$$P(response)=1-[1+dose\frac{2^{\frac{1}{a}}-1}{N^{50}} ]^{-a}$$

Description
Human/ Salmonella anatum strain II data [4]
Dose Positive stool culture No positive stool culture Total
89000 5 1 6
448000 4 2 6
1040000 6 0 6
3.9E+06 4 2 6
1E+07 6 0 6
2.39E+07 5 1 6
4.45E+07 6 0 6
6.73E+07 8 0 8

 

Goodness of fit and model selection
Model Deviance Δ Degrees 
of freedom		 χ20.95,1 
p-value 		χ20.95,m-k 
p-value
Exponential 62.8 54.2 7 3.84 
1.84e-13 		14.1 
4.22e-11
Beta Poisson 8.6 6 12.6 
0.198
Beta-Poisson fits better than exponential; cannot reject good fit for beta-Poisson.

 

Optimized parameters for the beta-Poisson model, from 10000 bootstrap iterations
Parameter MLE estimate Percentiles
0.5% 2.5% 5% 95% 97.5% 99.5%
α 2.3E-01 3.63E-02 5.29E-02 6.00E-02 5.51E-01 6.70E-01 1.40E+00
N50 4.91E+03 2.59E-10 2.73E-08 4.91E-06 8.72E+04 1.23E+05 2.29E+05

 

Parameter scatter plot for beta Poisson model ellipses signify the 0.9, 0.95 and 0.99 confidence of the parameters.

beta Poisson model plot, with confidence bounds around optimized model

# of Doses
8.00
Μodel
beta-Poisson
N50
4.91E+03
LD50/ID50
4.91E+03
Dose Units
CFU
Response
positive stool culture
Exposure Route
oral (with eggnog)
Contains Preferred Model
a
2.3E-01
Agent Strain
strain II
Experiment ID
233
Host type
human
Description
Human/ Salmonella anatum strain I & II pooled data [4]
Dose Positive stool culture No positive stool culture Total
12000 2 3 5
24000 3 3 6
66000 4 2 6
89000 5 1 6
93000 1 5 6
141000 3 3 6
256000 5 1 6
448000 4 2 6
587000 4 2 6
860000 6 0 6
1040000 6 0 6
3.9E+06 4 2 6
1E+07 6 0 6
2.39E+07 5 1 6
4.45E+07 6 0 6
6.73E+07 8 0 8

 

Goodness of fit and model selection
Model Deviance Δ Degrees 
of freedom		 χ20.95,1 
p-value 		χ20.95,m-k 
p-value
Exponential 138 119 15 3.84 
0 		25 
0
Beta Poisson 18.9 14 23.7 
0.168
Beta-Poisson fits better than exponential; cannot reject good fit for beta-Poisson.

 

Optimized parameters for the beta-Poisson model, from 10000 bootstrap iterations
Parameter MLE estimate Percentiles
0.5% 2.5% 5% 95% 97.5% 99.5%
α 3.18E-01 1.35E-01 1.71E-01 1.92E-01 5.70E-01 6.66E-01 9.56E-01
N50 3.71E+04 1.40E+03 5.50E+03 8.86E+03 8.82E+04 1.00E+05 1.30E+05

 

Parameter scatter plot for beta Poisson model ellipses signify the 0.9, 0.95 and 0.99 confidence of the parameters.

beta Poisson model plot, with confidence bounds around optimized model

# of Doses
16.00
Μodel
beta-Poisson
N50
3.71E+04
LD50/ID50
3.71E+04
Dose Units
CFU
Response
positive stool culture
Exposure Route
oral (with eggnog)
Contains Preferred Model
a
3.18E-01
Agent Strain
strain I
Experiment ID
232, 233
Host type
human
Description
Human / Salmonella anatum strain I data
Dose Positive stool culture No positive stool culture Total
12000 2 3 5
24000 3 3 6
66000 4 2 6
93000 1 5 6
141000 3 3 6
256000 5 1 6
587000 4 2 6
860000 6 0 6

 

Goodness of fit and model selection
Model Deviance Δ Degrees 
of freedom		 χ20.95,1 
p-value 		χ20.95,m-k 
p-value
Exponential 20.5 11 7 3.84 
0.000922 		14.1 
0.00457
Beta Poisson 9.53 6 12.6 
0.146
Beta-Poisson fits better than exponential; cannot reject good fit for beta-Poisson.

