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)-366-5910
Email: Bruce.H.Stanley-1@usa.dupont.com
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
http://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
Let:
Xi
X
s
Number of successes for replicate I
Average of Xis (i=1 to m)
Sample standard deviation of Xis
Mean
X  n p
Variance
Del. Chapter of the ASA
s 2  n  p  (1  p)
B. H. Stanley, 19 Jan 2006
Slide 8
Method of Moments Estimator (MME)
Binomial Parameter n
Xi
X
s
n Estimator
Let:
Number of successes for replicate i
Average of Xis (i=1 to m)
Sample standard deviation of Xis
2
X
n
2
X s
Conditions
1. X > 0
2. X >s2
3. n> Xmax
Del. Chapter of the ASA
Note: >2, since 2 = np(1-p) = (1-p)
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
Let:
Xi
X
s
Number of successes for replicate i
Average of Xis (i=1 to m)
Sample standard deviation of Xis
Mean
X  n p
Variance
Del. Chapter of the ASA
s 2  n  p  (1  p)
B. H. Stanley, 19 Jan 2006
Slide 12
Method of Moments Estimator (MME)
Negative Binomial Parameter n
Xi
X
s
n Estimator
Let:
Number of successes for replicate i
Average of Xis (i=1 to m)
Sample standard deviation of Xis
2
X
n 2
s X
Conditions
Del. Chapter of the ASA
_
1. X > 0_
2. S2 > X
3. n > Xmax
Note: 2> , since 2 = np(1+p) = (1+p)
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
Binomial Random Variates (n = 20, p = 0.1)
0
Bin(20,0.1) 10 reps
1
2
3
4
5
6
7
Bin(20,0.1) 20 reps
Bin(20,0.1) 100 reps
30
20
Percent
10
Bin(20,0.1) 200 reps
Bin(20,0.1) 1000 reps
30
0
1
2
3
4
5
6
7
0
20
10
0
0
Del. Chapter of the ASA
1
2
3
4
5
6
7
B. H. Stanley, 19 Jan 2006
Slide 18
Summary of Generated Data
n=20,p=0.1
Reps
Mean
Variance
10
1.7
2.0111
20
2.05
1.5237
100
2.05
1.9268
200
1.95
1.8266
1000
2.013
1.6785
Theoretical
2
1.8
Del. Chapter of the ASA
Est(n)
-9.289617
7.98499
34.1112
30.81442
12.11411
NB Est(n)
9.289617
-7.98499
-34.1112
-30.81442
-12.11411
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
Binomial Random Variates (n = 1000, p = 0.1)
72
n=10
80
88
96 104 112 120 128
n=20
n=100
20
15
10
Percent
5
n=200
20
n=1000
72
80
88
96 104 112 120 128
0
15
10
5
0
72
Del. Chapter of the ASA
80
88
96 104 112 120 128
B. H. Stanley, 19 Jan 2006
Slide 21
Summary of Generated Data
n=1000,p=0.1
Reps
Mean
Variance
10
102.5
161.17
20
98.5
57.421
100
102.22
87.3
200
100.05
82.15
1000
100.25
80.38
Theoretical
100
90
Del. Chapter of the ASA
Est(n)
-179.0736
236.1852
700.3303
559.218
505.7908
NB Est(n)
179.0736
-236.1852
-700.3303
-559.218
-505.7908
B. H. Stanley, 19 Jan 2006
Slide 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)
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)-366-5910
Email: Bruce.H.Stanley-1@usa.dupont.com
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”
Del. Chapter of the ASA
B. H. Stanley, 19 Jan 2006