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Keywords:

  • Food safety;
  • sampling plan

Abstract

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. MODEL, METHODS, AND ASSUMPTIONS
  5. 3. RESULTS
  6. 4. DISCUSSION
  7. 5. CONCLUSION
  8. DISCLAIMER
  9. Appendix
  10. REFERENCES

Much of the literature regarding food safety sampling plans implicitly assumes that all lots entering commerce are tested. In practice, however, only a fraction of lots may be tested due to a budget constraint. In such a case, there is a tradeoff between the number of lots tested and the number of samples per lot. To illustrate this tradeoff, a simple model is presented in which the optimal number of samples per lot depends on the prevalence of sample units that do not conform to microbiological specifications and the relative costs of sampling a lot and of drawing and testing a sample unit from a lot. The assumed objective is to maximize the number of nonconforming lots that are rejected subject to a food safety sampling budget constraint. If the ratio of the cost per lot to the cost per sample unit is substantial, the optimal number of samples per lot increases as prevalence decreases. However, if the ratio of the cost per lot to the cost per sample unit is sufficiently small, the optimal number of samples per lot reduces to one (i.e., simple random sampling), regardless of prevalence. In practice, the cost per sample unit may be large relative to the cost per lot due to the expense of laboratory testing and other factors. Designing effective compliance assurance measures depends on economic, legal, and other factors in addition to microbiology and statistics.

1. INTRODUCTION

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. MODEL, METHODS, AND ASSUMPTIONS
  5. 3. RESULTS
  6. 4. DISCUSSION
  7. 5. CONCLUSION
  8. DISCLAIMER
  9. Appendix
  10. REFERENCES

Much of the literature regarding food safety sampling plans fails to explicitly consider the impact of resource constraints and implicitly assumes that all lots entering commerce are tested.1 For example, Whiting et al.[1] do not consider resource limitations in the development of microbiological sampling plans for lot rejection. ICMSF[2] states that its “sampling plans were developed based on past experience, available data, practical constraints, and statistical considerations” but gives no clear indication of the role played by resource constraints in the development of its guidance. Codex[3] states that the “pragmatic” sampling procedures commonly used for the determination of compliance with maximum residue limits for pesticides and veterinary drugs are not statistically-based procedures. Under Codex,[4] for example, the minimum number of samples to be drawn from a lot is one. In practice, however, only a fraction of lots may be tested due to a budget constraint. In such a case, there is a tradeoff between the depth of sampling (samples per lot) and coverage (the number of lots tested). The appropriate balance is an empirical question with no single solution that is applicable under all circumstances. In some instances, as will be illustrated here, the statistical power of sampling under resource constraints may be maximized by limiting the depth of sampling to expand its coverage. Therefore, the optimal design of food safety sampling plans requires a broader approach than one that focuses exclusively on the simple statistical relationship between the probability of lot acceptance and lot quality, as in the classic operating characteristic curve.

2. MODEL, METHODS, AND ASSUMPTIONS

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. MODEL, METHODS, AND ASSUMPTIONS
  5. 3. RESULTS
  6. 4. DISCUSSION
  7. 5. CONCLUSION
  8. DISCLAIMER
  9. Appendix
  10. REFERENCES

2.1. Optimization Model for Fixed Prevalence

Food safety sampling for compliance with microbiological limits is often conducted under a budget constraint. Assuming that available resources are insufficient to perform 100% lot-by-lot acceptance sampling, only a fraction of lots can be tested. Assume a two-stage sampling process. In the first stage, m lots are selected for testing. In the second stage, a number of sample units are drawn from each of the m lots. For simplicity, assume that the m lots are randomly selected and that the number of sample units per lot (n) is the same for all lots. These simplifying assumptions would be applicable in cases where lot size is uniform and where prior information on observable attributes and sampling outcomes (e.g., plant compliance history) is inadequate to satisfactorily predict lots that will fail to conform to microbiological specifications. With random sampling of lots of uniform size, sampling of producers is proportional to production volume.

