Chapter X. Pathogenic Escherichia coli Kyle S. Enger, MPH

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1 Chapter X. Pathogenic Escherichia coli Kyle S. Enger, MPH X.1 Overview Escherichia coli usually exists as a commensal 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 (Kaper 2004, Nataro 1998): Enteropathogenic () o 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. o Causes an inflammatory response and diarrhea, but seldom in persons older than 2y; it can also be isolated from healthy older persons. o Primarily found in developing countries. Enterohemorrhagic (EHEC) o This is discussed in more detail in its own chapter. Enterotoxigenic () o Attaches to small intestinal wall. o 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 amino acids. o Primarily found in developing countries and is a major cause of diarrhea in weaned infants, as well as traveler s diarrhea. o Can be shed even by immune asymptomatic individuals. Enteroaggregative (EAEC) o Loosely classified group, some of which may be nonpathogenic. o Produces a thick biofilm ( stacked brick configuration) in the small or large intestines. o Thought to cause persistent diarrhea (lasting >14d). o Can produce many different secretory toxins and cytotoxins, but not ST or LT. Enteroinvasive (EIEC) o Actually invades the epithelial cells of the large intestine, where it multiplies. o Usually produces watery diarrhea similar to that of and, sometimes inflammatory colitis or dysentery. o 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) o Attaches to the small intestinal wall and induces formation of projections which wrap around the bacterium.

2 Enterotoxigenic Escherichia coli () is the most common type of diarrheagenic E. coli (Qadri 2005). 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 1y and almost ¼ of diarrheal episodes in 1-4 year olds (Wenneras 2004). It can also cause severe dehydrating cholera-like disease in adults (Qadri 2005). 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 (Wenneras 2004). 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 (Wenneras 2004). Feeding studies of or in healthy volunteers typically give 2-3g of, which neutralizes stomach acid and reduces the infectious dose (Levine et al. 1977). However, it has been suggested that food as a vehicle would have a similar acid-neutralizing effect, so feeding studies given may better represent natural foodborne infection.(levine et al. 1977) and generally have high ID 50, and partly as a consequence of this, they do not appear to be transmitted person-to-person; a study of -infected volunteers co-housed uninfected volunteers did not result in any transmission of infection.(levine 1980) Food was all served individually to the volunteers over the course of the experiment, so there was no opportunity for to spread via that route.(levine 1980) X.2 Summary of data and models There are many 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. In general, data exist spanning a wide range of doses and responses for disease. This is not the case for infection; most datasets 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 bicarbonate, which would neutralize some stomach acid and possibly increase infectivity. Haas, Rose, and Gerba (1999) fitted beta-poisson models to strains O111 (Ferguson et al. 1952) and O55 (June et al. 1953), as well as EIEC strains 4608 and 1624 (DuPont et al. 1971). Diarrhea was the response. However, it mixed data from experiments in which bacteria were given and out bicarbonate. The best available dataset using infection as a response comes from an experiment 3 dose levels, feeding EIEC to adult s (DuPont et al. 1971). Haas, Rose, and Gerba (1999) fitted a beta-poisson model to several pooled datasets describing the disease response from,, and EIEC. One strain ( O111) was found to differ from the rest, and was excluded. Powell et al. (2000) pooled 3 trial datasets (Levine et al. 1978, Beiber et al. 1998) for

3 to produce a beta-poisson model and a Weibull-gamma model. Additional pooling analyses for this chapter were conducted on the basis of pathotype ( or ), whether the dose was given 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 disease (buffered or unbuffered), disease (buffered), infection (unbuffered), and 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 (Haas, Rose, and Gerba 1999). Two experiments (DuPont et al. 1971) examining diarrhea from EIEC were also pooled. Table X.X: Summary of dose response data and models Experiment Reference Route Host Pathogen type type number Respons e of disease DuPont et al DuPont et al DuPont et al DuPont et al June et al Ferguson et al Coster et al Graham et al Levine et al B7A EIEC 4608 EIEC 1624 B2C O55 O111 B7A H10407 H10407 B7A milk milk milk milk Dose units cells Respon se diarrhea Best fit model None (2 doses) Optimized parameters ID 50 cells diarrhea Exponential k = 9.70E E+07 cells diarrhea Exponential k = 1.22E E+07 cells diarrhea cells diarrhea None (2 doses) Beta-Poisson (no significant trend) cells diarrhea Beta-Poisson CFU diarrhea CFU diarrhea CFU diarrhea None (2 doses) None (2 doses) cells diarrhea None (2 doses) α = 8.69E-02 N 50 = 1.94E+05 α = 2.61E-01 N 50 = 3.39E E E+06

