PB85-240455
Methodology for Characterization of Uncertainty in Exposure Aasesraent*
ftetearch Triangle Institute
Bftaearch Triangle Park, MC
Aug 85
U.S. DEPARTMENT OF COMMERCE
National Tichnictl Infomition Sirvlci
IMTIS
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EPA/600/8-85/009
August 1985
METHODOLOGY FOR CHARACTERIZATION OF UNCERTAINTY
IN EXPOSURE ASSESSMENTS
by
Roy W. Vhitaore
Research Triangle lostitutc
Research Triangle Park, NC 27709
' Contract No.: 68-01-6826
Task Manager: Jaaes V. Faleo
Project Officer: Jobn Varren
OFFICE OF HEALTH AND EHVUOWOKIAL ASSES9fENT
OFFICE OF RESEARCH AND DEVELOPMENT
W.S."ENVIRONMENTAL PROTECTION AGENCY
WASHINGTON, DC 20460
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EPA/600/8-85/009
Htthodolooy for Characterization of Uncertainty 1n
Exposure A
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DISCUIHER
Tbt information la thii report tut bctn fuadtd wholly or la part by the
Oaitod lutu toviroBMBUl Protection Aftncy mrftr Contract Bo. 66-01-6826
to tkt ItMtreh TtUaili laccltutt. It tut Wto wb>et of tht Aftncy't ptcr
oad alaiviatntivt nvltv. *ad it •«• btco cpprevid for pvbllcttioo •• aa IPX
torant. lantioB of trado amavs or coaMreUl product! dott aot eoaitltutt
ooJoraoMat or roeoawadatioa for ott.
11
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TABLE OF CONTENTS
Pige
IIST OF TABLES
LIST OF FIGURES
FOREWORD
ABSTRACT vi
ACKNOWLEDGMENTS vi"
1. nCTRODUCTION AND OVERVIEW 1
1.1 Whit it an Exposure Aasessnent? }
1.2 Characterisation of Uncertainty 5
2. ASSESSMENTS BASED UPON LIMITED DATA 12
2.1 Liaited Data for Directly Measured Exposures 12
2.2 Liaited Data for Bedel Input Variables 13
2.2.1 Population Exposure Models 13
2.2.2 Peraonal Exposure Models 14
3. ASSESSMENTS BASED UPON ESTIMATION OF INPUT VARIABLE
DISTRIBUTIONS 17
9.1 Estimation Based on Input Variable Distributions .... 17
3.2 Licitins Distributional Results 21
4. ASSESSMENTS BASED UPON DATA FOR MODEL INPUT VARIABLES .... 25
4.1 Interval Estimates of Exposure Percentiles: Input
Variable Distributions Not Known 25
4.2 Interval Estiaiates of Exposure Percentiles: Input
Variable Distributions Known 26
5. ASSESSMENTS BASED UPON EXPOSURE MEASUREMENTS 29
6. A06REGATING ACROSS SOURCES, PATHWAYS, AND/OR ROUTES OT
IXPOSURE 94
6.1 frimMnint Estimated Expoaure Distributions 94
6.2 Umitiof Distributional Result* 35
7. COMBINING OVER DISJOINT SUBPOFULATIONS 91
B. SUMMARY . . . «
REFERENCES *2
APPENDIX A A*J
111
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LIST OF TABUS
1 lOBury of Primary Methods for Chsracterisinf Uncertainty
for Exposure Assessments ................ 8
A.I Selected Percentiles for Estisuted Dove Resultinj free
Bquilly Likely Input Variable Combinations ....... A-8
Selected Perccntiles for the Do»e Dittributiea
tnm tht PretiMd Model «ad Loput Variables Biatributioai A- 12
A.9 telectod Perccatilei for Satiaattd Dose itaultiat froa
Alternative Bquilly likely Input Variable Coabia*tioni. . A-15
A.4 95 Percent Confidence Interval EatiMtes for Selected
Pcrccntilet of the Distribution of Dose (T) Baaed on
Model-Predicted Doses A-19
A.5 95 Percent Confidence Interval EatiMtes for Selected
Pereentilcs of the Dose Distribution (Y) Assuaiag Unifon
and Beta Distributions for IR and ED/600, Bespectivcly. . A-25
A.6 95 Percent Confidence Interval EstiMtes for Selected
Pereentiles of the Dose Distribution (T) Directly from
Exposure Data A-30
1v
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LIST OF FIGURES
Titure Pa££
1 Zategrated Exposure Asaessaent 2
ii
2 Baak AssessMBt/Risk Haaageaent Process 4
3 Hypothetical Jeiat Relative Trequeacies for Exposure
Duration (ED) aad lagestion Bate (XX) 18
A.I Dlstributioa of Estimated Dotes Resultia* fro* Equally
likely Zaput Variable Combination* A-7
A.2 Hypothetical Subjective Estisute of tke Distribution of
Exposure Duration (ED) A-9
A.3 Predicted Dose Distributioa Resultiag fro* the Presuaed
Model aad laput Variable Distributioas A-ll
A.A Distribution of Estimated Dose Resultiai froo Alternative
Equally likely laput Varisble Combination! A-U
A.5 listofrao of the Hypothetical Staple of Site 500 for
Exposure Duration (ED) A-17
A.6 HistofraB of the Hypothetical Staple of Siae 500 for
Xatestioa Rate (IR) A-18
A.7 95 Percent Coafidence Interval Estiaates for Selected
Perceatiles ef the Dose Distributioa Based en Model-
Predicted Doses A-20
A.B Coafidence Region for the Estiaated Beta Distribution
Paraaeters 6 tad $ A-23
A.9 95 Percent Confidence Interval Estiaates for Selected
Percentiles of the Dose Distribution Assuaiag Uniform and
Beta Diatributioas for XR aad ED/600, Respectively . . . A-26
A.10 95 Perceat Ceafideace laterval Estimates for Selected
Percentiles ef the Dose Distribution fro* Directly Meaaured
Doses A-31
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The Exposure Asiessacnc Group (EAC) of tht U.S. Eaviroaacntal Protection
Afency'i (EPA) Office of Research aad Developaeat has three a*io fuoctiou:
1) to aoaduct exposure assessaeats; 2) to reriev aaaesaaents and related
docaajeatai nd 3) to develop galdaliaoe for Afeney oipoeure aeeeieMBti. The
actiTltioe vader aaeh of tbeee throe fuaetioae are aupporced by and respond to
the Mode of the various IPA profraa offices.
Aa part of ita fvactioa to develop guidaliaae, IAC coaducte reaearch
projects for the purpose of developing or refialof toehaiques used in exposure
Aaeeeeveat. This docuateat presents the results of auch a project and serves as
a oupport doeuaent to EPA*a lateria Ouidelleei for Ispoaure Aaseesa«nt.
Specifically, thie report describee procedures vhich can be used to cherecteritc
•acortaiaty ie ostiasted oxposure aad coataiaa aa aaaaple to doaoastrete the
atility of the awst coaplez techaiques.
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ABSTRACT
Virtually all exposure •••ciMcats except these based upon Matured
exposure levela for a probability •••pic of population swabers rely 1900 a
•odel to predict exposure. The aodel aay be any aatheaatical function, aiaple
or coaplex, that estimates the population distribution of exposure or an
individual population amber's exposure «s a function of me or e»re input
variables. Whenever • Bedel that has aot been validated is wed as the basis
for aa exposure assessaent, the aaeertaiaty associated with the exposure
aMesaaent My be substantial. The priaary characterisation of uncertainty is
at least partly qualitative ia this case, i.e., it iacludes • description of
the assuaptioni iaberent ia the aodel aad their Justification. Plausible
alternative aodels ahould be discussed. Sensitivity of the exposure assess-
acnt to aodel foraulation can be investigated by replicating the •sscssaent
for plausible alternative aodels.
Vhen aa exposure assessaent is based opon directly aeasured exposure
levels for a probability aaaple of population a*abers, uncertainty can be
freatly reduced and described quantitatively. In this CMC, the priaary
•ources of uncertainty are aeasureoent errors end stapling errors. A thorough
quality assurance prograa should be designed into the study to ensure that the
awgnitude of •easureaent errors can be estiaated. The effects of all sources
of randoo error ehould be Measured quantitatively by confidence interval
eatiaatet of paraaeters of interest, e.g., perccatiles of the exposure
distribution. Moreover, the effect of randoa errors can be reduced by taking
• larger aaaple.
Vhenever the latter is aot feasible, it is aoaetistes possible to obtain
at least eeae data for exposure ead aodcl iaput variables. These dsts ahould
be ased to assess goodness of fit of the Model and/or presuaed distributions
af input variables. This substantially reduces the amount of quantitative
uncertainty for estimation of the distribution of exposure and is strongly
recoMaended. It is recognised, however, thst it a«y aot always be feaaible to
collect such data.
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ACKNOWLEDGMENTS
Ac rather would like to acknovled|e that tuny KTI staff ambers, act all
of whoa) eta bt Matleotd la this space, ceatributed te preparatioa ef thii
report, Several discussion teaaieas attended by various XII ataff •cabers
ceatribated freatly te the preparatiea ef this report. The anther thaaki Dr.
Bartia Haaso|lia fer May helpful euffeitioas aad for coatributioa ef Figures
1 aad 2, which he had prepared fer ether ITI Mseereh. Appreciation is
earteaded to Or. Jaates Chroay, Dr. Douglas Pnsainnil. aad Or. Clark Allen for
their austestioas aad constructive criticise. Thaaks is extended to Mr. Fred
IsBereun aad Mr. George Xircher fer their technical assistance ia preparation
ef Appcadix A. The author thanks the following peer reviewers for their
careful review ef the draft wersiea ef this doosaeat aad their suay helpful
aufttestions: Dr. Doaald Porcella, Or. Doaiaic Ditero, Dr. John Speafler, Dr.
J. Julian Chisel*, Mr. John L. Beaahaw; aad Dr. Vill M. Ollisoa. The
lesdersbip aad insightful direction provided by the EPA task Master, Dr.
Jaaes Falco, is also apprecisted. last, but act least, the author thanks his
aecretarial staff for their patient preparation ef several revisions of this
Mnuscript.
V111
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1. INTRODUCTION AND OVERVIEW
AD exposure assessaeat oftea progresses through several stage* of refine-
ment. The purpose of this document is to provide technical guidance for
characterisation ef uacertaiaty for assessments at various stages of refiae-
•
•eat, from assessments based upon limited initial data to those based upon
••tensive data.
Tali docmneat is written for the professional who is actively involved in
the performance ef exposure assessments. It ie intended to facilitate use ef
the latest statistical methods for characterising uncertainty for exposure
essessaeats.
This introductory chapter begins with a brief nectien that discusses the
nature of an exposure assessaeat. This section serves as a poiat of reference
for the remainder of this docuaent. It is else intended to aake this docuaeat
e. jeatially aelf-contained. An overview ef the characterisation of uncertain-
ty for exposure aaaesaaeats at various stages ef refinement is presented in
the final section of this chapter, which also serves es a guide to the regain-
ing chapters.
1.1 WAT IS AN EXPOSURE ASSESSMENT?
An exposure assessaeat quantitatively estiaates the coatact ef an af-
fected populatioa with a substance under investigation. Typically, the aagni-
tude, duration, sad/or frequency ef coatact ere estiaated. The U. S. En-
vironmental Protection Ageacy (EPA) has prepared a draft handbook [1] to
provide guidance pertinent to performing an exposure assessaeat. la addition,
t
refereaces [2] end (3] are valuable sources ef information with regard to the
••ethodology of exposure Msessaeats.
An integrated exposure sasessaeat quantifies the contact ef populatioa
•embers with tae substance under investigation via nil routes ef exposure
(inhalation, ingestion, nnd dermal) nnd all pathways from sources to exposed
individuals (e.g., from maaufacturiaf plants to plant workers snd/or consumers
•f manufactured products). As shown in Figure 1, an integrated exposure
Assessment also •atimates the sine of population affected by each exposure
roui«. nincc an integrated exposure assessment is eitea needed to enable
regulatory decisions/ characterisatioa of uncertainty is addressed in this
context.
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Figure 1. Integrated Bxporare AsMmwnt
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The overall uncertainty associated with both the exposure assessment and
the dose-response relationships Bust be addressed by risk assessments. Charac-
terization of uncertainty for exposure assessments is addressed by this docu-
ment; ether comparable documents address characterization of uncertainty for
risk assessments per ee. The primary purpose of an exposure assessment it to
e&able riak asaessaeat aad possible aeleetioa of a regulatory option. Regula-
tory decisions emit be capable of withstanding close acrutiay (perhaps even in
e court ef lav). Therefore, the characterization of uncertainty for aa expo-
sure assessment ahould be very thorough aod based upon state-of-the-art quan-
titative methods to the extent possible.
The type of exposure (internal dose, external dose, etc.) investigated by
an integrated exposure assessment depends open the input needed to enable risk
assessnent (see Figure 2). Zn most cases an exposure assessnent should ideal-
ly quantify the dose ef the substance mmder investigation that is absorbed by
individual body organs in erder to enable the most reliable estimates of
health effects. Presently it is practical to quantify absorbed organ-specific
doses only in certain circumstances--e.g., when the affected organ is the akin
(denul route) or when exposure is due to radiation. Moreover, the absorbed
•hole-body dose often cannot be quantified. Instead, indicators of the ab-
sorbed dose are often used by necessity in exposure assessments. These indi-
cators of absorbed doses may be either environmental exposure or body burden
(0)i (*))• Environmental exposure refers to the levels ef the substance
onder investigation contacted through the air, water, food, soil, etc. Body
burden refers to levels of the substance in body fluids or tissues, e.g.,
blood, Brine, breath, fatty tissues, hair, mails, etc. In the remainder of
this document, the ten "exposure" will usually be used in a generic sense to
represent either the absorbed internal dose er aa indicator ef the absorbed
dcae because both have been used aa the basis for exposure assessments.
Aa integrated exposure assessment generally partitions the affected
population into aubpopulatioas euch that ell embers ef e eubpopulation are
effected by the earn* eeurces ef exposure. Tor each aource, there can be ene
er more pathways by which the aubstaoce mader atudy travels from the aoarce to
the affected population. lor each aource and pathway, e population •ember can
be exposed via one er more ef three absorption routes: inhalation, ingestion,
end dermal. An integrated exposure assessment eften begins by estimating
exposure for a aubpopulatioa via a apecific combination ef source, pathway,
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Figure 2. Rink Acsesmwnt/Risk HaiMgewnt Proce**
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and route. Exposure is then aggregated across psthways and sources to esti-
mate exposure via a particular route for aubpopulation member*. Exposure may
also be aggregated across routes if this aggregate is relevant to estimation
ef risk. Finally, the aises of the aubpopulatieas should also be estimated.
An exposure assessment addresses the exposures experienced by the sub-
jects (human or aonhuman) of a specified population. The most ceaplete charac-
terisation ef these exposures is the population distribution of individual
exposures. If the population distribution ef expoaure was known, then all
summary parameters of the distribution would alee be known, such as the fol-
lowing:
1. Minimum of all possible exposures.
2. Kaxiaua of all possible exposures.
3. Mean, aedian, end ao*e ef the population exposures levels.
4. Standard deviation ef the expoaure distribution.
5. Proportion of subjects whose exposure exceeds any specified exposure
level.
6. Pcrcentiles of the exposure distribution.
Since knowledge of the diatributioa ef exposures implies knowledge of all the
above parameters that are ao useful for evaluation of risk, estimation of this
distribution is a primary goal ef.exposure asaeaaments. However, aome expo-
aure aasessaents Bay be limited by the available data to estimating summary
parameters of the exposure distribution, e.g., the mean exposure or certain
percentiles ef the exposure distribution.
1.2 CHARACTERIZATION OF BHCERTAim
Inference (5] addresses characterisation ef ancertainty for e source
severity index to be ased an an aid in regulatory decision aaking. Interval
•stiaatios fer the aesa value ef the severity index baaed ea "error propaga-
tion foxaulasn is discussed. Assuming that the imdex ia a function ef random
7*ii«wlcs for which direct confidence interval estimates ere available, a
first-order Taylor series epproximation ef the function ia recommended for
computation ef as approximate confidence interval estimate for the mean value
ef the index. A Taylor aeries approximation is alao recommended for deriva-
tion of bounds on the systematic error, or hiss, in estimation of the Bean
eeverity index. Tables of error propogstion formulas for random errors and
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systematic errors are provided. Reference (1] recommends these methods for
characteriaiag uncertainty for exposure assessments. However, these methods
have limited applicability for the following reasons: (1) The primary purpose
of am exposure assessment is usually to estimate the distribution of exposure,
mot a mean severity index, and (2) direct confidence interval estimation of
the mean index would be preferable to using "error propagation formulas." The
present document will concentrate on estimation of the distribution of expo-
aares aad on characterisation of the uncertainty associsted with this esti-
mated distribution.
A review af methodology for characteriaiag uncertainty is provided by
reference |f]. This article discusses uncertainty for a predicted output
variable T when the output can be expressed as a deterministic function of
several random variables. Since it is the uncertainty of the predicted output
that is of interest in this article, the probability distribution of the
output variable aad the variance of this distribution are advocated as primary
characterixations of the uncertainty associated with the predicted output.
Vithin this context, the estimated distribution of axposures could be regarded
as a characterixation of the uncertainty with regard to the prediction of en
individual population member's exposure level, T. However, estimation of the
population distribution of exposures is the primary goal of an exposure assess-
ment, aad it is the uncertainty with regard to this estimated distribution of
exposures that must be addressed. Thus, reference (6) is a valuable reference
with regard to characterisation o'f uncertainty, but the present document will
rely mainly upon somewhat different methods that more directly address charac-
terising uncertainty for an estimated distribution of exposures.