The assumed objective function is to maximize the number of nonconforming lots that are rejected (LR) by lot acceptance sampling subject to a budget constraint on total sampling cost.2 Assume a two-class lot acceptance sampling plan for attributes where the maximum number of nonconforming sampling units allowed per lot is zero (a zero acceptance number sampling plan).[2, 3, 5] For fixed prevalence, the constrained optimization problem is:

  • display math(1)

where CT is the budgeted total sampling cost ($), Cl is the cost per lot ($), Cn is the cost per sample ($), m is the number of lots selected for sampling, n is the samples per lot, p is the prevalence of nonconforming sample units, q = (1 – p), and 1 – qn = p (reject a lot). For illustrative purposes, consider values of p between 10−1 and 10−4.

To solve the constrained optimization problem, we substitute m defined in terms of the budget constraint into the objective function:

  • math image(2)

To obtain the optimal number of samples per lot (nopt) that maximizes Equation (2) subject to a budget constraint, sampling costs, and fixed prevalence, solve for ∂(LR)/∂(n) = 0, where:

  • math image(3)

To obtain nopt, Equation (3) was solved numerically for n using Microsoft Excel Solver by minimizing the logarithm of ∂(LR)/∂(n).3 The solutions were confirmed by calculating Equation (2) over the integer interval n = [1, 200] and checking that the solution obtained for nopt maximized LR (Equation (2)). (Under the range of input values considered, the maximum value of nopt = 199.)

In practice, some optimization algorithms have difficulty solving the derivative in (Equation (3)). For some combinations of input values, LR Equation (2) changes very slightly in response to changes in n in the region surrounding nopt. (The slope (Equation (3)) may be essentially flat over a large range of n values.) For example, the optimization algorithm failed to converge in many instances using the default options under the SAS NLP (nonlinear programming) procedure. Also, strictly speaking, the optimization problem solution for Equation (3) is constrained in that n and m take integer values, with n, m ≥ 1. However, we can solve Equation (3) for continuous n without loss of generality, considering that over a budget period (e.g., a month or quarter) the average of n or m can take continuous values. In practice, little may be lost by treating n as continuous in solving Equation (3) and then rounding off. Rounding down (up) results in more (fewer) lots tested under the budget constraint (Equation (1)). The constraint n ≥ 1 still applies.

An interesting general finding results from applying the Lagrangian method. The Lagrangian for Equation (1) is:

  • display math(4)

As shown in the Appendix, the solution of L is:

  • math image(5)

Therefore,

  • math image

2.2. Optimization Model for Variable Prevalence

For the more general case where within-lot prevalence varies among lots,

  • math image(6)

where inline image.4 For illustrative purposes, consider inline image = 10−1 to 10−4 (in order of magnitude decrements) and assume the coefficient of variation (cv = σp/μp) = 0.0 to 1.0 (in increments of 0.25). With cv = 0, the model reduces to fixed prevalence. For cv = 1, the 95% confidence intervals for the prevalence variability distributions span slightly more than two orders of magnitude.

The constrained optimization problem is:

  • display math(7)

Monte Carlo simulation was used to numerically integrate Equation (6) subject to the constraint on m in Equation (7) with 1,000 iterations (samples from g(p)) for each value on the integer interval n = [1, 200]. The optimum number of samples per lot (nopt) under each set of inputs satisfied Equation (7). Monte Carlo simulations were performed with Latin hypercube sampling using Palisades @Risk, Ver. 5.7.1, an add-on to Microsoft Excel.

2.3. Microbiological Sampling and Testing Assumptions

For lot acceptance sampling plans for attributes based on presence-absence tests, the prevalence of nonconforming sample units refers to the proportion of sample units in which one or more colony-forming units (cfu) of the microorganism(s) of concern are detected. Here, we ignore the issue of test sensitivity, as lot disposition decisions are based on apparent prevalence. For simplicity, specificity is assumed to be 100%. Other things being equal, the larger the size of the sample unit analyzed by the detection method, the higher the observed prevalence of contamination. In practice, a 25 g sample unit is commonly recommended for microbiological sampling plans based on presence-absence detection methods.[2] However, in some cases, the sample unit may be ≤10 g.[6] Therefore, the prevalence for a sampling plan based on a presence-absence microbial detection method must be specifically referenced to the size of the sample unit drawn from a lot.

