Section 8.0 Human Exposure Factors

8-1

8.0 Human Exposure Factors

Exposure factors are used in the Human Exposure Module and the Human Risk Module

of the 3MRA modeling system to calculate the dose of a chemical (in mg/kg of body weight/d)

based on contact with contaminated media or food, the duration of that contact, and the body

weight of the exposed individuals. This section describes the derivation and use of values for the

human exposure factors used in the 3MRA modeling system.

The Human Exposure Module calculates exposures to human receptors from media and

food concentrations calculated by other modules. The Human Exposure Module calculates

exposures for two basic receptor types: residential receptors (residents and home gardeners) and

farmers. Residential receptors also may be recreational fishers, in addition to being a resident or

home gardener. Farmers may be beef farmers or dairy farmers, and either type of farmer also

may be a recreational fisher. The subcategories within residential receptors and farmers differ in

the particular exposures they incur. For example, a resident (only) differs from a home gardener

in that home gardeners are exposed to contaminated produce, but residents are not. Within each

of the two basic receptor types, the Human Exposure Module calculates exposures for five age

cohorts: infants (ages 0 to 1 year), children ages 1 to 5 years, children ages 6 to 11 years,

children ages 12 to 19 years, and adults (ages 20 years or older).

The media inputs needed for the Human Exposure Module include ambient air

concentration (both vapor and particulate), soil concentration, ground water concentration,

exposed vegetable concentration, protected vegetable concentration, exposed fruit concentration,

protected fruit concentration, root vegetable concentration, beef concentration, milk

concentration, and fish filet concentration for trophic level 3 and trophic level 4 fish. For

vegetables and fruits, the terms “exposed” and “protected” refer to whether the edible portion of

the plant is exposed to the atmosphere.

Exposure to humans other than infants may occur through eight pathways: 

# Inhalation of ambient air

# Inhalation of shower air

# Ingestion of ground water

# Ingestion of soil

# Ingestion of fruits and vegetables

# Ingestion of beef

# Ingestion of milk

# Ingestion of fish.

However, not all receptors are exposed through all of these pathways. Residents are exposed

through inhalation of ambient air, inhalation of shower air, ingestion of ground water, and
Section 8.0 Human Exposure Factors

8-2

(8-1)

ingestion of soil. Home gardeners have the same exposures as residents, plus exposure through

ingestion of fruits and vegetables. All farmers are exposed through inhalation of ambient air,

inhalation of shower air, ingestion of ground water, ingestion of soil, and ingestion of fruits and

vegetables. In addition, beef farmers are exposed through ingestion of beef, and dairy farmers

are exposed through ingestion of milk. Recreational fishers have the same exposures as one of

the other receptor types, plus fish ingestion. Not all age cohorts are exposed through all

pathways–shower exposures are calculated only for adults and children aged 12 to 19 years.

The generalized equation for calculating dose based on exposure through a single

pathway (e.g., ingestion of contaminated drinking water or ingestion of contaminated produce) is

the following:

where

Dosei = mg/kg of body weight/d of contaminant taken into the body through exposure

pathway i

Ci = concentration (in mg/kg, mg/L, or mg/m3) of a contaminant in the media or

food product associated with exposure pathway i

CRi = contact rate (in L/d, g/d, or m3/d) with the media or food product associated

with exposure pathway i (e.g., ingestion rate of water per day or inhalation

rate of air per day)

Fi = fraction of the total contact rate that is associated with the contaminated media

or food (e.g., fraction of air breathed [m3/d] or water ingested [L/d] that is

contaminated)

BW = body weight for the exposed cohort (e.g., adult, infant, and so on).

Carcinogens include the additional dimension of the duration of the exposure averaged over a

lifetime.

The human exposure factors used in the 3MRA modeling system are based on national

data for these factors provided in the Exposure Factors Handbook (EFH) (U.S. EPA, 1997a,

1997b, and 1997c). Where appropriate, distributions were developed for use in the human

exposure model. This section explains the parameters for which data were obtained (Section

8.1), the data sources (Section 8.2), and the methods (Section 8.3) used to develop distributions

or fixed values for all the exposure factors used in the analysis (Section 8.4). Uncertainties and

issues associated with these variables are described in Section 8.5.
Section 8.0 Human Exposure Factors

8-3

8.1 Parameters Collected

The exposure factors used in the Human Exposure Module are shown in Table 8-1, along

with the data source and whether they are represented by a distribution or as a fixed value. 

These inputs address inhalation and ingestion exposure from contact with media and various

food items as well as duration of exposure for the receptor types modeled: residents, home

gardeners, beef and dairy farmers, and recreational fishers.

Age is a relevant covariate (i.e., an important determinant) for most environmental

exposure factors, and stratification on age in risk assessment simulations is commonly used to

reduce variability. Although there are no universally recognized standardized definitions of age

groups, the representative national data set uses the following: child1 (0 to 1 year old), child2 (1

to 5 years old), child3 (6 to 11 years old), child4 (12 to 19 years old), and adult (older than 19

years of age). These age groups were selected because a majority of the data in the EFH were

provided in a similar manner, and the use of these age groups in this analysis (for all receptor

types) reduced the need to manipulate the data sets. In addition, other human risk analyses

performed by the U.S. Environmental Protection Agency’s (EPA’s) Office of Solid Waste

(OSW) have used these same age cohort definitions.

Site-specific and regional data sets are not available for many of the human exposure

inputs. Although food consumption rates and exposure duration data are grouped by four

regions (Northeast, Midwest, South, and West) in the EFH, all age groups were combined when

these data were presented by region. The unavailability of similar regional distributions for

other human exposure inputs in addition to the loss of age-specific data indicated that deriving

regional distributions was not feasible. Therefore, all human exposure model inputs were

collected and processed on a national basis.

The human exposure parameters either are characterized by distributions (stochastic

variables) or are fixed values (constants). Table 8-1 shows this breakdown; national

distributions were developed for all factors with data that could be used to derive distributions. 

A few parameters were fixed based on central tendency values from the best available source,

either because limited variability was expected or because available data were not adequate to

generate national distributions.

8.2 Data Sources

The following documents are the primary data sources for the exposure factors used for

the representative national data set: 

# U.S. Environmental Protection Agency (EPA). 1997a. Exposure Factors Handbook,

Volume I, General Factors. Office of Research and Development, Washington, DC.

August.

# U.S. Environmental Protection Agency (EPA). 1997b. Exposure Factors Handbook,

Volume II, Food Ingestion Factors. Office of Research and Development,

Washington, DC. August.
Section 8.0 Human Exposure Factors

8-4

Table 8-1. Input Parameters and Data Sources: 3MRA Human Exposure Factors

Abbreviationa Parameter Data Source

Distributed (stochastic) variables

bodywt Body weight (adult, child1-4) U.S. EPA (1997a)

inhal Inhalation rate (adult, child1-4) U.S. EPA (1997a)

soilIng Ingestion rate: soil (adult, child2-4) U.S. EPA (1997a)

drinkH2O Ingestion rate: drinking water (adult, child1-4) U.S. EPA (1997a)

bmilk Breast milk consumption (child1) U.S. EPA (1997b)

expveg

Consumption rate for gardener: exposed vegetables (adult, child2-4) U.S. EPA (1997b)

Consumption rate for farmer: exposed vegetables (adult, child2-4) U.S. EPA (1997b)

rootveg

Consumption rate for gardener: root vegetables (adult, child2-4) U.S. EPA (1997b)

Consumption rate for farmer: root vegetables (adult, child2-4) U.S. EPA (1997b)

proveg

Consumption rate for gardener: protected vegetables (adult, child2-4) U.S. EPA (1997b)

Consumption rate for farmer: protected vegetables (adult, child2-4) U.S. EPA (1997b)

expfruit

Consumption rate for gardener: exposed fruit (adult, child2-4) U.S. EPA (1997b)

Consumption rate for farmer: exposed fruit (adult, child2-4) U.S. EPA (1997b)

profruit

Consumption rate for gardener: protected fruit (adult, child2-4) U.S. EPA (1997b)

Consumption rate for farmer: protected fruit (adult, child2-4) U.S. EPA (1997b)

fish Consumption rate for recreational fisher: fish (adult, child2-4) U.S. EPA (1997b)

beef Consumption rate for farmer: beef (adult, child2-4) U.S. EPA (1997b)

milk Consumption rate for farmer: milk (adult, child2-4) U.S. EPA (1997b)

showerT Shower contact time U.S. EPA (1997c)

cumTroom Total time in shower and bathroom U.S. EPA (1997c)

Fixed variables (constants)

- Exposure frequency (adult, child1-4) EPA policy

- Exposure duration (adult, child1-4) U.S. EPA (1997c)

- Fraction contaminated: soil EPA policy

- Fraction contaminated: drinking water EPA policy

- Fraction contaminated for recreational fisher (fish) U.S. EPA (1997b)

-

Fraction homegrown for gardener (exposed vegetables, root vegetables, protected

vegetables, exposed fruit, protected fruit) U.S. EPA (1997b)

-

Fraction homegrown for farmer (exposed vegetables, root vegetables, protected

vegetables, exposed fruit, protected fruit) U.S. EPA (1997b)

- Fraction contaminated (home-raised) for farmer (beef, dairy) U.S. EPA (1997b)

- Event frequency–showering U.S. EPA (1997c)

- Fraction of fat in maternal breast milk U.S. EPA (1997b)

- Fraction of T3 fish consumed U.S. EPA (1997b)

- Fraction of T4 fish consumed U.S. EPA (1997b)

- Human lifetime (used in carcinogenic risk calculation) U.S. EPA (1997a)

a

Abbreviations are used only for stochastic parameters.
Section 8.0 Human Exposure Factors

8-5

# U.S. Environmental Protection Agency (EPA). 1997c. Exposure Factors Handbook,

Volume III, Activity Factors. Office of Research and Development, Washington, DC.

August.

The EFH is the most important data source for most human exposure model inputs. It

summarizes data on human behaviors and characteristics related to human exposure from

relevant key studies and provides recommendations and associated confidence estimates on the

values of exposure factors. By providing a consistent set of exposure factors, the EFH serves as

a support document to EPA’s Guidelines for Exposure Assessment (U.S. EPA, 1992), developed

to promote consistency among various exposure assessment activities across different program

offices. For many factors, percentile data that can be used to develop distributions are provided

in the handbook.

EPA carefully reviews and evaluates data quality before inclusion in the EFH. 

Evaluation criteria include peer review, reproducibility, pertinence to the United States,

currency, adequacy of the data collection period, validity of the approach, representativeness of

the population, characterization of the variability, lack of bias in study design, and measurement

error (U.S. EPA, 1997a, 1997b, 1997c).

8.3 Methodology

The general methodology for collecting human exposure data relied upon data provided

by the EFH, which were used in one of three ways:

1. When EFH percentile data were adequate (most input variables), maximum

likelihood estimation was used to fit selected parametric models (gamma, lognormal,

Weibull, and generalized gamma) to the EFH data. The chi-square measure of

goodness of fit was then used to choose the best distribution to assume for HWIR.

Parameter uncertainty information (e.g., for averages, standard deviations) also was

derived using the asymptotic normality of the maximum likelihood estimate or a

regression approach.

2. For a few variable conditions when percentile data were not adequate for model

fitting, models were selected on the basis of results for other age cohorts or, if no

comparable information was available, by assuming lognormal as a default

distribution and reasonable coefficients of variation (CVs).

3. Other variables for which data were not adequate for either 1 or 2 above were fixed at

EFH-recommended mean values or according to established Agency policy.

The approach for developing national distributions (1 and 2) is described in Section 8.3.1,

followed by discussions of the development of fixed parameters (Section 8.3.2) and quality

assurance (QA) and quality control (QC) procedures (Section 8.3.3). Results by parameter are

provided in Section 8.4. 
Section 8.0 Human Exposure Factors

1

 In preliminary goodness-of-fit tests, the normal distribution fit best in only 3 cases out of 69.

2

 Gamma, Weibull, and lognormal are all special cases of the generalized gamma.

8-6

8.3.1 National Distributions

For most variables for which national distributions were developed, exposure factor data

from the EFH were analyzed to fit selected parametric models. Steps in this process include

preparing data, fitting models, assessing fit, and preparing parameters to characterize

distributional uncertainty in the human exposure module inputs.

Because the EFH data are always positive and almost always skewed to the right (i.e.,

have a long right tail),1 three two-parameter probability models commonly used to characterize

such data (gamma, lognormal, and Weibull) were selected. In addition, a three-parameter model

(generalized gamma) was used that unifies them2

 and allows for a likelihood ratio test of the fit

of the two-parameter models. However, only the two-parameter models were selected for use in

the analysis because the three-parameter generalized gamma model did not significantly improve

the goodness of fit over the two-parameter models in 58 of 59 cases at the 5 percent level of

significance.

This simple setup constitutes a considerable improvement over the common practice of

using a lognormal model in which adequate EFH data were available to support maximum

likelihood estimation. However, in a few cases (soil ingestion, breast milk consumption, and

inhalation rate), data were not adequate to fit a distribution, and the lognormal model was

assumed as the default.

8.3.1.1 Preparation of Percentile Data. For many exposure factors, EFH data include

sample sizes and estimates of the following parameters for specific receptor types and age

groups: mean, standard deviation, standard error, and percentiles corresponding to a subset of the

following probabilities–0.01, 0.02, 0.05, 0.10, 0.15, 0.25, 0.50, 0.75, 0.85, 0.90, 0.95, 0.98, and

0.99. These percentile data were used as a basis of fitting distributions where available.

Although in no case are all of these percentiles actually provided for a single factor, seven or

more are typically present in the EFH data. Therefore, using the percentiles is a fuller use of the

available information than simply fitting based on the method of moments (e.g., selecting models

that agree with the data mean and standard deviation).

For some factors, certain percentiles were not used in the fitting process because sample

sizes were too small to justify their use. Percentiles were used only if at least one data point was

in the tail of the distribution. For example, the 1st and 99th percentiles were used only when

sample sizes exceeded 100; when sample size data were unavailable, the 5th through 95th

percentiles were used.

If the EFH data repeated a value across several adjacent percentiles, only one value (the

most central) was used in most cases. For example, for the time in the shower for those aged

12 to 19, the 1st, 2nd, and 5th percentiles were all equal to 5 min. This means that 1 percent of the

data is below 5, 2 percent of the data are below 5, and 5 percent of the data are below 5, in

essence indicating that 5 percent of the data are below 5. In this case, only the 5th percentile (the
Section 8.0 Human Exposure Factors

8-7

most central or closest to the 50th) was used in the analysis. Similarly, because both the 98th and

99th percentiles are listed as 60 min, only the 98th value was used. In this case, the data summary

indicates that 2 percent of the data are above 60 min and 1 percent of the data is above 60 min. 

The first statement implies the second, so the 99th percentile was dropped from the analysis.