The present document 'is not iateaded to be a general review of method-
ology for characterising uncertainty. lastead, this document focuses on those
aethods for characteriaiag macartaiaty that are most appropriate for exposure
assessments, the determination of which methods are appropriate for charac-
terising the macerteiaty of an estimate depends apoa the underlying parameters
feeing estimated, the type aad extent of slats available, aad the estimation
freceduies •tilised. Therefore, this document is laraely organised eccording
to the types of 4ata aad estimation procedures that «ay reasonably form the
basis for en exposure essessmeat. As a result, e great deal of this document
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actually addresses estimation procedures for exposure assessments. The aethod-
elegy ceaaidcrcd most appropriate for characterizing the uncertainty associ-
ated vith aa exposure aaaessaent ia tbeo diacuaaed with regard to each eatiaw-
tiea procedure, for example, when the population diatribution of exposure is
being eatimated, characterization ef uncertainty eddreaaea the poaaible dif-
ferencea between the eatimated diatributioa ef expoaures «ad the true popula-
tion diatributioa of exposures.
An expoaure aaaeaameat ia eftea baaed ea eae er more expoaure aceaarios
•ad eaaociated mathematical models. Aa expoaure accaario ideally coasiders
•11 potential eoureca of the •ubataace that could ceme la contact vith popula-
tiea eubjecta. Expoaure predictioa models baaed opoo traaaportation aad fate
ef the eubataace would then be uaed to describe the pathways from aources to
coatact vith population Members. Theae aodels auy be very coaplex, uaiag data
•a levels ef the eubataaee at aources aad/or ambient monitoring aites aad the
geographic diatributioa ef the target populatioa to eatiaate the population
diatributioa ef expoaures (|1), (7], (g]( (9)). Alternatively, these aodels
caa be relatively aiaple models, auch aa amltiplicative factora, that predict
•a individual population aeaber'a peraoaal expoaure as a function of aeveral
input variables (aee, e.g., references (3] ead [10J, ead Appeadix A).
Table 1 provides an overview of the primary methods recoanended for
characterizing uncertainty for expoaure eaaeaaaenta. Both qualitative and
quantitative actbods for characterizing uncertainty are presented for each of
the following five types ef data: •
1. Measured personal expoaure levels for a eaaple of population
obers.
2. Measured model iaput variables for a aaaple of population mem-
bcrs, given a model that predicts peraoaal exposure aa a function
ef input variables.
3. Estixated Joint diatribution ef tjodel iaput variables, given a
ajodel that predicta peraoaal •xpoaure as • fuaction of the iaput
variables.
4. limited data for model iaput variables, given a model that pre-
dicta peraoaal •xpoaure «a a fuaction ef the iaput variablea.
S. Data en levels ef • eubstance et fixed aites (possibly including
eourccs) and geographic distribution ef pepulatioa meabera, given
a model that predicta the pepulatioa exposure diatribution frea
aucb input data.
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•- ••«-aea uiai tfte purpose ef the exposure asset.aent is to
estimate the distribution cf exposure and/or summary parameters thereof (e.g.,
mean, 5th and 95th perceotiles, Mxiaua possible exposure, proportion of
population with exposure exceeding • specified level, etc.).
The initial exposure assessae&t for a •ubstance it often based upon a
very limited amount of data. Under auch conditions, the primary characteriza-
tion of eBcsrtainty may be qualitative. Exposure assessments based upon
limited data are discussed in Chapter 2.
Chapter 3 then discusses assessments baaed upon the estisuted joint
probability distribution of model input variables, liven a model that predicts
ptnoaal opoaure as a function of the iaput variables. Depending open the
mtsnt of data available to validate the awdel as well as the input variable
distributions, this type of assessment has the potential for reducing uncer-
tainty relative to the Mthods discussed in Chapter 2. It alao has the poten-
tial for a more quantitative characterirition of the uncertainties.
The uncertainties associated vith an txpoaure assessment can potentially
be reduced further by basing the assesament on measured values of awdel input
variables for a sample of population embers, given a awdel that predicts
personal exposure aa a function of the input variables, as discussed in Chap-
tar 4. Finally, Chapter 5 discusses exposure assessments based on measured
exposure levels for a aample of population members, which has the potential
for minimising the uncertainties aaaociatcd vith an assessment.
Vhen the initial exposure assessment for a substance is performed, only
limited data relating to exposure will usually be available. As more re-
sources are devoted to an exposure assessment, a more refined assessment may
be generated. There may be many atages of refinement for sn exposure assess-
SKnt. The degree of refinement ultimately meeded for sn sxposure saaeaament
efcpeads upon the accuracy meeded to enable risk management decisions, sa well
ma the coat of obtaining adequate data, the time available for data acquisi-
tion, etc. ,. ,~>-*r>, : .« v • v
Vhen the exposure levels experienced by individual population subjects
sre measured, the measured exposures generally rtflect the total expoaure
across nil sources snd pathways, mud possibly across nil routes as well.
Bovrrer, mmen the modeling approach ia employed for sn exposure nsaeasment, a
ejodel often predicts sn individual population member's exposure due to s
specific combination of source, pathway, snd route. Methods for combining
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estiaated exposure distributions over sources, pathways, and routes are dis-
cussed in Chapter 6. Finally, Chapter 7 discusses coabining the estiaated
exposure distributions over disjoint subpopulations to produce sn estiaate of
the exposure distribution for the total population. Methods for characteriz-
ing the uncertainties associated with these estiaated exposure distributions
•re also discussed in Chapters 6 and 7.
A hypothetical czaaple of a subpopulation exposure assessment that pro-
gresses through several stages of refiaeaent is presented in Appendix A. The
sections of the exaaple in Appendix A correspond directly to the sections of
the first five chapters of this report end ere auabered accordingly (e.g.,
Section 4.1 of Appendix A corresponds directly to Section 4.1 of Chapter 4).
•11-
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2. ASSESSMENTS BASED UPON LIMITED DATA
The initial exposure assessment for • substance My be based upon limited
data for directly measured personal exposures, concentrations in the environ-
ment, and/or variables that can be used to predict personal exposure. These
data say be either extant data or data produced by • Mall-scale study.
Characterization of uncertainty for auch an initial exposure assessment de-
pends mainly upon whether the data ere direct measurements of exposure or
measurements of variables that can be used to predict exposure. The purpose
of this chapter is to discuss characterisation of uncertainty in each of these
situation! when data are limited. Characterisation ef uncertainty for an
assessment baaed upon directly measured exposures for a small sample of indi-
viduals is discussed first. Characterization of uncertainty is then discussed
for estimates of the distribution of exposure based upon limited dsta for
model input variables. Both the population exposure models that directly
estimate the population distribution of exposure and the personal exposure
models that estimate the exposure experienced by a sample of population mea-
bers are addressed.
2.1 UNITED DATA FOR DIRECTLY MEASURED EXPOSURES
Historically, the discovery of a new environmental pollutant often begins
when a physician recognizes a clustering of cases with common clinical fea-
tures that are unique and unexpected for the community in which the clustering
has occurred. If the physician • reports this situation to the appropriate
public health authorities, an initial study on a small sample of unusual cases
may be undertaken. Alternatively, if the physician reports the situation in
the medical literature, a followup study by public health authorities may
result. Under these circumstances, the initial assessment is likely to be
based on limited extant data, i.e., data for a sample ef the unusual cases and
standard baseline laboratory values. Zf a potential health problem was dis-
covered, a more refined assessment bssed upon a probability sample1 of popula-
tion members would almost always be needed to permit defensible risk manage-
ment decisions.
*A probability sample is a aample for which every populstion member has a
chance (i.e., positive probability) of inclusion in the sample. The simplest
example is a simple random sample, in which every population member has the
same positive probability of inclusion in the sample. See reference [11] for
an introduction to probability sampling methods.
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If an assessment is bated upon a Halted database of Measured personal
exposures, a priaary aspect of the uncertainty characterization is whether or
not the data result froa a probability sample. Characterization of uncertain-
ty for an assessment based upon a valid sample of directly Measured exposures
is discussed la Chapter 5. Statistics calculated from a small probability
sample generally have relatively large standard errors, i.e., a relatively
hith level of uncertainty. However, a small probability sample provides the
basis for more rigorous quantitative characterization of uncertainty than
would be possible for a small non-probability sample.
If the measured personal exposure levels for a small sample of population
•embers were not so low that health risks were known to be negligible, a more
refined exposure assessment might be pursued. Two ways of refining the expo-
sure assessment would be as follows:
1. Measure exposure levels for a large enough sample to make inferences
with a predetermined level of precision using the methods discussed
in Chapter 5.
2. Develop an exposure scenario and associated exposure prediction
model to be used ss the basis for an assessment ss discussed from
Section 2.2 through Chapter 4.
Since the exposure variable needed for predicting risk is often not amenable
to direct observation, the latter model-based approach is often pursued first,
if not exclusively. Hence, the model-based approach to refining an assessaeot
is discussed first in this document.
2.2 LIMITED DATA FOR MODEL INPUT VARIABLES
2.2.1 Population Exposu*-' Models
An initial exposure assessment is often based upon extant data or other
easily obtainable data regarding ambient concentrations and/or emission levels
for known sources of the substance in question. Such data may then be com-
bined with data on the geographic distribution of population members to esti-
mate the population distribution of exposures using an elaborate population
exposure model ([1], (7], (6], (9], (12]t [13], [U]). Such exposure assess-
ment? te«i t9 be subject to a high level of uncertainty. However, the charac-
terization of uncertainty for these assessments is often primarily qualitative
due to the nature of the underlying data and the estimation methods.
Characterisation of uncertainty for assessments bssed upon population
exposure models consists primarily of a description of the limitations of the
population exposure model and limitations of the data for the model input
•13-
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variables (see especially reference I 12]). In addition, sensitivity of the
predicted exposure distribution to plausible changes in the model and/or input
variable values can be quantified (see references I 13 J end | 14]). Additional
quantitative characterisation of uncertainty My not be possible, but valida-
tion of the exposure model, confidence interval estimates of relevant para-
meters of the exposure distribution, and estimation of the accuracy end pre-
cision of input variable measurements should all be pursued whenever possible.
2.2.2 Personal Exposure Models
An initial exposure assessment say be based open limited data (e.g.,
estimated ranges) for the input variables of e model that predicts personal
exposures. The personal exposure model would be bssed upon an exposure sce-
nario that considers all potential sources of the substance that could result
in contact with population members. Knowledge of transportation and fate of
the substance would be used to model the pathways from sources to contact with
population subjects. These models can be very simple, such es models involv-
ing only multiplicative .factors (see, e.g., references [3] aod (10], and
Appendix A). A personal exposure model is any model, simple or complex, that
expresses en individual populstion member's exposure es a function of one or
more input variables.
If the available data ere only sufficient to support estimates of the
ranges of the input variables, the exposure assessment might be limited to a
sensitivity analysis. Tne purpose of the sensitivity analysis would be to
identify influential aodel input variables and develop bounds on the distribu-
tion of exposure. A sensitivity anslysis would estimate the range of expo-
sures that would result •• individual model input variables were varied from
their miniauia to their maximus) possible values with the other input variables
held at fixed values, e.g., their mid-ranges. In addition, the overall mini-
mum end maxima possible exposures would usually be estimated.
The uncertainty for en exposure assessment ef this type would be charac-
terised by presenting the ranges of -exposure generated by the sensitivity
analysis, by inscribing the limitations of the data used to estimate plausible
ranges ot model input variables, end by discussing justification for the
model. Justification of the model should include s description of the expo-
sure scenario, choice of model input variables, end the functional form of the
model. Sensitivity to the model formulation should also be investigated by
replicating the sensitivity analysis for plausible alternative models.
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r w-r.._.w «.Jki.u.«.c. «, toe sensitivity analysis
presented no potentially significant health rick, there Bight be no aeed to
refine the assessaent. If both the aiaiaua tad aaxiaua exposures presented •
potentially significant health risk, it would be known that the exposure
oceaario represented a significant health problea without refining the assess-
•cat. van the aiaiaua does aot prtseat • potentially significant health risk
•ad the •BSiawa does, thea greater iaportanee is placed on choosing a ensnary
parameter of the exposure distribution (e.g., the Man or a apecific percent-
lie) aa the basia for a regulatory deciaioa. A a*re la-depth exposure assess-
•eat enabling estimation of the distribution of exposures ptnits selection of
•ay auaaary parameter (aiaiaua, aaxiaua, aeaa, or a percentile, etc.) as the
basis for a regulatory decision, and peraits the decision Baker to aeleet that
euaaary parameter after the fact (i.e., after the expoaure distribution hss
•MB eatiaated).
The aensitivity BBSlysis can be eahaaccd by cocputiag the predicted
exposures that result froa all possible iaput variable coabiaatioas. If each
Input variable has only a finite act of possible values, the aet of all possi-
ble coabiaatioas of the Iaput variables caa be foraed, and the predicted
exposure caa be coaputed for each coabiaatioa. The Iaput variable coabiaa-
tioaa resultiag ia txposures that coastitute poteatial health problaas can
thereby be Ideatified. These exposure predictioas caa be used to fora a
probability distribution of exposures, which will be referred to as the expo-
sure ocasitivity diatributioa, by coaputag the relative frequency of occurence
•f each exposure level or interval of exposures. The resulting exposure
eeasitivity distribution is equivslent to the population distribution of
exposures only if all coabiBstioas of the Iaput variable values are equally
likely to occur la the target populatioa. The exposure aeasitivity distribu-
tion caa also be derived for personal exposure aodels based open coatiauous
iaput vsriables with fiaite ranges by represeatiag theae variables by equally
•paced points, at the liait as the equal epaces beceae Infinitely aaall and
She BBBber of points becoacs Infinitely large, the procedure for eoaputing the
•xpoaure aensitivity diatributioa ia equivalent to eatiaating the probability
diatributioa of expoaurcs that results froa atatistically independent continu-
ous Input variables with Dnifor* diatributioas on the estiaated ranges. Thus,
the exposure aensitivity distribution can be derived analytically or by Hoate
Carlo eiaulatioa based upon independent Uniform laput variable distribution!.
•13-
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The Monte Carlo aethod consisit of raadoaly generating input variable values
•ad using these to eocpute corresponding txposure levels, genersting the
exposure teasitivity distribution via replication*.
Statistics based upon the exposure se&sitivity distribution should be
interpreted in tens of all possible iaf .* variable combinations. For exaa-
ple, tkt 13th perceatilc of the exposure sensitivity distribution would be the
exposure level exceeded by only 5 perceat of the exposures nsultiat froa all
possible ceabiaatiofts of input variable values. Although this distribution of
exposures is aot an estimate of the population distribution (oaless all possi-
ble coabiaations of the iaput variables actually arc equally likely), it has
nore otility for regulatory decision makiag than a standard sensitivity analy-
ois. For exaaple, if 99 pereeat of the computed axposures are below the level
that represents a significant health concern, the proportioa of the population
at risk is almost surely aaall (albeit possibly smaller or larger than 1
perceat).
Characterixation of aacertainty for the aahaaced oeasitivity aaalysis
vould include a discussion of liaitations of the data and justification for
the awdel as discussed above for the usual sensitivity aaalysis. Sensitivity
to ajodel fonulstioa should also be investigated by estimating the distribu-
tions of exposure that result froa using the sane Uaifora input variable
distributions with plausible alteraative models aad comparing the estiaated
perceatiles of the induced txposure distributions.
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3. ASSESSMENTS BASED UPON ESTIMATION OF INPUT
VARIABLE DISTRIBUTIONS
If • model has been formulated that expresses an individual'• personal
Upoiure as • function of oac or more input variable*, the population distri-
bution ef aposure CM be estimated analytically or by Monte Carle simulation
from an estimate of the Joint distribution ef the model input variables.' A
dlatribution ef exposure derived in this way ie not equivalent to one baaed
•pen Matured exposures free e personal monitoring study. The derived distri-
bution merely reflects the assumed personal exposure model nnd the probability
distributions need to represent the model input variables. All such assump-
tions nhould be explicitly stated and justified. Ideally, model input vari-
ables should be represented by empirically validated probability distribu-
tions. Bovever, the joint distribution of model input variable's can be esti-
mated fresi discussions with subject-cutter experts (e.g., vis nnivariate
histograms when the input variables may be treated ns statistically indepen-
dent). The estimated population distribution of exposures vill be equivalent
to the exposure sensitivity distribution discussed in Section 2.2.2 only for
the case when all the input variable distributions actually arc independent
Uniform distributions. Hence, the purpose of this chapter is to characterise
uncertainty for exposure assessments based mpon personal exposure models that
csn be nsed to estimate the population distribution of exposures froa informa-
tion concerning input variable distributions.
9.1 ESTIMATION BASED OK INPUT VARIABLE DISTRIBUTIONS
Vhen the data for model input variables are limited, discussions vitb
subject-matter experts c»o be nsed to formulate an estimate ef the joint
distribution of the input variables. Such discussions can be used to inves-
tigate independence ef the input variables and determine the approximate
location nnd shape of the input variable distributions. Qualitative knowledge
•f the input variable distributions can then be represented muantitatively by
input variable probability distributions. The input variable distributions
should be validated quantitatively usiai goodness-of-fit tests whenever suf-
ficient data nre available (see, e.g., Chapter • ef reference (13], er Section
4.3 nnd Chapter 6 ef reference
•Equivalently, one can estimate the mar-final distributions, i.e., the distri-
bution of each input variable ignoring the others, if the input variables can
be treated as independent.