For lot acceptance sampling plans for attributes based on a quantitative microbiological limit, prevalence refers to the proportion of sample units that exceed a concentration specified by the microbiological criteria (e.g., 100 cfu/g Listeria monocytogenes in ready-to-eat food that will not support growth). A sample unit may test positive for the target microorganism(s) but conform to the quantitative limit. For concentration-based lot acceptance sampling plans, the sampling variability in the estimated concentration increases as the size of the enumerated sample unit decreases:

  • math image(8)

where inline image is the estimated mean concentration (cfu/g) based on the observed counts (cfuobs), stest is the size of the enumerated sample unit (g), and λ is the Poisson parameter.[6] In practice, inline image can affect the proportion of lots rejected (1 – qn). For simplicity, assume that stest is sufficiently large that this effect is not appreciable. Also assume that the measurement error for the enumeration method is negligible, in which case the operating characteristics of lot acceptance sampling plans for attributes based on quantitative limits depend only on the prevalence of nonconforming sample units and not on the within-lot microbial concentration distribution (e.g., homogeneous or heterogeneous).[3]

2.4. Cost Assumptions

The cost of testing a lot in acceptance sampling (Cl + nCn) depends on the cost per lot (including time for travel and selecting a lot for testing), the cost per sample unit (including sample collection, shipping, and laboratory analysis), and the number of sample units drawn per lot. For food safety sampling, the cost per sample unit may be large relative to the cost per lot due to the expense of laboratory analysis, the labor-intensive nature of some sample collection procedures (e.g., excision samples), and the presence of natural bottlenecks in the food production and distribution system through which many lots must pass. The cost of laboratory analysis varies considerably. For example, according to a publicly available fee schedule, the cost of microbiological food safety sample analyses can range from less than $10 (for a simple coliform count) to more than $80 (for an Escherichia coli O157:H7 robust procedure with confirmation) depending on the microorganism(s), food product, testing, and confirmation requirements.[7] Analytical costs also depend on laboratory certification requirements and the volume of testing performed. Sample collection costs also vary. In New Zealand, where user fees fund food safety inspection costs, the fee schedule ranges from approximately $75 to $115 ($NZ96–$NZ144) per hour.[8] Examples of bottlenecks in the food production and distribution system include ports of entry and food business operations where numerous incoming shipments of raw materials are processed before entering commercial distribution channels (e.g., large meat and poultry slaughter plants). In some situations, food safety personnel are continuously present at a facility (or are present for an extended period) and carry out a variety of duties (e.g., HACCP (hazard analysis and critical control points) audits) such that the incremental cost of selecting a lot for testing is not appreciable.

Alternatively, the cost of selecting a lot may be substantial where food safety personnel must travel to a distant facility to select one or more lots for testing or where cargo containers must be unloaded to permit sampling. The cost varies, but at the Savannah/Brunswick shipyards, for example, the Georgia Ports Authority charges $904 for “devanning” (unloading) a cargo shipping container in preparation for government agency inspection.[9] Where laboratory analysis is conducted off-site, the cost of shipping food samples is also substantial and depends on the type and weight of sample, the urgency of the sample (overnight or ground transport), and the volume of shipments. At the Food Safety and Inspection Service, the average price per shipment ranges between $7.50 and $15.00.[10] Partitioning shipping costs between lots and sample units depends, however, on a variety of factors, including the shipping distance, the number of lots sampled at a location on a given occasion, and the number and weight of sample units drawn per lot. For illustrative purposes, consider the cost per lot to range from zero to twice the cost per sample unit (Cl/Cn = 0–2).