4 Tacket et al Donnenbe rg et al Clements et al Levine et al Levine et al Levine et al B7A E2528- C1 2348/ / / TD225- C4 H10407 B7A H10407 B7A H10407 cells diarrhea cells diarrhea CFU diarrhea CFU diarrhea CFU diarrhea CFU diarrhea CFU diarrhea CFU diarrhea CFU diarrhea CFU diarrhea milk cells diarrhea or vomitin g cells diarrhea cells diarrhea None (2 doses) None (2 doses) Beta-Poisson None (2 doses) cells diarrhea α = 2.50E-01 N 50 = 9.10E E+07

5 Bieber et al Levine et al Respons e of infection DuPont et al DuPont et al DuPont et al DuPont et al Levine et al Levine et al Levine et al Levine et al Tacket et al B /69 E851/17 1 E74/68 B7A EIEC 4608 EIEC 1624 B2C B7A B7A B7A E2528- C1 2348/69 cells diarrhea CFU diarrhea Exponential k = 1.97E E+08 CFU diarrhea CFU diarrhea CFU diarrhea milk milk milk milk cells cells cells cells cells cells cells cells CFU positive isolation positive isolation positive isolation positive isolation positive isolation positive isolation positive isolation positive isolation sheddin g in None (2 doses) None (poor fit) None (all responses negative) None (2 doses) No dose response trend Beta-Poisson None (2 doses) None (2 doses) α = 1.55E-01 N 50 = 2.11E E+006

6 Donnenbe rg et al Donnenbe rg et al Donnenbe rg et al Levine et al Levine et al Levine et al Levine et al / / /69 E851/71 E74/68 milk CFU CFU CFU cells CFU CFU CFU sheddin g in sheddin g in sheddin g in positive isolation positive isolation positive isolation positive isolation None (2 doses) All but 1 subject positive None (2 doses) All subjects positive All but 1 subject positive Pooled models 38, 39, 40, 42, 99, , 216, , 143, 144, 145, 147, 151, 161, 162, 163, 164, 168, 169, 170, , 42, 99, , 100, 166 See above; fit by Haas, Rose, and Gerba 1999 See above; fit by Powell 2000,, EIEC See above milk or See above (no ) See above (no CFU diarrhea Beta-Poisson CFU diarrhea Beta-Poisson CFU diarrhea Beta-Poisson CFU diarrhea Beta-Poisson CFU infectio n Beta-Poisson α = 1.78E-01 N 50 = 8.60E+07 α = 2.21E-01 N 50 = 6.85E+07 α = 7.54E-02 N 50 = 1.70E+06 α = 2.06E-01 N 50 = 1.28E+08 α = 3.75E-01 N 50 = 1.78E E E E E E+005

7 153, 157, 159, 214, 216, , 156, 158, 160, 219, 220, , 40 See above See above DuPont et al EIEC ) CFU diarrhea Beta-Poisson Exponential infectio CFU (no low dose n data) milk α = 1.62E-01 N 50 = 9.98E+07 k = 1.95E E E+005 cells diarrhea Exponential k = 1.07E E+007

8 X.3. Optimized models and fitting analysis X.3.1. Data and optimization output for experiment 39. Table 2. Model data for Escherichia coli (EIEC 4608) in the Dose Positive Negative Total Model Table 3. Best fitting model determination Devianc e Δ Degrees of freedom χ ,1 χ ,m-k 1 (10 4 ) (10 6 ) (10 8 ) DuPont et al. (1971) Exponential Beta-Poisson Exponential fits better than beta-poisson; cannot reject good fit for exponential. Table 4. Optimized parameter(k) and ID 50 for best fitting model (exponential); percentiles from 10 4 bootstrap iterations Parameter MLE estimate Percentiles 0.5% 2.5% 5% 95% 97.5% 99.5% k 9.70E E E E E E E-08 ID E E E E E E E+08 Figure 1. Histogram of k parameter for exponential model. Figure 2. Exponential model plot, confidence bounds around the optimized model.