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Discus.^ons with subject-Batter experts for the purpose of estinitins the
Joint distribution of aodel input variables should first detemine vhetber or
not the Bedel input variables can be treated as statistically independent for
the population of interest. If sjodcl input variables arc assisted to be inde-
pendent, the theoretical and/or eapirical justification of the aesuBcd inde-
pendence ahould bo discussed. If two swdel input variables cannot be treated
•a independent, their joint probability distribution can be estimated by
etetiCBimiag the relative frequency of occurrence for aroaa in the cross prod-
vet apace defined by the two variables. As a hypothetical saunple, the joint
'probability distribution of ingestion rate (IF) and exposure duration (CD)
•enld bo ostiMted for the aiodel in Appendix A by the relative frequencies of
occurrence shown in figure 3. (These relative frequencies are hypothetical
for the purpose of illustration only; these variables are treated as indepen-
dent variables in Appendix A.) In addition, the etartinal distribution of each
of the correlated variables, i.e., the diatribution *f each variable ignoring
(the others, should be derived fro* the joint distribution and reconciled vitb
available knowledge of the surginal distributions. Of course, if the input
variables are sot highly correlated or if the swdel is aot aensitive to the
assumption of independence, it suy be sufficient to treat the variables as
independent and to acknowledge Minor departures free) the independence assump-
tion as a aource of uncertainty.
figure 3. lypothetical Joint Xelative frequencies lor
bpoaure Duration (CD) and tegestion late (IK)
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The input variables of an exposure model that are statistically inde-
pendent of all ether input variables can be considered individually by sub-
ject-cutter experts, one variable at • tisje. The expert that is consulted
with retard to establishing a probability distribution for a factor in an
exposure model aeed set be conversant in statistics. A probability distribu-
tion can be estimated free auch information as the following general shape
•f the distribution, smallest possible value, largest possible value, aw it
likely value (mode), average value (swan), the median value aucb that half the
values of the variable are larger and half are smaller, and/or percentilea of
the probability distribution (e.g., the 5th aad fSth pcrceatiles). The sub-
ject-matter expert aeed not be able to specify all of these statistics in
order to estimate the diatribution. la sea* cases, the subject-matter txpert
Buy be able to provide a histogram to be used as the probability distribution
of a model input variable.
Such subjective probability distributions should be confined in discus-
sions with other subject-matter experts. The Delphi technique can be used to
obtain the most reliable consensus of opinion free a group of experts by using
a series of questionnaires interspersed with controlled feedback of group
opinion (17). If such discussions do not yield a consensus of opinion, sensi-
tivity of the model to the alternative input variable distributions suggested
by experts should be investigated. Alternatively, direct measurements of the
model input variables and/or exposure Itself may be used to estimate the
population distribution of exposures as discussed in Chapters 4 and 5.
Discussions with subject-matter experts, empirical data, and/or the-
oretical argumenta may auggest that a particular family of-probability distri-
butions is appropriate for representing a model input variable. For input
variables with e finite range of possible values, distributions such as the
Uniform, Beta, Truncated Vormal, or Truncated lognormal may be supported. The
parameters of auch distributions can usually be estimated from s measure of
central tendency (e.g., mean, median, or mode), a measure of variability
(e.g., •tamdard deviation or a mercantile), mnd an estimate of the finite
range of possible values. Reference (IB] provides guidance with regard to
estimation of parameters of continuous probability distributions.
The following considerations may be useful when trying to determine an
appropriate probability distribution to represent a model, input variable:
1. A histogram distribution may be s very useful initial estimate for
the distribution of a model input variable.
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If • Model input variable has a finite range of possible values and
if a smooth probability function is desired, a Beta distribution
(scaled to the dtsirtd range) My be appropriate. The Beta probabil-
ity distribution is very flexible; its probability function can
••SUM • variety of shapes by varying its parameters.
S. If «n input variable only assuats values grester than COM sjiaiaua
value (e.g., is positive-valued; then • GasjM, Veibull, or logaonal
tittributioa aay be appropriate. The fiaaM distribution is sjore
flexible, end it* probability function caa •••use a variety of
•hapes by varying it* parameter*.
4. If aa iaput variable has aa aarestrieted range of possible vsluei, a
Venal, Student's t, or Logistic distribution My be an appropriate
probability distribution. These distributions differ sjaialy in
their tails. la •dditioa, traacated versions ef these variables My
be ased in aituatioas 2 aad 3 described above.
The example in Appendix A vacs Hoate Carlo •iemlation to predict the
diatribution of exposure from presuaed independent distribution* for exposure
duration (ED) aad ingestion rate (IR), the a»dcl input variables. For •
population of V sjeabers, each person would have his ova iadividual value for
each of these variables. The actual Joint distribution of the input variables
would then be the distribution of the N paired values of ED and IX experienced
by the iadividual population embers. If the correlatioa of these variables
was aot negligible, the Joint distribution of these variables would have to be
•stiMted (e.g., using jeiat relative frequencies such as ahovn in Figure 3)
la order to produce a reliable estiMte of the population distribution of
exposure usiag the persoaal exposure aodel.
then qualitative knowledge of the input variable distributions is used to
•ntiMte the population distribution of exposure, aaccrtaiaty is characterized
by discussing justification for the presoMd swdel aad iaput variable distri-
butions. The choice of sjodcl iaput variables tad the faactioaal fora of the
•jodel ahould be discussed. Empirical data aad/or theoretical arguments that
aqpport the model ahould be preseated. Use of a ejodel aad iaput variable
distributions that have been empirically validated is recommended in order to
induce aacertaiaty. Alteraative models aad/or alternative iaput variable
attributions ahould also be discussed, teasitivity to these alterastives cac
be iavestitated by astiMtiag the distributions that result fro* plausible
alternatives aad comparing the perceatiles af the iaduced exposure distribu-
tions. All available data, even if the data are limited, ahould be used to
validate the presumed input variable distributions aad the predicted distribu-
tion of exposure.
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3.2 1IHITJNC DISTRIBUTIONAL RESULTS
Given a matheaatical model that expresses personal exposure as a function
of several input variables, the distribution of exposure can be derived ana-
lytically or by using Monte Carlo simulation •• described above. The distri-
bution of ejcposure is then induced by the distributions of the model input
variables. lower, as the number of input variables becomes large, it can be
•hem that the induced exposure distribution approaches a apecific limiting
distribution for certain classes of swdels. Ihe true exposure distribution
for a model based open several input variables Bay then be approximated by the
limit of a sequence of conceptual models with mere and more input variables.
for these classes of swdels, .the veana, variances, and covariaaeos are the
ssost important characteristics of the input variable distributions because
they are aufficient to determine the limit ing distribution. Other shape
characteristics of the input variable distributions then becoae of lesser
importance for models with aeveral input variables.
Limiting distributionsl results can easily be derived for models that can
be represented as a apecial case of the following model (such as the models
discussed in references [3] and [10], and Appendix A),
n d
i • n x, / n x, ,
i*n+l x
where all input variables, Xj through X^, are independent positive-valued
random variables (which includes positive constants as s apecisl case). Since
all input variables are positive, this model can aqulvalently be expressed as
n d
In Ys Z In X. - 2 In X, . (2)
i»l * i«n*l *
It then follows that if the independent random variables In Xj through In X^
are aacb that a central limit type of result applies (i.e., there is mo one
input vsrisble (In X^) whose variance is large relative to the variance of In
I), then the limiting distribution of the mxposore, Y, is lognomal
-------
rately reflects the true state of nature and so long as there is no single
iaput variable whose variance dominates the total variance. Data for exposure
aad/or Model iaput variables should be used to validate the aodel whenever
possible.
Siace the sjeaa aad variance are sufficient statistics for the logaoraal
distribution, estimation of these parameters will completely specify the
liatttiag logaonal distribution whoa the exposure model is of the fon given
by Equation (1). In fact, sny sjeasure of central tendency (e.f., acan, median,
•r aede) together with say acssure of dispersion (o.|., standard deviation or
a mercantile) can be aaed to estimate the parameters af the limiting lognormal
distribution. Vhen directly avaaured aiposuro data are available for indi-
vidual population subjects, those data say be asod to directly astiaate the
paraacters of the limiting, lofnoraal distribution. It is also possible to use
tike sjeaas, variances, aad covsrisaces of the aodel input variables to cstiaate
the parsaetera of the liaitinf lognormal distribution of exposure without
formulating apecific probability distributions for the aodel iaput variables,
an shove below.
A consistent estimate of the Bean of Y is the ratio
E m « n ECX.) / n ECX.),
x *
where E denotes an unbiased astiaate of the swan (sn ostiaate for which the
m
expected value over all possible staples is the populstion aean) snd EC is a
consistent estimate of the aean (an estiaate that converges to the population
Man as the sample site increases). Vhen the iaput vsrisbles can be treated
as independent, the numerator and denominator astiaatea are unbiased, but the
ratio is s bissod estimate of T. When the iaput vsrisbles cannot be treated
as independent, standard practice would be to continue to ase Equation (3)
anyway, which still yields a consistent astisute af T.
A consistent estimate af the variance (?e) af the exposure distribution
caa ae obtained by waiag a first-ardor Taylor aeries approximation. This
variance approximation say be expressed in general far a function af a raadom
variable* as -fellows: •
Tff.,....!)) • 2 I g (S)l» V (X )
C 1 B i«l Wi *
42 I 2 [ g (X)J I g (X)] Cav(Xi.X,)l (*)
wi " J J
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where JB if the partial derivative operator; £ is the mean of X., .... X ; V
•od Cov denote unbiased estimates of variance and eovariance; and V ii a
consistent estimste of variance (see, e.g., rcfcrtact |20] and Section 10.6 of
fftftrace 121]). Thii mult can bt applied directly to the model given by
Equation (1) because this vvdel it easily differentiate vita respect to the
input variables, Xi- Moreover, whenever the model input variables are all
independent, es is eften presumed to be true, the second ten in Equation (4)
becomes sero.
It ebould be noted that the Taylor aeries variance approximation, Equa-
Uon (4), is the basis for the "propagation of error formulas" found is refer-
ences (1] and |5J. Tbe application here is very different, Befereaces (1)
aad (5) use the Taylor series to approximate the variance of the mean of model
predictions as a characterisation of the uncertainty vith regard to the popu-
lation awan. The present application is for estimation of the population
variance of the exposure distribution when the bssic fen of the exposure
distribution, e.g., legnonal, has already been established. Thus, Equation
(4) is being used to facilitate estimation of the population distribution of
axposurc. The uncertainty vith regard to this estimated distribution is the
uncertainty that must be addressed, not that for the estimated mean of the
distribution.
Consistent estimates of the mean aad variance of T frost Equations (3) and
(4) can be used to fit the limiting lognormal distribution of Y directly. No
apecific distributional form need' be assumed for the model input variables,
Xj. A* noted earlier, any estimate of the central tendency of Y, together
vith any estimate of variability or a percentilc of the distribution of Y, can
also be used to directly estimate* the limiting distribution of Y. Moreover,
whenever the estimate sf risk simply involves multiplication of the exposure
by one or more positive factora, these results extend directly to estimation
of the probability distribution for risk.
The lognormal distribution for txposure T is attained for the exposure
sjodels given by Equation (1) as the number of model input variables becomes
infinitely large. Za practice, the number of factora in a model will always
be finite. The importance of the limiting lognomal result ie that the pre-
dicted exposure distribution vill be approximately lognormal when there are
neveral input veriables, none of which dominates the variability of exposures.
Thus, when the model has a sufficient number of factors, the means, variences,
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and covariance* of the input variable distributions arc typically sufficient
te determine the limiting distribution. Other shape characteristic! of the
input variable distribution! then beCOM Itss important.
Although the limiting distribution could be estimated directly for a
model vith several factors, research into the rate of conveyance to '-he
limitiag distribution would be aeeded for proper interpretation. Empirical
research would be needed to determine when • sjodel had a sufficient aumber of
factors to Me the limiting distribution as aa adequate approximation to the
true distribution. The results vould likely depend open the type of model and
iaput variables.
If locations (3) and (4) are ued to estimate the limitiag lognoraal
exposure distribution for a model of the type given by Equation (1), adequacy
af the approximation of the true exposure distribution by the limiting diitri-
bution ahould be discussed. The results ef an empirical atudy en convergence
to the Halting distribution Bight be cited. If data en measured exposures
for a aaaplc af population members were available, these data ahould be used
to test the adequacy of the limiting distribution. Characterisation of uncer-
tainty vith regard to the estimated exposure distribution vould also need to
address aacertaiaty vith regard to the estimated mesa and variance given by
Equations (3) and (4). If possible, confidence interval estimates ef the aean
aad variance ahould be derived. Sensitivity ef the predicted exposure distri-
bution to plausible ranges of values for these parameters ahould then be
investigated. Justification of the presumed exposure model ahould be dis-
cussed, as veil as limitations of the data used to estimate means and vari-
ances er to validate the model or limiting distribution.
.24.
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<•. ASSESSMENTS BASED UPON DATA FOR MODEL 2KPUT VARIABLES
The exposure asscssaeot bated upon » estiaate of the joint probability
distribution for tbc input variables of a personal exposure aodel can be
nfiaed by collecting saaplc survey data for all aodel input variables. The
population diatributioa of exposure can then be estimated by coaputing the
•odel-based exposure for each saaple aeaber. Tbese exposure values can be
wed to directly coapute confidence interval estiaates for perceotilcs of the
exposure distribution. If specific families of probability distributions are
accepted as rtpraseatint the joint distribution of the nwdel input variables,
staple survey data can be used to coapute joint confidence interval estimates
for the parameters of the input variable distributions. These interval esti-
mates for distribution paraa>eters can then be ased to cosjpute confidence
interval estiaatea for percentiles of tt». exposure distribution. Vsing either
•ethod of estiaation, interval estiaates for percentiles of the exposure
distribution are a useful quantitative characterisation of uncertainty for the
exposure assessment.
The purpose of this chapter is to discuss characterisation of uncertainty
for exposure assessaents based upon aodel input data collected for a saaple of
population aeabers. Coaputation of confidence interval estiaates for percen-
tiles of the exposure distribution directly froa aodel-based exposures for the
aaaple subjects is considered first. Derivation of equivalent confidence
intervals free confidence intervals on paraaetera of the input variable distri-
butions is then considered in Section 4.2.
4.1 INTERVAL ESTIMATES Of EXPOSURE PERCENTIIES: IKPUT VARIABLE DISTRIBUTIONS
DOT KNOWN
For suny personal exposure aodels, it suy be aore feasible and less
costly to collect data en the aodel input variables, cather than en the expo-
sure itaelf. The population aay be thought af as a aet af • subjects, each of
which experiences specific values ef the aodel input variables end resulting
exposure. If the input variable values have been recorded for a saaple of
population subjects (see, e.|., the hypothetical data in Attachaent A.I ef
appendix A), then a. aodel csn be ased to ceapute the predicted exposure for
•ach saaple aaaber. Treaties these ceaputed exposures es if they were en-
sured exposures, the aethods described in Section 10.18 of 122) end Section
12.9 ef (23] can be used to coapute confidence interval estiaates for percen-
tiles of the exposure distribution.
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Interval estimates for percentiles of the exposure distribution ire •
useful quantitative characterisation of uncertainty for the exposure assess-
•cat. lovever, interval estimates ce«puted •• distuned in this aection are
haaed tfm aeveral assumptions, primarily the presumed model for personal
opoaarea. luce, characterisation of uncertainty for the exposure assessment
requires JutificatioD of the model and ether assumptions. The empirical
and/or theoretical haaia for the model should he diacuaaed. Uae of a vali-
dated model always decreases the overall uncertainty. Design of the aample
amrvey that ia teed to produce the data ahould aloe he discussed, aa veil as
accuracy aad frociaioa of the meaaurmmeat processes. If a probability aample
{e.g., a aiso>le raadosi aaaplc) was not sjaed, the haaia for any inferences
hoyoad the individuals la the aaaple aust he discussed. Sensitivity to ewdel
foraulatioa caa he iaveatifated by aatiMtiag the distribution of exposure for
plaaaible altaraetive awdela if aaaple aurrey data have heen collected for the
input variables of the alternative Models.
4.2 IITTIRVAL ESTUUTES OF EXPOSURE 1EKCEVTIUS: HIFUT VARIABLE DISTRIBUTIONS
WOW
When the Joint probability distribution for the eicposure awdel input
variables can he represented by a apecific faadly of probability distribu-
tions, confidence interval estinates for perceatilea of the exposure distri-
bution can he coaputcd free) a joint confidence region for the parameters of
the input veriable distributions. In general, let the vector 6 denote the
peraaeters of the joint probability distribution for the model input vari-
ables. Let 6 denote a 100(1-a) percent confidence region for the aodel input
parameter •, i.e.,
Prob I § c 6 } • !•;.
A 100(l-«) percent confidence totarval estimate of the p-th perceatile of the
ejxfoaure distribution ia than given by
i *,(•> I**).
T (£) daaotea the p-th percaatile of the eopoaure diatribution imduced
by the penomal axposare aodal when the values of the Imput parameters are
f ir:f .ct .the «alue -«. That ia, the. «et of .all .fth fereaatilea ,• Ty( of the
induced exposure distributions generated by the aet of tee forms a 100(1-0)
pereent confidence interval estimate of the p-th percentile of the population
exposure distribution. Hence, the interval from the amalleat to the largest
•26-
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of these induced p-th percentiles, i.e.,*
[ In£ { Yp(6) | 6e6 }, Sup { Yp(6) | 6c6 } J,
constitutes,** interval estimate of the p-th percentile of the exposure distri-
bution haviag • confidence cetlficicnt of at least 100(1- •) percent. IB prac-
tice, It it set feasible to derive the exposure distributions corresponding to
•11 0e6. lower, when the personal exposure sjodel is mathematically tract-
able (e.g., continuous and monotonically increasing, it may be possible to
deduct those values of 6c6 that will generate the IBS1lest and-largest induced
p-th percentiles. la this case, only these two estimates of the p-th percen-
tile Deed to be computed ia order to geaerate the confidence interval esti-
•ate.