The EFH does not use standardized age cohorts across exposure factors. Therefore, to

obtain the percentiles for fitting the five standardized age cohorts (i.e., less than 1, 1 to 5, 6 to 11,

12 to 19, and more than 20), each EFH cohort-specific value for a given exposure factor was

assigned to one of these five cohorts. When multiple EFH cohorts fit into a single cohort, the

EFH percentiles were averaged within each cohort. If sample sizes were available, weighted

averages were used, with weights proportional to sample sizes. If sample sizes were not

available, equal weights were assumed (i.e., the percentiles were simply averaged). 

Appendix 8A contains the final EFH data used in the analysis, developed by applying the

methods described previously to the EFH data. All raw data used to develop each exposure

factor distribution also are summarized by parameter in Section 8.4.

8.3.1.2 Statistical Methods and Data Analysis. The following statistical methods were

used to fit distributions to the EFH percentile data, to determine the goodness of fit of these

distributions, and to develop the statistics needed to describe the distributions in the 3MRA

modeling system:

# Model Fitting–Lognormal, gamma, Weibull, and generalized gamma distributions

were fit to each factor data set using maximum likelihood estimation (Burmaster and

Thompson, 1998).

# Assessment of Fit–When sample sizes were available, the goodness of fit was

calculated for each of the four models using the chi-square test (Bickel and Doksum,

1977). For the two-parameter models, the goodness of fit was calculated against the

generalized gamma model using the maximized log likelihood (likelihood ratio) test

(Bickel and Doksum, 1977). When percentile data were available but sample sizes

were unknown, a regression F-test for the goodness of fit against the generalized

gamma model was used.

# Distributions for Parameter Uncertainty–For each of the two-parameter models,

parameter uncertainty information was provided as parameter estimates for a

bivariate normal distribution that could be used for simulating parameter values

(Burmaster and Thompson, 1998). The information necessary for such simulations

includes estimates of the two model parameters, their standard errors, and their

correlation. To obtain this parameter uncertainty information, one can use the

asymptotic normality of the maximum likelihood estimate (Burmaster and Thompson,

1998) when sample sizes are available, and a regression approach when sample sizes

are not available (Jennrich and Moore, 1975; Jennrich and Ralston, 1979). In either

case, uncertainty can be expressed as a bivariate normal distribution for the model

parameters. 
Section 8.0 Human Exposure Factors

8-8

The chi-square statistic and the maximized log likelihood both were used to compare the

3 two-parameter models (gamma, lognormal, and Weibull) for each parameter. The two

statistics agreed on which two-parameter model fit best in 61 of 69 cases. For testing goodness

of fit, the chi-square test of absolute fit against the data is more stringent than the likelihood ratio

test of relative fit against the generalized gamma model and was used for this analysis. 

Each test required a known sample size, which was available in 59 of the 69 cases. If a

p-value above 0.05 represents adequate fit, then adequate fit was obtained by 1 of the

two-parameter models in 46 of 59 cases using the likelihood ratio test versus 30 of 59 cases for

the chi-square test. Using the chi-square test at the 5 percent significance level, in only 1 of

59 cases did the generalized gamma model offer improved fit over the best two-parameter

model. However, if the likelihood ratio test was used, then the generalized gamma model

improved significantly on the best-fitting two-parameter model in 13 of 59 cases.

8.3.1.3 Analysis Results–Goodness of Fit and Parameterization of Distributions.

Table 8-2 and Appendices 8B and 8C summarize the results of this statistical analysis. The

tabulated results are applicable using any software tool capable of generating random variables

from lognormal, gamma, and Weibull distributions, such as Crystal Ball or @RISK.

Table 8-2 contains the results from the analysis based on the chi-square goodness-of-fit

test. Results from the likelihood ratio test are not shown but are available on request. In

Table 8-2, M is the number of percentiles used in the fitting process. The ranking of the

3 two-parameter models based on the chi-square statistic is indicated by columns labeled First,

Second, and Third. Values of the chi-square statistic also are shown for all three models. PGOF

denotes the p-value for the chi-square test of goodness of fit. Small values of PGOF, such as

PGOF <0.01, correspond to large chi-square values and tend to cast doubt on the fit of the

model.

Appendix 8B identifies the models selected for use in developing the human exposure

factors for the 3MRA modeling system representative national data set and provides the

estimated means and standard deviations for each of the two-parameter models fit to the

exposure factors data. These quantities characterize the models for use in risk assessment. For

instance, because the gamma model was the best-fitting model for the showerT variable in Table

8-2, we used the gamma distribution with a mean of 16.7 min and a standard deviation of 9.91

min for the 3MRA model inputs.

Appendix 8C has been provided for the convenience of risk assessors who want to use

the distributions with Crystal Ball Monte Carlo software or software requiring similar statistics. 

It contains parameter estimates for each model based on the parameterizations used in

Appendix B of the Crystal Ball manual. For instance, columns labeled LOG MEAN and LOG

SDEV contain the values of the lognormal parameters μ and F (page 255 of the Crystal Ball

manual). Similarly, the last four columns of Appendix 8C contain the values of the parameters "

(shape) and $ (scale) for each of the gamma and Weibull distributions. Note that in HWIR,

shape and scale are passed to the system for Weibull distributions instead of average and

standard deviation.
Section 8.0 Human Exposure Factors

8-9

Table 8-2. Chi-Square Statistics and p-Values for Goodness-of-Fit Tests for

Models Fit by Maximum Likelihood

Parameter Age Cohort N M First Second Third

CHISQ

GAM

CHISQ

LOG

CHISQ

WEI

PGOF

GAM

PGOF

LOG

PGOF

WEI

beef 6-11 38 7 lognormal gamma Weibull 12.77 4.86 15.01 0.047 0.562 0.02

beef 12-19 41 7 gamma Weibull lognormal 4.06 5.07 4.81 0.668 0.534 0.569

beef farmer 182 9 lognormal gamma Weibull 35.28 5.57 64.54 0 0.695 0

bodywt <1 356 9 gamma lognormal Weibull 20.21 22.23 37.05 0.01 0.005 0

bodywt 1-5 3,762 9 lognormal gamma Weibull 87.53 52.03 706.3000

bodywt 6-11 1,725 9 lognormal gamma Weibull 156.9 105.1 585.1000

bodywt 12-19 2,615 9 lognormal gamma Weibull 113.8 70.32 625.2000

bodywt 20+ 12,504 9 lognormal gamma Weibull 420.7 215 2584000

cumTroom all ages 6,661 8 lognormal gamma Weibull 626 426.6 1067000

drinkH2O <1 403 5 Weibull gamma lognormal 14.62 44.72 12.59 0.006 0 0.013

drinkH2O 1-5 3,200 9 gamma Weibull lognormal 60.84 290.5 113.3000

drinkH2O 6-11 2,405 9 gamma Weibull lognormal 38.77 243 55.76000

drinkH2O 12-19 5,801 9 gamma Weibull lognormal 93.37 625.2 131.8000

drinkH2O 20+ 13,394 9 gamma Weibull lognormal 172.2 624.3 467000

expfruit 1-5 49 6 gamma Weibull lognormal 6.38 8.38 6.79 0.271 0.137 0.236

expfruit 6-11 68 7 lognormal Weibull gamma 15.83 6.84 15.25 0.015 0.336 0.018

expfruit 12-19 50 7 lognormal Weibull gamma 16.71 14.11 16.52 0.01 0.028 0.011

expfruit farmer 112 9 lognormal gamma Weibull 13.43 7.7 17.11 0.098 0.463 0.029

expfruit home gard. 596 9 lognormal gamma Weibull 70.31 14.52 93.27 0 0.069 0

expveg 1-5 105 7 gamma Weibull lognormal 11.33 16.23 11.44 0.079 0.013 0.076

expveg 6-11 134 8 lognormal Weibull gamma 11.9 8.26 9.2 0.104 0.31 0.238

expveg 12-19 143 8 gamma Weibull lognormal 16.09 35.88 16.11 0.024 0 0.024

expveg farmer 207 8 lognormal gamma Weibull 19.03 18.92 20.18 0.008 0.008 0.005

expveg home gard. 1,361 9 Weibull gamma lognormal 81.63 98.94 77.22000

fish all ages 1,053 5 lognormal Weibull gamma 41.9 6.74 20.55 0 0.15 0

milk <1 20 6 Weibull lognormal a 22.51 8.63 a 0 0.125

milk 1-5 40 7 Weibull gamma lognormal 3.25 9.1 1.72 0.777 0.168 0.943

milk 6-11 20 7 Weibull gamma lognormal 0.71 3.08 0.25 0.994 0.798 1

milk 12-19 20 7 Weibull gamma lognormal 1.53 6.66 1.17 0.958 0.354 0.978

milk farmer 63 7 Weibull gamma lognormal 14.29 26.01 13.43 0.027 0 0.037

profruit 12-19 20 7 lognormal Weibull gamma 4.58 1.74 4.21 0.599 0.942 0.649

profruit 20+ 106 7 lognormal Weibull gamma 19.94 8.93 17.95 0.003 0.177 0.006

profruit all ages 173 9 lognormal Weibull gamma 68.45 12.82 65.92 0 0.118 0

profruit home gard. 146 9 lognormal Weibull gamma 38.27 14.05 36.12 0 0.08 0

proveg 1-5 53 7 lognormal gamma Weibull 15.51 11.09 19.95 0.017 0.086 0.003

proveg 6-11 63 7 lognormal gamma Weibull 8.63 6.35 11.91 0.196 0.385 0.064

proveg 12-19 51 7 lognormal gamma Weibull 13.29 6.71 17.2 0.039 0.349 0.009

proveg farmer 142 9 lognormal gamma Weibull 83.34 13.04 108.5 0 0.11 0

proveg home gard. 602 9 lognormal gamma Weibull 181.4 17.93 318.4 0 0.022 0

rootveg 1-5 45 7 lognormal Weibull gamma 8.79 4.22 7.72 0.186 0.647 0.259

rootveg 6-11 67 7 Weibull gamma lognormal 4.68 10.45 3.83 0.586 0.107 0.699

rootveg 12-19 76 7 Weibull lognormal a 23.43 6.82 a 0.001 0.337

rootveg farmer 136 9 lognormal gamma Weibull 43.62 11.16 59.84 0 0.193 0

rootveg home gard. 682 9 Weibull gamma lognormal 27.9 102 26.91 0 0 0.001

showerT all ages 3,547 10 gamma lognormal Weibull 485.2 524.4 935.2000

a Information on gamma distribution is not available because the nonlinear optimization routine (the SAS NLIN procedure) used for model

fitting would not converge on a solution.

CHISQ = chi-square; GAM = gamma; LOG = lognormal; PGOF = p-values for the goodness of fit; WEI = Weibull.
Section 8.0 Human Exposure Factors

3

 The implied Weibull cumulative distribution function (CDF) is, therefore, 1-exp (-" * x$).

8-10

Parameter uncertainty information is provided in Appendixes 8D, 8E, and 8F. This

information includes maximum likelihood estimates for location and scale parameters for each of

the 3 two-parameter models (Appendix 8D), as well as estimates of the standard errors for each

of these parameter estimates (Appendixes 8E and 8F) and estimates of the correlation between

the location and scale parameters (Appendix 8F). Model parameterizations based on location

and scale parameters are provided in Appendix 8G. 

To generate the required bivariate normal random variables, the estimated variances and

the estimated correlations between the location and scale parameters must be known. The

estimated correlations are in Appendix 8F. The estimated variances are the squares of the

standard errors in Appendixes 8E and 8F, which contain two sets of estimated standard errors.

The first set, generated using the assumed asymptotic normality of the maximum likelihood

estimate as in Burmaster and Thompson (1998), can be used when sample sizes are available.

The second employs a nonlinear regression method (Jennrich and Ralston, 1979) that uses the

number of percentiles but not the sample size in calculating statistics; it can be used when

sample sizes are not available. 

The regression method infers the precision of the estimates solely from the fit to the data

(i.e., from the regression mean squared error). It has two advantages: it produces statistics when

the sample size is unknown, and it absorbs model lack of fit in precision estimates. Therefore,

the regression method takes both model uncertainty and parameter uncertainty into account. 

Accordingly, the standard errors from the regression method tend to be much larger than those

based on the asymptotic normality of the maximum likelihood estimate. Although the regression

method has been cited as more valid and recommended for use in all cases (Huber, 1967; White,

1982), some users may opt for standard errors based on the asymptotic normality of the

maximum likelihood estimate when these are available, reserving the regression statistics only

for situations with unknown sample sizes.

8.3.1.4 Selection of Distributions and Measures of Uncertainty. For each variable

listed in Table 8-2, the distribution with the first chi-square rank was selected for use in the

human exposure module. For all selected distribution types except Weibull, the arithmetic

means and standard deviations in Appendix 8B were passed to the 3MRA modeling system; for

Weibull, shape (") and scale ($) parameters from Appendix 8C were used.3

 These statistics also

are provided by exposure factor variable in Section 8.4.

Parameters that had to be analyzed separately because of lack of adequate percentile data

include beef consumption for those aged 1 to 5 years; protected fruit consumption for those aged

1 to 5 years, those aged 6 to 11 years, and farmers; breast milk consumption; soil ingestion; and

inhalation rate. For these parameters, either distributions for similar parameters were assumed to

apply or a default lognormal model was selected. Methods and results for these parameters also

are provided in Section 8.4.

# Percentile data were not available for beef consumption for 1- to 5-year-olds. In this

case, the lognormal model was used because, among the other age groups, it was the

best-fitted model in all but one case.
Section 8.0 Human Exposure Factors

8-11

# Percentile data were not provided for consumption of protected fruit (child1,

child2, and farmer). Therefore, the lognormal model was considered the most

appropriate because lognormal fits the best in other age groups for protected fruit and

protected vegetables.

# The lognormal model was assumed for breast milk consumption (those aged 1 to

12 months) because no other applicable data were available. In the EFH data, the

population mean for breast milk consumption was 688 mL/d and the population

standard deviation (for CV=1.5) was 1,032 mL/d, both of which were used for this

analysis.

# Similarly, the lognormal model was used for soil consumption for all age groups

because of limited percentile data. Parameter estimates were obtained by assuming

CV=0.5, 1, and 1.5 (see Section 8.4.12).

# No percentile data were available for inhalation rate, and the default lognormal

model was assumed. An analysis of inhalation data by Myers et al. (1998) found that

for those younger than age 3, CV was close to 70 percent; for other age groups, it was

close to 30 percent. The lognormal distribution was fitted by using CV=70 percent

for the child1 age group; CV=50 percent [(30+70)/2] for the child2 age group; and

CV=30 percent for the child3, child4, and adult age groups (see Section 8.4.13).

8.3.2 Fixed Parameters

Certain parameters were fixed nationally, based on central tendency values from the best

available source (usually EFH recommendations), either because no regional variability was

expected or because the available data were not adequate to generate national distributions. 

