EPA/630/R-97/001
March 1997
Guiding Principles for Monte Carlo
Analysis
Technical Panel
Office of Prevention, Pesticides, and Toxic Substances
Michael Firestone (Chair) Penelope Fenner-Crisp
Office of Policy, Planning, and Evaluation
Timothy Barry
Office of Solid Waste and Emergency Response
David Bennett Steven Chang
Office of Research and Development
Michael Callahan
Regional Offices
AnneMarie Burke (Region I) Jayne Michaud (Region I)
Marian Olsen (Region II) Patricia Cirone (Region X)
Science Advisory Board Staff
Donald Barnes
Risk Assessment Forum Staff
William P. Wood Steven M. Knott
Risk Assessment Forum
U.S. Environmental Protection Agency
Washington, DC 20460
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DISCLAIMER
This document has been reviewed in accordance with U.S. Environmental Protection
Agency policy and approved for publication. Mention of trade names or commercial products
does not constitute endorsement or recommendation for use.
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TABLE OF CONTENTS
List of Figures iv
Preface v
Introduction 1
Fundamental Goals and Challenges 3
When a Monte Carlo Analysis Might Add Value to a Quantitative Risk Assessment 5
Key Terms and Their Definitions 6
Preliminary Issues and Considerations 9
Defining the Assessment Questions 9
Selection and Development of the Conceptual arid Mathematical Models 10
Selection and Evaluation of Available Data 10
Guiding Principles for Monte Carlo Analysis 11
Selecting Input Data and Distributions for Use in Monte Carlo Analysis 11
Evaluating Variability and Uncertainty 15
Presenting the Results of a Monte Carlo Analysis 17
Appendix A: Policy for Use of Probablistic Analysis in Risk Assessment at the U.S.
Environmental Protection Agency A-l
Appendix B: Probability Distribution Selection Issues B-l
References R-l
References Cited in Text R-l
References for Futher Reading R-3
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LIST OF FIGURES
la. Example Monte Carlo Estimate of the PDF for Lifetime Cancer Risk 20
Ib. Example Monte Carlo Estimate of the CDF for Lifetime Cancer Risk 20
2. Example Box and Whiskers Plot of the Distribution of Lifetime Cancer Risk 22
IV
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PREFACE
The U.S. Environmental Protection Agency (EPA) Risk Assessment Forum was
established to promote scientific consensus on risk assessment issues and to ensure that this
consensus is incorporated into appropriate risk assessment guidance. To accomplish this, the
Risk Assessment Forum assembles experts throughout EPA in a formal process to study and
report on these issues from an Agency-wide perspective. For major risk assessment activities,
the Risk Assessment Forum has established Technical Panels to conduct scientific reviews and
analyses. Members are chosen to assure that necessary technical expertise is available.
This report is part of a continuing effort to develop guidance covering the use of
probabilistic techniques in Agency risk assessments. This report draws heavily on the
recommendations from a May 1996 workshop organized by the Risk Assessment Forum that
convened experts and practitioners in the use of Monte Carlo analysis, internal as well as external
to EPA, to discuss the issues and advance the development of guiding principles concerning how
to prepare or review an assessment based on use of Monte Carlo analysis. The conclusions and
recommendations that emerged from these discussions are summarized in the report "Summary
Report for the Workshop on Monte Carlo Analysis" (EPA/630/R-96/010). Subsequent to the
workshop, the Risk Assessment Forum organized a Technical Panel to consider the workshop
recommendations and to develop an initial set of principles to guide Agency risk assessors hi the
use of probabilistic analysis tools including Monte Carlo analysis. It is anticipated that there will
be need for further expansion and revision of these guiding principles as Agency risk assessors
gain experience in their application.
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Introduction
The importance of adequately characterizing variability and uncertainty in fate, transport,
exposure, and dose-response assessments for human health and ecological risk assessments has
been emphasized in several U.S. Environmental Protection Agency (EPA) documents and
activities. These include:
the 1986 Risk Assessment Guidelines;
the 1992 Risk Assessment Council (RAC) Guidance (the Habicht memorandum);
the 1992 Exposure Assessment Guidelines; and
the 1995 Policy for Risk Characterization (the Browner memorandum}.
As a follow up to these activities EPA is issuing the policy presented hi Appendix A and
preliminary guidance on using probabilistic analysis. The policy documents the EPA's position
"that such probabilistic analysis techniques as Monte Carlo analysis, given adequate supporting
data and credible assumptions, can be viable statistical tools for analyzing variability and
uncertainty in risk assessments." The policy establishes conditions that are to be satisfied by risk
assessments that use probabilistic techniques. These conditions relate to the good scientific
practices of clarity, consistency, transparency, reproducibility, and the use of sound methods.
The EPA policy lists the following conditions for an acceptable risk assessment that uses
probabilistic analysis techniques. These conditions were derived from principles that are
presented later in this document and Appendix B. Therefore, after each condition, the relevant
principles are noted.
1. The purpose and scope of the assessment should be clearly articulated hi a "problem
formulation" section that includes a full discussion of any highly exposed or highly
susceptible subpopulations evaluated (e.g., children, the elderly, etc.). The questions
the assessment attempts to answer are to be discussed and the assessment endpoints
are to be well defined.
2. The methods used for the analysis (including all models used, all data upon which the
assessment is based, and all assumptions that have a significant impact upon the
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results) are to be documented and easily located in the report. This documentation is
to include a discussion of the degree to which the data used are representative of the
population under study. Also, this documentation is to include the names of the
models and software used to generate the analysis. Sufficient information is to be
provided to allow the results of the analysis to be independently reproduced.
(Principles 4, 5, 6, and 11)
3. The results of sensitivity analyses are to be presented and discussed in the report.
Probabilistic techniques should be applied to the compounds, pathways, and factors of
importance to the assessment, as determined by sensitivity analyses or other basic
requirements of the assessment. (Principles 1 and 2)
4. The presence or absence of moderate to strong correlations or dependencies between
the input variables is to be discussed and accounted for in the analysis, along with the
effects these have on the output distribution. (Principles 1 and 14)
5. Information for each input and output distribution is to be provided in the report.
This includes tabular and graphical representations of the distributions (e.g.,
probability density function and cumulative distribution function plots) that indicate
the location of any point estimates of interest (e.g., mean, median, 95th percentile).
The selection of distributions is to be explained and justified. For both the input and
output distributions, variability and uncertainty are to be differentiated where
possible. (Principles 3,7, 8,10,12, and 13)
6. The numerical stability of the central tendency and the higher end (i.e., tail) of the
output distributions are to be presented and discussed. (Principle 9)
7. Calculations of exposures and risks using deterministic (e.g., point estimate) methods
are to be reported if possible. Providing these values will allow comparisons between
the probabilistic analysis and past or screening level risk assessments. Further,
deterministic estimates may be used to answer scenario specific questions and to
facilitate risk communication. When comparisons are made, it is important to explain
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the similarities and differences in the underlying data, assumptions, and models.
(Principle 15).
8. Since fixed exposure assumptions (e.g., exposure duration, body weight) are
sometimes embedded in the toxicity metrics (e.g., Reference Doses, Reference
Concentrations, unit cancer risk factors), the exposure estimates from the probabilistic
output distribution are to be aligned with the toxicity metric.
The following sections present a general framework and broad set of principles important
for ensuring good scientific, practices in the use of Monte Carlo analysis (a frequently
encountered tool for evaluating uncertainty and variability). Many of the principles apply
generally to the various techniques for conducting quantitative analyses of variability and
uncertainty; however, the focus of the following principles is on Monte Carlo analysis. EPA
recognizes that quantitative risk assessment methods and quantitative variability and uncertainty
analysis are undergoing rapid development. These guiding principles are intended to serve as a
minimum set of principles and are not intended to constrain or prevent the use of new or
innovative improvements where scientifically defensible.
