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Delaware Chapter ASA

Delaware Chapter ASA. January 19, 2006. Tonight’s speaker: Dr. Bruce H. Stanley DuPont Crop Protection. “Applications of Binomial “n” Estimation, Especially when No Successes Are Observed”. Applications of Binomial “n” Estimation, Especially when No Successes Are Observed.

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Delaware Chapter ASA

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  1. Delaware Chapter ASA January 19, 2006 Tonight’s speaker: Dr. Bruce H. Stanley DuPont Crop Protection “Applications of Binomial “n” Estimation,Especially when No Successes Are Observed”

  2. Applications of Binomial “n” Estimation, Especially when No Successes Are Observed Dr. Bruce H. Stanley DuPont Crop Protection Stine-Haskell Research Center Newark, Delaware Tel: (302)-366-5910 Email: Bruce.H.Stanley-1@usa.dupont.com

  3. Applications of Binomial “n” Estimation, Especially when No Successes Are Observed • Dr. Bruce H. Stanley – Many processes, such as flipping a coin, follow a binomial process where there is one of two outcomes. The researcher often knows that both outcomes are possible, even if no events of one of the outcomes is observed. This talk presents techniques for estimating number of trials, e.g., number of flips, based upon the observed outcomes only, and focuses on the case where events of only one possibility are observed. Dr. Stanley then discusses applications of this methodology.

  4. Agenda • Introduction • Binomial processes • Replicated observations • Successes in at least one replicate • All replicates had no successes • All replicates are the same • Over and under dispersion • Some applications • Conclusion

  5. Example: Codling Moth (Cydia pomonella (L.)) in Apples From: New York State Integrated Pest Management Fact Sheet http://www.nysipm.cornell.edu/factsheets/treefruit/pests/cm/codmoth.html

  6. Typical Questions How many apples? How many “bad” apples?

  7. Binomial Moments Let: Xi Number of successes for replicate I Average of Xis (i=1 to m) s Sample standard deviation of Xis Mean Variance

  8. Method of Moments Estimator (MME)Binomial Parameter n Let: Xi Number of successes for replicate i Average of Xis (i=1 to m) s Sample standard deviation of Xis n Estimator Conditions • X > 0 • X >s2 • n> Xmax Note: >2, since 2 = np(1-p) = (1-p)

  9. What About Over-dispersion?

  10. Genesis “A simple model, leading to the negative binomial distribution, is that representing the number of trials necessary to obtain m occurrences of an event which has constant probability p of occurring at each trial.” (Johnson & Kotz 1969)

  11. Negative Binomial Moments Let: Xi Number of successes for replicate i Average of Xis (i=1 to m) s Sample standard deviation of Xis Mean Variance

  12. Method of Moments Estimator (MME)Negative Binomial Parameter n Let: Xi Number of successes for replicate i Average of Xis (i=1 to m) s Sample standard deviation of Xis n Estimator Conditions _ • X > 0 • S2 > X • n > Xmax _ Note: 2> , since 2 = np(1+p) = (1+p)

  13. Use the Var/Mean to Select a Method • If Mean > Variance use binomial • If Mean < Variance use negative binomial

  14. What if…X =0 ?

  15. Example Simulations

  16. Example: Minitab – binomial variates (n=20, p=0.1)

  17. Histograms of Generated Data

  18. Summary of Generated Data

  19. Example: Minitab – binomial variates (n=1000, p=0.1)

  20. Histograms of Generated Data

  21. Summary of Generated Data

  22. Key References Binet, F. E. 1953. The fitting of the positive binomial distribution when both parameters are estimated from the sample. Annals of Eugenics 18: 117-119. Blumenthal, S. and R. C. Dahiya. 1981. Estimating the binomial parameter n. JASA 76: 903 – 909. Olkin, I., A. J. Petkau and J. V. Zidek. 1981. A comparison of n estimators for the binomial distribution. JASA 76: 637 – 642. Johnson, N. L. and S. Kotz. 1969. Discrete Distributions. J. Wiley & Sons, NY 328 pp. (ISBN 0-471-44360-3)

  23. Conclusions • You can work backwards from binomial data to estimate the number of trials. • If data appear “over-dispersed”, try the negative binomial distribution approach. • Bias adjustments exist. • Methods exist to handle the case where no events are observed. However, one must assume something about the probability of an event.

  24. Thank You! Dr. Bruce H. Stanley DuPont Crop Protection Stine-Haskell Research Center Newark, Delaware Tel: (302)-366-5910 Email: Bruce.H.Stanley-1@usa.dupont.com

  25. Delaware Chapter ASA Next Meeting: Feb 16, 2006 Professor Joel BestAuthor of“LIES, DAMN LIES, AND STATISTICS”and “MORE LIES, DAMN LIES, AND STATISTICS”

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