3. RESULTS

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. MODEL, METHODS, AND ASSUMPTIONS
  5. 3. RESULTS
  6. 4. DISCUSSION
  7. 5. CONCLUSION
  8. DISCLAIMER
  9. Appendix
  10. REFERENCES

3.1. Optimization Model for Fixed Prevalence

Equation (5) produces the following general result. In the presence of any budget constraint that does not permit testing 100% of lots, the optimal number of samples per lot (nopt) for a given prevalence (p) depends only on the ratio of the cost per lot to the cost per sample unit (Cl/Cn). The size of the budget (CT) determines the absolute number of lots that can be tested m (or the frequency of lot inspection (1/m)) but not nopt. These results are consistent with Connelly,[11] who showed that in designing multisite clinical medical trials, the optimal balance between the number of sites and the number of patients per site (the size of sites) equates the cost ratio (cost per site/cost per patient) to the statistical marginal rate of substitution between these two inputs to the trial design.

Fig. 1 shows the relationship between nopt, p, and Cl/Cn under the assumed sampling objective. If the ratio of the cost per lot to the cost per sample unit is substantial, the optimal number of samples per lot increases rapidly as prevalence decreases. However, for a given prevalence, the optimum number of samples per lot decreases as the cost ratio (Cl/Cn) declines. If the ratio of the cost per lot to the cost per sample unit is sufficiently small (Cl << Cn), the optimal number of samples per lot reduces to n = 1 (i.e., simple random sampling without replacement of lots), regardless of prevalence. This results due to the diminishing returns to the probability of rejecting a lot (1 – qn) from increasing the number of samples per lot (n). Statistically, the maximum marginal return occurs by increasing n from zero (skipping the lot in inspection) to n = 1.

image

Figure 1. Relationship between the optimal number of samples per lot, fixed sample unit prevalence, and the ratio of cost per lot to cost per sample ($Cl/$Cn).

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The tradeoff between n and m under the constraint CT is illustrated by Fig. 2, which shows the relationship between n and LR (Equation (2)), with nopt = 44 for the case where Cl/Cn = 1 and p = 10−3. As n initially increases from n = 1, the increased probability of lot rejection (1 – qn) more than offsets the effect of reducing the number of lots inspected from its maximum possible value under the budget constraint (m = max at n = 1). As n increases beyond 44, the reduction in m offsets the increasing probability of lot rejection, and LR recedes from its maximum value. The absolute values of m and LR depend on the absolute values of CT, Cl, and Cn, but the value of nopt depends only on Cl/Cn and p.

image

Figure 2. Relationship between the number of samples per lot and the number of lots rejected as a proportion of the maximum (max) for the case $Cl/$Cn = 1 and p = 10−3.

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Fig. 2 also illustrates that in terms of maximizing LR (Equation (2)), the returns from increasing n can diminish rapidly. Under this scenario, LR reaches 95% of its maximum value for n = 10. Any sacrifice to the objective of maximizing the number of lots rejected is suboptimal with respect to maximizing LR. Under this scenario, however, a small sacrifice to maximizing LR could allow the number of lots inspected to increase by a factor of 4.1 (45/11). This sacrifice could permit, for example, facilities to be inspected more frequently under the budget constraint. Alternatively, LR also obtains 95% of its maximum value for n = 172. From this vantage point, a small sacrifice to maximizing LR could allow the probability of lot rejection to increase by a similar factor of 3.7 ((1–0.999172)/(1–0.99944)). This sacrifice could permit a greater assurance of detecting contamination in lots that undergo acceptance sampling under the budget constraint. These results illustrate that in designing an “optimal” sampling plan, the tradeoffs also may involve multiple, competing objectives for food safety sampling.

3.2. Optimization Model for Variable Prevalence

Fig. 3 shows the relationship between nopt, μp, cv, and Cl/Cn under the assumed sampling objective. For μp = 10−4 and 10−3, as cv increases from 0 to 1, nopt decreases by approximately 25–30% for Cl/Cn ≥ 0.01. For μp = 10−2 and 10−1, the same holds for Cl/Cn ≥ 0.1 and 0.5, respectively. Below these cost ratio values, the nopt curves are essentially flat over cv = 0–1. Consistent with the fixed prevalence model, if the ratio of the cost per lot to the cost per sample unit is sufficiently small, the optimal number of samples per lot reduces to n = 1, regardless of the prevalence distribution.

image

Figure 3. Relationship between the optimal number of samples per lot, mean sample unit prevalence, coefficient of variation, and the ratio of cost per lot to cost per sample ($Cl/$Cn).