Fundamental Goals and Challenges
In the context of this policy, the basic goal of a Monte Carlo analysis is to chatacterize,
quantitatively, the uncertainty and variability in estimates of exposure or risk. A secondary goal
is to identify key sources of variability and uncertainty and to quantify the relative contribution
of these sources to the overall variance and range of model results.
Consistent with EPA principles and policies, an analysis of variability and uncertainty
should provide its audience with clear and concise information on the variability in individual
exposures and risks; it should provide information on population risk (extent of .harm in the
exposed population); it should provide information on the distribution of exposures and risks to
highly exposed or highly susceptible populations; it should describe qualitatively and
quantitatively the scientific uncertainly in the models applied, the data utilized, and the specific
risk estimates that are used.
Ultimately, the most important aspect of a quantitative variability and uncertainty
analysis may well be the process of interaction between the risk assessor, risk manager and other
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interested parties that makes risk assessment into a dynamic rather than a static process.
Questions for the risk assessor and risk manager to consider at the initiation of a quantitative
variability and uncertainty analysis include:
Will the quantitative analysis of uncertainty and variability improve the risk
assessment?
What are the major sources of variability and uncertainty? How will variability
and uncertainty be kept separate in the analysis?
Are there time and resources to complete a complex analysis?
Does the project warrant this level of effort?
Will a quantitative estimate of uncertainty improve the decision? How will the
regulatory decision be affected by this variability and uncertainty analysis?
What types of skills and experience are needed to perform the analysis?
Have the weaknesses and strengths of the methods been evaluated?
How will the variability and uncertainty analysis be communicated to the public
and decision makers?
One of the most important challenges facing the risk assessor is to communicate,
effectively, the insights an analysis of variability and uncertainty provides. It is important for the
risk assessor to remember that insights will generally be qualitative in nature even though the
models they derive from are quantitative. Insights can include:
An appreciation of the overall degree of variability and uncertainty and the
confidence that can be placed in the analysis and its findings.
An understanding of the key sources ofvariability and key sources of uncertainty
and their impacts on the analysis.
An understanding of the critical assumptions and their importance to the analysis
and findings.
An understanding of the unimportant assumptions and why they are unimportant.
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An understanding of the extent to which plausible alternative assumptions or
models could affect any conclusions.
An understanding of key scientific controversies related to the assessment and a
sense of what difference they might make regarding the conclusions.
The risk assessor should strive to present quantitative results in a manner that will clearly
communicate the information they contain.
When a Monte Carlo Analysis Might Add Value to a
Quantitative Risk Assessment
Not every assessment requires or warrants a quantitative characterization of variability
and uncertainty. For example, it may be unnecessary to perform a Monte Carlo analysis when
screening calculations show exposures or risks to be clearly below levels of concern (and the
screening technique is known to significantly over-estimate exposure). As another example, it
may be unnecessary to perform a Monte Carlo analysis when the costs of remediation are low.
On the other hand, there may be a number of situations in which a Monte Carlo analysis
may be useful. For example, a Monte Carlo analysis may be useful when screening calculations
using conservative point estimates fall above the levels of concern. Other situations could
include when it is necessary to disclose the degree of bias associated with point estimates of
exposure; when it is necessary to rank exposures, exposure pathways, sites or contaminants;
when the cost of regulatory or remedial action is high and the exposures are marginal; or when
the consequences of simplistic exposure estimates are unacceptable.
Often, a "tiered approach" may be helpful in deciding whether or not a Monte Carlo
analysis can add value to the assessment and decision. In a tiered approach, one begins with a
fairly simple screening level model and progresses to more sophisticated and realistic (and
usually more complex) models only as warranted by the findings and value added to the decision.
Throughout each of the steps in a tiered approach, soliciting input from each of the interested
parties is recommended. Ultimately, whether or not a Monte Carlo analysis should be conducted
is a matter of judgment, based on consideration of the intended use, the importance of the
exposure assessment and the value and insights it provides to the risk assessor, risk manager, and
other affected individuals or groups.
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Key Terms and Their Definitions
The following section presents definitions for a number of key terms which are used
throughout this document.
Bayesian
The Bayesian or subjective view is that the probability of an event is the degree of belief
that a person has, given some state of knowledge, that the event will occur. In the classical or
frequentist view, the probability of an event is the frequency with which an event occurs given a
long sequence of identical and independent trials. In exposure assessment situations, directly
representative and complete data sets are rarely available; inferences in these situations are
inherently subjective. The decision as to the appropriateness of either approach (Bayesian or
Classical) is based on the available data and the extent of subjectivity deemed appropriate.
Correlation, Correlation Analysis
Correlation analysis is an investigation of the measure of statistical association among
random variables based on samples. Widely used measures include the linear correlation
coefficient (also called the product-moment correlation coefficient or Pearson's correlation
coefficient), and such non-parametric measures as Spearman rank-order correlation coefficient,
and Kendall's tau. When the data are nonlinear, non-parametric correlation is generally
considered to be more robust than linear correlation.
Cumulative Distribution Function (CDF)
The CDF is alternatively referred to in the literature as the distribution function,
cumulative frequency function, or the cumulative probability function. The cumulative
distribution function, F(x), expresses the probability the random variable X assumes a value less
than or equal to some value x, F(x) = Prob (X x). For continuous random variables, the
cumulative distribution function is obtained from the probability density function by integration,
or by summation in the case of discrete random variables.
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Latin Hypercube Sampling
In Monte Carlo analysis, one of two sampling schemes are generally employed: simple
random sampling or Latin Hypercube sampling. Latin hypercube sampling may be viewed as a
stratified sampling scheme designed to ensure that the upper or lower ends of the distributions
used in the analysis are well represented. Latin hypercube sampling is considered to be more
efficient than simple random sampling, that is, it requires fewer simulations to produce the same
level of precision. Latin hypercube sampling is generally recommended over simple random
sampling when the model is complex or when time and resource constraints are an issue.
Monte Carlo Analysis, Monte Carlo Simulation
Monte Carlo Analysis is a computer-based method of analysis developed in the 1940's
that uses statistical sampling techniques in obtaining a probabilistic approximation to the solution
of a mathematical equation or model.
Parameter
Two distinct, but often confusing, definitions for parameter are used. In the first usage
(preferred), parameter refers to the constants characterizing the probability density function or
cumulative distribution function of a random variable. For example, if the random variable W is
known to be normally distributed with mean n and standard deviation , the characterizing
constants ji and are called parameters. In the second usage, parameter is defined as the constants
and independent variables which define a mathematical equation or model. For example, in the
equation Z = X + Y, the independent variables (X,Y) and the constants (,) are all parameters.
Probability Density Function (PDF)
The PDF is alternatively referred to in the literature as the probability function or the
frequency function. For continuous random variables, that is, the random variables which can
assume any value within some defined range (either finite or infinite), the probability density
function expresses the probability that the random variable falls within some very small interval.
For discrete random variables, that is, random variables which can only assume certain isolated
or fixed values, the term probability mass junction (PMF) is preferred over the term probability
density function. PMF expresses the probability that the random variable takes on a specific
value.
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Random Variable
A random variable is a quantity which can take on any number of values but whose exact
value cannot be known before a direct observation is made. For example, the outcome of the toss
of a pair of dice is a random variable, as is the height or weight of a person selected at random
from the New York City phone book.
Representativeness
Representativeness is the degree to which a sample is characteristic of the population for
which the samples are being used to make inferences.
Sensitivity, Sensitivity Analysis
Sensitivity generally refers to the variation in output of a mathematical model with
respect to changes hi the values of the model's input. A sensitivity analysis attempts to provide a
ranking of the model's input assumptions with respect to their contribution to model output
variability or uncertainty. The difficulty of a sensitivity analysis increases when the underlying
model is nonlinear, nonmonotonic or when the input parameters range over several orders of
magnitude. Many measures of sensitivity have been proposed. For example, the partial rank
correlation coefficient and standardized rank regression coefficient have been found to be useful.