These fixed parameters include exposure frequency; shower frequency; and fraction

contaminated for drinking water, soil, exposed vegetables, root vegetables, protected vegetables,

exposed fruit, protected fruit, and fish. Fraction-contaminated values indicate the amount of

foodstuff grown or produced on a farm or in the garden in question and correspond to the

fraction homegrown or home-produced from the EFH (Table 13-71). Exposure duration was

fixed based on EPA policy. Human lifetime was set at a projected average value from the EFH

(Table 8-1, U.S. EPA, 1997a). Section 8.4.15 provides additional details on fixed exposure

factors.

8.3.3 Quality Assurance/Quality Control

The following QA/QC measures were used to minimize errors associated with data

collection, analysis, and processing:

# Raw Data Extraction and Entry–One hundred percent checks were made of all

exposure factor data extracted from the EFH and entered into spreadsheets for

statistical analysis.
Section 8.0 Human Exposure Factors

8-12

# Statistical Analysis–All statistical analyses were conducted in duplicate using

replicate data inputs to ensure reproducibility of results.

# Data Entry for the 3MRA Modeling System–Data entry was checked 100 percent

against the statistical analysis outputs. All conversions performed by the data

processing system were checked against hand calculations.

# Documentation–All hardcopy data sources are organized and maintained in a formal

filing system, along with documentation of QA/QC procedures.

Detailed QA/QC documentation can be made available upon request.

8.4 Results by Exposure Factor

Table 8-3 summarizes the exposure factor stochastic or distributed input data used for the

3MRA modeling system. The following sections describe how these data were collected and

processed by exposure factor. Each section includes a table showing the raw EFH data used to

develop the distributions, along with the final distributional statistics. Fixed factors are

discussed in Section 8.4.15.

8.4.1 Body Weight

EFH Data–Body Weight (kg) 3MRA Distributions

Age

Cohort N

Data

Mean

Data

SDev P05 P10 P15 P25 P50 P75 P85 P90 P95 Distribution

PopEstd

Mean

PopEstd

SDev

<1 356 9.102 1.287 7.053 7.451 7.852 8.252 9.151 9.752 10.4 10.65 11.15 gamma 9.09 1.23

1-5 3,762 15.52 3.719 12.5 13.1 13.45 14.03 15.26 16.67 17.58 18.32 19.45 lognormal 15.5 2.05

6-11 1,725 30.84 9.561 22.79 24.05 25.07 26.44 29.58 33.44 36.82 39.66 43.5 lognormal 30.7 5.96

12-19 2,615 58.45 13.64 43.84 46.52 48.31 50.94 56.77 63.57 68.09 71.98 79.52 lognormal 58.2 10.2

20+ 12,504 71.41 15.45 52.86 55.98 58.21 61.69 69.26 78.49 84.92 89.75 97.64 lognormal 71.2 13.3

N = number of samples; P05-P95 = percentiles; Pop-Estd = population-estimated; SDev = standard deviation.
8-13

Table 8-3. Summary of Human Exposure Factor Data: Stochastic Variables Code Description Units Distribution Type Mean Source SDev Source Min Sour

Bri_cr1 inhalation (breathing) rate (child1 resident) m3/d lognormal 4.5 LOG mean 3.15 LOG SDEV 0.5 EFH (newb

Bri_cr2 inhalation (breathing) rate (child2 resident) m3/d lognormal 7.55 LOG mean 3.78 LOG SDEV 1 0.5*(meanBri_cr3 inhalation (breathing) rate (child3 resident) m3/d lognormal 11.75 LOG mean 3.53 LOG SDEV 1 0.5*(meanBri_cr4 inhalation (breathing) rate (child4 resident) m3/d lognormal 14.0 LOG mean 4.2 LOG SDEV 1 0.5*(meanBri_r inhalation (breathing) rate (adult resident) m3/d lognormal 13.3 LOG mean 3.99 LOG SDEV 1 0.5*(meanBWa body weight (adult) kg lognormal 71.2 LOG mean 13.3 LOG SDEV 15 0.5*(meanBWc1 body weight (child1) kg gamma 9.09 GAM mean 1.23 GAM SDEV 2 0.5*(meanBWc2 body weight (child2) kg lognormal 15.5 LOG mean 2.05 LOG SDEV 4 0.5*(meanBWc3 body weight (child3) kg lognormal 30.7 LOG mean 5.96 LOG SDEV 6 0.5*(meanBWc4 body weight (child4) kg lognormal 58.2 LOG mean 10.2 LOG SDEV 13 0.5*(meanCRb_af consumption rate: beef (adult farmer) g WW/kg/d lognormal 2.5 LOG mean 2.69 LOG SDEV 0 CRb_cf_2 consumption rate: beef (child2 farmer) g WW/kg/d lognormal 3.88 LOG mean 4.71 LOG SDEV 0 CRb_cf_3 consumption rate: beef (child3 farmer) g WW/kg/d lognormal 3.88 LOG mean 4.71 LOG SDEV 0 CRb_cf_4 consumption rate: beef (child4 farmer) g WW/kg/d gamma 1.77 GAM mean 1.12 GAM SDEV 0 CRbm_cr_1 consumption rate: breast milk (child1) mL/d lognormal 688 EFH data 1,032 EFH data 0 Crfr_cf_2 consumption rate: exposed fruit (child2 farmer) g WW/kg/d gamma 2.25 GAM mean 1.89 GAM SDEV 0 CRfr_cf_3 consumption rate: exposed fruit (child3 farmer) g WW/kg/d lognormal 2.78 LOG mean 5.12 LOG SDEV 0 CRfr_cf_4 consumption rate: exposed fruit (child4 farmer) g WW/kg/d lognormal 1.54 LOG mean 2.44 LOG SDEV 0 CRfr_cg_2 consumption rate: exposed fruit (child2 gardener) g WW/kg/d gamma 2.25 GAM mean 1.89 GAM SDEV 0 CRfr_cg_3 consumption rate: exposed fruit (child3 gardener) g WW/kg/d lognormal 2.78 LOG mean 5.12 LOG SDEV 0 CRfr_cg_4 consumption rate: exposed fruit (child4 gardener) g WW/kg/d lognormal 1.54 LOG mean 2.44 LOG SDEV 0 
8-14

Code Description Units Distribution Type Mean Source SDev Source Min Sour

Table 8-3. (continued) CRfr_f consumption rate: exposed fruit (farmer) g WW/kg/d lognormal 2.36 LOG mean 3.33 LOG SDEV 0 

CRfr_g consumption rate: exposed fruit (gardener) g WW/kg/d lognormal 1.57 LOG mean 2.30 LOG SDEV 0 CRfs_a consumption rate: fish (adult) g/d lognormal 6.48 LOG mean 19.9 LOG SDEV 0 CRfs_c_2 consumption rate: fish (child2) g/d lognormal 6.48 LOG mean 19.9 LOG SDEV 0 CRfs_c_3 consumption rate: fish (child3) g/d lognormal 6.48 LOG mean 19.9 LOG SDEV 0 CRfs_c_4 consumption rate: fish (child4) g/d lognormal 6.48 LOG mean 19.9 LOG SDEV 0 CRl_cf_2 consumption rate: exposed vegetables (child2 farmer) g WW/kg/d gamma 2.55 GAM mean 2.58 GAM SDEV 0 CRl_cf_3 consumption rate: exposed vegetables (child3 farmer) g WW/kg/d lognormal 1.64 LOG mean 3.95 LOG SDEV 0 CRl_cf_4 consumption rate: exposed vegetables (child4 farmer) g WW/kg/d gamma 1.08 GAM mean 1.13 GAM SDEV 0 CRl_cg2 consumption rate: exposed vegetables (child2 gardener) g WW/kg/d gamma 2.55 GAM mean 2.58 GAM SDEV 0 CRl_cg3 consumption rate: exposed vegetables (child3 gardener) g WW/kg/d lognormal 1.64 LOG mean 3.95 LOG SDEV 0 CRl_cg4 consumption rate: exposed vegetables (child4 gardener) g WW/kg/d gamma 1.08 GAM mean 1.13 GAM SDEV 0 CRl_f consumption rate: exposed vegetables (adult farmer) g WW/kg/d lognormal 2.38 LOG mean 3.5 LOG SDEV 0 CRl_g consumption rate: exposed vegetables (gardener) g WW/kg/d Weibull 0.89 WEI shape 1.48 WEI scale 0 CRm_af consumption rate: milk (adult farmer) g WW/kg/d Weibull 1.25 WEI shape 17.45 WEI scale 0 CRm_cf_2 consumption rate: milk (child2 farmer) g WW/kg/d Weibull 1.7 WEI shape 26.47 WEI scale 0 CRm_cf_3 consumption rate: milk (child3 farmer) g WW/kg/d Weibull 1.56 WEI shape 14.82 WEI scale 0 CRm_cf_4 consumption rate: milk (child4 farmer) g WW/kg/d Weibull 1.14 WEI shape 6.52 WEI scale 0 CRpfr_cf_2 consumption rate: protected fruit (child2 farmer) g WW/kg/d lognormal 6.5 LOG mean 15.9 LOG SDEV 0 CRpfr_cf_3 consumption rate: protected fruit (child3 farmer) g WW/kg/d lognormal 6.5 LOG mean 15.9 LOG SDEV 0 CRpfr_cf_4 consumption rate: protected fruit (child4 farmer) g WW/kg/d lognormal 2.91 LOG mean 6.39 LOG SDEV 0 
8-15

Code Description Units Distribution Type Mean Source SDev Source Min Sour

Table 8-3. (continued) CRpfr_cg_2 consumption rate: protected fruit (child2 gardener) g WW/kg/d lognormal 6.5 LOG mean 15.9 LOG SDEV 0 

CRpfr_cg_3 consumption rate: protected fruit (child3 gardener) g WW/kg/d lognormal 6.5 LOG mean 15.9 LOG SDEV 0 CRpfr_cg_4 consumption rate: protected fruit (child4 gardener) g WW/kg/d lognormal 2.91 LOG mean 6.39 LOG SDEV 0 CRpfr_f consumption rate: protected fruit (adult farmer) g WW/kg/d lognormal 6.67 LOG mean 17.7 LOG SDEV 0 CRpfr_g consumption rate: protected fruit (adult gardener) g WW/kg/d lognormal 6.63 LOG mean 15.7 LOG SDEV 0 CRpl_cf_2 consumption rate: protected vegetables (child2 farmer) g WW/kg/d lognormal 1.88 LOG mean 1.98 LOG SDEV 0 CRpl_cf_3 consumption rate: protected vegetables (child3 farmer) g WW/kg/d lognormal 1.07 LOG mean 1.04 LOG SDEV 0 CRpl_cf_4 consumption rate: protected vegetables (child4 farmer) g WW/kg/d lognormal 0.77 LOG mean 0.69 LOG SDEV 0 CRpl_cg_2 consumption rate: protected vegetables (child2 gardener) g WW/kg/d lognormal 1.88 LOG mean 1.98 LOG SDEV 0 CRpl_cg_3 consumption rate: protected vegetables (child3 gardener) g WW/kg/d lognormal 1.07 LOG mean 1.04 LOG SDEV 0 CRpl_cg_4 consumption rate: protected vegetables (child4 gardener) g WW/kg/d lognormal 0.77 LOG mean 0.69 LOG SDEV 0 CRpl_f consumption rate: protected vegetables (adult farmer) g WW/kg/d lognormal 1.27 LOG mean 1.85 LOG SDEV 0 CRpl_g consumption rate: protected vegetables (adult gardener) g WW/kg/d lognormal 1.01 LOG mean 1.19 LOG SDEV 0 CRr_cf_2 consumption rate: root vegetables (child2 farmer) g WW/kg/d lognormal 2.31 LOG mean 6.05 LOG SDEV 0 CRr_cf_3 consumption rate: root vegetables (child3 farmer) g WW/kg/d Weibull 0.68 WEI shape 1.06 WEI scale 0 CRr_cf_4 consumption rate: root vegetables (child4 farmer) g WW/kg/d Weibull 0.84 WEI shape 0.91 WEI scale 0 

CRr_cg_2 consumption rate: root vegetables (child2 gardener) g WW/kg/d lognormal 2.31 LOG mean 6.05 LOG SDEV 0 CRr_cg_3 consumption rate: root vegetables (child3 gardener) g WW/kg/d Weibull 0.68 WEI shape 1.06 WEI scale 0 CRr_cg_4 consumption rate: root vegetables (child4 gardener) g WW/kg/d Weibull 0.84 WEI shape 0.91 WEI scale 0 CRr_f consumption rate: root vegetables (farmer) g WW/kg/d lognormal 1.45 LOG mean 2.06 LOG SDEV 0 CRr_g consumption rate: root vegetables (gardener) g WW/kg/d Weibull 0.87 WEI shape 1.07 WEI scale 0 
8-16

Code Description Units Distribution Type Mean Source SDev Source Min Sour

Table 8-3. (continued) CRs_cr2 ingestion rate: soil (child2 resident) kg/d lognormal 0.0001 LOG mean 1.5E-04 LOG SDEV 5E-07 Prof. Judgm

CRs_cr3 ingestion rate: soil (child3 resident) kg/d lognormal 5E-05 LOG mean 7.5E-05 LOG SDEV 5E-07 Prof. JudgmCRS_cr4 ingestion rate: soil (child4 resident) kg/d lognormal 5E-05 LOG mean 7.5E-05 LOG SDEV 5E-07 Prof. JudgmCRs_r ingestion rate: soil (adult resident) kg/d lognormal 5E-05 LOG mean 7.5E-05 LOG SDEV 5E-07 Prof. JudgmCRw_cr1 ingestion rate: drinking water (child1 resident) mL/d Weibull 1.16 WEI shape 318.60 WEI scale 0 Prof. JudgmCRw_cr2 ingestion rate: drinking water (child2 resident) mL/d gamma 698 GAM mean 406 GAM SDEV 26 0.5*(P01) CRw_cr3 ingestion rate: drinking water (child3 resident) mL/d gamma 787 GAM mean 430 GAM SDEV 34 0.5*(P01) CRw_cr4 ingestion rate: drinking water (child4 resident) mL/d gamma 965 GAM mean 574 GAM SDEV 33 0.5*(P01) CRw_r ingestion rate: drinking water (adult resident) mL/d gamma 1,383 GAM mean 703 GAM SDEV 104 0.5*(P01) t_bathroom time in bathroom after shower min Weibull 0.96 WEI shape 8.36 WEI scale 1 Prof. Judgmt_shower shower time min gamma 16.7 GAM mean 9.91 GAM SDEV 1 Prof. Judgm
Section 8.0 Human Exposure Factors

8-17

Body weight data were obtained from Tables 7-2, 7-3, 7-4, 7-5, 7-6, and 7-7 of the EFH

(U.S. EPA, 1997a). Data (in kg) were presented by age and gender. Weighted averages of

percentiles, means, and standard deviations were calculated for child1 (<1 year old), child2 (1 to

5 years old), child3 (6 to 11 years old), child4 (12 to 19 years old), and adult age groups; male

and female data were weighted and combined for each age group. These percentile data were

used as the basis of fitting distributions. These data were analyzed to fit parametric models

(gamma, lognormal, and Weibull) using maximum likelihood estimation. Measures of goodness

of fit were used to select the most appropriate model.

