Abstract
The proportion of pecky rice grains has been estimated empirically using composite sampling with subsampling. The procedure can be summarized as follows: (1) A fixed number of rice plants, n 1, are drawn at random in the paddy field; (2) all of the rice grains in the collected rice plants are mixed well to form a composite; (3) a portion of the grains, n 2, is drawn at random from the composite; and (4) the collected grains are examined by eye to estimate the proportion of pecky rice grains. We propose a method for determining the optimal sample size in estimating the proportion of defective items by this type of composite sampling with subsampling. Spatial heterogeneity in the proportion of defective items is included in the estimation. We use Taylor’s power law to describe the density-dependent change of spatial heterogeneity. In controlling the precision of the estimate, we use the relative precision, D, which is defined by the coefficient of variation of the estimated proportion. We propose a rejection procedure in which the product is rejected if the estimate of proportion with D=0.25 is larger than a predetermined tolerable threshold of proportion. We also consider another control criterion in which the consumer’s risk, \, is controlled by a zero-tolerance method. Finally, we examine the relationship between the two control criteria.
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Bar-Lev, S. K., Stadje,W., and Van der Duyn Schouten, F. A. (2006), “Group Testing Procedures With Incomplete Identification and Unreliable Testing Results,” Applied Stochastic Models in Business and Industry, 22, 281–296.
Bhattacharyya, G. K., Karandinos, M. G., and DeFoliart, G. R. (1979), “Point Estimates and Confidence Intervals for Infection Rates Using Pooled Organisms in Epidemiologic Studies,” American Journal of Epidemiology, 109, 124–131.
Boswell, M. T., Burnham, K. P., and Patil, G. P. (1988), “Role and Use of Composite Sampling and Capture-Recapture Sampling in Ecological Studies,” in Handbook of Statistics, Vol. 6, eds. P.R. Krishnaiah and C. R. Rao, North Holland: Elsevier Science, pp. 469–488.
Brookmeyer, R. (1999), “Analysis of Multistage Pooling Studies of Biological Specimens for Estimating Disease Incidence and Prevalence,” Biometrics, 55, 608–612.
Brown, G. H., and Fisher, N. I. (1972), “Subsampling a Mixture of Sampled Material,” Technometrics, 14, 663–668.
Burrows, P. M. (1987), “Improved Estimation of Pathogen Transmission Rates by Group Testing,” Phytopathology, 77, 363–365.
Chen, C. L., and Swallow, W. H. (1990), “Using Group Testing to Estimate a Proportion, and to Test the Binomial Model,” Biometrics, 46, 1035–1046.
Chiang, C. L., and Reeves, W. C. (1962), “Statistical Estimation of Virus Infection Rates in Mosquito Vector Populations,” American Journal of Hygiene, 75, 377–391.
Chick, S. E. (1996), “Bayesian Models for Limiting Dilution Assay and Group Test Data,” Biometrics, 52, 1055–1062.
Cochran, W. G. (1977), Sampling Techniques (3rd ed.), New York: Wiley.
Colón, S., Patil, G. P., and Taillie, C. (2001), “Estimating Prevalence Using Composites,” Environmental and Ecological Statistics, 8, 213–236.
Dorfman, R. (1943), “The Detection of Defective Members of Large Populations,” Annals of Mathematical Statistics, 14, 436–440.
Elder, R. S., Thompson, W. O., and Myers, R. H. (1980), “Properties of Composite Sampling Procedures,” Technometrics, 22, 179–186.
Emmanuel, J. C., Bassett, T. M., Smith, H. J., and Jacob, J. A. (1988), “Pooling of Sera for Human Immunodeficiency Virus (HIV) Testing: An Economical Method for Use in Developing Countries,” Journal of Clinical Pathology, 41, 582–585.
Federal Grain Inspection Service (1995), Grain Inspection Handbook, Book 1, Grain Sampling, Washington, DC: United States Department of Agriculture.
Gastwirth, J. L., and Johnson, W. O. (1994), “Screening With Cost-Effective Quality Control: Potential Applications to HIV and Drug Testing,” Journal of the American Statistical Association, 89, 972–981.