TWo advantages of this procedure with respect to that presented in Sec-
tion 4.1 are that eaaller aample aiaes may be aufficient and that the confi-
dence interval ostiMtes should be somewhat aarrover for a given sample size
because additional intonation has been osod (i.e., the assumed joint prob-
ability distribution of the input variables). A major disadvantage is that
this procedure is Bore difficult to implement.
for txaaple, suppose that an exposure model is based upon a independent
input vsrisbles. When the input variables are statistically independent, the
joint confidence level for the iaput variable distributions is the product of
the individual input variable confidence levels. Thus, 95 percent confidence
interval estimates for perceatiles'of the exposure distribution can be gene-
rated froa confidence interval estimates of the parameters of the a indepen-
dent variables, with the level of confidence being (0.95)1' for the param-
eters of each input variable. If in addition each iaput variable distribution
is iadeied by 1 parameters for which independent astiautors exist, the joint
(0.55)1/k confidence region for each iaput variable distribution is the cross
product apace defined by interval estimates for the 1 individual parameters
with a (0.9S)1'"* confidence level for each interval.4 The resulting joint 95
•Xoehaicslly^from the infimua (Inf) to the aupremua (Sup) of the induced p-th
percentiles corresponding to all values of 6 in 6.
tfioafideacc interval estimates of iaput variable parameters usually rely on a
model, e.g., approximate Normal distribution for a aamplc mean. Vben U ie
large,.this model may act be adequate for confidence interval estimates et the
(0.95)1'** confidence level.
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percent coafideace region for the k£ parameters of the input variable distri-
butions generates a set of eorreaponding probability distributions for expo-
sure via the model, baaed upoa the fc iadcpendcnt input variables. Given any
particular set of input variable distributions, the corresponding distribution
of exposures caa be generated analytically or by using Moate Carlo aimulation.
For any given pcretatilc of the exposure distribution, the 95 percent lover
•ad upper confidence limits arc the minimum tad maximum, rtipectively, of the
estimates of that percentile, baaed upon all exposure distributions generated
by the Joint 95 percent confidence region for the parameters of the input
variable distributions (see, e.g., Section 4.2 of Appendix A).
These interval estimates for percentiles of the exposure distribution are
s useful quantitative characterisation of uncertainty for the exposure assess-
swat. Bovever, they are baaed upon many assumptions. Characterisation of
uncertainty for the exposurs sssesssjcnt is not complete without s thorough
discussion of sll assumptions. Justification for the model should be dis-
cussed as veil as limitations of the data and design of the sample survey used
to collect the data. If s probability sample was not used, any inferences
beyond the sample subjects themselves must be justified. Any assumptions used
in computing the confidence interval estimates, such ss independence of model
input variables or estimators of parameters of the input variable distribu-
tions, should be explicitly stated and justified. Sensitivity to model formu-
lation caa be investigated by estimating the distribution of exposure for
•
plausible alternative models if sample survey data have been collected for the
input variables of the alternative models. Using validated input variable
distributions greatly reduces the uncertainty resulting from essuming par-
ticular input variable distributions. If exposure measurements are available,
the personal expoaurc model should be validated also.
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5. ASSESSMENTS BASED UPON EXPOSURE MEASUREMENTS
A major reduction in the uncertainty associated with an exposure assess-
ment can be achieved by directly measuring the exposure for a sufficiently
large representative sanple of aeabers of the affected population. This
reduction in uncertainty is achieved by eliminating the use of a model to pre-
dict exposure. Moreover, a siodel can be validated and model parameters can be
estimated if data are also collected for all potential model input variables.
The concept of directly measuring personal exposure to a substance is not
new. For example, occupational exposures to radiation and carbon monoxide
(CO) have been measured for years using personal exposure monitors (PEMs).
However, PEMs which measure concentrations that are expected to be an order of
magnitude lower than occupational concentrations have not previously been
readily available. In the past decade, advances in technology have enabled
manufacturers to miniaturize circuitry and components, thus making direct
measurement of personal exposures much more feasible [24]. As a result,
several personal exposure monitoring studies have been performed recently
([25], [26], [27]). These studies generally monitor an indicator of internal
dose, such as continuously monitored concentrations in the contsct zone,
rather than internal dose per se.
Given a database of measured exposure values for individual populatioo
subjects, such as the hypothetical sample in Attachment A.2 of Appendix A, the
population distribution of exposure may be estimated directly. In particular,
the sample exposure distribution estimates the population distribution. If a
probability sample design was used to select the sample members, a weighted
estimate of the exposure 4 ttribution should be computed (see, e.g., sample
survey inferences in [11], [21], [22], and [23]). Use of the sampling weights
is especially important if high-risk segments of the population (e.g., indi-
viduals with potential occupational exposure, or individuals expected to be
susceptible to health effects) have been oversampled. Typically, the sampling
weights account for unequal probabilities of selection and weight adjustments
to account for variations in response rate.
Tb* •••sured exposure values may also be used to directly compute confi-
dence interval estimates for percentiles of the exposure distribution (see,
e.g., Section 10.18 of [22], or Section 12.9 of [23]). These confidence
intervsl estimates for percentiles of the exposure distribution define an
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approximate confidence region for the exposure distribution. When the expo-
sure distribution is estimated from measured exposures for a probability
staple of population members, confidence interval estimates for percentiles of
the exposure distribution are the primary characterization of uncertainty for
the exposure assessment. Design of the survey used to collect the data should
also be discussed, ss well as accuracy and precision of the measurement tech-
niques. Ho mention of an exposure prediction model is necessary because the
exposure levels were measured directly, rather than calculated froo a model.
If, however, the measured exposures are not based on a probability sample, it
should be acknowledged that no strictly valid statistical inferences can be
•ade beyond the units actually in the sample. If inference procedures are
implemented for a nonprobability sample or if the sample design is ignored in
the analysis, the' assumptions upon which these inferences are based (e.g.,
treatment of the sample as if it was a simple random sample, or assumption of
an underlying model) should be explicitly stated and justified [28].
When personal exposures are being monitored, the exposure distribution is
often very skewed, with most of the sample members having exposure levels that
are less than the detection limit of the monitoring equipment, especially when
monitoring exposure to toxic substances ([27], [29]). In this situation,
estimation of the distribution of exposure may require a very large sample
size. Estimation of the proportion of population members with a detectable
level in specified analysis subgroups and in the total population may then
become an important analysis activity. Sometimes no sample members are found
to have a detectable level. When no sample members have detectable levels,
the power5 of the hypothesis test that the proportion of population members
with a detectable level exceeds 1 percent, 2 percent, etc., may become the
primary analysis (29]. It can then be inferred that the proportion of the
population with a detectable level does mot exceed seme fixed percentage for
which the power of the test is high. Qualitative characterization of uncer-
tainty for the assessment would be basically unchanged, but the power of the
hypothesis test for specified alternative hypotheses would be the primary
quantitative characterization of uncertainty.
•The power of the test is the probability of correctly rejecting the null
hypothesis that no population members have a detectable level when the alter-
native hypothesis (that the proportion of the population with a detectable
level exceeds 1 percent, 2 percent, etc.) is true.
•30-
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Whenever personal exposures are aeasured directly, quantification of
measurement errors becoaes an important aspect of the characterisation of
aacertainty. The tens "accuracy," "precision," and "bias" are used to de-
acribe specific aspects af measurement error ((3], 130], II.6-II.8 ef [31]).
She ten "bias" refers to systematic errors, i.e., errors that have a aoazero
expected value. Vaay chemical measurement techniques are kaown to produce
biaaed meaauremeats. Far example, the aaouat af a ceataaiaant ia a specimen
•ay be ceasisteatly greater than the measured amouat. The ten "precision"
refers to the repeatability ef a measurement. Bigh precision means there is
little variation between repeated measurements en the seme ebject, i.e., the
measurements exhibit a low coefficient ef variation (ratio ef the standard
deviation af repeated measurements divided by the mean). Finally, accuracy
refers to. the cumulative effect ef bias end precis ion. Bigh accuracy occurs
ealy when the bias ten ia eaall ead the precision ia high. Either high bias
er low precision results in low accuracy.
Vhen a survey collecting personal exposure data ia properly designed,
measurement error components can be quantified. Moreover, estimation proce-
dures can be developed that compensate for the estimated measurement biss. A
thorough quality assurance plan that eaables estiaation ef measurement error
components is presented in reference (32). The design presented ia reference
(32] enables estiaation ef everall precision, bias, aad accuracy ef a param-
eter estimate via the relative standard deviation (USD), relative bias (RB),
aad relative root mean square error (BMSE), respectively. These statistics
are defined es follows:
1. The BSD is calculated as the ratio af standard error ef a paraaeter
estimate divided by the parameter estimate itself.
2. The BB is calculated aa the ratio af the estimated absolute bias
divided by the peraaeter estimate.
3. USE • (BSD* + BB*A
•toother autistic that is aemetimes ased fcy the BPA as a measure af
•verall accaracy af a measurement procedure ia the "total error" defined aa
the sua af the absolute bias plus twice the coefficient af variation. IFA
guidelines specify that measurement accuracy is acceptable if the total error
is less than SO percent af the mean (33]. Vhen the measurement error ia
aaacceptable, improved •easurement techniques are aeeded before say reliable
easlyses can be performed [33]. Vhen the measurement error is acceptable, but
-31-
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the aeasureaent process has « nonzero bias, analysis technique* caa coapeasate
for the bias. Reference 132] discusses a survey design sod estiaatioa proce-
dure that enables estimation of the swan exposure, corrected for the estimated
MM bias. Beference (34] suggests th*t estimates of mesa levels bt multi-
plied by Ue relative biss to produce unbiased estimates of population waas.
finally, reference (35] tabulates the true significance level of a two-sided
Jqfpotheais Ust as a fuaction of the seminal significance level and bias for a
survey design that dees Bet allow estimation of the sjaaauroBcat bias. Uacer-
tatnty is then described qualitatively by eayiag that significant hypothesis
• test results would aot he affected smleae the hies was such greater than
expected.
If previous research aad/or theoretical derivetieas suggest that the
dspoaurc distribution should be a particular atatistical probability distri-
bution (e.g., Qatfora, Normal, or Logaoraal), the aoaiteriag data aay be used
(to ostiBate the parameters of this distributioa. A measure of caatrsl ten-
dency (e.g., swan, mediae, or Bode) together with a Beasure of variability
Ct.g., standard deviation or a parcentile) is usually aufficieat to sstiaate
»uch a distribution. Vhen the atatistical probability distribution has
liBited theoretical range, auch as the Unifon distribution, the saaple size
and the smallest and/or largest data values are usually needed ia order to
estiaate the extreme values of the distribution when they are not known a
priori. leference (IB] provides formulas for ostiBstioa of the paraaeters of
Best cesBwaly need probability distributions. Vhen a theoretical probability
distributioa has been fit to the data, a atatistical test of goodness of fit
. should be performed to determine whether or mot the theoretical distribution
provides a reasonably good fit to the monitoring mate (eee, e.g.., Chapter 8 of
|15], or lection 4.5 and Chapter 6 of |1«J).
Vhen total personal exposure vis a particular route, e.g., inhalation, is
•neitored, the mate provide • .direct estimate of the attribution of total
exposure for members of the sampled population. Since aggregation of ssti-
anted exposure distribution over aources and pathwaya ia mot required, the
swaitoring data provide a more defeasible vatimate of the diatributioa *l.
«sposttr< than the model-based estimates discussed la Chapters 2, 3, and 4.
Models that link observed exposure to emission aonrcea may he needed, however,
ia addition to Boaitoriag data, in order to astiaate the risk associated with
roguletery options. If data were collected for potential model input vari-
ables as well as exposure (see, e.g., the hypothetical exaaple in Attachment
-32-
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A.2 of Appendix A), then the paraaeters of a code! relating exposure to the
potential input variables can be estiaated using statistical regression tech-
aiques. Moreover, lack ef fit can be tested for • hypothetical sjodel, e.g., •
•odel bated open the physical transport and fate process. Such aodels linking
sources aad exposures ere very isjportant for assessing the potential iapact ef
regulatory actions. Using Models validated in this way greatly reduces the
tuaatitative uncertainty associated with exposure •aeesssjents. thfortuftstely,
the analytical SMasureaent difficulties «nd costs associated with collection
•f directly ewasued exposure levels often severly lix.it or even prevent col-
lection of eueh data.
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6. AGGREGATING ACROSS SOURCES, PATHWAYS, AND/OR ROUTES OF EXPOSURE
•
4.1 CQHBXiniG ESTIMATED EXPOSURE DISTRIBUTIONS
jeaerally wttMtei the distribution of teul expoaure via • particular route
(inhalation, iaaeation, «r dermal) for tat subjects of a nubpopulatioa ai
diecuasad ia Chapter 5. Bovever, an exposure eaaeaameat baaed 1900 a e»del
•anally estimates the diitributioo of exposure for subjects in a aubpopulatioa
for a particular combination mi .Aouxca, jXttOsjav, ^ad *x*ate. JnUcratioa of
tt tmaii ••Hifl -fa fjfrtet *m -«j»ti-
•ate tha tetei exposure distribution for an absorption route.
If tha index a, where s«l, 2, ..., S( indexes the unique combinations of
aources aad pathways for a particular route, r, and aubpopulatioa, t, then the
route r exposure for the i-th aubjoet in the aubpopulatioa can be expressed as
6
The distribution of T. »(i) can be derived analytically or by Monte Carlo
*»*
siawlatioa from the distributions ef Yf § t(i). In either case, an important
consideration ia whether or not the S aource-psthway combinations are indepen-
dent ef each ether. If not, the essessor ahould determine by sensitivity
testing whether the correlated cembinstioas must be simultaneously modeled so
ns to incorporate the correlation.' For example, if two pathways are strongly
correlated because they originate from the same source, the two pstbways would
value for ene pathway would then always be equal to the aource value for the
ether pathway for each iteration.
The total rente r exposure for the i-th subject in aubpopulatioa t can be
computed from Equation (7) by animation over the independent source-pathway
»inatioes indexed by s. The distribution of route r exposure may then be
xpeemre distributions. Thus, the Monte Carle method could be Impla-
nted by aimultaaeously ^taeratiAf-values for the independent source-pathway
combinations end their num. The result would be estimated exposure distribu-
tions for the independent aource-pathway combinations end theirsum, the total
«4iai
-------
In tone uses, the risk due to exposure to • subataace My be related to
total exposure over all routea (e.g., vbea there ia BO risk for expoaures
below sea* threshold level). Eatiaaticn of total exposure ia alao accessary
to judge whether regulation of • particular route of exposure ia an effieieat
approach to eoatrolling total population exposure. The total exposure experi-
enced by the 1-th subject ia aubpopulatioa t ia then given by
3 3 S
Tt(i) • 2 T (i) -2 2 Y (i) , (8)
* r»l *fl r»l ••! »•••*
•here r iadexes the three txpoaure routea (Inhalation, ingestion, and dermal).
Xa order to eatiaute the distribution of the total exposure, Tt(i), it
would be important to detcniae whether or not the exposures via the three
absorption routes vere independent randoa variablea. If act, the correlated
routea would have to he treated aiawltaaeoualy ia a Moate Carlo simulation, as
discussed above for correlated source-pathway combinations. When the suana-
tioa ia over iadependent routes, the Monte Carlo aianilation values for the
independent routea auy he added to produce a simulated value for the total
•xpoaure. The aianilatcd Monte Carlo diatributioa of total cspoaure can then
be generated by a large nunber of repetitioas of this tiaulatioa.
When the aubpopulatioa diatribution of exposure (either total exposure or
route r expoaure) has been estiauted aa described above, uncertainty concern-
log the individual route and/or souree-pathvay expoaure distributions should
he characterised aa described ia 'Chapters 3-S. Ia edditiea, these methods
should be applied, to the exteat possible, to the aggregated subpopulation
exposure diatribution. For example, if peraoaal •xpoaure has been measured
directly for varioua routea sod/or sourcc-pathvaya for a sample of members of
• suhpopulatioB, then the aggregate exposure has also been measured directly,
aad the sjethoda described la Chapter S csa be nsed to compute confidence
interval eatixutes for fercentiles of the aggregated exposure diatribution.
6.2 UOTING DISTRIBUTIONAL ttSULTS
When the total exposure given by Equation (7) or (•) is the ausnation of
•evere! tsrss, central liaiit theory provides iaiportant distributional results.
In either case, the'total expoaure is written ea the sm of the positive
rsndos) variables T . »(i). fence, if s central liaiit type of rcault appliea
—i—i—
to the variablea Tr t, then the liaiting expoaure distribution ia the Vernal
distribution aa the* nuaber of tena in the auamation in Equation (7) or (g)
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becomes large. ••••, when the total exposure is the SUB ever several sourcei,
psthwsys, and/or routes, and no one ten has a variance that ii large coapired
to the variance of the total, then the for. of the distribution for total
exposure iar lubjtets IB subpopulation t will be approximately Nonal (see,
e.g., toction 14-1.1 of (19]).
The MM and variance for the total ejrpoiurc, Tr t, can easily be esti-
eated fro* the Man, variance, and covariance for individual source-pathway
combinations as follows:
•l»rit(i)J • 2 I[Y (i)|, and (9)
*•* * *•••*
*»*
a*l
8
(i)] • X
8-1 8
2 2 2 Covtt (i), Y , (i)J, (10)
where E, Var, Cov repreaent the population mean, variance, and covariance,
respectively. Of course, when the aource-pathvay combinations iadexed by s
are assumed to be independent, the second ten ia Equation (10) is aero.
Substituting unbiased or consistent cstinatei directly into Equations (9) and
(10) provides corresponding unbiased or consistent estimates ef the Man and
variance ef the total exposure, Y t, for route r for members af subpopulation
t. These estimates may be used to .fit the limiting Normal distribution direct-
ly. In fact, any measure of central tendency (e.g., mean, median, or mode),
together with any measure of variability (e.g., etaadard deviation or a percen-
tile), can be used to estimate the limiting Venal distribution for total
exposure.
She limiting Bonal elistribmtiea far total exposure ia attained as the
amnmer ef tens in the Summation ia Equation (?) ar (8) becomes infinitely
large. In practice, the number ef tens mill always me finite. Xhe primary
importance ef the limiting Venal distribution is that the distribution for
total exyotare will be approximately Venal eftea there ere eeveral terms ia
the emanation, none ef which dominates the variability ef the total exposure.
.Ia this case the Mans, variances, and cevariaacea ef the tens in the aamma-
tiea an Mst important because they are aufficieat to determine the limiting
distribution. Other shape characteristics of the exposure distributions for
the individual tens are then less important.
