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

Description
human/type 14 strain SF 765model data 
Dose Infected Non-infected Total
0.5 1 8 9
1.5 4 6 10
5 4 6 10
15 6 4 10
150 27 13 40
300 10 2 12

 

Goodness of fit and model selection
Model Deviance Δ Degrees 
of freedom		 χ20.95,1 
p-value 		χ20.95,m-k 
p-value
Exponential 51.8 50.2 5 3.84 
1.42e-12 		11.1 
5.8e-10
Beta Poisson 1.68 4 9.49 
0.794
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 2.01E-01 7.76E-02 1.07E-01 1.22E-01 3.36E-01 3.69E-01 4.61E-01
N50 9.22E+00 1.41E+00 2.57E+00 3.40E+00 2.47E+01 3.14E+01 5.26E+01

 

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
6.00
Μodel
beta-Poisson
N50
9.22E+00
LD50/ID50
9.22E+00
Dose Units
TCID50
Response
infection
Exposure Route
intranasal
Contains Preferred Model
a
2.01E-01
Agent Strain
type 14
Experiment ID
64
Host type
human
Description
Dose response data 
Dose Infected Non-infected Total
48 0 6 6
454 2 4 6
4460 1 2 3
550000 2 1 3

 

Goodness of fit and model selection
Model Deviance Δ Degrees 
of freedom		 χ20.95,1 
p-value 		χ20.95,m-k 
p-value
Exponential 22.5 21.3 3 3.84 
3.97e-06 		7.81 
5.14e-05
Beta Poisson 1.22 2 5.99 
0.544
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 1.37E-01 1.34E-02 2.24E-02 4.22E-02 3.39E+00 2.21E+02 2.14E+03
N50 1.46E+04 5.09E+02 8.20E+02 1.06E+03 1.71E+08 8.25E+11 1.62E+18

 

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
1.46E+04
LD50/ID50
1.46E+04
Dose Units
Cysts
Response
infection
Exposure Route
oral
Contains Preferred Model
a
1.37E-01
Agent Strain
From infected humans
Experiment ID
47
Host type
beaver
Description
Humans model data 
Dose Infected Non Infected Total
1 1 7 8
10 3 7 10
100 2 2 4
1000 0 2 2
1E+04 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 27.8 22.8 4 3.84 
1.81e-06 		9.49 
 1.36e-05
Beta Poisson 5.04 3 7.81 
0.169
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 1.01E-01 1.20E-03 2.60E-02 3.61E-02 2.12E-01 2.57E-01 3.08E-01
N50 3.41E+02 5.02E+00 1.73E+01 2.61E+01 1.48E+05 2.08E+06 1.19E+10

 

 

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
5.00
Μodel
beta-Poisson
N50
3.41E+02
LD50/ID50
3.41E+02
Dose Units
Cysts
Response
infection
Exposure Route
oral
Contains Preferred Model
a
1.01E-01
Agent Strain
From an infected human
Experiment ID
45
Host type
human
Description
Escherichia coli (ETEC O111 (in paper as “E. coli 111, B4”)) in the human model data 
Dose Slight to severe illness No slight to severe illness Total
7E+06 7 4 11
5.31E+08 8 4 12
6.5E+09 11 0 11
9E+09 12 0 12

 

Goodness of fit and model selection
Model Deviance Δ Degrees 
of freedom		 χ20.95,1 
p-value 		χ20.95,m-k 
p-value
Exponential 39.8 33.4 3 3.84 
7.5e-09 		7.81 
1.19e-08
Beta Poisson 6.38 2 5.99 
0.0412
Neither the exponential nor beta-Poisson fits well; beta-Poisson is less bad.

 

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 2.63E-01 9.92E-04 7.95E-02 1.00E-01 4.71E-01 5.53E-01 1.48E+01
N50 3.56E+06 6.89E-01 1.15E+03 2.08E+03 1.85E+07 2.49E+07 4.18E+07
beta Poisson model plot, with confidence bounds around optimized model
beta Poisson model plot, with confidence bounds around optimized model

 

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.

