Article: Optimal definition of the limit of detection (LOD) in detecting genetically modified grains from heterogeneous grain lots

Author: Kohji Yamamura, Junichi Mano, Hiroyuki Shibaike (2019)

Journal: Quality Technology & Quantitative Management 16: 36-53 
DOI: https://doi.org/10.1080/16843703.2017.1347992

Flow of estimation

Appendix A: Estimation of the heterogeneity in a lot

The following R-function calculates the maximum likelihood estimates of the log(mean) and log(variance) of the proportion of defective items among increments in the increment sampling.

The program is applied to the data for the proportion of pecky rice grains as an example. The data are given in the Electronic Appendix of Yamamura and Ishimoto (2009): Optimal sample size for composite sampling with subsampling, when estimating the proportion of pecky rice grains in a field. J Agric Biol Environ Stat 14:135-153

R function to estimate the log(mean) and log(variance).

Appendix B: Estimation of the parameters of POD curve

It will be reasonable to consider that the detecion event is related to the quantity (q) of some materials which is given by a linear form, q = a + bx + e, where x is an explanatory variable, a and b are constant, e is a normal variability. If the detection occurs when the amount of q becomes larger than a threshold (t), then the probability of response (p) is given by probit(p) = (a + bx – t). If the threshold fluctuates among laboratories by following a normal distribution, we can use a probit binomial generalized linear mixed model (probit binomial GLMM). The following is an example of R-program to estimate the probability of detection curve (POD-curve) by assuming that the detection occurs when the amount of qbecomes larger than a threshold (t_W). POD increases with increasing proportion of GM grains by approximately following a logistic curve. Data from the inter-laboratory study of Kodama et al. (2011) are used as an example. A constant dispersion parameter is assumed.

R program to estimate POD curve.

Appendix C: Estimation of the optimal definition of LOD

The following R-function calculates the total cost for inspection for various definition of LOD.

R function to estimate the optimal LOD.

The results are given by a graph such as that shown below.

Appendix D: Estimation of the optimal sample size

The following R-function calculates the optimal combination of sample sizes for a given LOD.

R function to estimate the optimal sample size.