Statistical testing for contaminants in an agricultural product

Abstract

Agricultural wholesale products, including different cereal grains, are regularly tested for contaminants such as salmonella. To test for contaminants, individual samples are regularly drawn from the product and these samples are homogeneously mixed to form a single composite sample. A small amount of this composite sample is then selected and tested for the contaminant. Detailed procedures for testing samples for contamination are outlined and regulated by the FDA among other services. Ideally, failure to detect contamination would yield a statistically rigorous limit on the true amount of contamination present in the product. In this study, we use ideas from risk-limiting auditing and without-replacement sampling to derive a novel test for detecting contamination. We identify a set of conservative, worst-case assumptions that allow us to derive a closed-form probability for failing to detect a contaminant given a pre-specified proportion of contamination present in the product. We then use this probability to develop a risk-limiting statistical test for the null hypothesis that the amount of contamination present is beyond acceptable levels—if no contamination is found, this null is rejected, and our statistical test concludes that the amount of contamination is within a tolerable range. Furthermore, we compute the minimum sample size needed to ensure that, for a pre-specified significance level α, the test rejects the null hypothesis if no contamination is detected. We show that our approach is significantly more powerful than current methods for concluding that an agricultural product is not contaminated. The improvement of our method is especially large for when the amount of agricultural product being tested is small with respect to the size of the individual samples.