Scatter plots of the output against each of the model inputs can be a very effective tool for
identifying sensitivities, especially when the relationships are nonlinear. For simple models or
for screening purposes, the sensitivity index can be helpful.
In a broader sense, sensitivity can refer to how conclusions may change if models, data,
or assessment assumptions are changed.
Simulation
In the context of Monte Carlo analysis, simulation is the process of approximating the
output of a model through repetitive random application of a model's algorithm.
Uncertainty
Uncertainty refers to lack of knowledge about specific factors, parameters, or models.
For example, we may be uncertain about the mean concentration of a specific pollutant at a
contaminated site or we may be uncertain about a specific measure of uptake (e.g., 95th
percentile fish consumption rate among all adult males in the United States). Uncertainty
includes parameter uncertainty (measurement errors, sampling errors, systematic errors), model
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uncertainty (uncertainty due to necessary simplification of real-world processes, mis-
specification of the model structure, model misuse, use of inappropriate surrogate variables), and
scenario uncertainty (descriptive errors, aggregation errors, errors in professional judgment,
incomplete analysis).
Variability
Variability refers to observed differences attributable to true heterogeneity or diversity in
a population or exposure parameter. Sources of variability are the result of natural random
processes and stem from environmental, lifestyle, and genetic differences among humans.
Examples include human physiological variation (e.g., natural variation in bodyweight, height,
breathing rates, drinking water intake rates), weather variability, variation in soil types and
differences in contaminant concentrations in the environment. Variability is usually not
reducible by further measurement or study (but can be better characterized).
Preliminary Issues and Considerations
Defining the Assessment Questions
The critical first step in any exposure assessment is to develop a clear and unambiguous
statement of the purpose and scope of the assessment. A clear understanding of the purpose will
help to define and bound the analysis. Generally, the exposure assessment should be made as
simple as possible while still including all important sources of risk. Finding the optimum match
between the sophistication of the analysis and the assessment problem may be best achieved
using a "tiered approach" to the analysis, that is, starting as simply as possible and sequentially
employing increasingly sophisticated analyses, but only as warranted by the value added to the
analysis and decision process.
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Selection and Development of
the Conceptual and
Mathematical Models
To help identify and select plausible
models, the risk assessor should develop
selection criteria tailored to each assessment
question. The application of these criteria
may dictate that different models be used for
different subpopulations under study (e.g.,
highly exposed individuals vs. the general
population). In developing these criteria, the
risk assessor should consider all significant
assumptions, be explicit about the
uncertainties, including technical and
scientific uncertainties about specific
quantities, modeling uncertainties,
uncertainties about functional forms, and
should identify significant scientific issues
about which there is uncertainty.
At any step hi the analysis, the risk assessor should be aware of the manner in which
alternative selections might influence the conclusions reached.
Selection and Evaluation of Available Data
After the assessment questions have been defined and conceptual models have been
developed, it is necessary to compile and evaluate existing data (e.g., site specific or surrogate
data) on variables important to the assessment. It is important to evaluate data quality and the
extent to which the data are representative of the population under study.
Some Considerations in the Selection of
Models
. appropriateness of the model's assumptions vis-a-
vis the analysis objectives
. compatibility of the model input/output and linkages
to other models used in the analysis
. the theoretical basis for the model
. level of aggregation, spatial and temporal scales
. resolution limits
. sensitivity to input variability and input uncertainty
. reliability of the model and code, including peer
review of the theory and computer code
. verification studies, relevant field tests
. degree of acceptance by the user community
. friendliness, speed and accuracy
. staff and computer resources required
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Guiding Principles for Monte Carlo Analysis
This section presents a discussion of principles of good practice for Monte Carlo
simulation as it may be applied to environmental assessments. It is not intended to serve as
detailed technical guidance on how to conduct or evaluate an analysis of variability and
uncertainty.
Selecting Input Data and Distributions for Use in Monte Carlo
Analysis
1. Conduct preliminary sensitivity analyses or numerical experiments to identify model
structures, exposure pathways, and model input assumptions and parameters that
make important contributions to the assessment endpoint and its overall variability
and/or uncertainty.
The capabilities of current desktop computers allow for a number of "what if scenarios
to be examined to provide insight into the effects on the analysis of selecting a particular model,
including or excluding specific exposure pathways, and making certain assumptions with respect
to model input parameters. The output of an analysis may be sensitive to the structure of the
exposure model. Alternative plausible models should be examined to determine if structural
differences have important effects on the output distribution (in both the region of central
tendency and in the tails).
Numerical experiments or sensitivity analysis also should be used to identify exposure
pathways that contribute significantly to or even dominate total exposure. Resources might be
saved by excluding unimportant exposure pathways (e.g., those that do not contribute
appreciably to the total exposure) from full probabilistic analyses or from further analyses
altogether. For important pathways, the model input parameters that contribute the most to
overall variability and uncertainty should be identified. Again, unimportant parameters may be
excluded from full probabilistic treatment. For important parameters, empirical distributions or
parametric distributions may be used. Once again, numerical experiments should be conducted
to determine the sensitivity of the output to different assumptions with respect to the
distributional forms of the input parameters. Identifying important pathways and parameters
where assumptions about distributional form contribute significantly to overall uncertainty may
aid in focusing data gathering efforts.
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Dependencies or correlations between model parameters also may have a significant
influence on the outcome of the analysis. The sensitivity of the analysis to various assumptions
about known or suspected dependencies should be examined. Those dependencies or
correlations identified as having a significant effect must be accounted for in later analyses.
Conducting a systematic sensitivity study may not be a trivial undertaking, involving
significant effort on the part of the risk assessor. Risk assessors should exercise great care not to
prematurely or unjustifiably eliminate pathways or parameters from full probabilistic treatment.
Any parameter or pathway eliminated from full probabilistic treatment should be identified and
the reasons for its elimination thoroughly discussed.
2. Restrict the use of probabilistic assessment to significant pathways and parameters.
Although specifying distributions for all or most variables in a Monte Carlo analysis is
useful for exploring and characterizing the full range of variability and uncertainty, it is often
unnecessary and not cost effective. If a systematic preliminary sensitivity analysis (that includes
examining the effects of various assumptions about distributions) was undertaken and
documented, and exposure pathways and parameters that contribute little to the assessment
endpoint and its overall uncertainty and variability were identified, the risk assessor may
simplify the Monte Carlo analysis by focusing on those pathways and parameters identified as
significant. From a computational standpoint, a Monte Carlo analysis can include a mix of point
estimates and distributions for the input parameters to the exposure model. However, the risk
assessor and risk manager should continually review the basis for "fixing" certain parameters as
point values to avoid the perception that these are indeed constants that are not subject to change.
3. Use data to inform the choice of input distributions for model parameters.
The choice of input distribution should always be based on all information (both
qualitative and quantitative) available for a parameter. In selecting a distributional form, the risk
assessor should consider the quality of the information hi the database and ask a series of
questions including (but not limited to):
Is there any mechanistic basis for choosing a distributional family?
Is the shape of the distribution likely to be-dictated by physical or biological
properties or other mechanisms?
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Is the variable discrete or continuous?
What are the bounds of the variable?
Is the distribution skewed or symmetric?
If the distribution is thought to be skewed, in which direction?
What other aspects of the shape of the distribution are known1?
When data for an important parameter are limited, it may be useful to define plausible
alternative scenarios to incorporate some information on the impact of that variable in the overall
assessment (as done in the sensitivity analysis). In doing this, the risk assessor should select the
widest distributional family consistent with the state of knowledge and should, for important
parameters, test the sensitivity of the findings and conclusions to changes in distributional shape.
4. Surrogate data can be used to develop distributions when they can be appropriately
justified.