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Fig. 4 illustrates the tradeoff between n and m under the constraint CT when prevalence varies. For the case where μp = 10−3 and Cl/Cn = 1, nopt decreases from 44 to 31 as cv increases from 0 to 1. However, due to the diminishing returns from increasing n, LR meets or exceeds 95% of its maximum value for n = 10 over cv = 0–1. For the case where μp = 10−4 and Cl/Cn = 1, nopt decreases from 141 to 99 as cv increases from 0 to 1. However, LR achieves 95% of its maximum value for n = 14 over cv = 0–1. These results illustrate that the value of n that satisfies the LR objective may be not only substantially less than nopt but also insensitive to the degree of variability in prevalence among lots.

image

Figure 4. Relationship between the number of samples per lot and the number of lots rejected as a proportion of the maximum (max) for the case $Cl/$Cn = 1 with μp = 10−3 and 10−4.

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4. DISCUSSION

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. MODEL, METHODS, AND ASSUMPTIONS
  5. 3. RESULTS
  6. 4. DISCUSSION
  7. 5. CONCLUSION
  8. DISCLAIMER
  9. Appendix
  10. REFERENCES

It should be noted that in addition to the tradeoffs among competing objectives for food safety sampling under a budget constraint, there also may be nonbudgetary constraints that apply. For example, the optimal number of samples per lot indicated by the model may exceed the sample size required to achieve an importing country's appropriate level of protection for food safety. In this case, adopting the optimal sample size would be inconsistent with the legal obligations under the World Trade Organization Sanitary and Phytosanitary Agreement.[12] Cannon[13] cautions that for inspection under a budget restriction, the optimum is a guide, not the master.

In the food safety domain, sampling plans with a few samples per lot are commonly criticized for their lack of statistical rigor. Over 25 years ago, the National Research Council[14] expressed the still-conventional view that in order to be based on “sound statistical concepts,” food safety sampling plans need to “achieve a high degree of confidence in the acceptability of a lot.” As demonstrated here, however, sampling plans with a few samples per lot may represent a rational allocation of limited resources. It is worth noting, for example, that for p = 10−4 and n ≈ 200 (the optimal number of samples per lot under a budget constraint calculated for Cl/Cn = 2), the probability of lot rejection is just 2%. Considering economics in sampling and inspection is not novel. The economic design of statistical quality control measures has been a focus of research since the 1950s.[5] The economics of environmental pollution monitoring and regulatory compliance assurance was a particular research emphasis in the 1980s and 1990s.[15] In statistics, the tradeoffs between the number and size of clusters (such as food lots) have long been appreciated in the fields of experimental and survey design.

Some examples serve to place the current focus on optimal food safety sampling under a budget constraint within the context of the wider statistical literature. For example, in designing two-stage sampling surveys, the optimum within-cluster sample size to minimize the uncertainty about the population mean for a fixed total cost (or to minimize the total cost for a fixed precision) depends on the relative magnitude of the variance between and within clusters, the size of the clusters, and the relative costs of sampling a cluster and sampling an element within a cluster.[16] If the variance between clusters (e.g., households) is less than the variance within clusters (e.g., due to age and sex difference of household members), then it can be economically efficient to intensively sample within clusters. Alternatively, if members (e.g., birds) within clusters (e.g., flocks) are highly correlated (e.g., with respect to avian influenza status), then it can be efficient to sample more clusters less intensively. Similarly, in controlled experiments designed to evaluate the effects of maternal treatments on offspring (e.g., developmental toxicity from prenatal exposure), the statistical power for a fixed total sample size (a laboratory capacity constraint) is maximized by sampling single offspring from multiple litters rather than multiple offspring from a smaller number of litters. Here, mothers represent clusters, and litter siblings represent elements within clusters. The stronger the litter effect (intracluster correlation), the greater the gain in power by sampling more litters.[17] As these examples suggest, scarce resources should force us to consider the tradeoff between depth and coverage in acquiring data. Therefore, given limited resources for food safety sampling, it is mistaken to simply equate the number of samples per lot with the statistical rigor of a food safety sampling plan, and conclusions based solely on the probability of lot acceptance or rejection are limited.

