del. chapter of the asa b. h. stanley, 19 jan 2006 slide 1 delaware chapter asa january 19, 2006...
Post on 22-Dec-2015
213 views
TRANSCRIPT
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”
Del. Chapter of the ASA B. H. Stanley, 19 Jan 2006
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]
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 Sheethttp://www.nysipm.cornell.edu/factsheets/treefruit/pests/cm/codmoth.html
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
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
Del. Chapter of the ASA B. H. Stanley, 19 Jan 2006
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)
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
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
Del. Chapter of the ASA B. H. Stanley, 19 Jan 2006
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)
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 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 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)
Del. Chapter of the ASA B. H. Stanley, 19 Jan 2006
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
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 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)
Del. Chapter of the ASA B. H. Stanley, 19 Jan 2006
Slide 21
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
Del. Chapter of the ASA B. H. Stanley, 19 Jan 2006
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)
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. StanleyDuPont Crop Protection
Stine-Haskell Research Center
Newark, DelawareTel: (302)-366-5910
Email: [email protected]