Без темы <<  Apple 09 03 2015 итоги Apple 2013 июнь  >> “Applications of Binomial “n” Estimation, Especially when No Successes Applications of Binomial “n” Estimation, Especially when No Successes Applications of Binomial “n” Estimation, Especially when No Successes Agenda Example: Codling Moth (Cydia pomonella (L Typical Questions Binomial Moments Method of Moments Estimator (MME) Binomial Parameter n What About Over-dispersion Genesis Negative Binomial Moments Method of Moments Estimator (MME) Negative Binomial Parameter n If Mean > Variance use binomial If Mean < Variance use negative What if…X =0 Example Simulations Example: Minitab – binomial variates (n=20, p=0 Histograms of Generated Data Summary of Generated Data Example: Minitab – binomial variates (n=1000, p=0 Histograms of Generated Data Summary of Generated Data Key References Conclusions Thank You Professor Joel Best Author of “LIES, DAMN LIES, AND STATISTICS” and

Презентация: «Apple 19 января». Автор: STANLEB. Файл: «Apple 19 января.ppt». Размер zip-архива: 889 КБ.

## Apple 19 января

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1 ### “Applications of Binomial “n” Estimation, Especially when No Successes

Are Observed”

January 19, 2006

Tonight’s speaker: Dr. Bruce H. Stanley DuPont Crop Protection

Delaware Chapter ASA

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

X > 0 X >s2 n> Xmax

n Estimator

Conditions

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

_

X > 0 S2 > X n > Xmax

_

n Estimator

Conditions

Note: ?2> ?, since ?2 = np(1+p) = ?(1+p)

13 ### If Mean > Variance use binomial If Mean < Variance use negative

binomial

Use the Var/Mean to Select a Method

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 ### Professor Joel Best Author of “LIES, DAMN LIES, AND STATISTICS” and

“MORE LIES, DAMN LIES, AND STATISTICS”

Next Meeting: Feb 16, 2006

Delaware Chapter ASA

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