of ground beef collected at the end of production from approximately 5,000 kg lots.

The data used in this study spans the time period between January of 2007 and December 2009. The annual number of sampled collected was 9401, 10760, and 10774 for each of the three years. The number of positive samples observed in each year was 26,43 and 35, with the change in the enrichment media occurring in January of 2008.

The recorded data values are the result of two tests. The first being a qualitative test that provides a positive or negative result for each sample. The sample for this qualitative PCR test that consists of a 325 *g* subsample of the original 900 *g* sample that is incubated for 24 hours using an enrichment broth. The size of the sample implies a detection limit of 0.003 *cfu/g*.

The second test is only performed on the qualitatively positive samples. This test provides an estimate of the number of organisms per gram, derived from an additional 33.3 *g* sample of the remaining portion of the original 900 *g*. The estimate is derived using the most probable number (MPN) method (Harrigan, 1998) using three dilutions and three tubes per dilution. The dilutions used for MPN testing were 10, 1, and 0.1 *g*. Thus, the 33.3 *g* sample is divided into 3 10-*g*, 3 1-*g* and 3 0.1-*g* subsamples, incubated in a test tube, and tested for *E. coli* O157:H7 using the same PCR technique.

The two tests have different levels of detection and there are three possible outcomes for any sample (i.e., -/-,+/-, +/(estimated number or organisms per *g*)). Samples that are only positive on the first test will be referred to as qualitatively positive, while samples that were positive on both tests will be referred to as quantitatively positive.

The Food Safety Inspection Service (FSIS, 1998) provides additional descriptions of the testing methodology since the inception of the program.