 

Optimized parameters for the beta-Poisson model, from 10000 bootstrap iterations
Parameter MLE estimate Percentiles
0.5% 2.5% 5% 95% 97.5% 99.5%
α 2.91E-01 9.77E-04 9.77E-04 4.11E-02 1.68E+00 5.36E+00 1.20E+03
N50 4.44E+04 6.74E-02 8.03E+02 4.15E+03 1.64E+05 6.89E+06 9.07E+07

 

Parameter scatter plot for beta Poisson model ellipses signify the 0.9, 0.95 and 0.99 confidence of the parameters.

beta Poisson model plot, with confidence bounds around optimized model

# of Doses
8.00
Μodel
beta-Poisson
N50
4.44E+04
LD50/ID50
4.44E+04
Dose Units
CFU
Response
positive stool culture
Exposure Route
oral (with eggnog)
Contains Preferred Model
a
2.91E-01
Agent Strain
strain I
Experiment ID
232
Host type
human
Description
diabetic rat and 316c strain 
Dose DEATH NOT DEATH Total
3000 6 4 10
3E+04 7 3 10
3E+05 7 3 10
3E+06 10 0 10
3E+07 10 0 10

 

Goodness of fit and model selection
Model Deviance Δ Degrees 
of freedom		 χ20.95,1 
p-value 		χ20.95,m-k 
p-value
Exponential 43.4 39 4 3.84 
4.25e-10 		9.49 
8.61e-09
Beta Poisson 4.39 3 7.81 
0.222
Beta-Poisson fits better than exponential; cannot reject good fit for beta-Poisson.

 

Optimized parameters for the beta-Poisson model, from 500 bootstrap iterations
Parameter MLE estimate Percentiles
0.5% 2.5% 5% 95% 97.5% 99.5%
α 2.65E-01 7.56E-02 1.30E-01 1.54E-01 4.94E-01 5.69E-01 8.88E-01
N50 2.27E+03 7.58E-02 1.50E+01 5.32E+01 8.62E+03 1.11E+04 1.74E+04
Parameter scatter plot for beta Poisson model ellipses signify the 0.9, 0.95 and 0.99 confidence of the parameters.
Parameter scatter plot for beta Poisson model ellipses signify the 0.9, 0.95 and 0.99 confidence of the parameters.
beta Poisson model plot, with confidence bounds around optimized model
beta Poisson model plot, with confidence bounds around optimized model
# of Doses
5.00
Μodel
beta-Poisson
N50
2.27E+03
LD50/ID50
2.27E+03
Dose Units
CFU
Response
death
Exposure Route
intraperitoneal
Contains Preferred Model
a
2.65E-01
Agent Strain
316c
Experiment ID
23
Host type
diabetic rat
Experiment Dataset
Description
Model data for Shigella flexneri (2a, strain 2457T) in the human 
Dose Illness Not illness Total
180 9 27 36
5000 28 21 49
1E+04 52 36 88
1E+05 14 10 24

 

Goodness of fit and model selection
Model Deviance Δ Degrees 
of freedom		 χ20.95,1 
p-value 		χ20.95,m-k 
p-value
Exponential 154 153 3 3.84 
0 		7.81 
0
Beta Poisson 1.69 2 5.99 
0.429
Beta-Poisson fits better than exponential; cannot reject good fit for beta-Poisson.

 

Optimized parameters for the beta-Poisson model, from 10000 bootstrap iterations
Parameter MLE estimate Percentiles
0.5% 2.5% 5% 95% 97.5% 99.5%
α 1.35E-01 9.86E-04 1.22E-02 6.85E-02 2.06E-01 2.22E-01 2.58E-01
N50 3.11E+03 5.09E+01 4.31E+02 9.64E+02 7.49E+03 9.14E+03 1.89E+04

 

Parameter scatter plot for beta Poisson model ellipses signify the 0.9, 0.95 and 0.99 confidence of the parameters
Parameter scatter plot for beta Poisson model ellipses signify the 0.9, 0.95 and 0.99 confidence of the parameters
beta Poisson model plot, with confidence bounds around optimized model
beta Poisson model plot, with confidence bounds around optimized model
# of Doses
4.00
Μodel
beta-Poisson
N50
3.11E+03
LD50/ID50
3.11E+03
Dose Units
CFU
Response
illness
Exposure Route
oral (in milk)
Contains Preferred Model
a
1.35E-01
Agent Strain
2a (strain 2457T)
Experiment ID
223
Host type
human
Experiment Dataset
Description
EPEC disease in the human model data
Dose Diarrhea No diarrhea Total
1E+06 0 4 4
1E+06 1 4 5
1E+08 1 4 5
5E+08 3 2 5
2.5E+09 6 0 6
1E+10 3 2 5
1E+10 5 0 5
2E+10 2 0 2