9 X.3. Optimized models and fitting analysis X.3.1. Data and optimization output for experiment 40. Table 2. Model data for Escherichia coli (EIEC 1624) in the Dose Positive Negative Total Model Table 3. Best fitting model determination Devianc e Δ Degrees of freedom χ ,1 χ ,m-k 1 (10 4 ) (10 6 ) (10 8 ) DuPont et al. (1971) Exponential Beta-Poisson Exponential fits better than beta-poisson; cannot reject good fit for exponential. Table 4. Optimized parameter(k) and ID 50 for best fitting model (exponential); percentiles from 10 4 bootstrap iterations Parameter MLE estimate Percentiles 0.5% 2.5% 5% 95% 97.5% 99.5% k 1.22E E E E E E E-07 ID E E E E E E E+08 Figure 1. Histogram of k parameter for exponential model. Figure 2. Exponential model plot, confidence bounds around the optimized model.

10 X.3. Optimized models and fitting analysis X.3.1. Data and optimization output for experiment 42. Table 2. Model data for Escherichia coli ( O55 (in paper as type 55, B5 )) in the Dose Positive Negative Total 1.4 (10 8 ) (10 9 ) (10 9 ) (10 10 ) June et al. (1953) Model Table 3. Best fitting model determination Devianc e Exponential 53.3 Δ 48.7 Degrees of freedom Beta-Poisson χ ,1 χ ,m-k Beta-Poisson fits better than exponential; cannot reject good fit for beta-poisson. Table 4. Optimized parameters and LD 50 for best fitting model (beta-poisson); percentiles from 10 4 bootstrap iterations Parameter MLE estimate Percentiles 0.5% 2.5% 5% 95% 97.5% 99.5% α 7.23E E E E E E E-01 N E E E E E E E+008 Figure 1. Parameter scatter plot for beta-poisson model. Ellipses signify the 0.90, 0.95 and 0.99 confidence intervals. Figure 2. Beta-Poisson model plot, confidence bounds around the optimized model.

11 X.3. Optimized models and fitting analysis X.3.1. Data and optimization output for experiment 43. Table 2. Model data for Escherichia coli ( O111 (in paper as E. coli 111, B4 )) in the Dose Positive Negative Total 7 (10 6 ) (10 8 ) (10 9 ) (10 9 ) Ferguson et al. (1952) Model Table 3. Best fitting model determination Devianc e Exponential 51.4 Δ 45.0 Degrees of freedom Beta-Poisson χ ,1 χ ,m-k Neither the exponential nor beta-poisson fits well; beta-poisson is less bad. Table 4. Optimized parameters and LD 50 for best fitting model (beta-poisson); percentiles from 10 4 bootstrap iterations Parameter MLE estimate Percentiles 0.5% 2.5% 5% 95% 97.5% 99.5% α 2.61E E E E E E E+00 N E E E E E E E+007 Figure 1. Parameter scatter plot for beta-poisson model. Ellipses signify the 0.90, 0.95 and 0.99 confidence intervals. Figure 2. Beta-Poisson model plot, confidence bounds around the optimized model.

12 X.3. Optimized models and fitting analysis X.3.1. Data and optimization output for experiment 165. Table 2. Model data for Escherichia coli ( (ST)) in the Dose Positive Negative Total Model Table 3. Best fitting model determination Devianc e Δ Degrees of freedom χ ,1 χ ,m-k 1 (10 6 ) (10 8 ) (10 10 ) Levine et al. (1977) Exponential Beta-Poisson Beta-Poisson fits better than exponential; cannot reject good fit for beta-poisson. Table 4. Optimized parameters and LD 50 for best fitting model (beta-poisson); percentiles from 10 4 bootstrap iterations Parameter MLE estimate Percentiles 0.5% 2.5% 5% 95% 97.5% 99.5% α 2.50E E E E E E E+00 N E E E E E E E+008 Figure 1. Parameter scatter plot for beta-poisson model. Ellipses signify the 0.90, 0.95 and 0.99 confidence intervals. Figure 2. Beta-Poisson model plot, confidence bounds around the optimized model.