8.4.2 Drinking Water Intake

EFH Data–Drinking Water Intake (mL/d) 3MRA Distributions

Age

Cohort N

Data

Mean

Data

SDev P01 P05 P10 P25 P50 P75 P90 P95 P99 Distribution

PopEstd

Mean

PopEstd

SDev

< 1 403 302.7 258.5 100.3 255.4 413.2 666.3 780.3 Weibull 302 261

1-5 3,200 697.1 401.5 51.62 187.6 273.5 419.2 616.5 900.8 1,236 1,473 1,917 gamma 698 406

6-11 2,405 787 417 68 241 318 484 731 1,016 1,338 1,556 1,998 gamma 787 430

12-19 5,801 963.2 560.6 65.15 241.4 353.8 574.4 868.5 1,247 1,694 2,033 2,693 gamma 965 574

20+ 13,394 1,384 721.6 207.6 457.5 607.3 899.6 1,275 1,741 2,260 2,682 3,737 gamma 1,383 703

N = number of samples; P01-P99 = percentiles; Pop-Estd = population-estimated; SDev = standard deviation.

Drinking water intake data were obtained from Table 3-6 of the EFH (U.S. EPA, 1997a). 

Data (in mL/d) were presented by age groups. Weighted averages of percentiles, means, and

standard deviations were calculated for child1, child2, child3, child4, and adult age groups. 

Percentile data were used to fit parametric models (gamma, lognormal, and Weibull) using

maximum likelihood estimation. Measures of goodness of fit were used to select the most

appropriate model.

8.4.3 Exposed Fruit Consumption

EFH Data–Exposed Fruit Consumption (g WW/kg/d) 3MRA Distributions

Age

Cohort N

Data

Mean

Data

SDev P01 P05 P10 P25 P50 P75 P90 P95 P99 Distribution

PopEstd

Mean

PopEstd

SDev

1-5 49 2.6 3.947 0.373 1 1.82 2.64 5.41 6.07 gamma 2.25 1.89

6-11 68 2.52 3.496 0.171 0.373 0.619 1.11 2.91 6.98 11.7 lognormal 2.78 5.12

12-19 50 1.33 1.457 0.123 0.258 0.404 0.609 2.27 3.41 4.78 lognormal 1.54 2.44

farmer 112 2.32 2.646 0.072 0.276 0.371 0.681 1.3 3.14 5 6.12 15.7 lognormal 2.36 3.33

home

gard. 596 1.55 2.226 0.042 0.158 0.258 0.449 0.878 1.73 3.41 5 12.9 lognormal 1.57 2.3

N = number of samples; P01-P99 = percentiles; Pop-Estd = population-estimated; SDev = standard deviation.
Section 8.0 Human Exposure Factors

8-18

Data for consumption of homegrown exposed fruit were obtained from Table 13-61 of

the EFH (U.S. EPA, 1997b). Data (in g WW/kg/d) were presented by age groups and for

farmers and home gardeners (adults). For the child2 age group, data were only available for

those ages 3 to 5 years (not available for 1- to 2-year-olds); therefore, these data were used for

the entire 1- to 5-year-old age group (child2). Percentile data were used to fit parametric models

(gamma, lognormal, and Weibull) using maximum likelihood estimation. Measures of goodness

of fit were used to select the most appropriate model. The fraction of exposed fruit intake that is

home-produced is 0.116 for households that garden and 0.328 for households that farm

(Table 13-71, U.S. EPA, 1997b).

8.4.4 Protected Fruit Consumption

EFH Data–Protected Fruit Consumption (g WW/kg/d) 3MRA Distributions

Age

Cohort N

Data

Mean

Data

SDev P01 P05 P10 P25 P50 P75 P90 P95 P99 Distribution

PopEstd

Mean

PopEstd

SDev

1-5 ND ND lognormal 6.5 15.9

6-11 ND ND lognormal 6.5 15.9

12-19 20 2.96 4.441 0.16 0.283 0.393 1.23 2.84 7.44 11.4 lognormal 2.91 6.39

20+ 106 5.338 7.174 0.276 0.342 0.82 2.127 8.022 15.25 19.8 lognormal 6.67 17.7

all ages 173 5.74 8.221 0.15 0.266 0.335 0.933 2.34 7.45 16 19.7 47.3 lognormal 6.5 15.9

farmer ND ND lognormal 6.67 17.7

home gard. 146 5.9 8.422 0.117 0.265 0.335 1.16 2.42 7.46 16 19.1 47.3 lognormal 6.63 15.7

N = number of samples; P01-P99 = percentiles; Pop-Estd = population-estimated; SDev = standard deviation.

Data for consumption of homegrown protected fruit were obtained from Table 13-62 of

the EFH (U.S. EPA, 1997b). Data (in g WW/kg/d) were presented for those 12 to 19 years, 20 to

39 years, 40 to 69 years, all ages, and home gardeners. Available percentile data were used to fit

parametric models (gamma, lognormal, and Weibull) using maximum likelihood estimation. 

Measures of goodness of fit were used to select the most appropriate model.

Data were not available for farmers or those aged 1 to 2, 3 to 5, and 6 to 11 years. For

the child2 and child3 age groups, the lognormal model is most appropriate because lognormal

fits the best in other age groups for protected fruit and vegetables; the population estimated mean

and standard deviation for all age groups were used for the analysis (normalized to body weight). 

For farmers, the population estimated mean and standard deviation for those older than 20 years

(derived from the weighted average of means and standard deviations of those ages 20 to 39

years and those ages 40 to 69 years) were used for the analysis; lognormal also fits the percentile

data best for those older than 20 years. The fraction of protected fruit intake that is homeproduced is 0.094 for households that garden and 0.03 for households that farm (Table 13-71,

U.S. EPA, 1997b).
Section 8.0 Human Exposure Factors

8-19

8.4.5 Exposed Vegetable Consumption

EFH Data–Exposed Vegetable Consumption (g WW/kg/d) 3MRA Distributions

Age

Cohort N

Data

Mean

Data

SDev P01 P05 P10 P25 P50 P75 P90 P95 P99 Distribution

PopEstd

Mean

PopEstd

SDev

1-5 105 2.453 2.675 0.102 0.37 0.833 1.459 3.226 6.431 8.587 gamma 2.55 2.58

6-11 134 1.39 2.037 0.044 0.094 0.312 0.643 1.6 3.22 5.47 13.3 lognormal 1.64 3.95

12-19 143 1.07 1.128 0.029 0.142 0.304 0.656 1.46 2.35 3.78 5.67 gamma 1.08 1.13h

farmer 207 2.17 2.316 0.184 0.372 0.647 1.38 2.81 6.01 6.83 10.3 lognormal 2.38 3.5

home gard. 1,361 1.57 2.029 0.003 0.089 0.168 0.413 0.889 1.97 3.63 5.45 10.3 Weibull 1.57 1.76

N = number of samples; P01-P99 = percentiles; Pop-Estd = population-estimated; SDev = standard deviation.

Data for consumption of homegrown exposed vegetables were obtained from Table 13-

63 of the EFH (U.S. EPA, 1997b). Data (in g WW/kg/d) were presented for those ages 1 to 2,

3 to 5, 6 to 11, 12 to 19, 20 to 39, and 40 to 69 years, as well as farmers and home gardeners. 

Weighted averages of percentiles, means, and standard deviations were calculated for the child 2

age group (combining groups of those ages 1 to 2 years and 3 to 5 years). Percentile data were

used to fit parametric models (gamma, lognormal, and Weibull) using maximum likelihood

estimation. Measures of goodness of fit were used to select the most appropriate model.

8.4.6 Root Vegetable Consumption

EFH Data–Root Vegetable Consumption (g WW/kg/d) 3MRA Distributions

Age

Cohort N

Data

Mean

Data

SDev P01 P05 P10 P25 P50 P75 P90 P95 P99 Distribution

PopEstd

Mean

PopEstd

SDev

1-5 45 1.886 2.371 0.081 0.167 0.291 0.686 2.653 5.722 7.502 lognormal 2.31 6.05

6-11 67 1.32 1.752 0.014 0.036 0.232 0.523 1.63 3.83 5.59 Weibull 1.38 2.07

12-19 76 0.937 1.037 0.008 0.068 0.269 0.565 1.37 2.26 3.32 Weibull 0.99 1.19

farmer 136 1.39 1.469 0.111 0.158 0.184 0.365 0.883 1.85 3.11 4.58 7.47 lognormal 1.45 2.06

home gard. 682 1.15 1.494 0.005 0.036 0.117 0.258 0.674 1.5 2.81 3.64 7.47 Weibull 1.15 1.32

N = number of samples; P01-P99 = percentiles; Pop-Estd = population-estimated; SDev = standard deviation.

Homegrown root vegetable consumption data were obtained from Table 13-65 of the

EFH (U.S. EPA, 1997b). Data (in g WW/kg/d) were presented for those ages 1 to 2, 3 to 5, 6 to

11, 12 to 19, 20 to 39, and 40 to 69 years, as well as farmers and home gardeners. Weighted

averages of percentiles, means, and standard deviations were calculated for the child2 age group

(combining groups of those ages 1 to 2 and 3 to 5 years). Percentile data were used to fit

parametric models (gamma, lognormal, and Weibull) using maximum likelihood estimation. 

Measures of goodness of fit were used to select the most appropriate model.
Section 8.0 Human Exposure Factors

8-20

8.4.7 Protected Vegetable Consumption

EFH Data–Protected Vegetable Consumption (g WW/kg/d) 3MRA Distributions

Age

Cohort N

Data

Mean

Data

SDev P01 P05 P10 P25 P50 P75 P90 P95 P99 Distribution

PopEstd

Mean

PopEstd

SDev

1-5 53 1.76 1.79 0.265 0.408 0.829 1.397 2.066 3.053 6.812 lognormal 1.88 1.98

6-11 63 1.1 1.064 0.208 0.318 0.387 0.791 1.31 2.14 3.12 lognormal 1.07 1.04

12-19 51 0.776 0.622 0.161 0.239 0.354 0.583 0.824 1.85 2.2 lognormal 0.77 0.69

farmer 142 1.3 1.728 0.087 0.166 0.209 0.337 0.599 1.4 3.55 5.4 9.23 lognormal 1.27 1.85

home gard. 602 1.01 1.161 0.103 0.153 0.192 0.336 0.642 1.21 2.32 3.05 6.49 lognormal 1.01 1.19

N = number of samples; P01-P99 = percentiles; Pop-Estd = population-estimated; SDev = standard deviation.

Homegrown protected vegetable consumption data were obtained from Table 13-64 of

the EFH (U.S. EPA, 1997b). Data (in g WW/kg/d) were presented for those ages 1 to 2, 3 to 5,

6 to 11, 12 to 19, 20 to 39, and 40 to 69 years, as well as farmers and home gardeners. Weighted

averages of percentiles, means, and standard deviations were calculated for the child2 age group

(combining groups of those ages 1 to 2 and 3 to 5 years). Percentile data were used to fit

parametric models (gamma, lognormal, and Weibull) using maximum likelihood estimation. 

Measures of goodness of fit were used to select the most appropriate model. 

8.4.8 Dairy Products (Milk) Consumption

EFH Data–Milk Consumption (g WW/kg/d) 3MRA Distributions

Age

Cohort N

Data

Mean

Data

SDev P05 P10 P25 P50 P75 P90 P95 Distribution

PopEstd

Mean

PopEstd

SDev

<1 1 62.74 94.1 0.61 24.68 45.78 91.12 136.7 170.9 Weibull 65.4 78.7

1-5 2 23.71 35.86 2.98 7.47 13.56 21.5 32.22 42.63 49.62 Weibull 23.6 14.3

6-11 1 13.33 20 1.81 3.54 6.72 11.88 18.58 25.38 28.76 Weibull 13.3 8.7

12-19 1 6.293 9.44 0.27 0.61 2.31 5.29 9.2 12.75 15.12 Weibull 6.23 5.49

farmer 63 17.1 15.8 0.736 3.18 9.06 12.1 20.4 34.9 44 Weibull 16.3 13.1

N = number of samples; P05-P95 = percentiles; Pop-Estd = population-estimated; SDev = standard deviation.

Data were obtained from Tables 13-28 and 11-2 of the EFH (U.S. EPA, 1997b). Data for

consumption of home-produced dairy products (in g WW/kg/d) were presented for those 20 to

39 years old and farmers (Table 13-28). Per capita intake data for dairy products (including

store-bought products) were available for those younger than 1 year and those 1 to 2, 3 to 5, 6 to

11, and 12 to 19 years old (Table 11-2). Weighted averages of percentiles, means, and standard

deviations were calculated for the child2 age group (combining those 1 to 2 years old and those

3 to 5 years old). Percentile data were used to fit parametric models (gamma, lognormal, and

Weibull) using maximum likelihood estimation. Measures of goodness of fit were used to select

Weibull as the most appropriate model in all cases. 
Section 8.0 Human Exposure Factors

8-21

8.4.9 Breast Milk Consumption

Age

Cohort

Data Mean

(mL/d)

Data

SDev

Upper

Percentile Distribution

Pop-Estd

Mean

(mL/d)

Pop-Estd

SDev

(mL/d)

<1 688 ND 980 lognormal 688 1,032

1-5 ND ND ND ND ND ND

Pop-Estd = population-estimated; SDev = standard deviation.

The data mean and upper percentile for breast milk consumption in 1- to 12-month-olds

were 688 and 980 mL/d, respectively (Table 14-16, U.S. EPA, 1997b). The lognormal model

was used for breast milk consumption (12-month-olds) because no percentile or related data

were available. The EFH population mean for breast milk consumption was 688 mL/d, and the

population standard deviation for CV=1.5 was 1,032 mL/d.

8.4.10 Beef Consumption

EFH Data–Beef Consumption (g WW/kg/d) 3MRA Distributions

Age

Cohort N

Data

Mean

Data

SDev P01 P05 P10 P25 P50 P75 P90 P95 P99 Distribution

PopEstd

Mean

PopEstd

SDev

1-5 ND ND lognormal 3.88 4.71

6-11 38 3.77 3.662 0.663 0.753 1.32 2.11 4.43 11.4 12.5 lognormal 3.88 4.71

12-19 41 1.72 1.044 0.478 0.513 0.896 1.51 2.44 3.53 3.57 gamma 1.77 1.12

farmer 182 2.63 2.644 0.27 0.394 0.585 0.896 1.64 3.25 5.39 7.51 11.3 lognormal 2.5 2.69

N = number of samples; P01-P99 = percentiles; Pop-Estd = population-estimated; SDev = standard deviation.