The risk assessor should always seek representative data of the highest quality available.
However, the question of how representative the available data are is often a serious issue. Many
times, the available data do not represent conditions (e.g., temporal and spatial scales) in the
population being assessed. The assessor should identify and evaluate the factors that introduce
uncertainty into the assessment. In particular, attention should be given to potential biases that
may exist in surrogate data and their implications for the representativeness of the fitted
distributions.
When alternative surrogate data sets are available, care must be taken when selecting or
combining sets. The risk assessor should use accepted statistical practices and techniques when
combining data, consulting with the appropriate experts as needed.
Whenever possible, collect site or case specific data (even in limited quantities) to help
justify the use of the distribution based on surrogate data. The use of surrogate data to develop
distributions can be made more defensible when case-specific data are obtained to check the
reasonableness of the distribution.
5. When obtaining empirical data to develop input distributions for exposure model
parameters, the basic tenets of environmental sampling should be followed. Further,
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particular attention should be given to the quality of information at the tails of the
distribution.
As a general rule, the development of data for use in distributions should be carried out
using the basic principles employed for exposure assessments. For example,
Receptor-based sampling in -which data are obtained on the receptor or on the
exposure fields relative to the receptor;
Sampling at appropriate spatial or temporal scales using an appropriate
stratified random sampling methodology;
Using two-stage sampling to determine and evaluate the degree of error,
statistical power, and subsequent sampling needs; and
Establishing data quality objectives.
In addition, the quality of information at the tails of input distributions often is not as
good as the central values. The assessor should pay particular attention to this issue when
devising data collection strategies.
6. Depending on the objectives of the assessment, expert1 judgment can be included either
within the computational analysis by developing distributions using various methods or
by using judgments to select and separately analyze alternate, but plausible, scenarios.
When expert judgment is employed, the analyst should be very explicit about its use.
Expert judgment is used, to some extent, throughout all exposure assessments. However,
debatable issues arise when applying expert opinions to input distributions for Monte Carlo
analyses. Using expert judgment to derive a distribution for an input parameter can reflect
bounds on the state of knowledge and provide Insights into the overall uncertainty. This may be
particularly useful during the sensitivity analysis to help identify important variables for which
additional data may be needed. However, distributions based exclusively or primarily on expert
judgment reflect the opinion of individuals or groups and, therefore, may be subject to
1 According to NCRP (1996), an expert has (1) training and experience in the subject area resulting in
superior knowledge in the field, (2) access to relevant information, (3) an ability to process and effectively use the
information, and (4) is recognized by his or her peers or those conducting the study as qualified to provide
judgments about assumptions, models, and model parameters at the level of detail required.
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considerable bias. Further, without explicit documentation of the use of expert opinions, the
distributions based on these judgments might be erroneously viewed as equivalent to those based
on hard data. When distributions based on expert judgement have an appreciable effect on the
outcome of an analysis, it is critical to highlight this in the uncertainty characterization.
Evaluating Variability and Uncertainty
7. The concepts of variability and uncertainty are distinct. They can be tracked and
evaluated separately during an analysis, or they can be analyzed within the same
computational framework. Separating variability and uncertainty is necessary to
provide greater accountability and transparency. The decision about how to track
them separately must be made on a case-by-case basis for each variable.
Variability represents the true heterogeneity or diversity inherent in a well-characterized
population. As such, it is notreducible through further study. Uncertainty represents a lack of
knowledge about the population. It is sometimes reducible through further study. Therefore,
separating variability and uncertainty during the analysis is necessary to identify parameters for
which additional data are needed. There can be uncertainty about the variability within a
population. For example, if only a subset of the population is measured or if the population is
otherwise under-sampled, the resulting measure of variability may differ from the true population
variability. This situation may also indicate the need for additional data collection.
8. There are methodological differences regarding how variability and uncertainty are
addressed in a Monte Carlo analysis.
There are formal approaches for distinguishing between and evaluating variability and
uncertainty. When deciding on methods for evaluating variability and uncertainty, the assessor
should consider the following issues.
Variability depends on the averaging time, averaging space, or other dimensions
in which the data are aggregated.
Standard data analysis tends to understate uncertainty by focusing solely on
random error -within a data set. Conversely, standard data analysis tends to
overstate variability by implicitly including measurement errors.
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Various types of model errors can represent important sources of uncertainty.
Alternative conceptual or mathematical models are a potentially important source
of uncertainty. A major threat to the accuracy of a variability analysis is a lack of
representativeness of the data.
9. Methods should investigate the numerical stability of the moments and the tails of the
distributions.
For the purposes of these principles, numerical stability refers to observed numerical
changes in the characteristics (i.e., mean, variance, percentiles) of the Monte Carlo simulation
output distribution as the number of simulations increases. Depending on the algebraic structure
of the model and the exact distributional forms used to characterize the input parameters, some
outputs will stabilize quickly, that is, the output mean and variance tend to reach more or less
constant values after relatively few sampling iterations and exhibit only relatively minor
fluctuations as the number of simulations increases. On the other hand, some model outputs may
take longer to stabilize. The risk assessor should take care to be aware of these behaviors. Risk
assessors should always use more simulations than they think necessary. Ideally, Monte Carlo
simulations should be repeated using several non-overlapping subsequences to check for stability
and repeatability. Random number seeds should always be recorded. In cases where the tails of
the output distribution do not stabilize, the assessor should consider the quality of information in
the tails of the input distributions. Typically, the analyst has the least information about the input
tails. This suggest two points.
Data gathering efforts should be structured to provide adequate coverage at the
tails of the input distributions.
The assessment should include a narrative and qualitative discussion of the
quafity of information at the tails of the input distributions.
10. There are limits to the assessor's ability to account for and characterize all sources of
uncertainty. The analyst should identify areas of uncertainty and include them in the
analysis, either quantitatively or qualitatively.
Accounting for the important sources of uncertainty should be a key objective in Monte
Carlo analysis. However, it is not possible to characterize all the uncertainties associated with
the models and data. The analyst should attempt to identify the full range of types of uncertainty
impinging on an analysis and clearly disclose what set of uncertainties the analysis attempts to
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represent and what it does not. Qualitative evaluations of uncertainty including relative ranking
of the sources of uncertainty may be an acceptable approach to uncertainty evaluation, especially
when objective quantitative measures are not available. Bayesian methods may sometimes be
useful for incorporating subjective information into variability and uncertainty analyses in a
manner that is consistent with distinguishing variability from uncertainty.
Presenting the Results of a Monte Carlo Analysis
11. Provide a complete and thorough description of the exposure model and its equations
(including a discussion of the limitations of the methods and the results).
Consistent with the Exposure Assessment Guidelines, Model Selection Guidance, and
other relevant Agency guidance, provide a detailed discussion of the exposure model(s) and
pathways selected to address specific assessment endpoints. Show all the formulas used. Define
all terms. Provide complete references. If external modeling was necessary (e.g., fate and
transport modeling used to provide estimates of the distribution of environmental
concentrations), identify the model (including version) and its input parameters. Qualitatively
describe the major advantages and limitations of the models used.
The objectives are transparency and reproducibility - to provide a complete enough
description so that the assessment might be independently duplicated and verified.
12. Provide detailed information on the input distributions selected. This information
should identify whether the input represents largely variability, largely uncertainty,
or some combination of both. Further, information on goodness-of-fit statistics
should be discussed.
It is important to document thoroughly and convey critical data and methods that provide
an important context for understanding and interpreting the results of the assessment. This
detailed information should distinguish between variability and uncertainty and should include
graphs and charts to visually convey written information.
The probability density function (PDF) and cumulative distribution function (CDF)
graphs provide different, but equally important insights. A plot of a PDF shows possible values
of a random variable on the horizontal axis and their respective probabilities (technically, their
densities) on the vertical axis. This plot is useful for displaying:
17
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the relative probability of values;
the most likely values (e.g., modes);
the shape of the distribution (e.g., skewness, kurtosis); and
small changes in probability density.