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Although tbe Uniting distribution could be estimated directly when the
total exposure is tbe BUB of several terms, research into the rate of conver-
gence to the liaiting distribution would be Deeded for proper interpretation.
Empirical vesearch would be Deeded to detenine when the Dumber ef tens in
the summation was sufficiently large to use the limiting Venal distribution.
The results would likely depend upon tbe sources and pathways being summed and
their relative importance.
Xf Equations (9) and (10) arc used to estimate the liaiting Venal distri-
bution for total exposure, adequacy ef the approxiaation ef the true expoaure
distribution by the limiting distribution ahould be discussed. An empirical
Dtudy en convergence to tbe limiting distribution slight be Deeded. If total
exposure was measured for a sample ef population members, these data should be
used to teat the adequacy ef tbe limiting distribution. Characterisation ef
uncertainty with regard to the distribution ef total exposure would also Deed
to address uncertainty vith regard to the estiaated mean and variance given by
Equations (9) and (10). If possible, confidence interval estimates of tbe
•can and variance should be derived. Sensitivity ef the predicted exposure
distribution to plsusible rsnges of values for these parameters should slso be
investigated. In addition, limitations ef the data used to estimate aeans and
variances or to validate tbe distribution of total exposure should be dis-
cussed.
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7. COMBINING OVER DISJOINT SUBPOPULATIONS
Afltr the teul aubpopulatioo t distribution of axposure has been esti-
•ated for trery aubpopulatien, the «ixei of the aubpepulstions should be
astimated. The muBber of oubpopuUtion Mmbers experiencing • particular
level of exposure eao be estimated fro* ao estimate of the site of the subpop-
ttlation tad aa estimate of the properties of subpopulation members upcrieac-
ia| that exposure level. Moreover, the proportion! of the total population
•sobers IB the individual aobpopulations can be used to combine the estimated
•UstiibotiOM for the oubpopulationj to produce an estimate of the distribu-
Uoa of exposure for the total population.
Bnccrtainty concerning the oices of the oubpopulations sbould be ad-
dressed by diacuasini limitations of tbe data and methods need to estimate the
aobpopulation sixes. Confidence interval estimates of the subpopulation sizes
•bonld olso be produced whenever possible.
If the total affected population of V subjects is partitioned into T
disjoint anbpopulatiena that each contain >t aubjects, the estimated cumula-
tive probability distribution function for total route r axposure is given by
Fr(y) « Frob (Yr < y)
T .
• I (IT / ») Prob (Yr < y)
•el * r»l
• * Ptrr t <*>• (11)
•si • • • •
• *
where N and N are the estimated sizes of the t-th subpopulation and the total
population, respectively. Pt is the astimated proportion of the population in
aubpopulation t, and Ff t is the eatimatod cumulative probability distribution
function for route r axposure in aubpopulation t. Xf estimates of the route r
tmtpopnlatlen distribution functions, T>|t, are replaced by estimates of the
tistribution functions, ?t, for the total over all routes, Equation (11)
produces the astimated distribution function far total exposure over all
mates far the-total population.
Vhen the distribution of axposure mas been aggregated across subpopula-
tion* as described above, uncertainty concerning the individual aubpopulation
axposure distributions is characterized as described in Chapters 3-6. In
•38-
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addition, uncertainty concerning the total population exposure distribution
(cither for route r exposure or total txposure) can be addressed in terns of
percentilea of this distribution. The p-th percentile of the route r exposure
distribution is that txposure level, y . such that
p»Prtb
-------
can be iitiaated is follows
- •' (Pe .) 0-PC .)
V«(pC(t) « -^ Soi- . (17)
IB ftDcral, the variance expreaiien |iven by Equation (17) veuld be aultiplied
ty • 4tsi|o tffeet (or variaaee inflation factor) to reflect unequal aaaple
•fcleetioa probabilities and/or differential adjuataents to coatpeaaate for
•rcrrey aoartapoaae (ace taoplc aurvey aaalyaia in (11), (21], (22), (29)).
•40-
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8. SUMMARY
A floury of the priaary aethods recoaaended for characterising uncer-
tainty for exposure •••••uenti vas presented in Table 1. Virtually all
exposure asaessaents tictpt thoie baaad open Matured exposure levels for a
probability aaaple of population aeabers rely opoa a aodel to predict expo-
sure. The aodel aay be any aatheaatlcal function, staple or coaplex, that
astiaates the population distribution of exposure or an individual population
•saber's exposure as a function of one or aore input variables. Whenever a
eiodel that has aot been validated ia used aa the basis for an exposure aasess-
aent, the uncertainty associated vitb the exposure assesaaent stay be eubstan-
tial. The priaary characterisation of uncertainty ia at least partly qualita-
tive in this case, i.e., it includes a description of the assumptions inherent
in the aodel and their justification. Plausible alternative aodels ahould be
discussed. Sensitivity of the exposure asseaaaeat to aodel femulation can be
iaveatigated by replicating the assessaent for plausible alternative aodels.
When an exposure aasessaeat ia based upon directly aeasured exposure
levels for a probability aaaple of population aeabers, uncertainty can be
greatly reduced and described quantitatively. In this case, the priaary
aources of uncertainty are aeasureaeat errors and aaapling errors. A thorough
quality assurance prograa ahould be designed into the atudy to ensure that the
Mgnitude of aeasureaent errors can be estiaated. The effects of all sources
of randoa error ahould be aeasured quantitatively by confidence interval
eatiaates of paraaetera of interest, e.g., perceatiles of the exposure distri-
bution. Moreover, t>« effect of raadoa errors can be reduced by taking a
larger aaaple.
Whenever the latter ia not feasible, it ia aeaetiaes possible to obtain
at least eoae data for exposure and aodel input variables. These data ahould
be uaed te assess goodness of fit ef the aodel aad/or areauaed distributions
of input variables. This substantially reduces the eaount ef quantitative
uncertainty for estiaation of the distribution ef exposure end ia strongly
recoaaended. It ia recognised, however, that it aay not always be feasible to
collect such data. •
•41-
-------
REFERENCES
ID Callahan, H. A., Johnson, R. H., KcGinnity, J. L., HcUughlin, T. C.,
•warn, D. J., and 6lisak, P. V. (library, 1983). Draft Handbook
for Perforaiat Exposure Aaaeaaaenta. Prcp«rtd by the U. 8. Environmental
Prottctloa Agency, Office of Health and Environmental Assesscent.
|2) Executive Office of the President, Office of Science and Technology
Policy (Hay, 1984 Draft). Chcaical Carcinogens; Review of the Science
aad Ita Application*. Federal ietitter. Vol. 49, Me. 100, 21594-21637.
(3) Rational Heaearch Council, CotBittee en Prototype Explicit Analyse> for
Pesticides (1980). Retulatina Pesticides. Vatienal Acadesiy of Sciences,
VaahiBftoa, B.C.
(4) Calabreae, Cdvard J. (1982). The Role of Exposure Oats in Standard
Settint. Toxic Substances Journal. Vol. A, 12-22.
(5) 0. 8. Environmental Protection Agency (1978). Source Aaaesaaent: Analy-
ais of Uncertainty-Principles and Applications. EPA»600/2-76-004u.
FMO-131485.
[6] Cos, D. C.t and laybutt, P. (1981). Methods for Uncertainty Analysis: A
Comparative Survey. Risk Analysis. Vol. 1, No. A, 251-258.
(7] Hiller, C. (1978). Exposure Aaaeaaaent Hodelint: A State-of-the Art
Review. EPA-600/3-78-065.
(8] Drake, R. L. and Bar rafer, 8. H. (1979). iiathesMtical Models for Ataos-
pberic Pollutants. Electric Power Research Institute.
|9] Hoachandreas, D. J. and Stark, J. V. C. (1978). The 6EOMET Indoor-Out-
door Air Pollution Hodel. EPA-600/7-78-106, P
(10) EPA Enpoaure Aaaeaastent Group (Hay 1984). itochaatic Processes Applied
to Risk Analysis of TCDD Contaainated Soil; A Caae Study.
Ill) Ealton, 6. (1983). Introduction to Survey Saaolini. lage Publications,
•overly Hills.
112) Baamiaeu"i I*t Jones, A., Steifcrvald, B.( aad ThAapson, V. (Scptaaber
1984 Draft). The Magnitude aad Hature of the Air Tonics Problen in the
ites. OSEPA Office ol Air and Radiation, Office of Policy,
Evaluation.
113) Johnaoo, R. Jl.« Jr., iemhaxdt, ». .E.,-Haleon, *.-§.,..and Galley, H. «.,
Jr. (1973). Aaaeaastent cf Potential Radielotical Health Effects fro»
Radon la •aturaVGaa. EPA-520/1-73-OQ*.
114] U. 8. Environawntal Protection Agency (1983). Huaan Exposure to Ataes-
Pberic Concentrations of Selected Cheaieals. Office of Air Quality
Plannini and Standards.
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REFERENCES (coat.)
US] Daaiel, Wayne W. (1978). Applied RoaaarMetric Statistics. Routhton
Hifflia, Boston. '
(16] Cooover, W. J. (1971). Practical Roaparaaetric Statistics. John Wiley
and SOBS, Rev York.
(17] liaatoae, Rarold A., tad luroff, Hurray, eda. (1975). The Delphi Method:
Techniques and Applications. Addiaon-Wesley, leading, Massachusetts.
[18] Johnson, R. I., and Rotx, S. (1970). Distributions in Statistical Con-
tiauoua Caivariate Distributions. •oughton Miff HA, Boston.
(19] Burrill, Claude W. (1972). Measure, latetration. and Probability.
McGrav-Jlill, Rev York.
(20] Woodruff, Ralph S. (1971). Simple Method for Approximating the Variance
of • Complicated Estimate. Journal of the American Statistical Associ-
ation. Vol. 66, 411-414.
(21) Kendall, Maurice 6., and Stuart, A. (1968). The Advanced Theory of
Statistics. Voluae 3. Rafner, Rev York.
(22] Banaen, H. R., Burvitx, W. R., and Madow, W. C. (1953). Sample Survey
Methods and Theory. Voluae 1. Wiley, Rev York.
(23] Kish, I. (1965). Survey Sampling. Wiley, Rev York.
(24] Wallace, L. A., and Ott, W. R. (1682). Peraoaal Monitors: A State-of-
the-Art Survey, Journal of the Air Pollution Control Association. Vol.
32, 602-610. I
125] Speagler, J. D., and Soczek, M. 1. (1984). Evidence for Improved Ambient
Air Quality and the Reed for Personal Exposure Research, Environmental
Science and Technology. Vol. 18, Ro. 9, 268A-280A.
(26] lartvell, T. D., Claytoa, C. A., Michie, R. M., Wbitmore, R. W., Xeloa,
I. S.( Joaea, S. M., aad Wbitchurst, D. A. (1984). Study of Carbon Monox-
ide Expoaure of Resideats of Washiattoa. DC. aad Denver. Colorado.
IPA-600/S4-84-031, PBB4-183516.
(27] Pellissari, E. D., Bartvell, T. D., Sparacino, C. M., Sheldon, L. S.,
tbltsjort, R. W., lAialnier, C., aond Zeloa, I. S. (1984). Total EXPO-
ture Aasessaeat Hethodoletv (TEAM) Study; for** Beasea-Kortbern Rev
1392/03-35. Research Triaajle Institute, Research Triangle
PorkTRC.
*
(28] Banaen, H. H., Madow, W. C., tad Teppint, R. J. (1983). An Evaluation
•f Model-Dependent and Probability Sampling Inferences IB Sample Surveys.
Journal of the American Statistical Association. Vol. 78, Ro. 384, 776-
F9T
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(29] MasoB, X. 1., HcFadden, ». D., Ituaccaiooe, V. C., and HcGrath, C. S.
(1981). Survey cf DBCP Distribution in Ground Water Supplies and Sur-
ftct Hatar Fondi. KTI/W4/05-08J1. Xcaaarch Triaailc laititute. Kcaearcb
Triable Park, J»C.
(90] littahtrt, C. (1968). Expression of the ItocerUiotits of Final Btsults.
' ice, Vol. 160. 1201-1204.
131] ftdcrcr, V. T. (1973). lUtiiticB and locittv: Data Cellection and
lattrprttatioB. Hareel Dckkcr, Ktw York.
|S2] local, X. H.( Pitraea, S. A.. Byera, ». 1., ud Bandy, K. V. (1981).
Batioaal iuaun Adipose Tiaaue Surrey Quality Aaauraocc Project Plan.
RI1/1I64/21-11. Ktacarch Iriao«lc laatitutt, Xaaaarcb Trianflc Park, DC.
(33] Picrioa, S.f ud lucaa, X. (1983). Bitpanic HAKES Pilot Study; Hei-
i
pai_
t£f Volatile aad 8««iT8latil« Ortanic Ccapounds la Blood and
inc gpeciacaa. £PA 5»0/5-83-001.
(34J Bestafcld, J. M.. lueas, X. H., tt al. (1983). Uad and Cadniua Levels
IB Hoed of Baltimore. Maryland, tity Public School Teachers: World
health OrtaBiiatioB/llBited MatioBS EBviroaaeatal Protraat. Midwest
Xaaaarcb li
sacarcb lavtitute, Kaaaas City, Hiaaouri.
(35] Lncas, X. M.t Xaaaacchiene, V. 6., aad Helroy, D. X. (1982). Pelychlori-
ited Bipbecvls in HUMP Adipose Tiaaue and Mother*a Milk. XTI/1864/50-02F.
eaearcb Triangle Institute, Research Triangle Park, NC.
•at
Kei
-------
APPENDIX A
A HYPOTHETICAL EXAMPLE
A-l
-------
APPENDIX A: A HYPOTHETICAL EXAMPLE
A.I INTRODUCTION
The purpose of this appendix is to provide • tipple hypothetical example
of • aofuencc of exposure assessments with a charrctcriaetion of uncertainty
for each. The sequence begins with ao aaaeaament based 1900 limited initial
data and ends vith aa assessment based oa auffieiaat monitoring data to vali-
date and/or empirically estimate a personal exposure model. This example is
founded to clarify concepts presented in the text of this report. It is
oecofnittd that many expoaure aaaeaaMnts aay mot Nlatc directly to this
OBaapIe. mvvertheless, it is expected that this example vill provide a useful
Quide to characterization of uncertainty for exposure assessments snd clarifi-
cation of concepts discussed in this report.
The example is based open a recant case study by the EPA Exposure Assess-
ment Group.1 The cese study was concerned vith an assessment of human expo-
aure to TCDD (specifically 2,3,7,8-TCDD) resulting free) TCDD contaminated aoil
located in an insecure disposal aite auch as a aanitary landfill. It was
assumed that the aitc vas not located en residentisl or agricultural property
and that access was sufficiently restricted ao that aignificsnt human exposure
occurred only offsite. Two routes of exposure, dermal and infection, were
considered for exposure resulting from deposition of dust or eroded aoil en
rosidential areas. For the purposes of this illustration, only the soil
•
ingestion route vill be considered. The EPA case atudy focuied en the popu-
lation of individuals exposed to one part per billion of TCDD in the residen-
tisl aoil. A more refined assessment vould have accounted for variability in
the residential aoil contamination levels.
A.2 ASSESSBEKT BASED UPON LIMITED ZVITIAL DATA
The initial exposure assessment for a substance may be based upon limited
data for directly measured personal exposures and/or variables that can be
wed to predict personal exposure. Assessments based upon limited data for
directly measured exposure levels vill be considered first. Assessments based
vpeo limited data for the input varieties ef a personal exposure model vill
•hen be discussed in lection A.2.2.
SEPA Exposure Assessment ' Group (May 1984). Stochastic Processes Applied
to Riik Analviit af TCDD Conteminsted Soil: A Case Study.
A-2
-------
A.2.1 Halted Data for Directly Measured Exposures
Aa aaaessaent My begin vitb directly Matured exposure levels for a
Mall •••pit of population •cabers. The axpoaure data My be extant data,
a.|., data froa Mdieal records or routine Mnitoring data. A aaaple turvey
aight be deaifncd to coapile extant exposure data for aaaple aubjects. Selec-
Uen of the iaitial aaaple aight be purposive (e.g., o iiaple of individual*
with opecific health tffccta aad potential axpoaure to the iiibaunce aader
investigation). If a potential health problem was discovered, • sore refined
aaeesaMnt baaed apon a probability aaaple of populatioa ambers would alaost
always be Moded to enable risk SUM gown t decisions.
Uhea aa assessaeat ia baaed upon a limited database of awaaured peraoaal
exposures, a priaary aspect of the cbaracterisatioa of uncertainty is whether
or aot the data roault frost a probability aaaple. Characterisation of uncer-
tainty for aa aaaesaaeat baaed upon a probability aaaple of directly aeaaured
exposures ia discussed in Section 5. However, atatiatica calculated from a
•Mil probability aaaple |cnerally have relatively large ataadard errors,
i.e., a relatively high level of uncertainty.
If the Maaured peraonal cxpoaure levels for a aaall saaple of population
MBbers were not ao lew that health risks were known to be negligible, a more
refiaed exposure assessaent aight be pursued. Two waya of refining the expo-
sure assesaaent would be as follows:
1. Heaaure exposure levels for a large eaougb saaple to Bake inferences
with a predetermined level of precision using the aethods discussed
ia Section 5.
2. Develop an er?osure sceaario and associated exposure prediction
aodel to be used as the basis for aa assessaent as discussed in
Sections 2.2 through 4.2.
Since the exposure variable needed for predicting riak ia oftea aot aaeaable
to direct observstion, the latter aodel-based approach ia oftea puraued firat,
if aot axcluaively. trace, the awdel-baaod approach to refining the assess-
awnt ia preaeaud firat.