 

Goodness of fit and model selection
Model Deviance Δ Degrees 
of freedom		 χ20.95,1 
p-value 		χ20.95,m-k 
p-value
Exponential 31.8 20.6 7 3.84 
5.69e-06 		14.1 
4.37e-05
Beta Poisson 11.2 6 12.6 
0.0813
Beta-Poisson fits better than exponential; cannot reject good fit for beta-Poisson.

 

Optimized parameters for the beta-Poisson model, from 10000 bootstrap iterations
Parameter MLE estimate Percentiles
0.5% 2.5% 5% 95% 97.5% 99.5%
α 2.21E-01 9.31E-02 1.18E-01 1.27E-01 1.25E+00 7.41E+02 1.29E+04
N50 6.85E+07 6.14E+06 1.11E+07 1.52E+07 6.32E+08 7.69E+08 1.09E+09
Parameter scatter plot for beta Poisson model ellipses signify the 0.9, 0.95 and 0.99 confidence of the parameters.
Parameter scatter plot for beta Poisson model ellipses signify the 0.9, 0.95 and 0.99 confidence of the parameters.
beta Poisson model plot, with confidence bounds around optimized model
beta Poisson model plot, with confidence bounds around optimized model
# of Doses
8.00
Μodel
beta-Poisson
N50
6.85E+07
LD50/ID50
6.85E+07
Dose Units
CFU
Response
diarrhea
Exposure Route
oral (with NaHCO3)
Contains Preferred Model
a
2.21E-01
Agent Strain
EPEC B171-8 (serotype O11:NM)
Experiment ID
214, 216, 217
Host type
human
Experiment Dataset
Description
Pooled data of guinea pig and diabetic rat  
Dose DEATH NOT DEATH Total
44 1 4 5
440 3 2 5
3000 6 4 10
4400 4 1 5
3E+04 7 3 10
44000 5 0 5
3E+05 7 3 10
440000 5 0 5
3E+06 10 0 10
4.4E+06 4 1 5
3E+07 10 0 10

 

Goodness of fit and model selection
Model Deviance Δ Degrees 
of freedom		 χ20.95,1 
p-value 		χ20.95,m-k 
p-value
Exponential 163 152 10 3.84 
0 		18.3 
0
Beta Poisson 10.1 9 16.9 
0.343
Beta-Poisson fits better than exponential; cannot reject good fit for beta-Poisson.

 

Optimized parameters for the beta-Poisson model, from 10000 bootstrap iterations
Parameter MLE estimate Percentiles
0.5% 2.5% 5% 95% 97.5% 99.5%
α 2.13E-01 9.14E-02 1.26E-01 1.39E-01 3.38E-01 3.76E-01 5.60E-01
N50 4.77E+02 3.04E+00 1.63E+01 7.19E+01 2.16E+03 2.75E+03 4.44E+03
Parameter scatter plot for beta Poisson model ellipses signify the 0.9, 0.95 and 0.99 confidence of the parameters.
Parameter scatter plot for beta Poisson model ellipses signify the 0.9, 0.95 and 0.99 confidence of the parameters.

 

beta Poisson model plot, with confidence bounds around optimized model
beta Poisson model plot, with confidence bounds around optimized model
# of Doses
11.00
Μodel
beta-Poisson
N50
4.77E+02
LD50/ID50
4.77E+02
Dose Units
CFU
Response
death
Exposure Route
intraperitoneal
Contains Preferred Model
a
2.13E-01
Agent Strain
W294, 316c
Experiment ID
21,23
Host type
guinea pig and diabetic rat
Experiment Dataset
Description
Guinea Pigs W294 Strain Data 
Dose Dead Survived Total
44 1 4 5
440 3 2 5
4400 4 1 5
44000 5 0 5
440000 5 0 5
4.4E+06 4 1 5

 

Goodness of fit and model selection
Model Deviance Δ Degrees 
of freedom		 χ20.95,1 
p-value 		χ20.95,m-k 
p-value
Exponential 70.3 66.1 5 3.84 
4.44e-16 		11.1 
8.93e-14
Beta Poisson 4.14 4 9.49 
0.387
Beta-Poisson fits better than exponential; cannot reject good fit for beta-Poisson.