A plot of the cumulative distribution function shows the probability that the value of a random
variable is less than a specific value. These plots are good for displaying:
fractiles, including the median;
probability intervals, including confidence intervals;
stochastic dominance; and
mixed, continuous, and discrete distributions.
Goodness-of-fit tests are formal statistical tests of the hypothesis that a specific set of
sampled observations are an independent sample from the assumed distribution. Common tests
include the chi-square test, the Kolmogorov-Smirnov test, and the Anderson-Darling test.
Goodness-of-fit tests for normality and lognormality include Lilliefors' test, the Shapiro-Wilks'
test, and D'Agostino's test.
Risk assessors should never depend solely on the results of goodness-of-fit tests to select
the analytic form for a distribution. Goodness-of-fit tests have low discriminatory power and are
generally best for rejecting poor distribution fits rather than for identifying gpod fits. For small
to medium sample sizes, goodness-of-fit tests are not very sensitive to small differences between
the observed and fitted distributions. On the other hand, for large data sets, even small and
unimportant differences between the observed and fitted distributions may lead to rejection of the
null hypothesis. For small to medium sample sizes, goodness-of-fit tests should best be viewed
as a systematic approach to detecting gross differences. The risk assessor should never let
differences in goodness-of-fit test results be the sole factor for determining the analytic form of a
distribution.
Graphical methods for assessing fit provide visual comparisons between the experimental
data and the fitted distribution? Despite the fact that they are non-quantitative, graphical methods
18
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often can be most persuasive in supporting the selection of a particular distribution or in rejecting
the fit of a distribution. This persuasive power derives from the inherent weaknesses in
numerical goodness-of-fit tests. Such graphical methods as probability-probability (P-P) and
quantile-quantile (Q-Q) plots can provide clear and intuitive indications of goodness-of-fit.
Having selected and justified the selection of specific distributions, the assessor should
provide plots of both the PDF and CDF, with one above the other on the same page and using
identical horizontal scales. The location of the mean should be clearly indicated on both curves
[See Figures la and lb]. These graphs should be accompanied by a summary table of the relevant
data.
13. Provide detailed information and graphs for each output distribution.
In a fashion similar to that for the input distributions, the risk assessor should provide
plots of both the PDF and CDF for each output distribution, with one above the other on the
same page, using identical horizontal scales. The location of the mean should clearly be
indicated on both curves. Graphs should be accompanied by a summary table of the relevant
data.
14. Discuss the presence or absence of dependencies and correlations.
Co variance among the input variables can significantly affect the analysis output. It. is
important to consider covariance among the model's most sensitive variables. It is particularly
important to consider covariance when the foeus of the analysis is on the high end (i.e., upper
end) of the distribution.
When covariance among specific parameters is suspected but cannot be determined due
to lack of data, the sensitivity of the findings to a range of different assumed dependencies
should be evaluated and reported.
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Figure la. Example Monte Carlo Estimate of the PDF for Lifetime Cancer Risk
£
a
er
£
Mean
Risk
Point
Estimat
1.OJE-O8
1.0E-07
1.OE-O6 1.OE-OS 1.OE-O4
Lifetime Cancer Risk
1.OB-O3
Figure Ib: Example Monte Carlo Estimate of the CDF for Lifetime Cancer Risk
0.00
1.0E-O8
1.OB-07
1.OE-O6
L/fet/m,
1.0E-OS
Can cor Risk
1.OE-O4
1.0E-03
20
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15. Calculate and present point estimates.
Traditional deterministic (point) estimates should be calculated using established
protocols. Clearly identify the mathematical model used as well as the values used for each input
parameter in this calculation. Indicate in the discussion (and graphically) where the point
estimate falls on the distribution generated by the Monte Carlo analysis. Discuss the model and
parameter assumptions that have the most influence on the point estimate's position in the
distribution. The most important issue in comparing point estimates and Monte Carlo results is
whether the data and exposure methods employed in the two are comparable. Usually, when a
major difference between point estimates and Monte Carlo results is observed, there has been a
fundamental change in data or methods. Comparisons need to call attention to such differences
and determine their impact.
In some cases, additional point estimates could be calculated to address specific risk
management questions or to meet the information needs of the audience for the assessment.
Point estimates can often assist in communicating assessment results to certain groups by
providing a scenario-based perspective. For example, if point estimates are prepared for
scenarios with which the audience can identify, the significance of presented distributions may
become clearer. This may also be a way to help the audience identify important risks.
16. A tiered presentation style, in which briefing materials are assembled at various levels
of detail, may be helpful. Presentations should be tailored to address the questions
and information needs of the audience.
Entirely different types of reports are needed for scientific and nonscientific audiences.
Scientists generally will want more detail than non-scientists. Risk managers may need more
detail than the public. Reports for the scientific community are usually very detailed.
Descriptive, less detailed summary presentations and key statistics with their uncertainty
intervals (e.g., box and whisker plots) are generally more appropriate for non-scientists [See
Figure 2].
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Figure 2: Example Box and Whiskers Plot of the Distribution of Lifetime Cancer Risk
1E-04
44
.J2
1E-OS
o
a
CO
o
! JE-06
1E-O7
95th parcentlle
75th porcentllo
mac/ton
25f/7 porcontllu
5th ptrctnt/lm
O Point Estimate* Mean Risk
To handle the different levels of sophistication and detail needed for different audiences,
it may be useful to design a presentation in a tiered format where the level of detail increases
with each successive tier. For example, the first tier could be a one-page summary that might
include a graph or other numerical presentation as well as a couple of paragraphs outlining what
was done. This tier alone might be sufficient for some audiences. The next tier could be an
executive summary, and the third tier could be a full detailed report. For further information
consult Bloom et al., 1993.
Graphical techniques can play an indispensable role in communicating the findings from
a Monte Carlo analysis. It is important that the risk assessor select a clear and uncluttered
graphical style hi an easily understood format. Equally important is deciding which information
to display. Displaying too much data or inappropriate data will weaken the effectiveness of the
effort. Having decided which information to display, the risk assessor should carefully tailor a
graphical presentation to the informational needs and sophistication of specific audiences. The
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performance of a graphical display of quantitative information depends on the information the
risk assessor is trying to convey to the audience and on how well the graph is constructed
(Cleveland, 1994). The following are some recommendations that may prove useful for effective
graphic presentation:
Avoid excessively complicated graphs. Keep graphs intended for a glance (e.g.,
overhead or slide presentations) relatively simple and uncluttered. Graphs
intended for publication can include more complexity.
Avoid pie charts, perspective charts (3-dimensional bar and pie charts, ribbon
charts), pseudo-perspective charts (2-dimensional bar or line charts).
Color and shading can create visual biases and are very difficult to use effectively.
Use color or shading only when necessary and then, only very carefully. Consult
references on the use of color and shading in graphics.
When possible in publications and reports, graphs should be accompanied by a
table of the relevant data.
If probability density or cumulative probability plots are presented, present both,
with one above the other on the same page, with identical horizontal scales and
with the location of the mean clearly indicated on both curves with a solid point.
Do not depend on the audience to correctly interpret any visual display of data.
Always provide a narrative in the report interpreting the important aspects of the
graph.
Descriptive statistics and box plots generally serve the less technically-oriented
audience well. Probability density and cumulative probability plots are generally
more meaningful to risk assessors and uncertainty analysts.
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Appendix A: Policy for Use of Probabilistic Analysis in Risk
Assessment at the U.S. Environmental
Protection Agency
INTRODUCTION
The importance of adequately characterizing variability and uncertainty in risk assessments has
been emphasized hi several science and policy documents. These include the 1992 U.S.