13 X.3. Optimized models and fitting analysis X.3.1. Data and optimization output for experiment 214. Table 2. Model data for Escherichia coli ( B171-8 (serotype O11:NM)) in the Dose Positive Negative Total 5 (10 8 ) (10 9 ) (10 10 ) Bieber et al. (1998) Model Table 3. Best fitting model determination Devianc e Exponential 0.6 Δ NA Degrees of freedom Beta-Poisson NA 1 2 χ , χ ,m-k NA Exponential fits better than beta-poisson; cannot reject good fit for exponential. Table 4. Optimized parameter(k) and ID 50 for best fitting model (exponential); percentiles from 10 4 bootstrap iterations Parameter MLE estimate Percentiles 0.5% 2.5% 5% 95% 97.5% 99.5% k 1.97E E E E E E E-08 ID E E E E E E E+08 Figure 1. Histogram of k parameter for exponential model. Figure 2. Exponential model plot, confidence bounds around the optimized model.

14 X.3. Optimized models and fitting analysis X.3.1. Data and optimization output for experiment 98. Table 2. Model data for Escherichia coli (EIEC 1624) in the Dose Positive Negative Total Model Table 3. Best fitting model determination Devianc e Δ Degrees of freedom χ ,1 χ ,m-k 1 (10 4 ) (10 6 ) (10 8 ) DuPont et al. (1971) Exponential Beta-Poisson Beta-Poisson fits better than exponential; cannot reject good fit for beta-poisson. Table 4. Optimized parameters and ID 50 for best fitting model (beta-poisson); percentiles from 10 4 bootstrap iterations Parameter MLE estimate Percentiles 0.5% 2.5% 5% 95% 97.5% 99.5% α 1.55E E E E E E E+02 N E E E E E E E+140 Figure 1. Parameter scatter plot for beta-poisson model. Ellipses signify the 0.90, 0.95 and 0.99 confidence intervals. Figure 2. Beta-Poisson model plot, confidence bounds around the optimized model.

15 X.3. Optimized models and fitting analysis X.3.1. Data and optimization output, pooled experiments 38, 39, 40, 42, 99, 144. Table 2. Model data for E. coli disease (,, EIEC) in the Dose Positive Negative Total Model Table 3. Best fitting model determination Devianc e Δ Degrees of freedom χ ,1 χ ,m-k 1 (10 4 ) (10 4 ) (10 6 ) (10 6 ) (10 8 ) Exponential Beta-Poisson Beta-Poisson fits better than exponential; cannot reject good fit for beta-poisson. 1 (10 8 ) (10 8 ) (10 8 ) (10 8 ) (10 8 ) (10 9 ) (10 9 ) (10 10 ) (10 10 ) (10 10 ) DuPont et al. (1971), June et al. (1953), Graham et al. (1983) Table 4. Optimized parameters for best fitting model (beta-poisson); percentiles from 10 4 bootstrap iterations Parameter MLE estimate Percentiles 0.5% 2.5% 5% 95% 97.5% 99.5% α 1.78E E E E E E E-01 N E E E E E E E+008

16 Figure 1. Parameter scatter plot for beta-poisson model. Ellipses signify the 0.90, 0.95 and 0.99 confidence intervals. Figure 2. Beta-Poisson model plot, confidence bounds around the optimized model.