A.2.2 LiiiUd Data for Model Input Variablea
If liaitad initial data on exposure indicate that a auk stance My preaeat
a significant health concern, the exposure assessment My be refined by devel-
oping a peraoaal axposurc aodel beaed upon aa exposure scenario that coaaidera
all potential aources of the aubstaace that could reault in contact with
population Mabera. The aodel could be very aiaple, possibly involving only
A-3
-------
multiplicative factors, or be very cooplex. ID the present context, a per-
eonal exposure model is toy equation that expresses a population Member's
expoeure as • function of one or aero input variables.
for example, suppose that the follevinf model was adopted to predict the
dose of TCDD absorbed ia a lifetime * y the i-th populatioa member for the EPA
case study citad above:8
Ki) • (ftC(i)] lED(i)) IlJl(i)] (AF(i)J (c(i)], (l)
where T • Dose in Baits of aaaograms of TCDD absorbed in a lifetime,
1C • TCDD concentration in infested aoil
• 1 pat par billion (by definition in the IPA ease study),
ED • Exposure duration ia oaits of days of contaminated soil ingestion
par lifetime,
\
ZX • foil imgestioa rata for that portion of the lifetime during which
aoil is ingested in anits of grans of aoil ingested per day,
AT • Absorption fraction ia units of grans of TCDD absorbed per gran
of TCDD ingested, and
e » A positive randon arror ton with an oxpected value of one.'
For on initial exposure assessment, only limited data would typically be
available concerning the populatioa distribution of model input variables. For
•sample, the data may aopport estimation of mo more than approximate ranges of
values for the input variables. For the model (1), suppose that the following
estimates of the ranges of the input variables are based open very liaited
dats and/or aubjective considerations:4
•This model is adapted from that meed in the IPA case atudy. The case atudy
oodel included a potency factor ao that risk was estimated rather than expo-
Bare. The model alao otaadardiaed the exposure by dividing by the exposed
fforaoa'a body weight and lifetime. The body weight and lifetime.factors are
not presently considered for the aake of simplicity. Alao, lifetime, body
••weight, and exposure duration stay well -be correlated, especially Cor short
lifetimes, making the Monte Carlo aimulatiea much more complex.
*Ime raadoa error ton represents the cumulative multiplicative effect of all
.tsnobserved or latent variables not explicitly included in the model.
* •
4Xn practice, it My be relatively oaay to collect data with which to estimate
the population distribution of some model input variables, e.g., SC and AT for
oodel (1). The diatribution of other input variables, e.g., ID and IK, or
(their product—the total grama of aoil ingested per lifetime, may be much more
difficult to estimate. One may not be sble to improve upon expert subjective
estimates of the ranges of such variables.
A-4
-------
CD: 0 to 900 days of cootaaiaatcd toil iaieition per lifetime.
IR: 0 to 11 fract of soil ingested per day of ingest ion.
AT: 0.10 to 1.00 |rao> of TCDD absorbed per gram ingested.
Such data would typically verve ai the basis for performing a sensitivity
analysis for the Model. Tbe purpose of the sensitivity analysis would be to
identify influential model input variables and develop bounds en the distribu-
tion of lifetime absorbed dose. Tbe aaasitivity analysis Bight begin by
••tinting the maximua possible dose. In the worst case (ED»900, UNll.
AF«1.0), the ejcpected lifetime dose would be 9.900 aanograms of TCDD. If this
doae was not considered to be a potentially aignificant health concern, the
exposure assessment might be teninated. If one were sot confident that the
prtdieted dose truly represented the worst case, further investigation Bight
be dcsirsble.
If the worst case did present a aignificant health concern, the sensi-
tivity analysis Bight be pursued further. For example, one Bight use the
Bodel to estimate the range of dose that rtsults as oach input variable varies
over ita range of possible values, with the other variables fixed at their
Bidrange, i.e.,
ED: Dose varies from 0 to 4455 nanograms as CD vsries from 0 to 900,
with IR • 5.5 and AF • 0.90.
IB: Dose varies froo 0 to 4455 nanograas as IR varies from 0 to 11, with
CD * 450 and AF • 0.90.
AF: Dose varies from 1980 to 2475 nanograas as AF varies from 0.80 to
1.00, with CD « 450 and IR • 5.5.
Since the predicted dose is Best sensitive to changes in exposure duration
(ED) and aeil ingestion rate (IR), collection of additional data for these
variables, or their product—the lifetime aeil iagestion—would be warranted.
•incc the plausible variation in the absorption fraction has lass tffect on
the predicted dose, leas effort Bight he directed toward collecting data on
this factor; it Bight even he reasonable to treat this factor aa a known
consist fcr aubsequent modeling.
The acnsltivity analysis can he enhanced by computing the predicted doaes
that result from all poasible input variable combinations. If each input
variable had only a finite set of possible valves, the aet of all possible
combinations of the input variables could be formed, and the predicted dose
could be computed for each combination. These predicted doses could be used
A-5
-------
to fora • probability distribution of dose by computing the relative frequency
of occurrence of each dose or interval of doses. The resulting dose sensitiv-
ity distribution it equivalent to the population distribution of dose only if
all combinations of the input variable values arc equally likely to occur in
the target population. The doae sensitivity distribution can also be derived
for personal exposure models based upon continuous input variables with finite
ranges by representing these variables by equally-spaced points. At the
liait, na the equal spaces hecosie infinitely avail and the number of points
becomes infinitely large, the procedure for computing the doae sensitivity
distribution is equivalent to estimating the probability distribution of dose
that results fro* statistically independent continuous input variables vith
Oviform distributions on the estimated ranges. For the present example, the
dose sensitivity distribution was generated by the Monte Carlo method vitb
9,000 iterations using independent Uniform distributions for ED, IR, and AT on
the ranges 0 to 900, 0 to 11, and 0.80 to 1.00, respectively.1 The resulting
distribution of dose is shown in figure A.I. Estimates of selected percen-
tiles of this distribution arc shown in Tnblc A.I. Statistics bssed upon the
•lose sensitivity distribution are interpreted in tens* of all possible input
variable combinations, rstber than the population distributon of dose. For
example, the 95tb percentile of the dose sensitivity distribution (approxi-
mately 6400 nanograms of TCDD) ia an estimate of the lifetime dose exceeded by
only 5 percent of tbe doses resulting from all combinations of input variable
values.
The uncertainty for this initisl tsposure assessment would be charac-
terised by describing the limitations of the data used to estimate plausible
ranges of model input variables and by describing the basis for the exposure
model. A discussion of the limitations of tme model and justification for the
•jodel would me o major component of the character ins t ion of uncertainty.
flnusible alternative models should at least he discussed. Sensitivity to
^^ • ' »•«»•* »««•< '. * » + l—. .
model formulation should be sddressed by repeating the Honte Carle simulation
for plausible eltemative models. Tbe choice of a specific number iterations
"•Jbnte Carlo simulations were first ferformed vith 100, 1,000, 3,000, mnd
10,000 iterations. All simulations presented in this appendix ere based on
3,000 iterations because there was little difference in estimates based on
3,000 and 10,000 iterations. The additional cost of 10,000 iterations did not
appear to be justified.
A-6
-------
p
E
R 20-
C
E
N
T
10-
0
4000
PREDICTED DOSE CNG/LIFETIME3
Flgvre A.I. Diftritratien of Ettfmted Ooies ResiiltinR fro* Equally Likely tnpat VaHable Combinations
-------
Table A.I. SELECTED PERCENTILES FOR ESTIMATED DOSE RESULTING FROM
EQUALLY LIKELY INPUT VARIABLE COMBINATIONS
Pcrctatilc
•
O.01
0.02
0.03
O.O4
0.05
0.06
0.07
0.06
0.09
0.10
0. 15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
O.55
0.60
0.65
0.70
'0.75
0.60
0.85
0.90
0.91
.92
.93
.94
.95
.96
.97
.96
.99
Point
EstiMte
11.57
26.15
39.64
59.55
76.44
103.72
126.05
151. SI
172.36
194.40
314.61
467. 12
616.11
790.69
995.34
1202.6S
1406.98
1624.66
1909.63
2214.34
2571.43
2951.19
3S55.48
9924.13
4552.30
5316.73
5473. 16
5651.56
5865.06
4116.02
4379.43
4654.90
-->4939.50
7297.60
7828.86
Itproauctd from
Mii»v«iUbU copy.
A-8
-------
for the Monte Carle simulation should be justified, noting that more itera-
tieas would improve the precision of tstiMtcs at somewhat greater expense.
A.3 ASSESSMENT BASED UPON ESTIMATION OF IKPUT VARIABLE DISTRIBUTIONS
Discussions with subject-sutler experts (e.g., medical doctors) can be
used to fonulate an estimate of the joint distribution of the model input
variables, for example, auppose that discussions suggested that the input
variables for model (1) were all atatistically independent and that the fol-
lowing input variable distributions wore supported:
ID: Discussions auggested that the distribution of exposure durstion was
a symmetric continuous distribution with the following properties:
1. Minimum duration • 0 days.
2. Maximum duration • 600 days.
3. Mcsn duration • 300 days.
4. 10 percent of exposure durations are ISO days or less.
5. 10 percent of exposure durations are 450 days or more.
This information auggeats that the distribution of aspesure durstion
is as shown in Figure A.2. This distribution can be parameterized
by fitting a Beta probability distribution for ED/600.4 In particu-
lar, the Beta distribution with o « B • 3.1 is a good approximation
to the distribution of ED/600 shown in Figure A.2.
XR: Discussions auggested that the ingestion rate varied readonly from
0 to 5.0 grans per day over the population menbers, i.e., a
Unifen (0, 5.0) probability distribution was aupported for IR.
AT: Discussions suggested that tbe absorption frsction Bust be very
close to one, if not identically one.
Probability
Density
Figure A.2. Hypothetical Subjective Estimate of the Distribution of
Ixposure Duration (ED)
•See Johnson, *. 1., and 8. Koti (1970). Distributions in Statistics;
mou» Univariate Diatributions. Chapter 24 fo
distribution.
Contio-
or the parameterization of the Beta
A-9
-------
The estimated probability distributions for the model input variables were
used to generate an estimated distribution of lifeline dose, assuming tbe
•bore Joint distribution for the input variables for model (1) using a Monte
Carlo aiaulation baaed upon 3,000 iterations. The estimated distribution of
4ose ia ••own in Figure A.3. Selected perceatiles free this distribution as
•hewn in Table A.2.
She ptlBiary characterisation of oacertaiaty for this exposure assessaent
would be a discussion of the Justification fer the presumed model and input
variable distributions. Any available data,would be used to validate the
input variable distributions aad predicted distribution of dose. Discussions
vith subject-cutter experts are not aufficient to establish confidence inter-
val estimates for perceatiles of the estimated distribution of dose. However,
differences ia the expert opinions ceaceraiai the input variable distributions
ahould be discussed. Moreover, sensitivity of the estimated distribution of
dose to plausible alternative siodels ahould be discussed.
Coapariaon of Figure A.3 to Figure A.I illustrates the effect of discus-
aioaa with aubjeet-Mtter experts for this hypothetical aaposure asaessaent.
Several changes occurred ia the hypothetical aaaaple. The distribution of ED
was changed froa Unifora on the range (0,900) to a scaled Beta distribution on
(0,600). The range of the Uaifora IR distribution was modified froa (0,11) to
(0,5). Finally, the Uaifora diatribution of AF oa the range (O.8., 1.0) was
replaced with the fixed value unity. Such changes in the ranges of the input
variables Bight occur if conservative estimates of input variable ranges vere
aaed for the initial axpoaure aaaeaaaent. The aet effect of all these changes
an the worst caae acaaario ia rather dramatic. The worst case aceaario now
predicts the Maximum lifetime dose to be 3,000 aanograms of absorbed TCDD,
iaatead of 9,900 aanograms.
Vhcn comparing Figures A.I aad A.3, another important change ia that only
figure A.3 can be iaterpreted aa an aatimate af the population distribution of
dose. It can be ao iaterpreted becauae it ia based upon estimates of the
distributions af the aodel input variables. Figure A.I was generated by
treating all values within conservative estimates af the raages of the aodel
••impat variables as ooually likely; it represents the dose aaaaitivity distri-
bution, aot an estimate af the population distribution of dose.
In order to iaolate the affect of replacing the Vnifora distribution for
ED with a acaled Beta distribution, an additional simulation was performed
A-10
-------
40H
P
E
R 20-
C
E
N
T
o
2000 • 4000
PREDICTED DOSE CNG/LIFETIfO
Figure A.3. Predicted Dose Distribution Resulting from the Pretwwd Nedet and Inpat Variable Diatrihutlons
-------
Ttble A. 2. SELECTED FERCENTILES FOR THE DOSE DISTRIBUTION RESULTING FROM
THE FRESUKED MODEL AND INPUT VARIABLES DISTRIBUTIONS
Fercentile
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0. 10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
' 0.75
o.eo
o.es
0.90
0.91
0.92
0.93
0.94
0.95
0.96
C.97
0.98
0.99
Joint
SitiMtc
12.57
23.63
35.49
48.77
61.09
76.02
67.27
101.28
114.63
125.63
197.32
259.01
321.68
385.25
452.75
519.19
890. 15
668.44
751.23
628.63
919.94
1019.62
1116.64
1244.86
1383.62
1547.47
1592.77
1626.67
1664.63
1721.11
1767.70
1642.66
1921.01
2O34.71
8210.60
A-12
-------
using the aane input distributions that produced Figure A.3, except that a
Uniform (0,600) distribution was used for the ED distribution. Tbus, Figure
A.4 is based upon a Monte Carlo simulation with 3,000 iterations and indepen-
dent input variables with the following distributions: (1) Uniform (0,600)
for Er; (2) Uniform (0,5.0) for IK; and (3) AF • 1.00. Selected percaatiles
from the resulting distribution ef dose are presented in Table A.3 for compari-
aoa to Table A.2. Comparison of these tables reveals that the exposure distri-
bution baaed upon a Uniform distribution for ED (Table A.3) is flstter and
•ore disperse than that based upon the Beta distribution (Table A.2). This is
a result of the fact that the Uniform distribution is itself flatter and more
dispersed than the Beta distribution.
A. 4 ASSESSMENT BASED UPON DATA FOR MODEL IKPUT VARIABLES
The exposure assessment based upon an estimate of the joint probability
distribution for model input variables can be refined by collecting sample
surrey data for all personal exposure model input varisbles for a probability
•ample of population members. The population distribution of exposure can
then be estimated by computing the predicted exposure for each aample member
based upon tbe model. These predicted exposures can be used to directly
compute confidence interval estimates for percentiles of the exposure distri-
bution. If the joint distribution of the model input varisbles is hoove, the
aample survey data can be used to compute joint confidence interval estimates
for the parameters of tbe input vsrisble distributions, which can then be used
to compute confidence interval estimates for percentiles of the exposure
distribution. Using either method of computation, interval estimates for
percentiles of tbe exposure distribution are a useful quantitative charac-
terixation of uncertainty for an exposure assessment.
Development of confidence interval estimates for percentiles of the
exposure distribution would ordinarily require that sample survey data be
collected for exposure and model input variables. That is, a probability
•ample of population members would be •elected, and data would be collected
for these individuals for the exposure end model input variables. For the
example aader consideration, the exposure of interest is the lifetime absorbed
dose of TCDD via the ingestion rente and the deposition pathway. The model
input variables are the number of days of ingestion of contaminated aoil in a
lifetime, ED; the rate of soil ingestion, IB, in grams per day; and the absorp-
tion fraction, AF. Unfortunately, none of these variables are amensble to
A-13
-------
p
e
R
C
E
N
T
• • '
PREDICTEO DOSE CNG/LIFCTIMEJ
A.4. Dittritmtlofi e>f tstimted Doae Resulting tnm Alternative Cifially Likely Input Variable Co«blnation
-------
Table A.3. SELECTED PEXCENTILES FOR ESTIMATED DOSE RESULTING FROM
AITEXNATIVE EQUALLY LIKELY INPUT VARIABLE COMBINATIONS
Ptrccotile
0.01
0.02
0.03
O.04
0.05
0.06
O.07
o.oe
0.09
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
'0.55
0.60
0.65
0.70
0.75
0.60
O.65
0.90
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.96
0.99
Point
IstiMte
2.98
7.64
S3.57
19.66
27.51
as.se
43.09
49.62
86.27
44.57
101.46
154.39
206.61
266.19
331.07
401.69
464.14
852. 1&
645.oe
726.24
649.74
979.36
1134.76
1314.16
1503.40
1770.46
1642.22
1915.40
1960.76
2079.25
2156.24
2256.32
2328.70
2457.85
2626.13
I Itprorfuctd .from
I bttt
cop
A-15
-------
direct observation. For the take of illustration only, this example will be
coati&ued as if the absorption fraction were known to be unity and the other
variables could be measured dirtetly for a tuple of individuals from the
target population.
.4.1 Xmterval_Estim«tes ef Iicposure Percentiles: Input Variable Distribu-
>t Inovn
For many personal exposure models, it My be such easier end less costly
to collect data en the model input variables, rather than on the exposure
itself. Beace, given model (1), tappoae that data have been collected for the
exposure duration, ED, and the iagestion rate, ZR, for a probability Maple of
subjects from the target population. For illustration, hypothetical outcomes
for a simple random sample of aise 500 were generated using a symmetric Beta
(e * B * 3-D distribution for ED/600 sad an independent Uniform (0, 5.0)
distribution for XR. The 300 pairs of values generated for ED and IR are
listed in Attachment A.I. Histograms of the ED and XR distributions for this
hypothetical sample are shown in figures A.5 and A.6.
The predicted dose was computed for each of the 500 sample members as the
product of the ED and XR values shown in Attachment A.I in accordance with
model (1), assuming that AT is unity. Treating these 500 predicted doses as
if they were measured doses, equal tail area 95 percent confidence interval
estimates were computed for percentiles of the distribution of dose.9 These
confidence interval estimates are listed for selected percentiles in Table A.4
ond presented graphically in Figure*A.7.