Environmental Protection Agency (EPA) Exposure Assessment Guidelines, the 1992 EPA Risk
Assessment Council (RAC) Guidance, the 1995 EPA Policy for Risk Characterization, the EPA
Proposed Guidelines for Ecological Risk Assessment, the EPA Region 3 Technical Guidance
Manual on Risk Assessment, the EPA Region 8 Superfund Technical Guidance, the 1994
National Academy of Sciences "Science and Judgment in Risk Assessment," and the report by
the Commission on Risk Assessment and Risk Management* As part of the implementation of
the recommendations contained in these reports, the Agency is issuing guidance on the
appropriate use of an application for analyzing variability and uncertainty in Agency risk
assessments.
This policy and the guiding principles attached are designed to support the use of various
techniques for characterizing variability and uncertainty. Further, the policy defines a set of
Conditions for Acceptance. These conditions are important for ensuring good scientific practice
in quantifying uncertainty and variability. In accordance with EPA's 1995 Policy for Risk
Characterization, this policy also emphasizes the importance of clarity, transparency,
reasonableness, and consistency in risk assessments.
There are a variety of different methods for characterizing uncertainty and variability. These
methods cover a broad range of complexity from the simple comparison of discrete points to
probabilistic techniques like Monte Carlo analysis. Recently, interest in using Monte Carlo
analysis for risk assessment has increased. This method has the advantage of allowing the
analyst to account for relationships between input variables and of providing the flexibility to
investigate the effects of different modeling assumptions. Experience has shown that to benefit
fully from the advantages of such probabilistic techniques as Monte Carlo analysis, certain
standards of practice are to be observed: The Agency is issuing, therefore, this policy statement
and associated guiding principles. While Monte Carlo analysis is the most frequently
encountered probabilistic tool for analyzing variability and uncertainty in risk assessments, the
intent of this policy is not to indicate that Monte Carlo analysis is the only acceptable approach
for Agency risk assessments. The spirit of this policy and the Conditions for Acceptance
described herein are equally applicable to other methods for analyzing variability and
uncertainty.
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POLICY STATEMENT
It is the policy of the U.S. Environmental Protection Agency that such probabilistic analysis
techniques as Monte Carlo analysis, given adequate supporting data and credible assumptions,
can be viable statistical tools for analyzing variability and uncertainty in risk assessments. As
such, and provided that the conditions described below are met, risk assessments using Monte
Carlo analysis or other probabilistic techniques will be evaluated and utilized in a manner that is
consistent with other risk assessments submitted to the Agency for review or consideration. It is
not the intent of this policy to recommend that probabilistic analysis be conducted for all risk
assessments supporting risk management decisions. Such analysis should be a part of a tiered
approach to risk assessment that progresses from simpler (e.g., deterministic) to more complex
(e.g., probabilistic) analyses as the risk management situation requires. Use of Monte Carlo or
other such techniques in risk assessments shall not be cause, per se, for rejection of the risk
assessment by the Agency. For human health risk assessments, the application of Monte Carlo
and other probabilistic techniques has been limited to exposure assessments in the majority of
cases. The current policy, Conditions for Acceptance and associated guiding principles are not
intended to apply to dose response evaluations for human health risk assessment until this
application of probabilistic analysis has been studied further. In the case of ecological risk
assessment, however, this policy applies to all aspects including stressor and dose-response
assessment.
CONDITIONS FOR ACCEPTANCE
When risk assessments using probabilistic analysis techniques (including Monte Carlo analysis)
are submitted to the Agency for review and evaluation, the following conditions are to be
satisfied to ensure high quality science. These conditions, related to the good scientific practices
of transparency, reproducibility, and the use of sound methods, are summarized here and
explained more fully in the Attachment, "Guiding Principles for Monte Carlo Analysis."
1. The purpose and scope of the assessment should be clearly articulated in a "problem
formulation" section that includes a full discussion of any highly exposed or highly
susceptible subpopulations evaluated (e.g., children, the elderly). The questions the
assessment attempts to answer are to be discussed and the assessment endpoints are to
be well defined.
2. The methods used for the analysis (including all models used, all data upon which the
assessment is based, and all assumptions that have a significant impact upon the
results) are to be documented and easily located in the report. This documentation is
to include a discussion of the degree to which the data used are representative of the
population under study. Also, this documentation is to include the names of the
models and software used to generate the analysis. Sufficient information is to be
provided to allow the results of the analysis to be independently reproduced.
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3. The results of sensitivity analyses are to be presented and discussed in the report.
Probabilistic techniques should be applied to the compounds, pathways, and factors of
importance to the assessment, as determined by sensitivity analyses or other basic
requirements of the assessment.
4. The presence or absence of moderate to strong correlations or dependencies between
the input variables is to be discussed and accounted for in the analysis, along with the
effects these have on the output distribution.
5. Information for each input and output distribution is to be provided in the report.
This includes tabular and graphical representations of the distributions (e.g.,
probability density function and cumulative distribution function plots) that indicate
the location of any point estimates of interest (e.g., mean, median, 95th percentile).
The selection of distributions is to be explained and justified. For both the input and
output distributions, variability and uncertainty are to be differentiated where
possible.
6. The numerical stability of the central tendency and the higher end (i.e., tail) of the
output distributions are to be presented and discussed.
7. Calculations of exposures and risks using deterministic (e.g., point estimate) methods
are to be reported if possible. Providing these values will allow comparisons between
the probabilistic analysis and past or screening level risk assessments. Further,
deterministic estimates may be used to answer scenario specific questions and to
facilitate risk communication. When comparisons are made, it is important to explain
the similarities and differences in the underlying data, assumptions, and models.
8. Since fixed exposure assumptions (e.g., exposure duration, body weight) are
sometimes embedded in the toxicity metrics (e.g., Reference Doses, Reference
Concentrations, unit cancer risk factors), the exposure estimates from the probabilistic
output distribution are to be aligned with the toxicity metric.
LEGAL EFFECT
This policy and associated guidance on probabilistic analysis techniques do not establish or
affect legal rights or obligations. Rather, they confirm the Agency position that probabilistic
techniques can be viable statistical tools for analyzing variability and uncertainty in some risk
assessments. Further, they outline relevant Conditions for Acceptance and identify factors
Agency staff should consider in implementing the policy.
The policy and associated guidance do not stand alone; nor do they establish a binding norm that
is finally determinative of the issues addressed. Except where otherwise provided by law, the
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Agency's decision on conducting a risk assessment in any particular case is within the Agency's
discretion. Variations in the application of the policy and associated guidance, therefore, are not
a legitimate basis for delaying action on Agency decisions.
IMPLEMENTATION
Assistant Administrators and Regional Administrators are responsible for implementation of this
policy within their organizational units. The implementation strategy is divided into immediate
and follow-up activities.
Immediate Activities
To assist EPA program and regional offices with this implementation, initial guidance on the use
of one probabilistic analysis tool, Monte Carlo analysis, is provided in the Attachment, "Guiding
Principles for Monte Carlo Analysis" (EPA/630/R-97/001). The focus of this guidance is on
Monte Carlo analysis because it is the most frequently encountered technique in human health
risk assessments. Additional information may be found in the "Summary Report for the
Workshop on Monte Carlo Analysis" (EPA/630/R-96/010). This report summarizes discussions
held during the May 1996 Risk Assessment Forum sponsored workshop that involved leading
experts in Monte Carlo analysis.
Follow-Up Activities
To prepare for the use and evaluation of probabilistic analysis methods, including Monte Carlo
analysis, within the next year, EPA's Risk Assessment Forum (RAF) will develop illustrative
case studies for use as guidance and training tools. Further, the RAF will organize workshops or
colloquia to facilitate the development of distributions for selected exposure factors. EPA's
National Center for Environmental Assessment (NCEA) will develop an Agency training course
on probabilistic analysis methods, including Monte Carlo analysis for both risk assessors and risk
managers which will become available during Fiscal Year (FY) 1997 or FY 1998. Also, NCEA
will develop detailed technical guidance for the quantitative analysis of variability and
uncertainty.