 

 

# of Doses
4.00
Μodel
beta-Poisson
N50
3.56E+06
LD50/ID50
3.56E+06
Dose Units
CFU
Response
slight to severe illness
Exposure Route
oral
Contains Preferred Model
a
2.63E-01
Agent Strain
ETEC O111 (in paper as "E. coli 111, B4")
Experiment ID
43
Host type
human
Experiment Dataset
Description
Escherichia coli (ETEC O55 (in paper as “type 55, B5”)) in the human model data 
Dose Slight to severe illness No slight to severe illness Total
1.43E+08 6 2 8
1.73E+09 5 2 7
5.33E+09 6 2 8
1.6E+10 7 1 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 35 34.5 3 3.84 
4.2e-09 		7.81 
1.21e-07
Beta Poisson 0.486 2 5.99 
0.784
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 8.7E-02 9.87E-04 1.02E-03 1.67E-02 3.57E-01 4.67E-01 7.83E-01
N50 2.05E+05 3.40E-09 4.31E-06 2.31E-04 1.30E+08 2.09E+08 5.13E+08
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
2.05E+05
LD50/ID50
2.05E+05
Dose Units
CFU
Response
slight to severe illness
Exposure Route
oral
Contains Preferred Model
a
8.7E-02
Agent Strain
ETEC O55 (in paper as “type 55, B5”)
Experiment ID
42
Host type
human
Experiment Dataset
Description
ETEC disease, unbuffered, in the human model data
Dose Mild to severe diarrhea No mild to severe diarrhea Total
1E+06 0 4 4
1E+08 1 4 5
1E+08 2 3 5
1E+08 3 2 5
1.43E+08 6 2 8
1.73E+09 5 2 7
5.33E+09 6 2 8
1E+10 4 1 5
1E+10 3 2 5
1E+10 4 1 5
1.6E+10 7 1 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 68.4 62.9 10 3.84 
2.22e-15 		18.3 
8.97e-11
Beta Poisson 5.53 9 16.9 
0.786
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 2.06E-01 1.75E-02 1.17E-01 1.32E-01 3.79E-01 4.29E-01 5.57E-01
N50 1.28E+08 3.53E+05 3.17E+07 4.09E+07 3.83E+08 4.65E+08 6.86E+08

 

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
1.28E+08
LD50/ID50
1.28E+08
Dose Units
CFU
Response
mild to severe diarrhea
Exposure Route
oral (in milk)
Contains Preferred Model
a
2.06E-01
Agent Strain
ETEC B7A
Experiment ID
38, 42, 99, 165
Host type
human
Experiment Dataset
Description
E. coli disease (ETEC, EPEC, EIEC) in the human model data
Dose Mild to severe diarrhea No mild to severe diarrhea Total
1E+04 0 5 5
1E+04 0 5 5
1E+06 0 5 5
1E+06 1 8 9
1E+08 1 4 5
1E+08 5 3 8
1E+08 3 2 5
1E+08 2 3 5
1.43E+08 6 2 8
2.7E+08 9 7 16
1.73E+09 5 2 7
5.33E+09 6 2 8
1E+10 4 1 5
1E+10 3 2 5
1.6E+10 7 1 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 119 113 14 3.84 
0 		23.7 
0
Beta Poisson 6.56 13 22.4 
0.924
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 1.78E-01 9.39E-02 1.09E-01 1.19E-01 3.21E-01 3.63E-01 4.77E-01
N50 8.6E+07 1.75E+07 2.62E+07 3.25E+07 2.63E+08 3.23E+08 5.21E+08
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
15.00
Μodel
beta-Poisson
N50
8.6E+07
LD50/ID50
8.6E+07
Dose Units
CFU
Response
mild to severe diarrhea
Exposure Route
oral (in milk)
Contains Preferred Model
a
1.78E-01
Agent Strain
ETEC B7A
Experiment ID
38, 39, 40, 42, 99, 144
Host type
human
Experiment Dataset
Description

Please Note

The results from this experiment are not recommended for use, rather they are present for reference purposes. The confidence intervals are not complete for this experiment since the data is not allowing for a stable bootstrap outcome. It is recommended to use the recommended model as this data is likely just capable of passing the test of trend, but still likely suboptimal for dose response modeling purposes.