At very low acceptable contamination levels, the direct, curative effect of food sampling by itself can be an inefficient means of controlling product safety because the required sample sizes are uneconomically large, technically infeasible, or both. An exclusive focus on the direct, curative effect of sampling is misplaced, however. More broadly, food safety sampling also has an indirect, preventive effect. Sampling serves to assure that food safety control systems are performing as intended and as an enforcement tool to assure compliance with regulatory measures or private contract specifications. Food safety verification sampling creates economic incentives for food producing firms to develop, implement, and maintain effective control measures that limit the probability and degree of noncompliance with regulatory measures or private contract specifications.[18, 19] Even at low sampling levels, the economic incentives may be substantial as they are a function of both the probability of detecting noncompliance via the sampling levels and the consequences of noncompliance, given detection.

Although the curative effects of a lot acceptance sampling program may be weak, any analysis of its food safety impact that ignores the effect of economic pressures exerted by sampling is incomplete. Food safety inspection problems seem to call for a game-theoretic treatment.[20, 21] However, the economic incentives can vary among firms, geographically, and over time and may be subject to legal constraints that vary by jurisdiction and to uncertain liability exposures. A major challenge would be to specify how the heterogeneous players behave and in choosing the rules they use to make decisions. Ultimately, a complete analysis may prove intractable. However, a partial and crude proxy for the economic pressures exerted by a sampling program is simply the number of nonconforming lots rejected.

5. CONCLUSION

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. MODEL, METHODS, AND ASSUMPTIONS
  5. 3. RESULTS
  6. 4. DISCUSSION
  7. 5. CONCLUSION
  8. DISCLAIMER
  9. Appendix
  10. REFERENCES

As a practical matter, food safety personnel are typically concerned with allocating resources based on externally imposed resource constraints. It should be noted, however, that assuming a budget constraint raises the far more difficult question of what the size of the budget should be. In conclusion, recognizing the broader impacts of food safety sampling as well as the tension among multiple, competing objectives highlights that designing effective compliance assurance measures depends not only on microbiology and statistics but also on economic, legal, and other factors that are beyond the purview of science.

DISCLAIMER

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. MODEL, METHODS, AND ASSUMPTIONS
  5. 3. RESULTS
  6. 4. DISCUSSION
  7. 5. CONCLUSION
  8. DISCLAIMER
  9. Appendix
  10. REFERENCES

The opinions expressed herein are the views of the author and do not necessarily reflect the official policy or position of the U.S. Department of Agriculture. Reference herein to any specific commercial products, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the U.S. government.

Appendix

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. MODEL, METHODS, AND ASSUMPTIONS
  5. 3. RESULTS
  6. 4. DISCUSSION
  7. 5. CONCLUSION
  8. DISCLAIMER
  9. Appendix
  10. REFERENCES

LAGRANGIAN SOLUTION

The constrained optimization problem is:

  • display math

inline image

  • display math(A1)
  • math image(A2)
  • math image(A3)
  • math image(A4)
  • math image(A5)
  • math image(A6)
  • math image(A7)
  • math image(A8)
  1. 1

    Sampling plans for inspection by attributes have been formulated for lot-by-lot inspection (ISO 2859-1), isolated lot inspection (ISO 2859-2), and skip lot inspection (ISO 2859-3). In practice, each type of inspection has played a role in food safety sampling.

  2. 2

    Rejected lots may be subject to return to supplier, diversion to treatment (e.g., cooking), disposal, or other disposition.

  3. 3

    Because f(x)[RIGHTWARDS ARROW]0 as log(f(x))[RIGHTWARDS ARROW]–∞, numerically minimizing log(∂(LR)/∂(n)) is equivalent to finding the root for ∂(LR)/∂(n) = 0. This generally resulted in a better numerical approximation than solving directly for ∂(LR)/∂(n) = 0.

  4. 4

    Here, we define the beta distribution in terms of the mean (μ) and variance (σ2). It is more common to parameterize the beta distribution in terms of α and β, with α = μ2(1 – μ)/σ2μ and β = α(1 – μ)/μ.