Hsu, L. (2005), “Group Testing With a Goal in Estimating the Number of Defects Under Imperfect Environmental Stress Screen Levels,” Communications in Statistics—Theory and Methods, 34, 1363–1377.
Hughes-Oliver, J. M., and Swallow, W. H. (1994), “A Two-Stage Adaptive Group-Testing Procedure for Estimating Small Proportions,” Journal of the American Statistical Association, 89, 982–993.
ISO (1990), ISO 542, Oilseeds—Sampling, Genève: International Organization for Standardization.
— (1999), ISO 13690, Cereals, Pulses and Milled Products— Sampling of Static Batches, Geève: International Organization for Standardization.
— (2000), ISO 10725, Acceptance Sampling Plans and Procedures for the Inspection of Bulk Material, Genève: International Organization for Standardization.
— (2002), ISO 6644, Flowing Cereals and Milled Cereal Products—Automatic Sampling by Mechanical Means, Genève: International Organization for Standardization.
— (2003), ISO 11648-1, Statistical Aspects of Sampling From Bulk Materials—Part 1: General Principles, Genève: International Organization for Standardization.
Iwasaki, M. (2005), “Rule of 3 and Related Topics,” in Proceedings of the 2005 Symposium of the Biometric Society of Japan, Tokyo: Biometric Society of Japan, pp. 1–2 (in Japanese).
Japan Plant Protection Association (2003), Handbook of Experiments for Practical Use of Pesticides, Tokyo: Japan Plant Protection Association (in Japanese).
Japanese Ministry of Health Labor and Welfare (2001), Inspection Method of Food Using Genetically Modified Organisms, Notification No. 110, Annex, Tokyo: Japanese Ministry of Health, Labor and Welfare (in Japanese).
Johnson, G. D., and Patil, G. P. (2001), “Cost Analysis of Composite Sampling for Classification,” Environmental and Ecological Statistics, 8, 91–107.
Johnson, N. L., Kotz, S., and Kemp, A. W. (2005), Univariate Discrete Distrbutions (3rd ed.), New York: Wiley.
Jovanovic, B. D., and Levy, P. S. (1997), “A Look at the Rule of Three,” American Statistician, 51, 137–139.
Kiritani, K. (2006), “Predicting Impacts of Global Warming on Population Dynamics and Distribution of Arthropods in Japan,” Population Ecology, 48, 5–12.
Kuno, E. (1986), Research Methods for Population Dynamics of Animals: Estimation Methods of Populations, Tokyo: Kyoritsu (in Japanese).
— (1991), “Verifying Zero-Infestation in Pest Control: A Simple Sequential Test Based on the Succession of Zero Sample,” Researches on Population Ecology, 33, 29–32.
Lancaster, V., and Keller-McNulty, S. (1998), “A Review of Composite Sampling Methods,” Journal of the American Statistical Association, 93, 1216–1230.
Lohr, S. L. (1999), Sampling: Design and Analysis, Pacific Grove: Duxbury Pr.
Lovinson, G., Gore, S. D., and Patil, G. P. (1994), “Design and Analysis of Composite Sampling Procedures: A Review,” in Handbook of Statistics, Vol. 12, eds. G. P. Patil and C. R. Rao, Amsterdam: Elsevier, pp. 103–166.
Ministry of Agriculture Forestry and Fisheries (2001), “Regulations for the Agricultural Product Standards (Notification No. 244 of the Ministry of Agriculture, Forestry and Fisheries of Japan),” http://www.kokuji.maff. go.jp/kokujituti/ top.asp (in Japanese).
Minotani, C. (2003), Handbook of Statistical Distributions, Tokyo: Asakura (in Japanese).
Patil, G. P. (2002), “Composite Sampling,” in Encyclopedia of Environmetrics, Vol. 1, eds. A. H. El-Shaarawi and W. W. Piegorsch, Chichester: Wiley, pp. 387–391.
Perry, J. N. (1981), “Taylor’s Power Law for Dependence of Variance on Mean in Animal Populations,” Journal of the Royal Statistical Society, Ser. C, 30, 254–263.
