Escherichia coli: Dose Response Models

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Pathogenic Escherichia coli

Author: Kyle S. Enger

Overview

Escherichia coli typically resides as a symbiotic bacterium in the mammalian large intestine, benefiting itself as well as the host. However, there are several well-established pathotypes of disease-causing E. coli[1][2]:

  • Enteropathogenic (EPEC)
    • Attaches to small intestinal wall and produces ‘attaching and effacing lesions’, in which microvilli are destroyed and the bacteria become perched on pedestals on the surface of the epithelial cell. This ability is encoded on the locus of enterocyte effacement (LEE) pathogenicity island.
    • Causes an inflammatory response and diarrhea, but seldom in persons older than 2y; it can also be isolated from healthy older persons.
    • Primarily found in developing countries.
  • Enterohemorrhagic (EHEC)
    • This is discussed in more detail in its own chapter.
  • Enterotoxigenic (ETEC)
    • Attaches to small intestinal wall.
    • Produces a heat-labile (LT) and/or heat-stable (ST) toxins, both of which cause secretion from the small intestinal wall, leading to mild to severe watery diarrhea. LT is an immunogenic multisubunit protein similar to cholera enterotoxin. ST is a nonimmunogenic polypeptide containing 18-19 amino acids.
    • Primarily found in developing countries and is a major cause of diarrhea in weaned infants, as well as traveler’s diarrhea.
    • Can be shed even by immune asymptomatic individuals.
  • Enteroaggregative (EAEC)
    • Loosely classified group, some of which may be nonpathogenic.
    • Produces a thick biofilm (‘stacked brick’ configuration) in the small or large intestines.
    • Thought to cause persistent diarrhea (lasting >14 days).
    • Can produce many different secretory toxins and cytotoxins, but not ST or LT.
  • Enteroinvasive (EIEC)
    • Actually invades the epithelial cells of the large intestine, where it multiplies.
    • Usually produces watery diarrhea similar to that of EPEC and ETEC, sometimes inflammatory colitis or dysentery.
    • Particularly closely related to Shigella sp. (which are now thought to be subgroups of E. coli); much pathogenesis from EIEC and Shigella sp. is mediated by the pWR100 virulence plasmid.
  • Diffusely adherent (DAEC)
    • Attaches to the small intestinal wall and induces formation of projections which wrap around the bacterium.

Enterotoxigenic Escherichia coli (ETEC) is the most common type of diarrheagenic E. coli.[3] It may also be the most common cause of childhood diarrhea in the developing world, responsible for approximately 1/7 of diarrheal episodes in children aged less than 1 year and almost 1/4 of diarrheal episodes in 1-4 year olds.[4] It can also cause severe dehydrating cholera-like disease in adults.[3] Diagnosis is complicated since many other Gram-negative bacteria produce similar toxins, so toxins as well as the E. coli bacterium must be tested for in order to yield accurate results.[4]

ETEC can often be detected in apparently healthy people. In developing countries among healthy 0-11 month olds, and 1-4 year olds, 11.7% and 7.1%, respectively, are estimated to be colonized with ETEC.[4]

Feeding studies of ETEC or EPEC in healthy volunteers first gave 2-3g of NaHCO3, which neutralizes stomach acid and reduces the infectious dose.[5] However, it has been suggested that food as a vehicle has a similar acid-neutralizing effect, so feeding studies given with NaHCO3 may better represent natural foodborne infection.[5] ETEC and EPEC generally have high ID50, and partly as a consequence of this, they do not appear to be transmitted person-to-person; a study of ETEC-infected volunteers co-housed with uninfected volunteers did not result in any transmission of infection.[6] Food was all served individually to the volunteers over the course of the experiment, so there was no opportunity for ETEC to spread via that route.[6]

http://www.cdc.gov/ecoli/index.html


Summary of data and models

There are many human feeding studies of various E. coli types and strains, which can be pooled in various ways to yield different dose response models. Many of these have small sample sizes and cannot be used on their own to reliably fit a dose response model. Most datasets for E. coli infections describe high levels of infection resulting from high doses. Lower doses remain to be investigated, and dose response models for infection are therefore uncertain. Another important factor is whether the dose was given with bicarbonate, which would neutralize some stomach acid and possibly increase infectivity.

Haas, Rose, and Gerba (1999)[7] fitted beta-Poisson models to several pooled datasets describing the disease response from ETEC, EPEC and EIEC. Among these datasets were the EPEC strains O111[8] and O55, as well as EIEC strains 4608 and 1624[9] with diarrhea as the end point. However, it mixed data from experiments in which bacteria were given with and without bicarbonate.