 

Human Inaba Strain 569B 
Dose Diarrhea or culture positive No diarrhea or culture positive Total
1E+04 0 2 2
1E+06 0 4 4
1E+07 0 4 4
1E+08 2 2 4
1E+09 1 1 2
1E+10 0 1 1
1E+11 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 15.1 10.4 6 3.84 
0.00126 		12.6 
0.0193
Beta Poisson 4.72 5 11.1 
0.451
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 1.98E-01 9.92E-04 9.92E-04 1.04E-03 3.20E-01 3.65E-01 4.07E-01
N50 6.36E+08 8.26E+05 8.26E+05 1.18E+08 5.33E+207 5.33E+207 8.87E+254

 

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
7.00
Μodel
beta-Poisson
N50
6.36E+08
LD50/ID50
6.36E+08
Dose Units
CFU
Response
diarrhea or culture positive
Exposure Route
oral (no NaHCO3)
Contains Preferred Model
a
1.98E-01
Agent Strain
Inaba 569B
Experiment ID
35
Host type
human
Description
TITLE 
Dose INFECTION NOT

INFECTION || Total
0.05 0 4 4
0.15 1 4 5
0.5 3 14 17
1.5 4 10 14
5 4 11 15
50 28 14 42
Goodness of fit and model selection
Model Deviance Δ Degrees 
of freedom		 χ20.95,1 
p-value 		χ20.95,m-k 
p-value
Exponential 29.2 26.2 5 3.84 
{{{pbPbetter}}} 		11.1 
2.16e-05
Beta Poisson 2.93 4 9.49 
0

selection
Beta-Poisson fits better than exponential; cannot reject good fit for beta-Poisson.
Optimized parameters for the beta-Poisson model, from 10000 bootstrapiterations
Parameter MLE estimate Percentiles
0.5% 2.5% 5% 95% 97.5% 99.5%
α 2E-01 7.86E-02 9.79E-02 1.11E-01 4.65E-01 5.58E-01 8.66E-01
N50 1.05E+01 3.00E+00 3.98E+00 4.65E+00 3.67E+01 5.32E+01 1.22E+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
6.00
Μodel
beta-Poisson
N50
1.05E+01
LD50/ID50
1.05E+01
Dose Units
TCID50
Response
infection
Exposure Route
intranasal
Contains Preferred Model
a
2,00E-01
Agent Strain
type 39
Experiment ID
313
Host type
human
Description
TITLE 
Dose INFECTION NOT

INFECTION || Total
0.05 0 4 4
0.15 1 4 5
0.5 0 4 4
0.5 3 14 17
1.5 2 3 5
1.5 4 10 14
5 1 4 5
5 4 11 15
15 4 3 7
50 28 14 42
150 13 8 21
300 5 2 7

"

Goodness of fit and model selection
Model Deviance Δ Degrees 
of freedom		 χ20.95,1 
p-value 		χ20.95,m-k 
p-value
Exponential 78.6 72.3 11 3.84 
{{{pbPbetter}}} 		19.7 
2.8e-12
Beta Poisson 6.22 10 18.3 
0.796
{{{interpretation}}}

"

Optimized parameters for the beta-Poisson model, from 10000 bootstrapiterations
Parameter MLE estimate Percentiles
0.5% 2.5% 5% 95% 97.5% 99.5%
α 1.82E-01 1.04E-03 1.03E-01 1.14E-01 3.04E-01 3.38E-01 4.09E-01
N50 1.38E+01 2.98E+00 5.64E+00 6.67E+00 3.77E+01 4.98E+01 1.32E+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
12.00
Μodel
beta-Poisson
N50
1.38E+01
LD50/ID50
1.38E+01
Dose Units
TCID50
Response
infection
Exposure Route
intranasal
Contains Preferred Model
a
1.82E-01
Agent Strain
type 14
Experiment ID
312, 313
Host type
human
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