Computation of the interval estimates for perccatiles of the distribution
of dose shown in Table A.4 and Figure A.7 is based upon: (1) the exposure
model (1); (2) sample survey .data on the input variables (Attachment A.I); and
(3) the assumption that the absorption fraction (AT) is unity. The interval
estimates of p«rcontilcs of the distribution are a mseful fuantitative charac-
terisation of uncertainty for the exposure assessment. Bovever, chsracterisa-
tion of mneertainty for the exposure assessment is mot •complete without a
thorough discussion of all mssumptioas, especially Justification of the model
matd U compute the predicted doses. The design of the sample survey used to
collect the -data should also-fee discussed. 'Boreovcr, •tasitivity to the
Y8ee Bansen, M. H., V. H. Burwits, and V. C. lUdov (1953), Sample Survey
Methods and Theory. Section 10.18; or Kish, Leslie (1965), Survey Saapling.
Section 12.9.
A-16
-------
I5.O-
12.5-
P
E
R
C
E
N
T
7.5-
S.O-
2.5-
50 100
Pigare A.S.
150 200 250 300 350 400 450 500 55O 60O
EXPOSURE DURATIONCDAYS)
M of the Rypethettcal Savple of Size 500 for Exposure Duration (CD)
-------
12.
10.
P
E
R
C
E
7.5-
S T 5.0-
o.o-
0.5 1.0 1.5 2.0 2.5 9.0 3.5 4.O 4.5 5.8
INGESTION RATTCGRAHS/OAY)
Figure A.6. RistoRrM or the Hypothetical Sanple of Size 500 for Inge*lion Rate (IP)
-------
Table A.4. 95 PERCENT CONFIDENCE INTERVAL ESTIMATES FOR SELECTED PERCENTILES
OF TEE DISTRIBUTION OF DOSE (Y) BASED ON MODEL-PREDICTED DOSES
j^^^te^
^EfTt^' ff'i* • * *^S 0>w01
|P0|jfeh$ f?»~...&~j 8). 02
tey^^/j?'.. •' J j| Q,03
*-^^^ ' *^*7~""O.04
1:"!'i 0.05
"'- .""* O.06
.07
.06
.09
.10
.15
.20
.25
.30
.35
' .40
0.45
0.50
0.85
0.60
0.65
0.70
O.75
0.60
O.85
0.90
0.91
O.92
O.93
0.94
0.95
* O.96
.-.*..< 0.97
'• — _
W ' • ' •.*»
lover
i Limit
• ^^t
•80 •
4). 80
t.43
i3.26
23.79
88.36
48.86
86.38
47.01
78.97
186.50
196.79
257.81
285.10
•39.16
400.67
462.55
859.40
424.04
494.09
776.75
•85.71
981.62
1086.97
1243.56
1416.62
1443.15
1829.68
1899. 17
1448.46
1749.26
1859.73
1097.96
3)007.93
9143.18
Poiat
UtiMte
7^^
••V
10.49
93.12
88.46
49.83
41.23
74.49
•1.32
114.38
137.07
106.11
935.09
276.87
•32.24
991.30
452.37
848.56
408.42
482.09
781.23
•73.85
955.25
1085.00
1173.99
1331.26
1850.54
1430.07
1468.86
1747.71
1847.36
1880.79
1941.24
9027.12
9142.16
8980.78
Upper
limit
• A 4K
• V.29
95.28
44.49
87.01
72.48
•2.15
124.46
141.70
149.59
162.53
212.77
266.90
311.68
373.27
445.83
834.44
404.82
476.24
748.56
•61.61
949. 18
1020.47
1135.79
1291.32
1428.95
1725.20
1779.50
1666.79
1885.84
1944.57
2018.62
2127.76
2171.92
9295.75
9428.30
A-19
-------
i.e-l
c
u
N
U
L
A
T
X
V
E
P
C
R
C
E
N
T
0.4-
0.2-
0.
** ••
0
500
1000 1500
EXPOSURE CMS/LIFETIME*
2500
Pigvre A.?. 95 Perccat Coafldmce Interval Catinte* for Selected Pereentllea ef the BMC DUtrlbatim
on Model-Predicted Doses
-------
essuaed model should be investigated by computing the predicted doses and
confidence iaterval estimates for plausible alternative models. If any direct-
ly measured exposure data are available (data on absorbed doses for the pret-
ant hypothetical example), these data should be ased to validate the predicted
distribution of dose.
A.4.2 Interval Estimates of Exposure Percentilet; Input Vsrisble Distribu-
tions Known
Coafideace iaterval estimates vere derived for perceatiles of the expo-
sure distribution ia Section A.4.1 primarily by computing the doses predicted
by model (1), given dsta for ED sad I*. When the joint probability distribu-
tion for the personsl exposure model input vsriablcs is presumed to be haovn,
confidence iaterval estimates for perceatiles of the exposure distribution can
be generated from a joint confidence region for the parameters of the joint
distribution of all model input variablea. The level of coafideace essocisted
vith intervsl estimates of exposure distribution perceatiles is then the level
of confidence associsted vith the joint coafideace region for the psrameters
•f the model Input variables. Vhca the input variables are statistically
independent, the joint confidence level is the product of the confidence
levels for the parameters of the iadividusl input variables. For example, if
the exposure model has oaly two input variables that are independent, then 95
percent confidence iaterval estimates of the exposure distribution percentilet
arc generated from the iadividual 97.5 percent (V0.9S) confidence regions for
the parameters of the two iaput variables.
Thus, for model (1)» 95 perceat confidence iaterval estimates for percen-
tiles of the distribution of lifetime abaorbed dose vill be geaersted fron
97.S perceat confidence regions for the parameters of the ED end XR distribu-
tions, assuming that the variables are atatistically independent and that the
absorption fraction, AT, is fixed at one. For the purpose of illustration,
assume that expert opinion supports the Bets distribution as the true prob-
ability distribution for ID/600, and that the Vmifon (0,6) distribution is
•apperted as the population distribution for IB. Ae data ia Attachment A.I
vill then me msed to determine 97.5 perceat coafideace regions for the param-
eters of the ED aad 1R distributions ia accordance vith these assumptions.
First, a 97.5 percent confidence iaterval estimate vill be derived for the
parameter 6 of the IR distribution. Second, the dsts for ED vill be used to
A-21
-------
determine an approximate 97.5 percent joint confidence region for the param-
•ttrs of the Beta distribution for ID/600. The 95 percent joint confidence
teflon for the input variable parameters will then be used to generate 95
percent confidence interval estimates for percentiles of the exposure distri-
bution.
Since the presumed probability distribution for IR is Uniform (0,6), the
Unique Minimum Variance Unbiased Estimator (UMVUE) for the parameter 6 is8
t • (a+1) 1*^ / n,
•here fl^a> !• the largest iagestion rate observed in • simple randosj sample
•f aiae a. Using this estimator, the UMVUE for the pa***eter 6 based upon the
daU for IF in Attachment A.I is 5.005. Moreover, the standard trror (i.e.,
the estimated standard deviation) of this estimate can be shown to be
•E (6) » 21^ / VaTaTn • 0.010.
Since the maximum of a simple random sample of site 500 vill have approxi-
nately a formal probability distribution, an approximate equal tail area 97.5
percent (V0.95) confidence interval estimate of 0 is from 4.982 to 5.027.
([Rote that the true value 6, 6 » 5.0, which would mot be known with real data,
Lies within this interval.)
The data for exposure duration (ED) in Attachment A.I were used to com-
pute maximum likelihood estimates ef the parameters a and 0 for the Beta
probability distribution for CD/600.9 The maximum likelihood estimates of
these parameters were: fi • 2.97 and 9 • 2.93. A joint 97.5 percent (V0.95)
confidence region for these parameters was then approximated by independent
*
OS. 7 percent (4Vo795) -confidence interval estimate* for 6 end B.to Using
«*ee ratel, Jagdlsb t. (1973). Complete Sufficient Statistics aad HVU Isti-
oaters. Cemmunicstiona in Statistics. Vol. 2, Vo. 4, pp. 327-336.
°*ee I. 1. Johnson aad S. loti (1970), Pistributioni in Statistics; Contin-
nous Vaivarlste Distributions, Section 24.4.
fi°The joint confidence level it 97.5 percent (VoTw) assuming independence,
Diace tke joint confidence level for independent estimates is the product of
the individual coafidence levels.
A-22
-------
the Halting Normal diitributieo for these maximum likehihood estimates, the
equal uil area 98.7 percent confidence interval estimates vere computed to be
from 2.54 to 3.40 for 6 and 2.51 to 3.35 for 0. (Vote that the true values,
0 « 8 « 3.1ft, which would Dot be known with rtal data, lie^ within these inter-
vals.) If the maximum likelihood estimators for ft and 0 were statistically
imdepeadent, the croas product space defined by parameter values within the
above ranges would be • joint 97.5 percent (VOJ) confidence region for these
parameters, lastoad, the estimators ars positively correlated (correlation ~
0.15), and the joint confidence ration is shaped something like the ellipsoid
•haded in figure A.8. Since analytic specification of the 97.5 percent confi-
dence region shown in this figure is very complex, the joint 97.5 percent
confidence interval estimate for the Beta distribution parameters o and 6 was
approximated by the cross product apace defined by the Interval estimates of
the individual parameters, i.e., the rectangle in Figure A.8. This is a
conservative estimate since the probability content of the rectangle is neces-
sarily greater than that of the ellipsoid.
2.51
2.54 3.40
Figure A.8. Confidence region for the estimated Beta distribution
parameters 6 and 0
The cross product space defined by the 97.5 percent (V0.95) confidence
iaterval estimate ef the parameter t of the IX distribution and the joist 97.5
perceat confidence region for the parameters a and 0 of the ZD distribution
femeratea a corresponding act ef dose distributions via model (1). fince the
joint level of confidence associated with this cross product space is approxi-
mately 95 percent, an approximate lower (upper) 95 percent confidence limit
for any mercantile of the distribution of dose is given by the minimum (maxi-
mum) ef that percentile for ell dose distributions generated by input variable
parameters within this cross product space.
A-23
-------
Since the predicted dote it computed as tae product of EC tod IK for the
model under consideration, the lover (upper) confidence limits for the dote
diitribution ere generated by the ID and U distributions that give the most
probability to nail (large) values of ED and IB. Hence, the lover limit of
the parameter estimate for the IR distribution, 6 • 4.982, was used to gene-
rate lower confidence limits on the dose distribution, and the upper limit, 6
• 5.027, MS uod to generate upper confidence limits. Since the mean of the
Beta distribution varies directly with o, the lower liait « • 2.54, was used
to generate the lower confidence limits for dose, cad the upper limit, o *
3.40, MI Mtd to generate upper limits. Since the parameter 0 affects euioly
the variance of the Beta distribution for ID/600, it MS accessary to consider
both the upper and lover limits on 0 for generating each oet of dose confi-
dence limits, both lover and upper, fence, the following procedure was used
to compute the 95 percent confidence limits on the distribution of dose. To
geaerate the lower confidence limits, two Monte Carlo aimulations were per-
formed as follows:
1. The first simulation consisted of 3,000 iterations coaputed using
model (1) with the following inputs: AT • 1.0; IR distributed as a
uniform variate with 0 • 4.982; ED/600 distributed as a Beta variate
with o • 2.54 and 6 » 2.51.
2. The second simulation consisted of 3,000 iterations computed using
model (1) with the following inputs: AT • 1.0; IR distributed as a
Uniform variate with 6 • 4.982; ED/600 diatributed as a Beta variate
with a • 2.54 and 0 » 3-35.
•
Fercentiles were estimated for each of the dose distributions generated by the
Hoate Carlo method. The lover 95 percent confidence limit estimate was the
•mailer of the two estimates produced in this way for «ach percentile. The
same procedure was used to generate the upper 95 percent confidence limits lor
etosc, except that 6 • 9.027 and 0 • 9.40 was Mad for the Monte Carlo simula-
tions, and the larger of the two estimates MS used for each perccotile.
.. Aeae 95 percent comfidamca interval estimates are listed for selected
mercentilas ef the distribution ef dese in Table A.9 and praacnted graphically
in Figure A.9. The point estimates of percentilea nhown in Table A.5 and
figure A.f were computed directly from the 900 doses predicted by model (1)
Iron the data in Atuchment A.I. Thus, the point eatimatea in Table A.5 are
identical to the point estimates ahown in Table A.4. The confidence interval
estimates ere similar, but mot identical, because they are computed using
A-24
-------
Table A.5. 95 PERCENT CONFIDENCE INTERVAL ESTIMATES FOR SELECTED PERCEKTILES
OF THE DOSE DISTRIBUTION (Y) ASSUMING UNIFORM AKD BETA DISTRI-
BUTIONS FOR XR AND ED/600, RESPECTIVELY
Perceatile
0.01
0.02
.03
.04
.05
.06
.07
.06
.09
0.10
0.15
0.20
0.25
0.30
O.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
o.6o
0.85
0.90
0.91 .
0.92
.93
.94
.95
.96
.f7
.98
.99
lever
liflit
4.53
12.72
21.22
91.21
42.48
53.09
40.77
70.27
90.10
90.51
136.23
175.91
222.60
278.46
330.77
390.97
451.73
515.17
563.62
655.95
741.83
022. 19
931.30
1045.89
1160.18
1329.36
1371.03
1399.65
1444.76
1513.55
1945.16
1439.32
1794.95
1624.22
1993.04
.Point
Eitiaite
7.96
10.49
23.12
95.66
49.53
41.23
74.49
91.32
114.38
137.07
166.11
235.69
276.87
332.24
991.90
452.37
846.96
406.42
462.09
751.23
973.95
955.25
1055.00
1173.99
1331.26
1550.54
1430.07
1446.66
1747.71
1847.38
1980.79
1941.24
2027.12
2142.16
2250.78
Upper
LiBit
20.39
40.14
93.15
72.43
90.36
101.55
117.74
135.16
150.55
167.92
255.08
326.73
997.25
476.95
554.75
439.71
726.46
612.43
900.41
994.06
1098.52
1204.34
1309.51
1445.75
1590.58
1782.67
1616.17
1663.04
1914.19
1964.64
2036.44
2122.65
2224.31
2348.62
2486.36
A-25
-------
t.l
c
u
M
U
L
A
T
I
V
E
0,
0,
£ P
^M ^»
R
C
E 0.2-
N
T
e.0-
» •
t »
i » '
• » «
i » i
400
60O
120O
I60O
EXPOSURE CMC/LIFETIMES
Figure A.9. 93 Percent Cmfldmce Interval CstlMte* for Selected Percent! let of the fto«e Diatributien
A«*u»!nft Unifora and Beta Diatribntlons for 1R and ED/600, Reopectively
-------
different procedures, and the interval estimates shown in Table A.5 sake use
of the additional assumptions that • Uaifera (0,6) distribution is accepted as
the probability distribution for IK and an independent Beta distribution is
accepted for D/600. The interval estimates shown in Table A.5 should theo-
retically to aarrower than those ahovn in Table A.4 due to the use of these
additional assumptions, However, the opposite is true. This auy be due to
the conservative nature of the procedures used to estimate the joint confi-
dence region for the parameters of the ED and TR input variable distributions.
Moreover, the >ormal distribution say BOt^me • food approximation of the
distribution! of the parameter estimators (0, a, and 0) vhen such high levels
of confidence, 97.5 and 98.7, are necessary.
Computation of the interval estimate for percentiles of the distribution
of dose shown in Table A.5 and Figure A.9 is based open: (1) the exposure
model (1); (2) sample survey data oa the input variables (Attachment A.I);
(3) the assumption that the absorption fraction (AT) is unity; (4) the assump-
tion that the exposure duration (ED) aad aoil ingestion rate (IX) are indepen-
dent random variables; (5) the assumption that the distribution of ingestion
rate is Uniform (0,6); and (6) the assumption that the distribution of the
scaled exposure duration variable ED/600 can be characterised by the Beta
probability distribution. The resulting interval estimates for percentiles of
the dose distribution are a useful quantitative characterization of uncertain*
ty for the exposure assessment. However, characterisation of uncertainty for
the exposure assessment is not complete without a thorough discussion of the
justification for all assumptions used in deriving these interval estimates.
The assumption that a particular family of probability distributions can be
used to represent a model input variable, e.g., the Beta family of distribu-
tions for ED, can be tested by standard goodness of fit tests.11 limitations
of the data aad design of the aample survey that produced the data should be
discussed. limitations of the procedures used to compute joint confidence
interval estimates of the parameters of the input variable distributions
should also to discussed. The number of iterations used in the Bonte Carlo
slr^lstions should to justified, aclBovledgiag that sjort iterations would
produce mere precise estimates at higher cost, finally, sensitivity to the
"See Daniel, Wayne V. (1978), Applied Henparameteric Statistics. Chapter 8;
or Conover, V. J. (1971), Practical Honparametric Statistics. Section 4.3 and
Chapter 6.
A-27
-------
assuaed model should be investigated by confuting the predicted doses and
confidence Interval estimates for plausible alternative models.
a.S. aJSESSttXT BASED UPON EXPOSURE MEASUREMENTS
Zhe iiaal atage of refinement for an exposure •••tsucnt ii to collect
Maple eurvey data en exposure for a probability aaaiple of population members.
Such a database can greatly docraase the uncertainty eaaociated vith estima-
tion ef the population distribution of exposure if a sufficiently large sample
la selected. Ibis reduction in uncertainty is achieved mainly by eliminating
the mac ef a sjodal to predict exposure. Moreover, a model, which My be
meeded to relate exposures to sources, can be validated and its parameters can
be estimated, if data arc alao collected for ell model input variables.
Suppose that BO alternatives to model (1) were under consideration, end
data were collected for absorbed dose (Y), exposure duration (ED), and iages-
tion rate (XX), aesumiag that the absorption fraction (AT) was unity, for
illustration, hypothetical outceaes for a eimplc raadom sample of site11 500
wera generated using a symmetric Beta (a • B «3.1) distribution for ED/600 and
ea independent Uaifon (0, 5.0) distribution for IK. Bather than assuming
that sjodel (1) predicts an individual's dose exactly, the dose, Y(i), absorbed
by the i-th population member was generated fro* the model
In (Y(i)] • 1.10 In (ED(i)] -» 0.90 In (IF(i)] + ln(t(i)), (2)
•here la denotes the aatural legarithsi, and the error ten, ln(c), follows a
Bonal probsbility distribution vith a Man of xero and a atandard deviation
•
ef one-third.1* touivalently, the dose absorbed by the i-th population swaber
vas generated as
I(i) • lED(i))1'10 UR(i))0-90 C(i) , (3)
•here c is a random error-ten with e legaenal distribution having a ewdian
•f uity. The raadosj error tana in tjodels (2) aad (3) can be thought ef as
the collective effect of siiaor explanatory variables act explicitly included
"In practice, the sample aise aaeded to achieve acceptable precision within
•pacified cent constraints awst be derived. Sample aiaes much larger than 500
. may often be meeded.