Home-produced beef consumption data were obtained from Table 13-36 of the EFH

(U.S. EPA, 1997b). Data (in g WW/kg/d) were presented for farmers and those 6 to 11, 12 to 19,

20 to 39, and 40 to 69 years old. Percentile data were used to fit parametric models (gamma,

lognormal, and Weibull) using maximum likelihood estimation. Measures of goodness of fit

were used to select the most appropriate model. 

Data were not available for those 1 to 2 and 3 to 5 years old. For beef consumption for

those 1 to 5 years old, the lognormal model was used because, among the other age groups, it

was the best-fitted model in all but one case. The population estimated mean and standard

deviation for 6- to 11-year-olds were used for the analysis (normalized for body weight) and are

supported by data in Table 11-3 (per-capita intake for beef, including store-bought products),

which indicate that those 1 to 2, 3 to 5, and 6 to 11 years old have the highest consumption rate

of beef on a g/kg/d basis.
Section 8.0 Human Exposure Factors

8-22

8.4.11 Fish Consumption

EFH Data–Fish Consumption (g/d) 3MRA Distributions

Age

Cohort N

Data

Mean

Data

SDev P50 P66 P75 P90 P95 Distribution

Pop-Estd

Mean

Pop-Estd

SDev

1-5 ND ND lognormal 6.48 19.9

6-11 ND ND lognormal 6.48 19.9

12-19 ND ND lognormal 6.48 19.9

20+ ND ND lognormal 6.48 19.9

all ages 1,053 6.4 2 4 5.8 13 26 lognormal 6.48 19.9

N = number of samples; P50-P95 = percentiles; Pop-Estd = population-estimated; SDev = standard deviation.

Fish consumption data were obtained from Table 10-64 of the EFH (U.S. EPA, 1997b). 

Data (in g/d) were available only for adult freshwater anglers in Maine. Age-specific data for

children were not available; children were assumed to consume the same amount of fish as

adults. Percentile data were used to fit parametric models (gamma, lognormal, and Weibull) and

measures of goodness of fit were used to select lognormal as the most appropriate model. 

8.4.12 Soil Ingestion

EFH Data–Soil Ingestion Data and Distributions

Age

Cohort

Data

Mean

(mg/d) Distribution

Pop-Estd

Mean

 (mg/d)a

Pop-Estd

SDev

(CV=0.5)

Pop-Estd

SDev

(CV=1)

Pop-Estd

SDev

(CV=1.5)a

1-5 100 lognormal 100 50 100 150

6-11 ND lognormal 50 25 50 75

12-19 ND lognormal 50 25 50 75

adult 50 lognormal 50 25 50 75

a

HWIR distributions.

Pop-Estd = population-estimated; SDev = standard deviation.

Mean soil ingestion rates were cited as 100 mg/d for children (400 mg/d = upper

percentile), 200 mg/d for children (conservative estimate), 50 mg/d for adults, and 10 g/d for

pica children (Table 4-23, U.S. EPA, 1997a). No percentile data were recommended for use in

the EFH. The lognormal model was used for soil consumption for all age groups. Parameter

estimates were obtained by assuming CV=0.5, 1, and 1.5.

Population standard deviations based on a CV of 1.5 were used for the human exposure

module. Adult data were used for the child3 and child4 variables. 
Section 8.0 Human Exposure Factors

8-23

8.4.13 Inhalation Rate

Inhalation Rate

Age

Cohort Distribution

Population-Estimated

Mean (m3

/d)

PopulationEstimated

SDev (m3/d)

<1 lognormal 4.5 3.15

1-5 lognormal 7.55 3.78

6-11 lognormal 11.75 3.53

12-19 lognormal 14.0 4.2

adult lognormal 13.3 3.99

SDev = standard deviation.

No percentile data were available for the inhalation rate, and the default lognormal model

was assumed. In an analysis of inhalation data, Myers et al. (RTI, 1998) found that for those

younger than 3 years, CV was close to 70 percent; for other age groups, it was close to 30

percent. The lognormal distribution was fitted by using CV=70 percent for the child1 age group;

CV=50 percent [(30+70)/2] for the child2 age group; and CV=30 percent for the child3, child4,

and adult age groups.

8.4.14 Shower Parameters

EFH Data–Shower Parameters (minutes) 3MRA Distributions

Parameter

Age

Cohort N P02 P05 P10 P25 P50 P75 P90 P95 P98 P99 Distribution

PopEstd

Mean

PopEstd

SDev

showerT all ages 3,547 4 5 10 15 20 30 35 50 60 gamma 16.7 9.91

t_bathroom all ages 3,533 1 3 5 10 20 30 40 50 Weibull 0.96 8.36

N = number of samples; P02-P99 = percentiles; Pop-Estd = population-estimated; SDev = standard deviation.

Percentile data for time spent taking a shower (showerT) and cumulative time spent in

the bathroom after a shower (t_bathroom) were provided in the EFH (U.S. EPA, 1997c). 

Percentile data were used to fit parametric models (gamma, lognormal, and Weibull) using

maximum likelihood estimation. Measures of goodness of fit were used to select the most

appropriate model for each age variable.

8.4.15 Fixed Parameters

Fixed parameters are shown in Table 8-4 along with the value selected for the 3MRA

modeling system representative national data set and its data source. These constants include

variables for which limited or no percentile data were provided in the EFH: exposure frequency,

showering frequency, breastmilk parameters, and fraction contaminated for the various media
Section 8.0 Human Exposure Factors

8-24

and foodstuffs considered in the 3MRA modeling system. Most of these values were extracted

directly from the EFH. The fraction contaminated for various foodstuffs was assumed to be

equivalent to the fraction of food intake that is home-produced from Table 13-71 (U.S. EPA,

1997b). The fraction of consumed trophic level 3 (T3) and trophic level 4 (T4) fish was

determined from data in Table 10-66 of the EFH (U.S. EPA, 1997b), which contains the only

fish consumption data reported in the handbook with an adequate species breakdown to make

this distinction. Based on EPA policy, exposure duration for residents and farmers was assumed

to be equivalent to the average population mobility (Table 15-176, U.S. EPA, 1997c) and an

average projected U.S. life span (76.5 years) was used as human lifetime in the carcinogenic risk

calculations (Table 8-1, U.S. EPA, 1997a). 

Table 8-4. Summary of Human Exposure Factor Data: Constants

Model

Code Description Units Average Source

BF Event frequency (shower) event/d 1 EFH, Table 15-176

Vshower Shower volume m3 2 McKone, 1987

Vbath Bathroom volume m3 10 McKone, 1987

Rshower Shower rate L/min 5.5 Professional judgment

VRsb Shower-to-bathroom ventilation rate L/min 100 Professional judgment

VRbh Bathroom-to-house ventilation rate L/min 300 Professional judgment

Vn Droplet terminal velocity cm/s 400 Professional judgment

Hn Nozzle height cm 180 Professional judgment

DD Droplet diameter cm 0.1 Professional judgment

ffm Fraction of mother's weight that is fat Fraction 0.3 U.S. EPA, 1998, 2000a

fai Fraction of contaminant ingested by the infant that is absorbed Fraction 0.9 U.S. EPA 1998, 2000b

Fam Fraction of contaminant ingested by mother that is absorbed Fraction 1 U.S. EPA 1998

Fbl Fraction of contaminant in whole blood compartment Fraction 0 U.S. EPA 1998

Ff Fraction of contaminant stored in maternal fat Fraction 0.9 U.S. EPA, 1998, 2000a

Krbc Concentration proportionality constant between red blood cells

and plasma

Unitless 1 U.S. EPA, 1998

t_halfb Biological half-life of chemical in lactating women d 2555 U.S. EPA, 1998, 2000a

Caqueous Concentration in aqueous phase of maternal milk mg/kg 0 U.S. EPA, 1998

fmbm Fraction of fat in maternal breast milk fraction 0.04 US EPA, 1998, 2000a

Ffr_f Fraction homegrown: exposed fruit (farmer) fraction 0.328 EFH, Table 13-71

Ffr_g Fraction homegrown: exposed fruit (gardener) fraction 0.116 EFH, Table 13-71

Fl_f Fraction homegrown: exposed vegetables (farmer) fraction 0.42 EFH, Table 13-71

Fl_g Fraction homegrown: exposed vegetables (gardener) fraction 0.233 EFH, Table 13-71

Fpfr_f Fraction homegrown: protected fruit (farmer) fraction 0.03 EFH, Table 13-71

Fpfr_g Fraction homegrown: protected fruit (gardener) fraction 0.094 EFH, Table 13-71

(continued)
Table 8-4. (continued)

Section 8.0 Human Exposure Factors

Model

Code Description Units Average Source

8-25

Fpl_f Fraction homegrown: protected vegetables (farmer) fraction 0.394 EFH, Table 13-71

Fpl_g Fraction homegrown: protected vegetables (gardener) fraction 0.178 EFH, Table 13-71

Fr_f Fraction homegrown: root vegetables (farmer) fraction 0.173 EFH, Table 13-71

Fr_g Fraction homegrown: root vegetables (gardener) fraction 0.106 EFH, Table 13-71

Fb_f Fraction contaminated (home-raised): beef (farmer) fraction 0.485 EFH, Table 13-71

Fm_f Fraction contaminated (home-produced): milk (farmer) fraction 0.254 EFH, Table 13-71

Ff_s Fraction contaminated fish (recreational fisher) fraction 0.325 EFH, Table 13-71

FT3fish Fraction of T3 fish consumed fraction 0.36 EFH, Table 10-66

FT4fish Fraction of T4 fish consumed fraction 0.64 EFH, Table 10-66

Fs Fraction contaminated: soil fraction 1 EPA policy

Fw Fraction contaminated: drinking water fraction 1 EPA policy

EFr Exposure frequency (adult resident) d/yr 350 EPA policy

ExDur Exposure duration for carcinogens: residents and farmers yr 9 EFH, Table 15-176

Lifetime Human lifetime (used in carcinogenic risk calculations) yr 76.5 EFH, Table 8-1

Source: EFH (U.S. EPA, 1997a, 1997b, and 1997c)

The fraction contaminated for soil and drinking water was assumed to be 1 (i.e., all soil

and drinking water available for consumption at a site is potentially contaminated), with actual

concentrations depending on fate and transport model results. Thus, households for which the 

drinking water pathway was analyzed were assumed to get 100 percent of their drinking water

from ground water. Exposure frequency is set to 350 days per year in accordance with EPA

policy, assuming that residents take an average of 2 weeks’ vacation time away from their homes

each year.

The 3MRA modeling system evaluates the breast milk ingestion pathway only for dioxin,

which is lipophilic, and the component of the breast milk model designed to model

concentrations for nonlipophilic constituents (i.e., the component that projects constituent

concentrations in the aqueous phase of breast milk) is not used.

The constituent concentration in maternal milkfat is dependent on the biological

elimination constant for the contaminant in nonlactating women, which can be related to the

biological half-life of the contaminant in lactating women (t_halfb), which is chemical-specific

and empirically derived. For the 3MRA modeling system, a half-life of 2,555 days (7 years) was

used for dioxin. U.S. EPA (2000a) reports a half-life ranging from 5 to 7 years for

2,3,7,8-TCDD in lactating women. The 3MRA modeling system used the upper bound of the

identified range. In general, a higher half-life reflects a longer time to steady state and results in

a higher overall dioxin concentration in maternal fat and, consequently, in maternal breast milk. 

Because steady-state conditions were assumed in modeling the breast milk pathway, using the

maximum 7-year half-life is conservative.
Section 8.0 Human Exposure Factors

8-26

8.5 Issues and Uncertainties

Many difficult choices must be made for a comprehensive risk assessment. In this

section, some caveats and possible refinements are discussed and organized as follows: source

data, models, regional distributions, fixed parameters, estimation methods, goodness-of-fit tests,

parametric versus nonparametric approaches, and uncertainty issues.

8.5.1 Source Data Uncertainties

For most exposure factors addressed, data analyses involved fitting distributions of data

summaries from the EFH (U.S. EPA, 1997a, 1997b, 1997c), in most cases by fitting distributions

to selected percentiles. In our opinion, little information is lost by fitting to percentiles versus

fitting to raw data. However, some believe that such analyses should always be based on raw

data, synthesizing all credible sources. The question can be settled scientifically because there is

a formal statistical definition of information. We suggest pursuing this activity as a scientific

support activity for selected parameters. We also suggest that EPA include a broader range of

percentiles in future editions of the EFH, especially for the lower (<0.50) percentiles (e.g., 0.01,

0.02, 0.03, 0.05, 0.10, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80, 0.90, 0.95, 0.97, 0.98, and 0.99), to

provide better data for determining the best fit.

Similarly, the percentiles for fitting the standardized age cohorts could have been

obtained by fitting distributions to the original groups, generating simulated data from the fitted

distributions, mixing the simulated data in proportion to the subgroup sizes, and then fitting the

distributions again. Mixing proportions would best be determined by the demographics of the

population (e.g., using U.S. Bureau of the Census data) of interest for the risk assessment, rather

than the original study sample sizes. Resources prevented us from pursuing this option for the

representative national data set, but it could be considered as a science support activity to test the

uncertainty of the method used. 

The data sets for time spent in shower [showerT] and time spent in the bathroom after

showering [t_bathroom] clearly are affected by rounding and grouping of data. The fitting

methods do not account for these sources of inaccuracy but could be developed and explored

depending upon the significance of these input variables.

8.5.2 Model (Distribution) Uncertainty

Three standard two-parameter probability models (gamma, lognormal, and Weibull) were

used for this analysis. These distributions are special cases of a three-parameter model

(generalized gamma) that contains them and allows for a likelihood ratio test of the fit of the

two-parameter models. Other models are possible (e.g., Myers et al., 1998), but we believe this

simple setup offers a considerable improvement over using a lognormal model in all cases and is

appropriate for this analysis. In support of this conclusion, the three-parameter generalized

gamma module did not significantly improve on goodness of fit over the two-parameter models

in 58 of 59 cases at the 5 percent level of significance.

In the few cases where fixed values were assumed because of lack of percentile data,

nondegenerate probability distributions can be assigned. Although the assumption of a point
Section 8.0 Human Exposure Factors

8-27

estimate (i.e., mean or median) is typically used in risk assessments when data are not adequate

to develop a distribution, this does imply a degree of certainty about the value when, in fact, the

uncertainty in that value may be quite large. For variables that can have significant variability

and impact on the analysis results (i.e., significantly affect risk estimates), specifying a minimum

positive standard deviation would improve uncertainty estimation for the analysis. One possible

approach is to assume a reasonable minimum CV (based on available data) and use a default

distribution type such as lognormal or gamma. For example, except for body weight, every

population CV of human exposure factors analyzed exceeds 50 percent (see Appendix 8B). 

