### Abstract

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. MODEL, METHODS, AND ASSUMPTIONS
- 3. RESULTS
- 4. DISCUSSION
- 5. CONCLUSION
- DISCLAIMER
- Appendix
- 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

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. MODEL, METHODS, AND ASSUMPTIONS
- 3. RESULTS
- 4. DISCUSSION
- 5. CONCLUSION
- DISCLAIMER
- Appendix
- 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.

### 4. DISCUSSION

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. MODEL, METHODS, AND ASSUMPTIONS
- 3. RESULTS
- 4. DISCUSSION
- 5. CONCLUSION
- DISCLAIMER
- Appendix
- 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 *C*_{l}/*C*_{n} = 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.