Rohde, C. (1976), “Composite Sampling,” Biometrics, 32, 273–282.
Rohlf, F. J., Akçakaya, H. R., and Ferraro, S. P. (1996), “Optimizing Composite Sampling Protocols,” Environmental Science and Technology, 30, 2899–2905.
Shimizu, K. (2006), Insurance Risk Models: Calculating Claims Distributions, Tokyo: Kyoritsu (in Japanese).
Sterrett, A. (1957), “On the Detection of Defective Members of Large Populations,” Annals of Mathematical Statistics, 28, 1033–1036.
Swallow, W. H. (1985), “Group Testing for Estimating Infection Rates and Probabilities of Disease Transmission,” Phytopathology, 75, 882–889.
— (1987), “Relative Mean Squared Error and Cost Considerations in Choosing Group Size for Group Testing to Estimate Infection Rates and Probabilities of Disease Transmission,” Phytopathology, 77, 1376–1381.
Taylor, L. R. (1961), “Aggregation, Variance and the Mean,” Nature, 189, 732–735.
— (1984), “Assessing and Interpreting the Spatial Distribution of Insect Populations,” Annual Review of Entomology, 29, 321–357.
Taylor, L. R., Woiwod, I. P., and Perry, J. N. (1978), “The Density-Dependence of Spatial Behaviour and the Rarity of Randomness,” Journal of Animal Ecology, 47, 383–406.
— (1979), “The Negative Binomial as a Dynamic Ecological Model for Aggregation, and the Density Dependence of k,” Journal of Animal Ecology, 48, 289–304.
Tebbs, J.M., Bilder, C. R., and Moser, B. K. (2003), “An Empirical Bayes Group-Testing Approach to Estimating Small Proportions,” Communications in Statistics—Theory and Methods, 32, 983–995.
Thompson, K. H. (1962), “Estimation of the Proportion of Vectors in a Natural Population of Insects,” Biometrics, 18, 568–578.
Thompson, S. K. (2002), Sampling (2nd ed.), New York: Wiley.
United States Environmental Protection Agency (2000), Guidance for Choosing a Sampling Design for Environmental Data Collection (EPA QA/G-5S), Washington, DC: United States Environmental Protection Agency.
van Belle, G. (2002), Statistical Rules of Thumb, New York: Wiley.
Watanabe, T., and Higuchi, H. (2006), “Recent Occurrence and Problem of Rice Bugs,” Shokubutsu Boeki (Plant Protection), 60, 201–203 (in Japanese).
Yamamura, K. (2000), “Colony Expansion Model for Describing the Spatial Distribution of Populations,” Population Ecology, 42, 161–169.
— (2001), “Theoretical Studies on the Relationship Between the Spatial Distribution of Insects and Population Dynamics,” Bulletin of the National Institute for Agro-Environmental Sciences, 19, 1–60 (in Japanese with English summary).
Yamamura, K., and Hino, A. (2007), “Estimation of the Proportion of Defective Units by Using Group Testing Under the Existence of a Threshold of Detection,” Communications in Statistics—Simulation and Computation, 36, 949–957.
Yamamura, K., and Sugimoto, T. (1995), “Estimation of the Pest Prevention Ability of the Import Plant Quarantine in Japan,” Biometrics, 51, 482–490.
Yamamura, K., Yokozawa, M., Nishimori, M., Ueda, Y., and Yokosuka, T. (2006), “How to Analyze Long-Term Insect Population Dynamics Under Climate Change: 50-Year Data of Three Insect Pests in Paddy Fields,” Population Ecology, 48, 31–48.
Zenios, S. A., and Wein, L. M. (1998), “Pooled Testing for HIV Prevalence Estimation: Exploiting the Dilution Effect,” Statistics in Medicine, 17, 1447–1467.
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Yamamura, K., Ishimoto, M. Optimal sample size for composite sampling with subsampling, when estimating the proportion of pecky rice grains in a field. JABES 14, 135–153 (2009). https://doi.org/10.1198/jabes.2009.0009
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DOI: https://doi.org/10.1198/jabes.2009.0009