The best available dataset using infection as a response comes from an experiment (98) with 3 dose levels, feeding EIEC to adult humans.[9]

Powell et al. (2000)[10] pooled 3 human trial datasets[11][12] for EPEC to produce a beta-Poisson model and a Weibull-gamma model.

Additional pooling analyses for this chapter were conducted on the basis of pathotype (ETEC or EPEC), whether the dose was given with bicarbonate, and the nature of the response (disease or infection), incorporating more data from the literature than the previous two published models. Since some combinations of these factors lacked data, analyses could only be done for ETEC disease (buffered or unbuffered), EPEC disease (buffered), ETEC infection (unbuffered), and EPEC infection (buffered). The pooled datasets for infection contained mostly positive responses, and therefore their behavior at low doses is very uncertain. For the pooled analyses describing disease, datasets were excluded if they contributed significantly (P < 0.05) to the -2 log likelihood of the model given the data.[7] Two experiments[9] examining diarrhea from EIEC were also pooled.


Experiment serial number Reference Host type Agent strain Route # of doses Dose units Response Best fit model Optimized parameter(s) LD50/ID50
98* [9] human EIEC 1624 oral (in milk) 3 CFU positive stool isolation beta-Poisson α = 1.55E-01 , N50 = 2.11E+06 2.11E+06
39 [9] human EIEC 4608 oral (in milk) 3 CFU mild to severe diarrhea exponential k = 9.7E-09 7.14E+07
40 [9] human EIEC 1624 oral (in milk) 3 CFU mild to severe diarrhea exponential k = 1.22E-08 5.7E+07
42 [13] human ETEC O55 (in paper as “type 55, B5”) oral 4 CFU slight to severe illness beta-Poisson α = 8.7E-02 , N50 = 2.05E+05 2.05E+05
43 [8] human ETEC O111 (in paper as "E. coli 111, B4") oral 4 CFU slight to severe illness beta-Poisson α = 2.63E-01 , N50 = 3.56E+06 3.56E+06
165 [5] human ETEC 214-4 (ST) oral (in milk) 3 CFU diarrhea or vomiting beta-Poisson α = 2.5E-01 , N50 = 9.1E+07 9.1E+07
38, 39, 40, 42, 99, 144 [9][13][14][7] human ETEC B7A oral (in milk) 15 CFU mild to severe diarrhea beta-Poisson α = 1.78E-01 , N50 = 8.6E+07 8.6E+07
214, 216, 217 [12][11][10] human EPEC B171-8 (serotype O11:NM) oral (with NaHCO3) 8 CFU diarrhea beta-Poisson α = 2.21E-01 , N50 = 6.85E+07 6.85E+07
142, 143, 144, 145, 147, 151, 161, 162, 163, 164, 168, 169, 170, 172 [15][14][16][17][18][6] human ETEC B7A oral (with NaHCO3) 19 CFU diarrhea beta-Poisson α = 7.54E-02 , N50 = 1.7E+06 1.7E+06
38, 42, 99, 165 [9][13][5] human ETEC B7A oral (in milk) 11 CFU mild to severe diarrhea beta-Poisson α = 2.06E-01 , N50 = 1.28E+08 1.28E+08
153, 157, 159, 214, 216, 217 [19][12][11] human EPEC E2348/69 (O127:H6) oral (w. 2g NaHCO3) 11 CFU diarrhea beta-Poisson α = 1.62E-01 , N50 = 9.98E+07 9.98E+07
154, 156, 158, 160, 219, 220, 221 [19][20][11] human EPEC E2348/69 (O127:H6) oral (w. 2g NaHCO3) 13 CFU shedding in feces exponential k = 1.95E-06 3.56E+05
39, 40 [9] human EIEC 4608 oral (in milk) 6 CFU mild to severe diarrhea exponential k = 1.07E-08 6.5E+07
96, 100, 166 [21] human ETEC B7A oral (in milk) 7 CFU positive stool isolation beta-Poisson α = 3.75E-01 , N50 = 1.78E+05 1.78E+05
*This model is preferred in most circumstances. However, consider all available models to decide which one is most appropriate for your analysis.