Therefore, it might be reasonably assumed that an unknown CV is (at least) 50 percent for fixed

parameters whose uncertainty could have a significant impact on the analysis results.

8.5.3 Methods of Estimation

The maximum likelihood estimate method of estimating uncertainty parameters is

generally considered the best approach currently available for most situations. There may be

room for improvement in certain cases, however. Appendix 8B shows that the maximum

likelihood estimates for the means and standard deviations agree with the data means and

standard deviations much better for the gamma and Weibull models than for the lognormal

model. For example, note that even in cases where the lognormal model fits best, the gamma

estimate of the mean is often closer to the data mean than is the lognormal mean (i.e., the

lognormal maximum likelihood estimates of the mean and standard deviation can be biased). 

Truncation might reduce this problem, but if applied, truncated models should be fit to the data

rather than fitting a model and then truncating the distribution.

8.5.4 Testing Goodness of Fit

Although they offer significant improvement in objectivity over visual estimation,

goodness-of-fit tests are subject to some uncertainty that should to be considered in their

application. One area of concern is our uncertainty about how the survey statistics in the EFH

(U.S. EPA, 1997a, 1997b, 1997c) were calculated.

All of the statistics that have been used to assess goodness of fit here assume a random

sample, which may or may not be a valid assumption for EFH data. Specifically, many of the

EFH data sources are surveys that, in many cases, do not involve purely random samples. 

Rather, they use clustering and stratification, primarily for economic reasons. In such cases, the

calculation of estimates and their standard errors, as well as test statistics, should use the survey

weights and should take the study design (e.g., clustering and stratification) into account. The

EFH mentions that the SAS system was used for calculation of statistics. If the SAS

UNIVARIATE procedure was used to calculate percentiles, then the percentiles are unweighted. 

If the random sample assumption is not valid, the likelihood ratio test used in this

analysis may be more valid than the chi-square test. Valid chi-square statistics can be devised,

but they require raw data and information on the survey design and weights. One way to avoid

some of the difficult goodness-of-fit issues is to use empirical distributions (bootstrapping) when

the raw data are available. This nonparametric approach, however, is less convenient for risk

assessment simulations than using simple parametric probability models. 
Section 8.0 Human Exposure Factors

8-28

8.5.5 Treatment of Uncertainty

Regarding statistical treatment of uncertainty, the situation is less clear than for

estimation, where there is a fairly clear consensus in favor of the maximum likelihood estimate. 

Relying either on the asymptotic normality of the maximum likelihood estimate or on likelihoodbased contours to get parameter uncertainty distributions can be problematic when data do not

exactly fit the model, which is unfortunately always the case (Box, 1976). A partial remedy for

this problem was pointed out by Huber (1967); White (1982) offers a more recent and more

readable account. According to White (1982), if the model is correct, then the maximum

likelihood estimate is asymptotically normal, with the correct mean and variance/covariance

matrix given by the so-called information matrix (i.e., by the expectation of the negative second

derivatives of the log likelihood function with respect to the parameters). Still assuming the

model is correct, the information matrix is equal to the variance of the score vector (the score

vector is the gradient of the log likelihood function with respect to the parameters). 

If the model is false, however, then the equality of these two matrices fails, and a

sandwich estimator that combines them both is a better (robust) estimate of the

variance/covariance matrix (White, 1982). The pursuit of this theme leads to variance estimates

that absorb (are inflated by) model lack of fit and that, therefore, automatically take some

account of model uncertainty as well as parameter uncertainty. This finding has led to a

substantial body of recent practical statistical methodology under the name of robust sandwich

estimators of variance. As far as this analysis has seen, this area of study has received little

attention in the risk assessment community. In one sense, it is a treatment for an affliction

mentioned by Hattis and Burmaster (1994): “The application of standard statistical methodology

to a single data set will nearly always reveal only a trivial proportion of the overall uncertainty.” 

Regression estimates of uncertainty may be investigated as a means of addressing this problem.

8.5.6 Breast Milk Pathway Issues and Uncertainties

All of the variables used in modeling the breast milk pathway are subject to some

parameter uncertainty, and most are also subject to variability from differences in physiology or

physical attributes between modeled individuals. However, the 3MRA modeling system does

not consider between-individual variability or uncertainty in these variables (i.e., none are

specified as distributions and thereby treated stochastically). Uncertainty associated with

modeling the breast milk pathway within the human exposure module could be reduced if a

stochastic approach were adopted for modeling this pathway. Specifically, parameter-specific

variability and uncertainty distributions could be developed and incorporated into the 3MRA

modeling system Monte Carlo simulation.

The 3MRA modeling system assumes an exposure duration of 9 years for the mother,

which is close to the projected half-live for dioxin, and further assumes that breast feeding

occurs at the end of that exposure duration. Under these assumptions, maternal body burdens

should approach steady-state concentrations at the point when breast feeding occurs, which

should minimize the amount of error introduced into the analysis from the steady-state

assumption. Similarly, because the maternal body burdens should approach steady-state

concentrations at the end of a 9-year exposure period, the amount of error introduced into the

analysis by not accounting for reductions in maternal body burdens resulting from breast-feeding
Section 8.0 Human Exposure Factors

8-29

losses is expected to be small as well. As shown in the sensitivity analysis presented in U.S.

EPA (1998), considering maternal body burden losses from breast feeding impacts model results

primarily during the initial stages of maternal exposure (i.e., body burdens are low when the

breast milk loss mechanism is most significant).

8.6 References

Bickel, P.J., and K.A. Doksum. 1977. Mathematical Statistics. Holden-Bay, San Francisco,

CA.

Box, G.E.P. 1976. Science and statistics. Journal of the American Statistical Association,

71(356):791-799. December

Burmaster, D.E, and K.M. Thompson. 1998. Fitting second-order parametric distributions to

data using maximum likelihood estimation. Human and Ecological Risk Assessment

4(2):319-339.

Hattis, D., and D. Burmaster. 1994. Assessment of variability and uncertainty distributions for

practical risk analyses. Risk Analysis 14(5):713-730.

Huber, P.J. 1967. The behavior of maximum likelihood estimates under nonstandard

conditions. In: Proceedings of the Fifth Berkeley Symposium in Mathematical Statistics,

University of California Press.

Jennrich, R.I., and R.H. Moore. 1975. Maximum likelihood estimation by nonlinear least

squares. In: Statistical Computing Section Proceedings of American Statistical

Association. pp. 57-65.

Jennrich, R.I., and M.L. Ralston. 1979. Fitting nonlinear models to data. Ann Rev Biophys

Bioeng 8:195-238.

McKone, T. E. 1987. Human exposure to volatile organic compounds in household tap water:

the indoor inhalation pathway. Environmental Science and Technology, 21(12):1194-

1201.

Myers, L., J. Lashley, and R. Whitmore. 1998. Development of Statistical Distribution for

Exposure Factors. Prepared for U.S. EPA’s Office of Research and Development. 

Research Triangle Institute, Research Triangle Park, NC.

RTI (Research Triangle Institute). 1998. Development of Statistical Distributions for Exposure

Factors. (Final Report). U.S. Environmental Protection Agency, Office of Research and

Development. Research Triangle Institute, RTP, NC. March 18.

U.S. Environmental Protection Agency (EPA). 1992. Guidelines for Exposure Assessment.

Office of Research and Development, Office of Health and Environmental Assessment,

Washington, DC.
Section 8.0 Human Exposure Factors

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U.S. Environmental Protection Agency (EPA). 1997a. Exposure Factors Handbook, Volume I,

General Factors. EPA/600/P-95/002Fa. Office of Research and Development,

Washington, DC. August. 

U.S. Environmental Protection Agency (EPA). 1997b. Exposure Factors Handbook, Volume II,

Food Ingestion Factors. EPA/600/P-95/002Fb. Office of Research and Development,

Washington, DC. August. 

U.S. Environmental Protection Agency (EPA). 1997c. Exposure Factors Handbook, Volume

III, Activity Factors. EPA/600/P-95/002Fc. Office of Research and Development,

Washington, DC. August. 

U.S. Environmental Protection Agency (EPA). 1998. Methodology for Assessing Health Risks

Associated with Multiple Pathways of Exposure to Combustor Emissions.

EPA/600/P-98/137. National Center for Environmental Assessment, Cincinnati, OH. 

U.S. Environmental Protection Agency (EPA). 2000a. Exposure and Human Health

Reassessment of 2,3,7,8-Tetrachlorodibenzo-p-Dioxin (TCDD) and Related Compounds

and Related Compounds. Part I: Estimating Exposure to Dioxin-Like Compounds. 

Volume 3: Properties, Environmental Levels, and Background Exposures. Draft Final

Report. EPA/600/P-00/001Bc. Exposure Assessment and Risk Characterization Group,

National Center for Environmental Assessment, Office of Research and Development,

Washington, DC. September 2000.

U.S. Environmental Protection Agency (EPA). 2000b. Exposure and Human Health

Reassessment of 2,3,7,8-Tetrachlorodibenzo-p-Dioxin (TCDD) and Related Compounds

and Related Compounds. Part II: Health Assessment for 2,3,7,8-Tetrachlorodibenzo-pdioxin (TCDD) and Related Compounds. Chapters 1 through 7. Draft Final Report. 

EPA/600/P-00/001Be. Exposure Assessment and Risk Characterization Group, National

Center for Environmental Assessment, Office of Research and Development,

Washington, DC. September 2000.

White, H. 1982. Maximum likelihood estimation of misspecified models. Econometrica

50(1):1-25.
Appendix 8A Human Exposure Factors

8-31

Appendix 8A

Human Exposure Factors

Table 8A-1. 3MRA Exposure Factor Raw Data: Descriptive Statistics by Standardized 

Age Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-33
Appendix 8A Human Exposure Factors

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8-33

Table 8A-1. 3MRA Exposure Factor Raw Data: Descriptive Statistics by Standardized Age 

Parameter Age Cohort N Avg SDev Units P01 P02 P05 P10 P15 P25 P50 P75 P85 P9

beef 6-11 38 3.77 3.662 g WW/kg/d 0.663 0.753 1.32 2.11 4.43 1beef 12-19 41 1.72 1.044 g WW/kg/d 0.478 0.513 0.896 1.51 2.44 3beef farmer 182 2.63 2.644 g WW/kg/d 0.27 0.394 0.585 0.896 1.64 3.25 5bodywt <1 356 9.102 1.287 kg 7.053 7.451 7.852 8.252 9.151 9.752 10.4 10bodywt 1-5 3,762 15.52 3.719 kg 12.5 13.1 13.45 14.03 15.26 16.67 17.58 18bodywt 6-11 1,725 30.84 9.561 kg 22.79 24.05 25.07 26.44 29.58 33.44 36.82 39bodywt 12-19 2,615 58.45 13.64 kg 43.84 46.52 48.31 50.94 56.77 63.57 68.09 71bodywt 20+ 12,504 71.41 15.45 kg 52.86 55.98 58.21 61.69 69.26 78.49 84.92 89drinkH2O <1 403 302.7 258.5 mL/d 100.3 255.4 413.2 66

drinkH2O 1-5 3,200 697.1 401.5 mL/d 51.62 187.6 273.5 419.2 616.5 900.8 1drinkH2O 6-11 2,405 787 417 mL/d 68 241 318 484 731 1016 1drinkH2O 12-19 5,801 963.2 560.6 mL/d 65.15 241.4 353.8 574.4 868.5 1247 1drinkH2O 20+ 13,394 1384 721.6 mL/d 207.6 457.5 607.3 899.6 1275 1741 2expfruit 1-5 49 2.6 3.947 g WW/kg/d 0.373 1 1.82 2.64 5expfruit 6-11 68 2.52 3.496 g WW/kg/d 0.171 0.373 0.619 1.11 2.91 6expfruit 12-19 50 1.33 1.457 g WW/kg/d 0.123 0.258 0.404 0.609 2.27 3expfruit farmer 112 2.32 2.646 g WW/kg/d 0.072 0.276 0.371 0.681 1.3 3.14 expfruit home gard. 596 1.55 2.226 g WW/kg/d 0.042 0.158 0.258 0.449 0.878 1.73 3expveg 1-5 105 2.453 2.675 g WW/kg/d 0.102 0.37 0.833 1.459 3.226 6.expveg 6-11 134 1.39 2.037 g WW/kg/d 0.044 0.094 0.312 0.643 1.6 3expveg 12-19 143 1.07 1.128 g WW/kg/d 0.029 0.142 0.304 0.656 1.46 2expveg farmer 207 2.17 2.316 g WW/kg/d 0.184 0.372 0.647 1.38 2.81 6expveg home gard. 1,361 1.57 2.029 g WW/kg/d 0.003 0.089 0.168 0.413 0.889 1.97 3milk <1 20 62.74 12.52 g WW/kg/d 0.61 24.68 45.78 91.12 13
8-34

Parameter Age Cohort N Avg SDev Units P01 P02 P05 P10 P15 P25 P50 P75 P85 P9

Table 8A-1. (continued) milk 1-5 40 23.71 3.838 g WW/kg/d 2.98 7.47 13.56 21.5 32.22 42

milk 6-11 20 13.33 1.181 g WW/kg/d 1.81 3.54 6.72 11.88 18.58 25milk 12-19 20 6.293 0.657 g WW/kg/d 0.27 0.61 2.31 5.29 9.2 12milk farmer 63 17.1 15.8 g WW/kg/d 0.736 3.18 9.06 12.1 20.4 3profruit 12-19 20 2.96 4.441 g WW/kg/d 0.16 0.283 0.393 1.23 2.84 7profruit 20+ 106 5.338 7.174 g WW/kg/d 0.276 0.342 0.82 2.127 8.022 15profruit all ages 173 5.74 8.221 g WW/kg/d 0.15 0.266 0.335 0.933 2.34 7.45 profruit home gard. 146 5.9 8.422 g WW/kg/d 0.117 0.265 0.335 1.16 2.42 7.46 proveg 1-5 53 1.76 1.79 g WW/kg/d 0.265 0.408 0.829 1.397 2.066 3.proveg 6-11 63 1.1 1.064 g WW/kg/d 0.208 0.318 0.387 0.791 1.31 2proveg 12-19 51 0.776 0.622 g WW/kg/d 0.161 0.239 0.354 0.583 0.824 1proveg farmer 142 1.3 1.728 g WW/kg/d 0.087 0.166 0.209 0.337 0.599 1.4 3proveg home gard. 602 1.01 1.161 g WW/kg/d 0.103 0.153 0.192 0.336 0.642 1.21 2rootveg 1-5 45 1.886 2.371 g WW/kg/d 0.081 0.167 0.291 0.686 2.653 5.rootveg 6-11 67 1.32 1.752 g WW/kg/d 0.014 0.036 0.232 0.523 1.63 3rootveg 12-19 76 0.937 1.037 g WW/kg/d 0.008 0.068 0.269 0.565 1.37 2rootveg farmer 136 1.39 1.469 g WW/kg/d 0.111 0.158 0.184 0.365 0.883 1.85 3rootveg home gard. 682 1.15 1.494 g WW/kg/d 0.005 0.036 0.117 0.258 0.674 1.5 2showerT all ages 3,547 min 3 4 5 10 15 20 Avg = average; N = number of samples; P01-P99 = percentiles; SDev = standard deviation. Source: Exposure Factors Handbook (U.S. EPA, 1997a, 1997b, 1997c).
Appendix 8B Population-Estimated Averages, Standard Deviations, and Coefficients of Variation