In the longer term, various Regions, Programs and the Office of Research and Development
(ORD) may need to modify existing or develop new guidelines or models to facilitate use of such
techniques as Monte Carlo analysis. Also, the NCEA will revise or update the Exposure Factors
Handbook to include distributional information. ORD's National Exposure Research Laboratory
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(NERL) has formed a modeling group that may provide assessment and analysis advice to
Program and Regional Offices. The issue of using probabilistic techniques, including Monte
Carlo analysis in the dose response portion of human health risk assessments requires further
study. NCEA will conduct research in this area and additional guidance will be provided if
necessary.
Fred Hansen
Deputy Administrator
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Appendix B: Probability Distribution Selection
Issues
Surrogate Data, Fitting Distributions, Default Distributions
Subjective Distributions
Identification of relevant and valid data to represent an exposure variable is prerequisite
to selecting a probability distribution However, often the data available are not a direct measure
of the exposure variable of interest. The risk assessor is often faced with using data taken in
spatial or temporal scales that are significantly different from the scale of the problem under
consideration. The question becomes whether or not or how to use marginally representative or
surrogate data to represent a particular exposure variable. While there can be no hard and fas.t
rules on how to make that judgment, there are a number of questions risk assessors need to ask
when the surrogate data are the only data available.
/s there Prior Knowledge about Mechanisms? Ideally, the selection of candidate probability
distributions should be based on consideration of the underlying physical processes or
mechanisms thought to be key in giving rise to the observed variability. For example, if the
exposure variable is the result of the product of a large number of other random variables, it
would make sense to select a Ipgnormal distribution for testing. As another example, the
exponential distribution would be a reasonable candidate if the stochastic variable represents a
process akin to inter-arrival times of events that occur at a constant rate. As a final example, a
gamma distribution would be a reasonable candidate if the random variable of interest was the
sum of independent exponential random variables.
Threshold Question - Are the surrogate data of acceptable quality and representativeness to
support reliable exposure estimates?
What uncertainties and biases are likely to be introduced by using surrogate data? For
example, if the data have been collected in a different geographic region, the contribution of
factors such as soil type, rainfall, ambient temperature, growing season, natural sources of
exposure, population density, and local industry may have a significant effect on the exposure
concentrations and activity patterns. If the data are collected from volunteers or from hot spots,
they will probably not represent the distribution of values hi the population of interest. Each
difference between the survey data and the population being assessed should be noted. The
effects of these differences on the desired distribution should be discussed if possible.
How are the biases likely to affect the analysis and can the biases be corrected? The risk
assessor may be able to state with a high degree of certainty that the available data over-estimates
or under-estimates the parameter of interest. Use of ambient air data on arsenic collected near
smelters will almost certainly over-estimate average arsenic exposures in the United States.
However, the smelter data carf probably be used to produce an estimate of inhalation exposures
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that falls within the high end. In other cases, the assessor may be unsure how unrepresentative
data will affect the estimate as in the case when data collected by a particular State are used in a
national assessment. In most cases, correction of suspected biases will be difficult or not
possible. If only hot spot data are available for example, only bounding or high end estimates
may be possible. Unsupported assumptions about biases should be avoided. Information
regarding the direction and extent of biases should be included in the uncertainty analysis.
How should any uncertainty introduced by the surrogate data be represented?
In identifying plausible distributions to represent variability, the risk assessor should examine
the following characteristics 6f the variable:
1. Nature of the variable.
Can the variable only take on discrete values (e.g., either on or off; either heads or tails) or is
the variable continuous over some range (e.g., pollutant concentration; body weight; drinking
water consumption rate)? Is the variable correlated with or dependent on another variable?
2. Bounds of the variable.
What is the physical or plausible range of the variable (e.g., takes on only positive values;
bounded by the interval [a,b]). Are physical measurements of the variable censored due to limits
of detection or some aspect of the experimental design?
3. Symmetry of the Distribution.
Is distribution of the variable known to be or thought to be skewed or symmetric? If the
distribution is thought to be skewed, in which direction? What other aspects of the shape of the
distribution are known? Is the shape of the distribution likely to be dictated by
physical/biological properties (e.g., logistic growth rates) or other mechanisms?
4. Summary Statistics.
Summary statistics can sometimes be useful in discriminating among candidate distributions.
For example, frequently the range of the variable can be used to eliminate inappropriate
distributions; it would not be reasonable to select a lognormal distribution for an absorption
coefficient since the range of the lognormal distribution is (0,) while the range of the absorption.
coefficient is (0,1)- If the coefficient of variation is near 1.0, then an exponential distribution
might be appropriate. Information on skewness can also be useful. For symmetric distributions,
skewness = 0; for distributions skewed to the right, skewness > 0; for distributions skewed to the
left, skewness < 0.
5. Graphical Methods to Explore the Data.
The risk assessor can often gain important insights by using a number of simple graphical
techniques to explore the data prior to numerical analysis. A wide variety of graphical methods
have been developed to aid in this exploration including frequency histograms for continuous
distributions, stem and leaf plots, dot plots, line plots for discrete distributions, box and whisker
plots, scatter plots, star representations, glyphs, Chernoff faces, etc. [Tukey (1977); Conover
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(1980); du Toit et al. (1986); Morgan and Henrion, (1990)]. These graphical methods are all
intended to permit visual inspection of the density function corresponding to the distribution of
the data. They can assist the assessor in examining the data for skewness, behavior in the tails,
rounding biases, presence of multi-modal behavior, and data outliers.
Frequency histograms can be compared to the fundamental shapes associated with standard
analytic distributions (e.g., normal, lognormal, gamma, Weibull). Law and Kelton (1991) and
Evans et al. (1993) have prepared a useful set of figures which plot many of the standard analytic
distributions for a range of parameter values. Frequency histograms should be plotted on both
linear and logarithmic scales and plotted over a range of frequency bin widths (class intervals) to
avoid too much jaggedness or too much smoothing (i.e., too little or too much data aggregation).
The data can be sorted and plotted on probability paper to check for normality (or log-normality).
Most of the statistical packages available for personal computers include histogram and
probability plotting features, as do most of the spreadsheet programs. Some statistical packages
include stem and leaf, and box and whisker plotting features.
After having explored the above characteristics of the variable, the risk assessor has three
basic techniques for representing the data hi the analysis. In the first method, the assessor can
attempt to fit a theoretical or parametric distribution to the data using standard statistical
techniques. As a second option, the assessor can use the data to define an empirical distribution
function (EDF). Finally, the assessor can use the data directly in the analysis utilizing random
resampling techniques (i.e., bootstrapping). Each of these three techniques has its own benefits.
However, there is no consensus among researchers (authors) as to which method is generally
superior. For example, Law and Kelton (1991) observe that EDFs may contain irregularities,
especially when the data are limited and that when an EDF is used in the typical manner, values
outside the range of the observed data cannot be generated. Consequently, when the data are
representative of the exposure variable and the fit is good, some prefer to use parametric
distributions. On the other hand, some authors prefer EDFs (Bratley, Fox and Schrage, 1987)
arguing that the smoothing which necessarily takes place in the fitting process distorts real
information. In addition, when data are limited, accurate estimation of the upper end (tail) is
difficult. Ultimately, the technique selected will be a matter of the risk assessor's comfort with
the techniques and the quality and quantity of the data under evaluation.
The following discussion focuses primarily on parametric techniques. For a discussion of the
other methods, the reader is referred to Efron and Tibshirani (1993), Law & Kelton (1991), and
Bratley et al (1987).
Having selected parametric distributions, it is necessary to estimate numerical values for the
intrinsic parameters which characterize each of the analytic distributions and assess the quality
of the resulting fit.
