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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
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
Agenda
Example: Codling Moth (Cydia pomonella (L
Example: Codling Moth (Cydia pomonella (L
Typical Questions
Typical Questions
Binomial Moments
Binomial Moments
Method of Moments Estimator (MME) Binomial Parameter n
Method of Moments Estimator (MME) Binomial Parameter n
What About Over-dispersion
What About Over-dispersion
Genesis
Genesis
Negative Binomial Moments
Negative Binomial Moments
Method of Moments Estimator (MME) Negative Binomial Parameter n
Method of Moments Estimator (MME) Negative Binomial Parameter n
If Mean > Variance use binomial If Mean < Variance use negative
If Mean > Variance use binomial If Mean < Variance use negative
What ifX =0
What ifX =0
Example Simulations
Example Simulations
Example: Minitab  binomial variates (n=20, p=0
Example: Minitab binomial variates (n=20, p=0
Histograms of Generated Data
Histograms of Generated Data
Summary of Generated Data
Summary of Generated Data
Example: Minitab  binomial variates (n=1000, p=0
Example: Minitab binomial variates (n=1000, p=0
Histograms of Generated Data
Histograms of Generated Data
Summary of Generated Data
Summary of Generated Data
Key References
Key References
Conclusions
Conclusions
Thank You
Thank You
Professor Joel Best Author of LIES, DAMN LIES, AND STATISTICS and
Professor Joel Best Author of LIES, DAMN LIES, AND STATISTICS and

: Apple 19 . : STANLEB. : Apple 19 .ppt. zip-: 889 .

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Apple 19 .ppt
1 Applications of Binomial n Estimation, Especially when No Successes

Applications of Binomial n Estimation, Especially when No Successes

Are Observed

January 19, 2006

Tonights speaker: Dr. Bruce H. Stanley DuPont Crop Protection

Delaware Chapter ASA

2 Applications of Binomial n Estimation, Especially when No Successes

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

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

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

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

Typical Questions

How many apples? How many bad apples?

7 Binomial Moments

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

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

What About Over-dispersion

10 Genesis

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

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

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

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

binomial

Use the Var/Mean to Select a Method

14 What ifX =0

What ifX =0

15 Example Simulations

Example Simulations

16 Example: Minitab  binomial variates (n=20, p=0

Example: Minitab binomial variates (n=20, p=0

1)

17 Histograms of Generated Data

Histograms of Generated Data

18 Summary of Generated Data

Summary of Generated Data

19 Example: Minitab  binomial variates (n=1000, p=0

Example: Minitab binomial variates (n=1000, p=0

1)

20 Histograms of Generated Data

Histograms of Generated Data

21 Summary of Generated Data

Summary of Generated Data

22 Key References

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

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

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

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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