Exponential and betapoisson model.jpg

Optimization Output for experiment 98

Escherichia coli (EIEC 1624) in the human model data [9]
Dose Positive stool isolation No positive stool isolation Total
1E+04 0 5 5
1E+06 5 4 9
1E+08 3 2 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 28.6 27.2 2 3.84
1.82e-07
5.99
6.18e-07
Beta Poisson 1.38 1 3.84
0.24
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 1.55E-01 1.26E-03 1.26E-03 2.84E-02 1.84E+01 1.29E+02 1.90E+02
N50 2.11E+06 1.73E+05 2.95E+05 2.95E+05 7.85E+08 9.22E+140 9.22E+140


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


Optimization Output for experiment 39

Escherichia coli (EIEC 4608) in the human model data [9]
Dose Mild to severe diarrhea No mild to severe diarrhea Total
1E+04 0 5 5
1E+06 0 5 5
1E+08 5 3 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 0.0986 -0.00188 2 3.84
1
5.99
0.952
Beta Poisson 0.1 1 3.84
0.751
Exponential is preferred to beta-Poisson; cannot reject good fit for exponential.


Optimized k parameter for the exponential model, from 10000 bootstrap iterations
Parameter MLE estimate Percentiles
0.5% 2.5% 5% 95% 97.5% 99.5%
k 9.7E-09 2.85E-09 2.86E-09 4.66E-09 2.04E-08 2.04E-08 5.07E-08
ID50/LD50/ETC* 7.14E+07 1.37E+07 3.40E+07 3.40E+07 1.49E+08 2.43E+08 2.44E+08
*Not a parameter of the exponential model; however, it facilitates comparison with other models.


Parameter histogram for exponential model (uncertainty of the parameter)
Exponential model plot, with confidence bounds around optimized model


Optimization Output for experiment 40

Escherichia coli (EIEC 1624) in the human model data [9]
Dose Mild to severe diarrhea No mild to severe diarrhea Total
1E+04 0 5 5
1E+06 1 8 9
1E+08 3 2 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 2.99 2.98 2 3.84
0.0845
5.99
0.224
Beta Poisson 0.0156 1 3.84
0.901
Exponential is preferred to beta-Poisson; cannot reject good fit for exponential.


Optimized k parameter for the exponential model, from 10000 bootstrap iterations
Parameter MLE estimate Percentiles
0.5% 2.5% 5% 95% 97.5% 99.5%
k 1.22E-08 1.97E-09 2.19E-09 4.40E-09 4.03E-08 1.17E-07 2.50E-07
ID50/LD50/ETC* 5.7E+07 2.77E+06 5.92E+06 1.72E+07 1.58E+08 3.17E+08 3.52E+08
*Not a parameter of the exponential model; however, it facilitates comparison with other models.


Parameter histogram for exponential model (uncertainty of the parameter)
Exponential model plot, with confidence bounds around optimized model


Optimization Output for experiment 42

Escherichia coli (ETEC O55 (in paper as “type 55, B5”)) in the human model data [13]
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.
beta Poisson model plot, with confidence bounds around optimized model


Optimization Output for experiment 43

Escherichia coli (ETEC O111 (in paper as “E. coli 111, B4”)) in the human model data [8]
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


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


Optimization Output for experiment 165

Escherichia coli (ETEC 214-4 (ST)) in the human model data [5]
Dose Diarrhea or vomiting No diarrhea or vomiting Total
1E+06 0 4 4
1E+08 3 2 5
1E+10 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 15.9 15.3 2 3.84
9e-05
5.99
0.000359
Beta Poisson 0.531 1 3.84
0.466
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 2.5E-01 9.94E-04 9.94E-04 9.94E-04 1.76E+02 7.32E+02 7.32E+02
N50 9.1E+07 3.05E+04 4.09E+04 5.00E+04 3.20E+08 6.91E+08 6.91E+08


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


Optimization Output for experiment 38, 39, 40, 42, 99, 144

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


Optimization Output for experiment 214, 216, 217

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


Optimization Output for experiment 142, 143, 144, 145, 147, 151, 161, 162, 163, 164, 168, 169, 170, 172

EPEC disease in the human model data
Dose Diarrhea No diarrhea Total
1E+06 3 3 6
1E+07 6 9 15
1E+08 6 1 7
1E+08 7 4 11
1E+08 7 5 12
1E+08 9 3 12
1E+08 6 3 9
1E+08 4 2 6
1E+08 3 1 4
1E+08 4 0 4
2.7E+08 9 7 16
5E+08 19 8 27
1E+09 5 3 8
1E+09 7 1 8
1E+09 5 5 10
1E+10 8 0 8
1E+10 5 4 9
1E+10 9 3 12
1E+10 9 5 14