17 X.3. Optimized models and fitting analysis X.3.1. Data and optimization output for pooled experiments 214 and 216 and 217. Table 2. Model data for disease in the Dose Positive Negative Total Model Table 3. Best fitting model determination Devianc e Δ Degrees of freedom χ ,1 χ ,m-k 1 (10 6 ) (10 6 ) (10 8 ) (10 8 ) (10 9 ) Exponential Beta-Poisson Beta-Poisson fits better than exponential; cannot reject good fit for beta-poisson. 2 (10 10 ) (10 10 ) Levine et al. (1978), Bieber et al. (1998) Table 4. Optimized parameters for best fitting model (beta-poisson); percentiles from 10 4 bootstrap iterations Parameter MLE estimate Percentiles 0.5% 2.5% 5% 95% 97.5% 99.5% α 2.21E E E E E E E+03 N E E E E E E E+009

18 Figure 1. Parameter scatter plot for beta-poisson model. Ellipses signify the 0.90, 0.95 and 0.99 confidence intervals. Figure 2. Beta-Poisson model plot, confidence bounds around the optimized model.

19 X.3. Optimized models and fitting analysis X.3.1. Data and optimization output for pooled experiments 142, 143, 144, 145, 147, 151, 161, 162, 163, 164, 168, 169, 170, 172. Table 2. Model data for disease, buffered, in the Dose Positive Negative Total Model Table 3. Best fitting model determination Deviance Δ Degrees of freedom χ ,1 χ ,m-k 1 (10 6 ) (10 7 ) (10 8 ) (10 8 ) (10 8 ) Exponential Beta-Poisson Beta-Poisson fits better than exponential; cannot reject good fit for beta-poisson. 1 (10 8 ) (10 8 ) (10 8 ) (10 8 ) (10 8 ) (10 8 ) (10 8 ) (10 9 ) (10 9 ) (10 9 ) (10 10 ) (10 10 ) (10 10 ) (10 10 ) Coster et al. (2007), Graham et al. (1983), Levine et al. (1979), Levine et al. (1980), Clements et al. (1981), Levine et al. (1982) Table 4. Optimized parameters & ID 50 for best fitting model (beta-poisson); percentiles from 10 4 bootstrap iterations Parameter MLE estimate Percentiles 0.5% 2.5% 5% 95% 97.5% 99.5% α 7.54E E E E E E E-01 N E E E E E E E+007

20 Figure 1. Parameter scatter plot for beta-poisson model. Ellipses signify the 0.90, 0.95 and 0.99 confidence intervals. Figure 2. Beta-Poisson model plot, confidence bounds around the optimized model.

21 X.3. Optimized models and fitting analysis X.3.1. Data and optimization output for pooled experiments 38, 42, 99, 165. Table 2. Model data for disease, unbuffered, in the Dose Positive Negative Total Model Table 3. Best fitting model determination Devianc e Δ Degrees of freedom χ ,1 χ ,m-k 1 (10 6 ) (10 8 ) (10 8 ) (10 8 ) (10 8 ) Exponential Beta-Poisson Beta-Poisson fits better than exponential; cannot reject good fit for beta-poisson. 1.7 (10 9 ) (10 9 ) (10 10 ) (10 10 ) (10 10 ) (10 10 ) DuPont et al. (1971), June et al. (1953), Levine et al. (1977) Table 4. Optimized parameters and ID 50 for best fitting model (beta-poisson); percentiles from 10 4 bootstrap iterations Parameter MLE estimate Percentiles 0.5% 2.5% 5% 95% 97.5% 99.5% α 2.06E E E E E E E-01 N E E E E+07 2E E E+008

22 Figure 1. Parameter scatter plot for beta-poisson model. Ellipses signify the 0.90, 0.95 and 0.99 confidence intervals. Figure 2. Beta-Poisson model plot, confidence bounds around the optimized model.

