Delaware Chapter ASA

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. StanleyDuPont Crop Protection

“Applications of Binomial “n” Estimation,Especially when No Successes Are Observed”


Slide 2

Applications of Binomial “n” Estimation, Applications of Binomial “n” Estimation,

Especially when No Successes Are ObservedEspecially when No Successes Are Observed

Dr. Bruce H. StanleyDuPont Crop Protection

Stine-Haskell Research Center

Newark, Delaware

Tel: (302)-366-5910

Email: [email protected]


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.


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


Slide 5

Example: Codling Moth (Cydia pomonella (L.)) in Apples

From: New York State Integrated Pest Management Fact Sheethttp://www.nysipm.cornell.edu/factsheets/treefruit/pests/cm/codmoth.html


Slide 6

Typical Questions

How many apples?

How many “bad” apples?


Slide 7

Binomial Moments

Variance

pnX

)1(2 ppns

Mean

Let:

Xi Number of successes for replicate IAverage of Xis (i=1 to m)

s Sample standard deviation of XisX


Slide 8

Method of Moments Estimator (MME)Binomial Parameter n

Conditions

2

2

sX

Xn

n Estimator

Let:Xi Number of successes for replicate i

Average of Xis (i=1 to m)s Sample standard deviation of Xis

X

1. X > 02. X >s2

3. n> Xmax

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


Slide 9

What AboutOver-dispersion?


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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)


Slide 11

Negative Binomial Moments

Variance

pnX

)1(2 ppns

Mean

Let:

Xi Number of successes for replicate iAverage of Xis (i=1 to m)

s Sample standard deviation of Xis

X


Slide 12

Method of Moments Estimator (MME)Negative Binomial Parameter n

Conditions

Xs

Xn

2

2n Estimator

Let:Xi Number of successes for replicate i

Average of Xis (i=1 to m)s Sample standard deviation of Xis

X

1. X > 02. S2 > X3. n > Xmax

__

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


Slide 13

Use the Var/Mean to Select a Method

• If Mean > Variance use binomial

• If Mean < Variance use negative binomial


Slide 14

What if…X =0 ?


Slide 15

ExampleSimulations


Slide 16

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


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Histograms of Generated DataPerc

ent

76543210

76543210

30

20

10

0

76543210

30

20

10

0

Bin(20,0.1) 10 reps Bin(20,0.1) 20 reps Bin(20,0.1) 100 reps

Bin(20,0.1) 200 reps Bin(20,0.1) 1000 reps

Binomial Random Variates (n = 20, p = 0.1)


Slide 18

Summary of Generated Data

n=20,p=0.1Reps Mean Variance Est(n) NB Est(n)

10 1.7 2.0111 -9.289617 9.28961720 2.05 1.5237 7.98499 -7.98499

100 2.05 1.9268 34.1112 -34.1112200 1.95 1.8266 30.81442 -30.81442

1000 2.013 1.6785 12.11411 -12.11411Theoretical 2 1.8


Slide 19

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


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Histograms of Generated DataPerc

ent

12812011210496888072

12812011210496888072

20

15

10

5

0

12812011210496888072

20

15

10

5

0

n=10 n=20 n=100

n=200 n=1000

Binomial Random Variates (n = 1000, p = 0.1)


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Summary of Generated Data

n=1000,p=0.1Reps Mean Variance Est(n) NB Est(n)

10 102.5 161.17 -179.0736 179.073620 98.5 57.421 236.1852 -236.1852

100 102.22 87.3 700.3303 -700.3303200 100.05 82.15 559.218 -559.218

1000 100.25 80.38 505.7908 -505.7908Theoretical 100 90


Slide 22

Key ReferencesBinet, 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)


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.


Slide 24

Thank You!Dr. Bruce H. StanleyDuPont Crop Protection

Stine-Haskell Research Center

Newark, DelawareTel: (302)-366-5910

Email: [email protected]


Slide 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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Delaware Chapter ASA