Goodness of fit and model selection
Model Deviance Δ Degrees
of freedom
χ20.95,1
p-value
χ20.95,m-k
p-value
Exponential 412 393 18 3.84
0
28.9
0
Beta Poisson 19.1 17 27.6
0.322
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 7.54E-02 8.28E-03 1.11E-02 1.51E-02 1.46E-01 1.59E-01 1.84E-01
N50 1.7E+06 2.80E-11 1.91E-06 1.44E-03 2.34E+07 3.09E+07 4.67E+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


Optimization Output for experiment 38, 42, 99, 165

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


Optimization Output for experiment 153, 157, 159, 214, 216, 217

ETEC infection, unbuffered, 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 9 1 10
1E+10 9 5 14
1E+10 3 2 5
1E+10 5 0 5
2E+10 2 0 2
2.3E+10 14 5 19


Goodness of fit and model selection
Model Deviance Δ Degrees
of freedom
χ20.95,1
p-value
χ20.95,m-k
p-value
Exponential 57.8 43.4 10 3.84
4.51e-11
18.3
9.36e-09
Beta Poisson 14.4 9 16.9
0.108
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 1.62E-01 8.23E-02 9.98E-02 1.08E-01 3.69E-01 4.21E-01 5.36E-01
N50 9.98E+07 7.29E+06 1.55E+07 2.20E+07 8.50E+08 1.06E+09 1.57E+09


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


Optimization Output for experiment 154, 156, 158, 160, 219, 220, 221

EPEC, infection, buffered, in the human model data
Dose Shedding in feces No shedding in feces Total
1E+06 3 1 4
1E+06 5 0 5
1E+06 4 1 5
1E+08 5 0 5
1E+08 5 0 5
9E+08 8 0 8
1E+10 10 0 10
1E+10 9 0 9
1E+10 14 0 14
1E+10 5 0 5
1E+10 5 0 5
1E+10 5 0 5
2.3E+10 19 0 19


Goodness of fit and model selection
Model Deviance Δ Degrees
of freedom
χ20.95,1
p-value
χ20.95,m-k
p-value
Exponential 1.98 2.71e-06 12 3.84
0.999
21
0.999
Beta Poisson 1.98 11 19.7
0.999
Exponential is preferred to beta-Poisson; cannot reject good fit for exponential.


Optimized k parameter for the exponential model, from 10000 bootstrap iterations
Parameter MLE estimate Percentiles
0.5% 2.5% 5% 95% 97.5% 99.5%
k 1.95E-06 8.47E-07 1.03E-06 1.25E-06 2.64E-06 2.64E-06 2.64E-06
ID50/LD50/ETC* 3.56E+05 2.63E+05 2.63E+05 2.63E+05 5.53E+05 6.73E+05 8.18E+05
*Not a parameter of the exponential model; however, it facilitates comparison with other models.


Parameter histogram for exponential model (uncertainty of the parameter)
Exponential model plot, with confidence bounds around optimized model


Optimization Output for experiment 39, 40

EIEC disease, unbuffered, 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 5 3 8
1E+08 3 2 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 3.19 2.24 5 3.84
0.134
11.1
0.67
Beta Poisson 0.951 4 9.49
0.917
Exponential is preferred to beta-Poisson; cannot reject good fit for exponential.


Optimized k parameter for the exponential model, from 10000 bootstrap iterations
Parameter MLE estimate Percentiles
0.5% 2.5% 5% 95% 97.5% 99.5%
k 1.07E-08 3.63E-09 4.79E-09 5.77E-09 2.04E-08 2.28E-08 3.21E-08
ID50/LD50/ETC* 6.5E+07 2.16E+07 3.04E+07 3.40E+07 1.20E+08 1.45E+08 1.91E+08
*Not a parameter of the exponential model; however, it facilitates comparison with other models.


Parameter histogram for exponential model (uncertainty of the parameter)
Exponential model plot, with confidence bounds around optimized model


Optimization Output for experiment 96, 100, 166

Dose response data
Dose Positive stool isolation No positive stool isolation Total
1E+06 3 1 4
1E+08 4 1 5
1E+08 5 0 5
1E+08 5 0 5
1E+10 5 0 5
1E+10 5 0 5
1E+10 5 0 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 64.1 61.4 6 3.84
4.77e-15
12.6
6.67e-12
Beta Poisson 2.71 5 11.1
0.745
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 3.75E-01 1.29E-01 1.34E-01 1.34E-01 9.97E+00 9.97E+00 1.07E+01
N50 1.78E+05 3.63E-01 3.63E-01 3.63E-01 2.46E+06 3.48E+06 6.09E+06


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


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