8-35

Appendix 8B

Population-Estimated Averages, Standard

Deviations, and Coefficients of Variation

Table 8B-1. Population-Estimated Averages, Standard Deviations, and Coefficients of

Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-37
Appendix 8B Population-Estimated Averages, Standard Deviations, and Coefficients of Variation

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8-37

Table 8B-1. Population-Estimated Averages, Standard Deviations, and Coefficients of Var

Parameter Age Cohort N First Data Mean GAM Mean LOG Mean WEI Mean Data SDev GAM SDev LOG SDev WEI CV Data CV 

beef 6-11 38 lognormal 3.77 3.83 3.88 3.86 3.66 3.48 4.71 3.67 0.97 beef 12-19 41 gamma 1.72 1.77 1.82 1.76 1.04 1.12 1.41 1.07 0.61 beef farmer 182 lognormal 2.63 2.47 2.5 2.49 2.64 2.02 2.69 2.09 1.01 bodywt <1 356 gamma 9.1 9.09 9.09 9.04 1.29 1.23 1.25 1.31 0.14 bodywt 1-5 3,762 lognormal 15.5 15.5 15.5 15.4 3.72 2.05 2.05 2.35 0.24 bodywt 6-11 1,725 lognormal 30.8 30.7 30.7 30.4 9.56 5.94 5.96 6.87 0.31 bodywt 12-19 2,615 lognormal 58.5 58.1 58.2 57.7 13.6 10.2 10.2 11.6 0.23 bodywt 20+ 12,504 lognormal 71.4 71.2 71.2 70.7 15.5 13.2 13.3 14.8 0.22 drinkH2O <1 403 Weibull 303 304 333 302 259 271 404 261 0.85 

drinkH2O 1-5 3,200 gamma 697 698 719 698 401 406 510 390 0.58 drinkH2O 6-11 2,405 gamma 787 787 808 787 417 430 530 408 0.53 drinkH2O 12-19 5,801 gamma 963 965 1,000 964 561 574 739 546 0.58 drinkH2O 20+ 13,394 gamma 1,384 1,383 1,405 1,382 722 703 821 688 0.52 

expfruit 1-5 49 gamma 2.6 2.25 2.46 2.25 3.95 1.89 2.91 1.84 1.52 expfruit 6-11 68 lognormal 2.52 2.63 2.78 2.63 3.5 2.9 5.12 3.16 1.39 expfruit 12-19 50 lognormal 1.33 1.43 1.54 1.44 1.46 1.44 2.44 1.51 1.1 expfruit farmer 112 lognormal 2.32 2.24 2.36 2.24 2.65 2.1 3.33 2.18 1.14 expfruit home gard. 596 lognormal 1.55 1.51 1.57 1.52 2.23 1.47 2.3 1.58 1.44 expveg 1-5 105 gamma 2.45 2.55 3.06 2.56 2.68 2.58 5.61 2.65 1.09 expveg 6-11 134 lognormal 1.39 1.4 1.64 1.39 2.04 1.66 3.95 1.81 1.47 
8-38

Parameter Age Cohort N First Data Mean GAM Mean LOG Mean WEI Mean Data SDev GAM SDev LOG SDev WEI CV Data CV 

Table 8B-1. (continued) expveg 12-19 143 gamma 1.07 1.08 1.32 1.08 1.13 1.13 2.69 1.15 1.05 

expveg farmer 207 lognormal 2.17 2.22 2.38 2.22 2.32 2.13 3.5 2.18 1.07 expveg home gard. 1,361 Weibull 1.57 1.57 1.95 1.57 2.03 .68 4.27 1.76 1.29 fish all ages 1,053 lognormal 6.4 5.24 6.48 5.45 8.3 19.9 9.79 milk <1 20 Weibull 62.7 172 65.4 12.5 1025 78.7 0.2 milk 1-5 40 Weibull 23.7 23.9 25.8 23.6 3.84 16 23.4 14.3 0.16 milk 6-11 20 Weibull 13.3 13.4 14.5 13.3 1.18 9.51 14 8.7 0.09 milk 12-19 20 Weibull 6.29 6.28 8.17 6.23 0.66 5.9 14.9 5.49 0.1 milk farmer 63 Weibull 17.1 16.4 19.8 16.3 15.8 13.9 28.3 13.1 0.92 profruit 12-19 20 lognormal 2.96 2.62 2.91 2.62 4.44 3.05 6.39 3.36 1.5 profruit 20+ 106 lognormal 5.34 5.46 6.67 5.49 7.17 6.59 17.7 7.28 1.34 profruit all ages 173 lognormal 5.74 5.76 6.5 5.7 8.22 6.83 15.9 7.46 1.43 profruit home gard. 146 lognormal 5.9 5.78 6.63 5.75 8.42 6.72 15.7 7.29 1.43 proveg 1-5 53 lognormal 1.76 1.81 1.88 1.82 1.79 1.46 1.98 1.53 1.02 proveg 6-11 63 lognormal 1.1 1.04 1.07 1.04 1.06 0.8 1.04 0.82 0.97 proveg 12-19 51 lognormal 0.78 0.76 0.77 0.77 0.62 0.56 0.69 0.58 0.8 proveg farmer 142 lognormal 1.3 1.32 1.27 1.32 1.73 1.35 1.85 1.46 1.33 proveg home gard. 602 lognormal 1.01 1.01 1.01 1.01 1.16 0.88 1.19 0.93 1.15 rootveg 1-5 45 lognormal 1.89 1.95 2.31 1.95 2.37 2.37 6.05 2.63 1.26 rootveg 6-11 67 Weibull 1.32 1.35 2.3 1.38 1.75 1.78 10.6 2.07 1.33 
8-39

Parameter Age Cohort N First Data Mean GAM Mean LOG Mean WEI Mean Data SDev GAM SDev LOG SDev WEI CV Data CV 

Table 8B-1. (continued) rootveg 12-19 76 Weibull 0.94 1.7 0.99 1.04 5.97 1.19 1.11 

rootveg farmer 136 lognormal 1.39 1.39 1.45 1.39 1.47 1.31 2.06 1.36 1.06 rootveg home gard. 682 Weibull 1.15 1.15 1.49 1.15 1.49 1.26 3.61 1.32 1.3 showerT all ages 3,547 gamma 16.7 16.9 16.8 9.91 11.8 10.1 CV = coefficient of variation; CV = SDev/avg. GAM = gamma; LOG = lognormal; N = number of samples; SDev = standard deviation; WEI = Weibull.
8-40

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Appendix 8C Crystal Ball-Estimated Location and Scale Parameters

8-41

Appendix 8C

Crystal Ball-Estimated Location and

Scale Parameters

Table 8C-1. Crystal Ball-Estimated Location and Scale Parameters . . . . . . . . . . . . . . . . . . . 8-43
Appendix 8C Crystal Ball-Estimated Location and Scale Parameters

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Appendix 8C Crystal Ball-Estimated Location and Scale Parameters

8-43

Table 8C-1. Crystal Ball-Estimated Location and Scale Parameters

Parameter

Age

Cohort N First

LOG

Mean

LOG

SDev

GAMMA

SCALE

ALPHA

GAMMA

SHAPE

BETA

WEI

SCALE

ALPHA

WEI

SHAPE

BETA

beef 6-11 38 lognormal 3.88 4.71 3.15 1.22 3.94 1.05

beef 12-19 41 gamma 1.82 1.41 0.71 2.47 1.97 1.7

beef farmer 182 lognormal 2.5 2.69 1.66 1.49 2.64 1.19

bodywt <1 356 gamma 9.09 1.25 0.17 54.22 9.59 8.2

bodywt 1-5 3,762 lognormal 15.5 2.05 0.27 57.14 16.36 7.75

bodywt 6-11 1,725 lognormal 30.7 5.96 1.15 26.63 33.1 5.08

bodywt 12-19 2,615 lognormal 58.2 10.2 1.78 32.76 62.32 5.75

bodywt 20+ 12,504 lognormal 71.2 13.3 2.43 29.23 76.53 5.51

drinkH2O <1 403 Weibull 333 404 242.5 1.25 318.6 1.16

drinkH2O 1-5 3,200 gamma 719 510 236.55 2.95 785.66 1.86

drinkH2O 6-11 2,405 gamma 808 530 235.09 3.35 887.76 2.02

drinkH2O 12-19 5,801 gamma 1,000 739 341.82 2.82 1,084.4 1.83

drinkH2O 20+ 13,394 gamma 1,405 821 356.85 3.88 1,560.8 2.11

expfruit 1-5 49 gamma 2.46 2.91 1.58 1.43 2.4 1.23

expfruit 6-11 68 lognormal 2.78 5.12 3.2 0.82 2.39 0.84

expfruit 12-19 50 lognormal 1.54 2.44 1.45 0.99 1.41 0.95

expfruit farmer 112 lognormal 2.36 3.33 1.96 1.14 2.27 1.03

expfruit home gard. 596 lognormal 1.57 2.3 1.44 1.05 1.49 0.96

expveg 1-5 105 gamma 3.06 5.61 2.62 0.97 2.51 0.96

expveg 6-11 134 lognormal 1.64 3.95 1.97 0.71 1.21 0.78

expveg 12-19 143 gamma 1.32 2.69 1.19 0.91 1.05 0.94

expveg farmer 207 lognormal 2.38 3.5 2.05 1.08 2.24 1.02

expveg home gard. 1,361 Weibull 1.95 4.27 1.81 0.87 1.48 0.89

fish all ages 1,053 lognormal 6.48 19.9 13.15 0.4 3.54 0.59

milk <1 20 Weibull 172 1025 59.39 0.83

milk 1-5 40 Weibull 25.8 23.4 10.67 2.24 26.47 1.7

milk 6-11 20 Weibull 14.5 14 6.74 1.99 14.82 1.56

milk 12-19 20 Weibull 8.17 14.9 5.55 1.13 6.52 1.14

milk farmer 63 Weibull 19.8 28.3 11.84 1.38 17.45 1.25

profruit 12-19 20 lognormal 2.91 6.39 3.56 0.74 2.28 0.79

profruit 20+ 106 lognormal 6.67 17.7 7.95 0.69 4.68 0.76

(continued)
Table 8C-1. (continued)

Appendix 8C Crystal Ball-Estimated Location and Scale Parameters

Parameter

Age

Cohort N First

LOG

Mean

LOG

SDev

GAMMA

SCALE

ALPHA

GAMMA

SHAPE

BETA

WEI

SCALE

ALPHA

WEI

SHAPE

BETA

8-44

profruit all ages 173 lognormal 6.5 15.9 8.11 0.71 4.91 0.77

profruit home gard. 146 lognormal 6.63 15.7 7.81 0.74 5.05 0.8

proveg 1-5 53 lognormal 1.88 1.98 1.17 1.54 1.93 1.2

proveg 6-11 63 lognormal 1.07 1.04 0.61 1.7 1.13 1.29

proveg 12-19 51 lognormal 0.77 0.69 0.41 1.86 0.84 1.33

proveg farmer 142 lognormal 1.27 1.85 1.39 0.95 1.25 0.9

proveg home gard. 602 lognormal 1.01 1.19 0.77 1.3 1.04 1.09

rootveg 1-5 45 lognormal 2.31 6.05 2.89 0.67 1.64 0.75

rootveg 6-11 67 Weibull 2.3 10.6 2.35 0.57 1.06 0.68

rootveg 12-19 76 Weibull 1.7 5.97 0.91 0.84

rootveg farmer 136 lognormal 1.45 2.06 1.24 1.12 1.4 1.02

rootveg home gard. 682 Weibull 1.49 3.61 1.39 0.83 1.07 0.87

showerT all ages 3,547 gamma 16.9 11.8 5.89 2.83 18.78 1.71

LOG = lognormal; N = number of samples; SDev = standard deviation; WEI = Weibull.
Appendix 8D Population-Estimated Location and Scale Parameters

8-45

Appendix 8D

Population-Estimated Location and

Scale Parameters

Table 8D-1. Population-Estimated Location and Scale Parameters . . . . . . . . . . . . . . . . . . 8-47
Appendix 8D Population-Estimated Location and Scale Parameters

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Appendix 8D Population-Estimated Location and Scale Parameters

8-47

Table 8D-1. Population-Estimated Location and Scale Parameters

Parameter

Age

Cohort N First

GAM

 LOC

LOG

 LOC

WEI

 LOC

GAM

SCALE

LOG

SCALE

WEI

SCALE

beef 12-19 41 gamma 0.4 0.36 0.68 0.58 0.69 0.59

beef farmer 182 lognormal 0.65 0.53 0.97 0.72 0.88 0.84

bodywt <1 356 gamma 2.2 2.2 2.26 0.14 0.14 0.12

bodywt 1-5 3,762 lognormal 2.73 2.73 2.79 0.13 0.13 0.13

bodywt 6-11 1,725 lognormal 3.41 3.41 3.5 0.19 0.19 0.2

bodywt 12-19 2,615 lognormal 4.05 4.05 4.13 0.17 0.17 0.17

bodywt 20+ 12,504 lognormal 4.25 4.25 4.34 0.18 0.18 0.18

drinkH2O <1 403 Weibull 5.42 5.35 5.76 0.77 0.95 0.86

drinkH2O 1-5 3,200 gamma 6.4 6.37 6.67 0.54 0.64 0.54

drinkH2O 6-11 2,405 gamma 6.54 6.52 6.79 0.51 0.6 0.5

drinkH2O 12-19 5,801 gamma 6.72 6.69 6.99 0.55 0.66 0.55

drinkH2O 20+ 13,394 gamma 7.12 7.1 7.35 0.48 0.54 0.47

expfruit 1-5 49 gamma 0.55 0.46 0.88 0.73 0.94 0.82

expfruit 6-11 68 lognormal 0.57 0.28 0.87 0.89 1.22 1.2

expfruit 12-19 50 lognormal 0.01 -0.2 0.34 0.84 1.12 1.05

expfruit farmer 112 lognormal 0.49 0.31 0.82 0.79 1.05 0.97

expfruit home gard. 596 lognormal 0.08 -0.12 0.4 0.82 1.07 1.04

expveg 1-5 105 gamma 0.58 0.38 0.92 0.84 1.21 1.04

expveg 6-11 134 lognormal -0.1 -0.47 0.19 0.94 1.39 1.28

expveg 12-19 143 gamma -0.3 -0.54 0.04 0.86 1.28 1.07

expveg farmer 207 lognormal 0.47 0.29 0.81 0.81 1.07 0.98

expveg home gard. 1,361 Weibull 0.07 -0.21 0.39 0.88 1.32 1.12

fish all ages 1,053 lognormal 1.03 0.7 1.27 1.12 1.53 1.69

milk <1 20 Weibull NA 3.35 4.08 NA 1.9 1.2

milk 1-5 40 Weibull 2.99 2.95 3.28 0.61 0.77 0.59

milk 6-11 20 Weibull 2.39 2.34 2.7 0.64 0.81 0.64

milk 12-19 20 Weibull 1.52 1.37 1.88 0.8 1.21 0.88

milk farmer 63 Weibull 2.53 2.43 2.86 0.74 1.06 0.8

(continued)
Table 8D-1. (continued)