REFERENCES

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. MODEL, METHODS, AND ASSUMPTIONS
  5. 3. RESULTS
  6. 4. DISCUSSION
  7. 5. CONCLUSION
  8. DISCLAIMER
  9. Appendix
  10. REFERENCES
  • 1
    Whiting R, Rainosek A, Buchanan R, Miliotis M, LaBarre D, Long W, Ruple A, Schaub S. Determining the microbiological criteria for lot rejection from the performance objective or food safety objective. International Journal of Food Microbiology, 2006; 110:263267.
  • 2
    ICMSF (International Commission on Microbiological Specifications for Foods). Micro-Organisms in Foods 7: Microbiological Testing in Food Safety Management. New York: Springer, 2002.
  • 3
    Codex (Codex Alimentarius Commission). General Guidelines on Sampling. Rome: FAO, 2004.
  • 4
    Codex (Codex Alimentarius Commission). Recommended Methods of Sampling for the Determination of Pesticide Residues for Compliance with MRLs. Rome: FAO, 1999.
  • 5
    Montgomery DC. Introduction to Quality Control, 5th ed. Hoboken, NJ: John Wiley & Sons, Inc., 2005.
  • 6
    Haas CN, Rose JB, Gerba CP. Quantitative Microbial Risk Assessment. New York: John Wiley & Sons, Inc., 1999.
  • 7
    Washington State University. Avian Health and Food Safety Laboratory Fee Schedule, 2012. Available at: http://www.vetmed.wsu.edu/depts_waddl/avian.aspx#Fees, Accessed on June 26, 2012.
  • 8
    New Zealand Parliamentary Counsel Office. Food (Fees and Charges) Regulations 1997, 2008. Available at: http://www.legislation.govt.nz/regulation/public/1997/0100/latest/DLM232769.html?search=ts_regulation_food_resel&sr=1, Accessed on June 26, 2012.
  • 9
    Georgia Ports Authority. Rule 34-525 Government Agency Inspections, 2012. Available at: http://www.gaports.com/tabid/244/xmid/3373/xmview/2/xmmid/741/Default.aspx, Accessed on December 12, 2012.
  • 10
    Esteban E. Food Safety and Inspection Service, 2013, personal communication.
  • 11
    Connelly LB. Balancing the number and size of sites: An economic approach to the optimal design of cluster samples. Controlled Clinical Trials, 2003; 24:544559.
  • 12
    WTO (World Trade Organization). Agreement on the Application of Sanitary and Phytosanitary Measures. Geneva: WTO, 1994.
  • 13
    Cannon R. Inspecting and monitoring on a restricted budget—Where best to look? Preventive Veterinary Medicine, 2009; 92:163174.
  • 14
    National Research Council. An Evaluation of the Role of Microbiological Criteria for Foods and Food Ingredients. Washington, DC: National Academies Press, 1985.
  • 15
    Cohen MA. Empirical research on the deterrent effect of environmental monitoring and enforcement. Environmental Law Reporter, 2000; 30(4):1024510252.
  • 16
    Cochran WG. Sampling Techniques, 3rd ed. New York: John Wiley & Sons, 1977.
  • 17
    National Toxicology Program. National Toxicology Program's Report of the Endocrine Disruptors Low-Dose Peer Review. Research Triangle Park, NC: National Toxicology Program, 2001.
  • 18
    Starbird SA. Designing food safety regulations: The effect of inspection policy and penalties for noncompliance on food processor behavior. Journal of Agricultural and Resource Economics, 2000; 25(2):616635.
  • 19
    Starbird AS. Moral hazard, inspection policy, and food safety. American Journal of Agricultural Economics, 2005; 87(1):1527.
    Direct Link:
  • 20
    Harrington W. Enforcement leverage when penalties are restricted. Journal of Public Economics, 1988; 37:2953.
  • 21
    Bier VM, Shi-Woei L. Should the model for risk-informed regulation be game theory rather than decision theory? Risk Analysis, 2013; 33(2):281291.