 

Optimized parameters for the beta-Poisson model, from 10000 bootstrap iterations
Parameter MLE estimate Percentiles
0.5% 2.5% 5% 95% 97.5% 99.5%
α 2.67E-01 3.36E-02 8.80E-02 1.16E-01 1.05E+01 3.45E+02 3.55E+03
N50 2.55E+02 4.80E-07 2.49E+00 1.48E+01 1.35E+03 1.80E+03 3.22E+03
Parameter scatter plot for beta Poisson model ellipses signify the 0.9, 0.95 and 0.99 confidence of the parameters.
Parameter scatter plot for beta Poisson model ellipses signify the 0.9, 0.95 and 0.99 confidence of the parameters.
beta Poisson model plot, with confidence bounds around optimized model
beta Poisson model plot, with confidence bounds around optimized model
# of Doses
6.00
Μodel
beta-Poisson
N50
2.55E+02
LD50/ID50
2.55E+02
Dose Units
CFU
Response
death
Exposure Route
intraperitoneal
Contains Preferred Model
a
2.67E-01
Agent Strain
W294
Experiment ID
21
Host type
guinea pig
Experiment Dataset
Description
Moredunn isolate data 
Dose Infected Non-infected Total
100 2 2 4
300 2 3 5
1000 1 2 3
3000 3 1 4

 

Goodness of fit and model selection
Model Deviance Δ Degrees 
of freedom		 χ20.95,1 
p-value 		χ20.95,m-k 
p-value
Exponential 7.37 6.16 3 3.84 
0.0131 		7.81 
0.0611
Beta Poisson 1.21 2 5.99 
0.546
Beta-Poisson fits better than exponential; cannot reject good fit for beta-Poisson.

 

Optimized parameters for the beta-Poisson model, from 10000 bootstrap iterations
Parameter MLE estimate Percentiles
0.5% 2.5% 5% 95% 97.5% 99.5%
α 1.14E-01 9.79E-04 9.81E-04 9.82E-04 1.17E+03 2.25E+03 5.52E+03
N50 4.55E+02 2.13E-09 2.19E-06 1.55E-05 5.62E+05 3.59E+09 1.43E+16

Parameter scatter plot for beta Poisson model ellipses signify the 0.9, 0.95 and 0.99 confidence of the parameters.

beta Poisson model plot, with confidence bounds around optimized model.

# of Doses
4.00
Μodel
beta-Poisson
N50
4.55E+02
LD50/ID50
4.55E+02
Dose Units
oocysts
Response
infection
Exposure Route
oral
Contains Preferred Model
a
1.14E-01
Agent Strain
Moredun isolate
Experiment ID
183
Host type
human
Description
TU502 data 
Dose Diarrhea Not diarrhea Total
10 2 3 5
30 3 2 5
100 5 2 7
500 3 1 4

 

Goodness of fit and model selection
Model Deviance Δ Degrees 
of freedom		 χ20.95,1 
p-value 		χ20.95,m-k 
p-value
Exponential 11.6 11.5 3 3.84 
0.000708 		7.81 
0.00894
Beta Poisson 0.119 2 5.99 
0.942
Beta-Poisson fits better than exponential; cannot reject good fit for beta-Poisson.

 

Optimized parameters for the beta-Poisson model, from 10000 bootstrap iterations
Parameter MLE estimate Percentiles
0.5% 2.5% 5% 95% 97.5% 99.5%
α 2.7E-01 9.83E-04 9.85E-04 2.03E-03 6.60E+00 6.70E+02 3.19E+03
N50 1.68E+01 3.41E-16 9.86E-09 1.68E-06 7.15E+01 9.76E+01 6.59E+02

Parameter scatter plot for beta Poisson model ellipses signify the 0.9, 0.95 and 0.99 confidence of the parameters.

beta Poisson model plot, with confidence bounds around optimized model.

# of Doses
4.00
Μodel
beta-Poisson
N50
1.68E+01
LD50/ID50
1.68E+01
Dose Units
oocysts
Response
diarrhea
Exposure Route
oral
Contains Preferred Model
a
2.7E-01
Agent Strain
*C. hominis*, TU502
Experiment ID
181
Host type
human
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