Parameter Estimation. Parameter estimation is generally accomplished using conventional
statistical methods, the most popular of which include the method of maximum likelihood,
method of least squares, and the method of moments. See Johnson and Kotz (1970), Law
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and Kelton (1991), Kendall and Stewart (1979), Evans et al. (1993), Ang and Tang (1975),
Gilbert (1987), and Meyer (1975).
Assessing the Representativeness of the Fitted Distribution. Having estimated the
parameters of the candidate distributions, it is necessary to evaluate the "quality of the fit"
and, if more than one distribution was selected, to select the "best" distribution from among
the candidates. Unfortunately, there is no single, unambiguous measure of what constitutes
best fit. Ultimately, the risk assessor must judge whether or not the fit is acceptable.
Graphical Methods for Assessing Fit. Graphical methods provide visual comparisons
between the experimental data and the fitted distribution. Despite the fact that they are non-
quantitative, graphical methods often can be most persuasive in supporting the selection of a
particular distribution or hi rejecting the fit of a distribution. This persuasive power derives
from the inherent weaknesses in numerical goodness-of-fit tests. Commonly used graphical
methods include: frequency comparisons which compare a histogram of the experimental
data with the density function of the fitted data; probability plots compare the observed
cumulative density function with the fitted cumulative density function. Probability plots are
often based on graphical transformations such that the plotted cumulative density function
results in a straight line; probability-probability plots (P-P plots) compare the observed
probability with the fitted probability. P-P plots tend to emphasize differences in the middle
of the predicted and observed cumulative distributions; quantile-quantile plots (Q-Q plots)
graph the ith-quantile of the fitted distribution against the ith quantile data. Q-Q plots tend to
emphasize differences hi the tails of the fitted and observed cumulative distributions; and box
plots compare a box plot of the observed data with a box plot of the fitted distribution'.
Goodness-of-Fit Tests. Goodness-of-fit tests are formal statistical tests of the hypothesis that
the set of sampled observations are an independent sample from the assumed distribution.
The null hypothesis is that the randomly sampled set of observations are independent,
identically distributed random variables with distribution function F. Commonly used
goodness-of-fit tests include the chi-square test, Kolmogorov-Smirnov test, and Anderson-
Darling test. The chi-square test is based on the difference between the square of the
observed and expected frequencies. It is highly dependent on the width and number of
intervals chosen and is considered to have low power. It is best used to reject poor fits. The
Kolmogorov-Smirnov Test is a non-parametric test based on the maximum absolute
difference between the theoretical and sample Cumulative Distribution Functions (CDFs).
The Kolmogorov-Smirnov test is most sensitive around the median and less sensitive in the
tails and is best at detecting shifts hi the empirical CDF relative to the known CDF. It is less
proficient at detecting spread but is considered to be more powerful than the chi-square test.
The Anderson-Darling test is designed to test goodness-of-fit in the tails of a Probability
Density Function (PDF) based on a weighted-average of the squared difference between the
observed and expected cumulative densities.
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Care must be taken not to over-interpret or over-rely on the findings of goodness-of-fit tests.
It is far too tempting to use the power and speed of computers to run goodness-of-fit tests
against a generous list of candidate distributions, pick the distribution with the "best"
goodness-of-fit statistic, and claim that the distribution that fit "best" was not rejected at
some specific level of significance. This practice is statistically incorrect and should be
avoided [Bratley et al, 1987, page 134]. Goodness-of-fit tests have notoriously low power
and are generally best for rejecting poor distribution fits rather than for identifying good fits.
For small to medium sample sizes, goodness-of-fit tests are not very sensitive to small
differences between the observed and fitted distributions. On the other hand, for large data
sets, even minute differences between the observed and fitted distributions may lead to
rejection of the null hypothesis. For small to medium sample sizes, goodness-of-fit tests
should best be viewed as a systematic approach to detecting gross differences.
Tests of Choice for Normality and Lognormality. Several tests for normality (and
lognormality when log-transformed data are used) which are considered more powerful than
either the chi-square or Komolgarov-Smirnoff (K-S) tests have been developed: Lilliefbrs'
test which is based on the K-S test but with "normalized" data values, Shapiro-Wilks test (for
sample sizes 50), and D'Agostino's test (for sample sizes 50). The Shapiro-Wilks and
D'Agostino tests are the tests of choice when testing for normality or lognormality.
If the data are not well-fit by a theoretical distribution, the risk assessor should consider the
Empirical Distribution Function or bootstrapping techniques mentioned above.
For those situations in which the data are not adequately representative of the exposure
variable or where the quality or quantity of the data are questionable the following
approaches may be considered.
Distributions Based on Surrogate Data. Production of an exposure assessment often
requires that dozens of factors be evaluated, including exposure concentrations, intake rates,
exposure times, and frequencies. A combination of monitoring, survey, and experimental
data, fate and transport modeling, and professional judgment is used to evaluate these factors.
Often the only available data are not completely representative of the population being
assessed. Some examples are the use of activity pattern data collected in one geographic
region to evaluate the duration of activities at a Superfund site in another region; use of
national intake data on consumption of a particular food item to estimate regional intake; and
use of data collected from volunteers to represent the general population.
In each such case, the question of whether to use the unrepresentative data to estimate the
distribution of a variable should be carefully evaluated. Considerations include how to
express the possible bias and uncertainty introduced by the unrepresentativeness of the data
and alternatives to using the data. In these situations, the risk assessor should carefully
evaluate the basis of the distribution (e.g., data used, method) before choosing a particular
surrogate or before picking among alternative distributions for the same exposure parameter.
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The following table indicates exposure parameters for which surrogate distributions may be
reasonable and useful.
Table 1 Examples of exposure parameters for which
distributions based on surrogate data might be reasonable
Receptor Physiological
Parameters
body weight
height
total skin surface area
exposed skin - hands, forearms, head, upper
body
Behavioral
Receptor
Time-Activity
Patterns
residency periods - age, residency type
weekly work hours
time since last job change
showering duration
Receptor
Contact Rates
soil ingestion rates
soil adherence
food ingestion - vegetables, freshwater finfish,
saltwater finfish, shellfish, beef
water intake - total water, tapwater
inhalation rates
Rough Characterizations of Ranges and Distributional Forms. In the absence of
acceptable representative data or if the study is to be used primarily for screening, crude
characterizations of the ranges and distributions of the exposure variable may be adequate.
For example, physical plausibility arguments may be used to establish ranges for the
parameters. Then, assuming such distributions as the uniform, log-uniform, triangular and
log-triangular distributions can be helpful hi establishing which input variables have the
greatest influence on the output variable. However, the risk assessor should be aware that
there is some controversy concerning the use of these types of distributions in the absence of
data. Generally, the range of the model output is more dependant on the ranges of the input
variables than it is on the actual shapes of the input distributions. Therefore, the risk assessor
should be careful to avoid assigning overly-restrictive ranges or unreasonably large ranges to
variables. Distributional assumptions can have a large influence on the shapes of the output
distribution. When the shape of the output distribution must be estimated accurately, care
and attention should be devoted to developing the input distributions.
Distributions Based on Expert Judgment. One method that has seen increasing usage in
environmental risk assessment is the method of subjective probabilities in which an expert or
experts are asked to estimate various behaviors and likelihoods regarding specific model
variables or scenarios. Expert elicitation is divided into two categories: (1) informal
elicitation, and (2) formal elicitation. Informal elicitation methods include self assessment,
bramstorming, causal elicitation (without structured efforts to control biases), and taped
group discussions between the project staff and selected experts.
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Formal elicitation methods generally follow the steps identified by the U.S. Nuclear
Regulatory Commission (USNRC, 1989; Oritz, 1991; also see Morgan and Henrion, 1990;
IAEA, 1989; Helton, 1993; Taylor and Burmaster; 1993) and are considerably more elaborate
and expensive than informal methods.
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