23 X.3. Optimized models and fitting analysis X.3.1. Data and optimization output for pooled experiments 96, 100, 166. Table 2. Model data for infection, unbuffered, in the Dose Positive Negative Total Model Table 3. Best fitting model determination Devianc e Δ Degrees of freedom χ ,1 χ ,m-k 1 (10 6 ) (10 8 ) (10 8 ) (10 8 ) (10 10 ) Exponential Beta-Poisson Beta-Poisson fits better than exponential; cannot reject good fit for beta-poisson. 1 (10 10 ) (10 10 ) DuPont et al. (1971), Levine et al. (1977) Table 4. Optimized parameters and ID 50 for best fitting model (beta-poisson); percentiles from 10 4 bootstrap iterations Parameter MLE estimate Percentiles 0.5% 2.5% 5% 95% 97.5% 99.5% α 3.75E E E E E E E+01 N E E E E E E E+006

24 Figure 1. Parameter scatter plot for beta-poisson model. Ellipses signify the 0.90, 0.95 and 0.99 confidence intervals. Figure 2. Beta-Poisson model plot, confidence bounds around the optimized model.

25 X.3. Optimized models and fitting analysis X.3.1. Data and optimization output for pooled experiments 153, 157, 159, 214, 216, 217. Table 2. Model data for disease, buffered, in the Dose Positive Negative Total Model Table 3. Best fitting model determination Devianc e Δ Degrees of freedom χ ,1 χ ,m-k 1 (10 6 ) (10 6 ) (10 8 ) (10 8 ) (10 9 ) Exponential Beta-Poisson Beta-Poisson fits better than exponential; cannot reject good fit for beta-poisson. 1 (10 10 ) (10 10 ) (10 10 ) (10 10 ) (10 10 ) (10 10 ) Tacket et al. (2000), Donnenberg et al. (1998), Bieber et al. (1998), Levine et al. (1978) Table 4. Optimized parameters and ID 50 for best fitting model (beta-poisson); percentiles from 10 4 bootstrap iterations Parameter MLE estimate Percentiles 0.5% 2.5% 5% 95% 97.5% 99.5% α 1.62E E E E E E E-01 N E E E E E E E+009

26 Figure 1. Parameter scatter plot for beta-poisson model. Ellipses signify the 0.90, 0.95 and 0.99 confidence intervals. Figure 2. Beta-Poisson model plot, confidence bounds around the optimized model.

27 X.3. Optimized models and fitting analysis X.3.1. Data and optimization output for pooled experiments 154, 156, 158, 160, 219, 220, 221. Table 2. Model data for, infection, buffered, in the Dose Positive Negative Total Model Table 3. Best fitting model determination Devianc e Δ Degrees of freedom χ ,1 χ ,m-k 1 (10 6 ) (10 6 ) (10 6 ) (10 8 ) (10 8 ) Exponential Beta-Poisson Exponential fits better than beta-poisson; cannot reject good fit for exponential. 9 (10 8 ) (10 10 ) (10 10 ) (10 10 ) (10 10 ) (10 10 ) (10 10 ) (10 10 ) Tacket et al. (2000), Donnenberg et al. (1993), Levine et al. (1978) Table 4. Optimized parameter(k) and ID 50 for best fitting model (exponential); percentiles from 10 4 bootstrap iterations Parameter MLE estimate Percentiles 0.5% 2.5% 5% 95% 97.5% 99.5% k 1.95E E E E E E E-06 ID E E E E E E E+05

28 Figure 1. Histogram of k parameter for exponential model. Figure 2. Exponential model plot, confidence bounds around the optimized model.

29 X.3. Optimized models and fitting analysis X.3.1. Data and optimization output for pooled experiments 39 and 40. Table 2. Model data for EIEC disease, unbuffered, in the Dose Positive Negative Total Model Table 3. Best fitting model determination Devianc e Δ Degrees of freedom χ ,1 χ ,m-k 1 (10 4 ) (10 4 ) (10 6 ) (10 6 ) (10 8 ) Exponential Beta-Poisson Exponential fits better than beta-poisson; cannot reject good fit for exponential. 1 (10 8 ) DuPont et al. (1971) Table 4. Optimized parameter(k) and ID 50 for best fitting model (exponential); percentiles from 10 4 bootstrap iterations Parameter MLE estimate Percentiles 0.5% 2.5% 5% 95% 97.5% 99.5% k 1.07E E E E E E E-08 ID E E E E E E E+08 Figure 1. Histogram of k parameter for exponential model. Figure 2. Exponential model plot, confidence bounds around the optimized model.

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