Appendix 8D Population-Estimated Location and Scale Parameters

Parameter

Age

Cohort N First

GAM

 LOC

LOG

 LOC

WEI

 LOC

GAM

SCALE

LOG

SCALE

WEI

SCALE

8-48

profruit 12-19 20 lognormal 0.53 0.19 0.83 0.93 1.33 1.27

profruit 20+ 106 lognormal 1.25 0.86 1.54 0.95 1.44 1.31

profruit all ages 173 lognormal 1.31 0.9 1.59 0.94 1.39 1.29

profruit home gard. 146 lognormal 1.33 0.95 1.62 0.92 1.37 1.26

proveg 1-5 53 lognormal 0.34 0.26 0.66 0.71 0.86 0.83

proveg 6-11 63 lognormal -0.19 -0.27 0.12 0.68 0.81 0.77

proveg 12-19 51 lognormal -0.49 -0.55 -0.18 0.66 0.76 0.75

proveg farmer 142 lognormal -0.08 -0.33 0.23 0.85 1.07 1.11

proveg home gard. 602 lognormal -0.28 -0.42 0.04 0.75 0.93 0.92

rootveg 1-5 45 lognormal 0.21 -0.19 0.49 0.95 1.44 1.33

rootveg 6-11 67 Weibull -0.21 -0.72 0.06 1.01 1.76 1.46

rootveg 12-19 76 Weibull NA -0.76 -0.1 NA 1.61 1.19

rootveg farmer 136 lognormal 0.01 -0.18 0.34 0.8 1.05 0.98

rootveg home gard. 682 Weibull -0.26 -0.56 0.07 0.89 1.39 1.15

showerT all ages 3,547 gamma 2.66 2.63 2.93 0.55 0.63 0.59

GAM = gamma; LOC = location; LOG = lognormal; N = number of samples; NA = not available; WEI = Weibull.
Appendix 8E Population-Estimated Standard Errors of Location Parameters

8-49

Appendix 8E

Population-Estimated Standard Errors of

Location Parameters

Table 8E-1. Population-Estimated Standard Errors of Location Parameters . . . . . . . . . . . . . 8-51
Appendix 8E Population-Estimated Standard Errors of Location Parameters

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Appendix 8E Population-Estimated Standard Errors of Location Parameters

8-51

Table 8E-1. Population-Estimated Standard Errors of Location Parameters

Parameter

Age

Cohort N First

MLE ASY

GAM LOC

STD ERR

REGRESS

GAM LOC

STD ERR

MLE ASY

LOG LOC

STD ERR

REGRESS

LOG LOC

STD ERR

MLE ASY

WEI LOC

STD ERR

REGRESS

WEI LOC

STD ERR

beef 6-11 38 lognormal 0.157 0.252 0.159 0.207 0.169 0.272

beef 12-19 41 gamma 0.104 0.25 0.11 0.283 0.1 0.235

beef farmer 182 lognormal 0.062 0.102 0.066 0.054 0.067 0.119

bodywt <1 356 gamma 0.007 0.012 0.007 0.013 0.007 0.013

bodywt 1-5 3,762 lognormal 0.002 0.007 0.002 0.006 0.002 0.019

bodywt 6-11 1,725 lognormal 0.005 0.021 0.005 0.017 0.005 0.038

bodywt 12-19 2,615 lognormal 0.003 0.014 0.003 0.011 0.004 0.03

bodywt 20+ 12,504 lognormal 0.002 0.012 0.002 0.009 0.002 0.028

drinkH2O <1 403 Weibull 0.047 0.09 0.05 0.16 0.047 0.084

drinkH2O 1-5 3,200 gamma 0.01 0.047 0.011 0.275 0.01 0.038

drinkH2O 6-11 2,405 gamma 0.011 0.05 0.012 0.277 0.011 0.027

drinkH2O 12-19 5,801 gamma 0.008 0.048 0.009 0.286 0.008 0.032

drinkH2O 20+ 13,394 gamma 0.004 0.025 0.005 0.085 0.004 0.044

expfruit 1-5 49 gamma 0.125 0.145 0.138 0.195 0.127 0.147

expfruit 6-11 68 lognormal 0.14 0.26 0.151 0.146 0.157 0.267

expfruit 12-19 50 lognormal 0.15 0.278 0.164 0.252 0.162 0.3

expfruit farmer 112 lognormal 0.092 0.183 0.101 0.109 0.099 0.18

expfruit home gard. 596 lognormal 0.041 0.311 0.045 0.082 0.046 0.241

expveg 1-5 105 gamma 0.103 0.139 0.121 0.213 0.109 0.146

expveg 6-11 134 lognormal 0.106 0.238 0.122 0.131 0.119 0.177

expveg 12-19 143 gamma 0.09 0.111 0.109 0.258 0.096 0.117

expveg farmer 207 lognormal 0.069 0.127 0.076 0.137 0.074 0.136

expveg home gard. 1,361 Weibull 0.03 0.106 0.037 1.22 0.033 0.102

fish all ages 1,053 lognormal 0.051 0.149 0.058 0.072 0.061 0.123

milk <1 Weibull 0.882 0.41

milk 1-5 Weibull 0.086 0.189 0.056

milk 6-11 Weibull 0.058 0.144 0.033

milk 12-19 Weibull 0.11 0.296 0.099

milk farmer 63 Weibull 0.111 0.2 0.137 0.388 0.109 0.191

profruit 12-19 20 lognormal 0.273 0.308 0.304 0.219 0.307 0.317

profruit 20+ 106 lognormal 0.123 0.231 0.144 0.202 0.138 0.237

profruit all ages 173 lognormal 0.094 0.219 0.108 0.164 0.106 0.207

(continued)
Table 8E-1. (continued)

Appendix 8E Population-Estimated Standard Errors of Location Parameters

Parameter

Age

Cohort N First

MLE ASY

GAM LOC

STD ERR

REGRESS

GAM LOC

STD ERR

MLE ASY

LOG LOC

STD ERR

REGRESS

LOG LOC

STD ERR

MLE ASY

WEI LOC

STD ERR

REGRESS

WEI LOC

STD ERR

8-52

profruit home gard. 146 lognormal 0.1 0.226 0.116 0.188 0.112 0.217

proveg 1-5 53 lognormal 0.116 0.184 0.121 0.134 0.124 0.202

proveg 6-11 63 lognormal 0.1 0.173 0.105 0.138 0.105 0.196

proveg 12-19 51 lognormal 0.108 0.142 0.11 0.111 0.115 0.167

proveg farmer 142 lognormal 0.089 0.234 0.091 0.114 0.101 0.238

proveg home gard. 602 lognormal 0.037 0.174 0.039 0.051 0.04 0.176

rootveg 1-5 45 lognormal 0.191 0.252 0.22 0.188 0.216 0.257

rootveg 6-11 67 Weibull 0.169 0.172 0.221 0.274 0.193 0.169

rootveg 12-19 76 Weibull 0.19 0.491 0.147 0.174

rootveg farmer 136 lognormal 0.083 0.144 0.092 0.116 0.09 0.152

rootveg home gard. 682 Weibull 0.043 0.088 0.054 0.321 0.047 0.083

showerT all ages 3,547 gamma 0.01 0.092 0.011 0.093 0.011 0.12

GAM = gamma; LOC = location; LOG = lognormal; MLE = maximum likelihood estimation; N = number of samples; REGRESS =

regression; STD ERR = standard error; WEI = Weibull.
Appendix 8F Population-Estimated Standard Errors of Scale Parameters

8-53

Appendix 8F

Population-Estimated Standard Errors of Scale

Parameters and Estimated Correlations Between

Location and Scale Parameters

Table 8F-1. Population-Estimated Standard Errors of Scale Parameters and Estimated

Correlations Between Location and Scale Parameters . . . . . . . . . . . . . . . . . . 8-55
Appendix 8F Population-Estimated Standard Errors of Scale Parameters

8-54

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Table 8F-1. Population-Estimated Standard Errors of Scale Parameters and Estimated Correlations Between Location and Scale Parameters Parameter Age Cohort N First MLE ASY GAM SCA STD ERR REGRESS GAM SCA STD ERR MLE ASY LOG SCA STD ERR REGRESS LOG SCA STD ERR MLE ASY WEI SCA STD ERR REGRESS WEI SCA STD ERR 

C

beef 6-11 38 lognormal 0.239 0.384 0.132 0.171 0.144 0.232 beef 12-19 41 gamma 0.237 0.572 0.128 0.327 0.14 0.328 beef farmer 182 lognormal 0.101 0.165 0.055 0.044 0.061 0.109 bodywt <1 356 gamma 0.082 0.132 0.041 0.071 0.045 0.087 bodywt 1-5 3,762 lognormal 0.025 0.085 0.013 0.033 0.014 0.12 bodywt 6-11 1,725 lognormal 0.037 0.162 0.019 0.066 0.021 0.157 bodywt 12-19 2,615 lognormal 0.03 0.121 0.015 0.047 0.017 0.139 bodywt 20+ 12,504 lognormal 0.014 0.101 0.007 0.036 0.008 0.125 drinkH2O <1 403 Weibull 0.083 0.161 0.046 0.148 0.048 0.086 

drinkH2O 1-5 3,200 gamma 0.025 0.113 0.013 0.322 0.014 0.053 drinkH2O 6-11 2,405 gamma 0.029 0.128 0.016 0.347 0.017 0.041 

drinkH2O 12-19 5,801 gamma 0.018 0.111 0.01 0.324 0.011 0.045 drinkH2O 20+ 13,394 gamma 0.012 0.069 0.006 0.115 0.007 0.072 

expfruit 1-5 49 gamma 0.21 0.242 0.118 0.167 0.126 0.146 expfruit 6-11 68 lognormal 0.164 0.305 0.094 0.091 0.104 0.176 expfruit 12-19 50 lognormal 0.194 0.361 0.111 0.17 0.122 0.227 expfruit farmer 112 lognormal 0.126 0.251 0.071 0.076 0.08 0.144 
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Parameter Age Cohort N First MLE ASY GAM SCA STD ERR REGRESS GAM SCA STD ERR MLE ASY LOG SCA STD ERR REGRESS LOG SCA STD ERR MLE ASY WEI SCA STD ERR REGRESS WEI SCA STD ERR 

C

Table 8F-1. (continued) expfruit home gard. 596 lognormal 0.054 0.406 0.031 0.056 0.034 0.18 

expveg 1-5 105 gamma 0.13 0.176 0.077 0.135 0.083 0.111 expveg 6-11 134 lognormal 0.113 0.254 0.065 0.07 0.072 0.107 expveg 12-19 143 gamma 0.109 0.135 0.064 0.153 0.069 0.084 expveg farmer 207 lognormal 0.094 0.174 0.053 0.095 0.058 0.107 expveg home gard. 1,361 Weibull 0.035 0.124 0.021 0.691 0.022 0.07 fish all ages 1,053 lognormal 0.061 0.179 0.037 0.046 0.037 0.074 milk <1 Weibull 0.413 0.307 milk 1-5 Weibull 0.18 0.194 0.076 milk 6-11 Weibull 0.113 0.138 0.041 milk 12-19 Weibull 0.154 0.195 0.092 milk farmer 63 Weibull 0.175 0.315 0.103 0.292 0.108 0.189 profruit 12-19 20 lognormal 0.304 0.343 0.176 0.127 0.194 0.2 profruit 20+ 106 lognormal 0.131 0.246 0.078 0.11 0.086 0.147 profruit all ages 173 lognormal 0.099 0.23 0.057 0.086 0.063 0.124 profruit home gard. 146 lognormal 0.107 0.241 0.062 0.1 0.069 0.133 proveg 1-5 53 lognormal 0.197 0.313 0.106 0.117 0.119 0.194 proveg 6-11 63 lognormal 0.182 0.316 0.098 0.129 0.108 0.202 proveg 12-19 51 lognormal 0.205 0.269 0.11 0.111 0.121 0.175 
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Parameter Age Cohort N First MLE ASY GAM SCA STD ERR REGRESS GAM SCA STD ERR MLE ASY LOG SCA STD ERR REGRESS LOG SCA STD ERR MLE ASY WEI SCA STD ERR REGRESS WEI SCA STD ERR 

C

Table 8F-1. (continued) proveg farmer 142 lognormal 0.112 0.294 0.063 0.078 0.069 0.164 

proveg home gard 602 lognormal 0.056 0.264 0.03 0.04 0.034 0.149 rootveg 1-5 45 lognormal 0.199 0.263 0.118 0.1 0.13 0.154 rootveg 6-11 67 Weibull 0.156 0.159 0.097 0.12 0.105 0.092 rootveg 12-19 76 Weibull . . 0.094 0.244 0.099 0.116 . rootveg farmer 136 lognormal 0.116 0.2 0.064 0.08 0.071 0.12 rootveg home gard 682 Weibull 0.049 0.099 0.029 0.171 0.031 0.055 showerT all ages 3,547 gamma 0.024 0.216 0.013 0.108 0.014 0.156 CORR = correlation; GAM = gamma; LOC = location; LOG = lognormal; MLE = maximum likelihood estimate; N = number of samples; REGRESS = r

ERR = standard error; WEI = Weibull. 
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Appendix 8G Parameterization Model

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Appendix 8G

Parameterization Model
Appendix 8G Parameterization Model

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Appendix 8G Parameterization Model

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(AG-1)

(AG-2)

Appendix 8G. Parameterization Model

Denote the lognormal location and scale parameters by ML and LSL, and let SL =

exp(LSL). Then the lognormal variate is exp(X), where X has a normal distribution with mean

ML and standard deviation SL.

Denote the gamma location and scale parameters by MG and LSG, let a = exp(LSG), and

let b = [exp(MG)]/a. Then the gamma pdf is

Denote the Weibull location and scale parameters by MW and LSE, let SW = exp(LSW),

let a = exp(-MW/SW), and let b = 1/SW. Then the Weibull CDF is
Appendix 8G Parameterization Model

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