Del. Chapter of the ASA B. H. Stanley, 19 Jan 2006 Slide 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”

Del. Chapter of the ASA B. H. Stanley, 19 Jan 2006 Slide 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)

Del. Chapter of the ASA B. H. Stanley, 19 Jan 2006 Slide 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.

Del. Chapter of the ASA B. H. Stanley, 19 Jan 2006 Slide 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

Del. Chapter of the ASA B. H. Stanley, 19 Jan 2006 Slide 5 Example: Codling Moth (Cydia pomonella (L.)) in Apples From: New York State Integrated Pest Management Fact Sheet

Del. Chapter of the ASA B. H. Stanley, 19 Jan 2006 Slide 6 Typical Questions How many apples? How many “bad” apples?

Del. Chapter of the ASA B. H. Stanley, 19 Jan 2006 Slide 7 Binomial Moments Variance Mean Let: X i Number of successes for replicate I Average of X i s (i=1 to m) sSample standard deviation of X i s

Del. Chapter of the ASA B. H. Stanley, 19 Jan 2006 Slide 8 Method of Moments Estimator (MME) Binomial Parameter n Conditions n Estimator Let: X i Number of successes for replicate i Average of X i s (i=1 to m) sSample standard deviation of X i s 1.X > 0 2.X >s 2 3.n> X max Note:  >  2, since  2 = np(1-p) =  (1-p)

Del. Chapter of the ASA B. H. Stanley, 19 Jan 2006 Slide 9 What About Over-dispersion?

Del. Chapter of the ASA B. H. Stanley, 19 Jan 2006 Slide 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)

Del. Chapter of the ASA B. H. Stanley, 19 Jan 2006 Slide 11 Negative Binomial Moments Variance Mean Let: X i Number of successes for replicate i Average of X i s (i=1 to m) sSample standard deviation of X i s

Del. Chapter of the ASA B. H. Stanley, 19 Jan 2006 Slide 12 Method of Moments Estimator (MME) Negative Binomial Parameter n Conditions n Estimator Let: X i Number of successes for replicate i Average of X i s (i=1 to m) sSample standard deviation of X i s 1.X > 0 2.S 2 > X 3.n > X max _ _ Note:  2> , since  2 = np(1+p) =  (1+p)

Del. Chapter of the ASA B. H. Stanley, 19 Jan 2006 Slide 13 Use the Var/Mean to Select a Method If Mean > Variance use binomial If Mean < Variance use negative binomial

Del. Chapter of the ASA B. H. Stanley, 19 Jan 2006 Slide 14 What if…X =0 ?

Del. Chapter of the ASA B. H. Stanley, 19 Jan 2006 Slide 15 Example Simulations

Del. Chapter of the ASA B. H. Stanley, 19 Jan 2006 Slide 16 Example: Minitab – binomial variates (n=20, p=0.1)

Del. Chapter of the ASA B. H. Stanley, 19 Jan 2006 Slide 17 Histograms of Generated Data

Del. Chapter of the ASA B. H. Stanley, 19 Jan 2006 Slide 18 Summary of Generated Data

Del. Chapter of the ASA B. H. Stanley, 19 Jan 2006 Slide 19 Example: Minitab – binomial variates (n=1000, p=0.1)

Del. Chapter of the ASA B. H. Stanley, 19 Jan 2006 Slide 20 Histograms of Generated Data

Del. Chapter of the ASA B. H. Stanley, 19 Jan 2006 Slide 21 Summary of Generated Data

Del. Chapter of the ASA B. H. Stanley, 19 Jan 2006 Slide 22 Key References Binet, F. E The fitting of the positive binomial distribution when both parameters are estimated from the sample. Annals of Eugenics 18: Blumenthal, S. and R. C. Dahiya Estimating the binomial parameter n. JASA 76: 903 – 909. Olkin, I., A. J. Petkau and J. V. Zidek A comparison of n estimators for the binomial distribution. JASA 76: 637 – 642. Johnson, N. L. and S. Kotz Discrete Distributions. J. Wiley & Sons, NY 328 pp. (ISBN )

Del. Chapter of the ASA B. H. Stanley, 19 Jan 2006 Slide 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.

Del. Chapter of the ASA B. H. Stanley, 19 Jan 2006 Slide 24 Thank You! Dr. Bruce H. Stanley DuPont Crop Protection Stine-Haskell Research Center Newark, Delaware Tel: (302)

Del. Chapter of the ASA B. H. Stanley, 19 Jan 2006 Slide 25 Delaware Chapter ASA Next Meeting: Feb 16, 2006 Professor Joel Best Author of “LIES, DAMN LIES, AND STATISTICS” and “MORE LIES, DAMN LIES, AND STATISTICS”