"It ia recognized that data generated by this model ere much eimpler and more
veil-behaved than any real exposure data. Lacking real data with which to
illustrate the proposed procedures, a aimple hypothetical database has been
generated.
A-28
-------
in the model. The data generated for i hypothetical simple random aaople of
500 Mabers of the target population are presented in Attachment A.2.
Given • database of Matured exposure values such as the hypothetical
doses ahowc in Attachment A.2, the population distribution of exposure may be
estimated directly. The cample exposure distribution is a direct estimate of
the population distribution of exposure. The Measured exposure values My
also be oaed to directly confute confidence interval tstiMtes for percentiles
of the exposure distribution ea ehovn in Table A.6 end Figure A. 10." These
confidence interval estimates for ptrcontilos of the exposure distribution
define an approximate confidence region fer the expoaure distribution. When
the catiMte ef the diatribution of expoaure ia baaed upon Masured exposures
for a probability aample of population members, confidence interval estimates
for percentiles ef the exposure distribution are the priaury characterization
ef oaccrtainty for the expoaure essesse«nt. liaiitations of the data and
design of the aurvey naed to collect the data ehould alao be diacussed. No
mention ef e model ia necessary because the exposure levels were Masured
directly rather than calculated from e ewdel.
If a database such as that shown in Attachment A.2 is available, the
parameters ef e Model relating peraonal expoaure to explanatory variables for
which data were collected My be estimated using etatistical regression tech-
niques. Moreover, lack ef fit can be tested for a hypothetical model based
upon a postulated exposure scenario, auch as model (1). Both of these tech-
niques will be illustrated below. '
Suppose that model (1), with an absorption fraction, AT, ef unity, is
expected to predict an individual'a lifetime abaorbed dose. This model csn be
expreased equivalently aa
la T(i) • In ED(i) * la XX(i) + la «(i), (4)
where T(i) ia the dose experienced by the i-th population member. It would
then be reaaonable to ase the data to eatimatc the coefficients e^ end o^ ef
the regression equation
la T(i) • Oj In DCi) * fl2 In IR(i) * I*, (5)
"See Bans en, H. B., Burvitc, V. »., end Madow, V. 6. (1953), Basn>le 8
Methods and Theory. Section 1.18; or Xiah, 1. (1965), Survey Sampling.
tction 12.9.
A-29
-------
Table A.6. 95 PERCENT CONFIDENCE INTERVAL ESTIMATES FOR SELECTED
KRCENTIiES OF THE DOSE DISTRIBUTION (Y) DIRECTLY FROM
EXPOSURE DATA
lover Peiat Upper
Ptrctttile Uait IitiMte limit
0.01 8.46 2O.30 31.70
0.02 19.74 84.18 78.86
0.03 23.96 78.18 97.05
0.04 88.31 90.82 128.52
05 78.39 105.47 137.30
.06 90.06 129.99 151.25
.07 100.43 137.32 191.73
.08 127.78 149.44 206.15
.09 134.82 181.19 228.47
.10 143.55 204.65 256.13
.15 239.87 319.20 971.06
0.20 954.23 994.30 480.99
0.25 414.71 . 820.98 898.39
0.30 834.03 422.84 476.32
0.35 436.54 703.46 804.16
0.40 720.13 830.11 912.92
0.45 846.09 932.71 1066.57
0.50 948.75 1099.23 1176.51
0.55 1101.77 1181.76 1353.50
0.60 1212.82 1363.13 1477.58
O.65 1377.45 1489.36 1633.91
0.70 1830.40 1490.40 1600.59
0.75 1714.53 1853.71 1982.21
0.60 1893.70 2075.07 2312.22
O.85 2202.19 2402.35 2667.51
0.90 2626.97 2966.36 8233.86
0.91 2727.32 9084.35 9305.52
0.92 2916.67 9169.80 9408.31
O.93 9081.00 9297.61 9592.00
0.94 9185.46 9937.49 9831.06
O.95 9296.06 9464.11 9889.16
0.96 9)397.85 8809.79 4033.09
•.97 ' "8)472.42 -^894.85 •"tf*253.92
0.98 8881.38 4092.06 4692.71
0.99 4106.45 4447.16 4466.43
I
fttproduced from
••»» tvaiUbU copy.
A-30
-------
0.8-
c o.o-
u
N
U
L
A
T
I
V
E
0.4-
P
e
R
C
e 0.2-
N
T
0,
0
IOO8 2O0O 3000
EXPOSURE CNG/LIFETIME5
A.W. 95 P«rceat Confidence Interval Cttioates for Selected Perceetile* «f the I>Me Distribution
fron Directly Heaaured Do*e»
-------
where C* is • Morally distributed random error tern with Dean zero, Using
the data la Attachment A.2, the estimated regression equation is
la T(i) • 1.10 la £D(i) * 0.91 la XR(i). (6)
The analysis of variance Uble for this regression equation is presented as
ftiiplsy A,i. the multiple coefficient of deteniaation, Ra, is a useful
measure of goodness of fit of the model.11 The value of R2 is very large for
•the present example (R*«0.998) because the hypothetical data were geaerated
from a model of the type bciag fitted, la practice, ouch nailer values of
1*, e.g., 1* > 0.60, may bo regarded as Indicating a reasonably good fit.
laving fitted the regression model (6), adequacy of the model (4) can be
tooted by toatiag the hypothesis that the regression coefficients, 0j and Oj,
arc both uaity. Za this case, the P-value of the hypothesis test is aeen in
Dioplay A.I to bo essentially aero. That ia, there is almost no chance that
the estimated values of C. aad o. can differ froa unity by the observed aaount
or sjore when the true values of o. and o2 are both unity. Thus, the hypo-
thesis that model (1) generated the data can be rejected based upon this
hypothetical aaaple of SCO measured exposures when the true state of nature is
described by awdel (3). Of course, the true atate of nature is never known
for a real exposure assessment.
»»The multiple .coefficient of determination, R1, «ay he interpreted os the
mwportion of variability ia the observed exposures taplaimed by the model.
It ia the square of the multiple correlation coefficient, which mcssures the
partial correlation between the dependent variable and the independent vari-
ables.
A-32
-------
DISPLAY A.i: ANALYSIS OF VALANCE TAME f
REGRESSION EQUATION
DEF VARIABLE! LOG. DOSE
sun or MEAN
SOURCE BF SQUARES 6QUAF.E F VALUE FRO* F
MODEL 2 23484.986 11742.493 112743.629 O.OOC1
ERROR 498 81.867778 0.104152
U TOTAL 900 23536.854
WOT HSE 0.322726 ft-SOUARE 0.9978
SCP MEAN 4*774368 ADJ R-SQ 0*9978
C.V. 4.76393
NOTE: NO INTERCEPT TERM xs USED. R-SOUARE xs REDEFXNED.
PARAMETER STANDARD T FOR HO:
VARIABLE BF CSTXHATE CRROR PARAHETER-0 PROK > IT! TYPE XX 6S
LOG.XR 1 0.911218 0.014461 43.010 0.0001 413.514
LOG.ED 1 1.102879 0.003020861 365.088 0.0001 13882.336
TEST: NUMERATOR: ?o,803e BF: 2 r VALUE: 679.eno
BENONINATOR: 0.104152 IT: 498 PROII >F : o.oooi
A-33
-------
ATTACHMENT A.It EXPOSURE DURATION (ED) AND 1NGESTJON RATE (IB) FOR
A HYPOTHETICAL SIMPLE lANOOH IAHPIC OP SCO PEOPLE
THE TAI6ET POPULATION
It
II
ID
XI
ID
ID
IR
10
11
12
IS
14
IS
14
17
18
19
20
11
12
15
14
25
24
17
to
19
10
11
12
13
14
IS
14
17
38
2)9
40
Ml
42
41
44
48
44
47
48
49
00
890.43
814.40
408.44
447.18
ISO. 84
188.89
182.29
420.17
459.12
450.48
155.22
100.14
418.41
194.13
228.30
114.87
842.12
129.49
149.24
148.81
74.99
404.88
273.09
130.12
19J.45
252.09
407.47
114.19
197.10
140.30
243.93
158.40
420.39
174.70
145.24
141.91
203.41
148.00
J04.ll
488.84
480.21
444.08
•80.99
94.11
244.08
111.11
198.21
417.01
82.99 <
115.47 1
.92
.25
.43
.43
.99
.19
.78
.44
.99
.43
.41
.19
.42
.70
.74
.48
.49
.11
.73
.11
.99
.02
.89
.42
.19
.14
.40
.91
.04
.42
.72
.48
.49
.40
.84
.21
.43
.19
.11
.44
.01
.17
.90
.17
.tl
.tl
.18
.90
>.84
t.49
tl
82
83
84
15
t4
17
18
89
80
41
82
85
84
45
84
87
88
89
70
71
72
73
74
75
74
77
78
79
80
81
82
83
84
85
•4
87
88
•9
90
91
92
91
94
91
94
97
98
99
100
171.17
447.14
170.43
188.41
144.11
190.10
187.03
87.41
841.90
144.93
192.84
190.85
182.49
111.24
139.34
442.77
124.05
108.89
278.44
102.18
142.25
185.15
242.74
179.92
111.84
88.94
118.18
1 20. :14
123.07
105.42
197.14
279.84
85.84
470.18
211.74
804.74
177.40
180.88
188.10
140.47
407.41
427.99
147.81
171.84
144.71
199.91
140.08
124.41
H. 48
.49
.71
.15
.55
.94
.88
.19
.99
.03
.24
.12
.43
.44
.74
.24
.83
.27
.80
.75
.44
.71
.21
.05
.73
.40
.14
.80
.48
.25
.75
.41
.51
.13
.45
.84
.43
.17
.77
.97
.71
.17
.98
.84
.89
.44
.91
.44
.72
.81
"?S
101
102
103
104
108
104
107
108
109
110
111
112
111
114
115
114
117
110
119
120
121
122
123
124
12S
124
127
128
129
130
111
132
133
134
135
114
117
118
119
140
141
142
141
144
148
144
147
148
149
ISO
118.54
142.14
179.78
119.49
73.44
147.19
120.48
171.22
75.28
148.33
282.11
413.10
114.39
121.84
362.24
184.95
170.34
855.35
272.88
144.87
415.49
149.05
197.44
144.39
185.87
489.48
133.19
802.49
145.89
144.77
73.24
407.04
174.11
133.77
112.18
83.97
189.57
177.97
t20.ll
101.79
179.42
til. 94
til. 04
110.04
198.25
142.18
172.41
123.74
147.22 j
141.94 j
.84
.40
.77
.75
.34
.24
.43
.54
.07
.93
.03
.04
.18
.77
.93
.79
.39
.90
.25
.78
.14
.24
.15
.59
.43
.47
.01
.41
.39
.90
.92
.93
.01
.52
.40
.44
.28
.29
.44
.17
.89
.39
.81
.15
.47
.13
.89
.79
1.89
1.44
A-34
-------
ATTACHMENT A.It IXPOSURE DUIATIDN (ID) AND IN6E5TION IATE (It) FOR
A HYPOTHETICAL SIMPLE lANDON SAMPLE Of 500 PEOPLE
MOM THE TARGET POPULATION
ID
ID
II
ID
ID
XI
ID
ID
It
151
152
185
184
ISS
ISt
157
158
159
ItO
Itl
142
its
144
its
Itt
U7
ItO
It*
170
171
172
175
174
175
174
177
178
179
180
181
182
185
184
185
184
187
188
189
1*0
191
1*1
Ifl
1*4
1*5
1*8
1*7
1*8
J99
00
190.71
189.82
148.1*
140.80
814.24
95.49
174.07
119.01
ItS. 22
129.18
855.89
125.87
454.11
150.00
•18.90
419. *7
112.51
487.82
45S.14
418.85
70.77
115.04
454.19
172.28
244. t9
429.04
140.99
177.84
440.08
495. Ot
lit. 97
•08.29
It?. IS
117.45
151.75
111.02
•15.42
•18.45
197. SI
185.18
144.45
188. *8
141. IS
184.00
418.10
182.88
17*. IS
155.84
142.78 !
72.40 <
.45
.82
.1?
.05
.14
.75
.25
.17
.48
.02
.47
.22
.85
.84
.44
.84
.78
.85
.88
.7*
.42
.74
.84
.20
.10
.88
.52
.93
.80
.88
.77
.84
.48
.10
.97
.85
.59
.17
.84
.48
.19
.44
.45
.•8
.40
.91
.82
.78
!:»?
101
202
105
104
105
104
10?
108
109
110
111
112
115
114
111
Sit
117
118
119
120
121
222
223
124
125,
124
127
128
229
ISO
151
252
ISS
1S4
155
ISt
IS?
til
159
•48
•41
•42
•41
•44
•45
244
•47
•48
{49
50
102. tO
500.53
843.97
404.74
417.91
442.45
ItS. 25
144.84
77.55
•25.80
185.14
172.14
178.15
241.54
482.47
•20.72
218.50
404.58
•20. Ot
175.50
244.42
42*. 92
810.91
157. tl
120.54
194.92
253.80
445. tt
255. t9
•58. It
212.74
420.2?
415.19
419.05
119.04
410.10
1*8.42
•12.41
lit. 12
•72.80
274.48
409.85
•11.10
115.15
425.52
179.17
511.55
.52
.53
.20
.52
.55
.00
.74
.05
.92
.85
.82
.46
.85
.1?
.89
.59
.18
.79
.84
.75
.51
.It
.04
.34
.85
.21
.58
.78
.82
.80
.75
.45
.71
.54
.12
.IS
.4*
.09
.54
.14
.11
.18
.40
.11
.58
.SI
.17
111.74 .»
ill:!! I'M
tsi
152
111
II*
tss
IS*
157
ase
tst
1(0
s«i
it*
its
•44
its
It*
It?
tta
It*
170
I?l
172
I7S
174
175
I7t
177
178
271
ISO
181
182
•SS
184
185
ISt
117
188
18*
•to
III
Itl
its
1*4
t»S
1*8
1*7
IK
loo
101. Ot
SOS. 82
178.10
434.31
**.S8
ItS. 57
147.10
457.74
181. tl
145.17
154.54
414. S*
81.05
ISt.ll
121.41
190.25
457.18
•St. 72
100.71
ISS.2*
148.02
S10.59
.450. IS
IS1.7S
422.33
ISt. 88
252.25
217.10
247.fl
108.27
140.13
535.47
144.51
115.77
182.70
120.81
107.27
129.08
KS.Ot
409.41
151.94
814.78
I4S.4S
MS. 87
14*. SO
ISS. 18
140.74 .
152.71
188.48 <
185.47 0
.11
.09
.49
.tl
.1?
.44
.41
.92
.75
.57
.80
.9}
.02
.57
.IS
.42
.81
.IS
.95
.SS
.07
.44
.82
.7)
.91
.18
.90
.55
.24
.75
.47
.94
.77
.77
.15
.SB
.24
.11
.20
.51
.27
.48
.95
.IS
.84
.02
.78
.7*
.94
.42
A-35
-------
ATTACHMENT A.It IXPOSUU OUftATlON CIO) AND 1NOESTION IATE (II)
A HYPOTHETICAL SIMPLE IANDOM SAMPLE OP 500 PIOPlE
MOM TNC TAI6ET POPULATION
ID 10
II
10
10
tl
10
10
II
IS1
102
101
•04
JOS
104
107
108
lOf
110
111
111
us
114
111
114
117
IIS
119
120
121
122
12 S
124
12 S
124
127
12B
129
ISO
1S1
1*2
1SS
114
1SS
ISI
1S7
ISO
•19
140.
141
•42
•41
r,t.
•41
144
•49
•48
149
SO
S9S.4S
144.82
112.91
189.78
•80.82
194.89
114.22
2*2.17
127. SS
171. 78
417.80
144.05
90.1*
180.47
140.14
194. 91
144.48
18*. 90
144.81
1*7.49
4*1.71
•41.98
145.85
147.18
•19.42
492.89
144.89
118.92
•0.74
til. 79
441.88
140.11
120.99
114.12
114.07
124.19
477.19
191.11
117. SO
470.8*
178.41
421. 4S
•••.17
.19
.19
.11
.11
.92
.89
.99
.04
.19
.OS
.SO
.17
.81
.12
.44
.12
.42
.14
.40
.91
.01
.87
.21
.44
.04
.15
.8*
.41
.00
.OS
.21
.9.
.14
.99
.OS
.41
.28
.17
.12
.72
.17
.IS
.88
19S.41 1.81
418. 8) 1-44
•44.74 t.10
177.44 4.99
•12. SS 1.12
1SS.S8 4.04
4*5. 86 1.42
SSI
152
US
•14
SSS
154
117
158
189
140
141
142
143
144
I4S
84 4
147
148
149
170
171
172
171
174
17S
174
177
178
179
180
181
182
•85
••4
•85
•84
•87
•88
••9
•90
•91
•91
89 >
•94
*9S
•94
197
198
299
00
•21.42
115.00
170.04
•91.11
100.10
108.90
119.49
99.45
197.01
197.05
•15.05
411.10
78.44
149.01
178.78
US. 21
•04.72
•58.84
401.45
408.44
12S.77
244.44
455.21
199.11
414.78
•81.40
12*4.25
124.18
415.04
•08.01
484.95
170.74
•14.94
•11.14
418.94
427.27
148.89
8*2.81
•14.8*
488.11
114.81
••4. SO
•98.81
45*. 49
•88.11
441.41
171.44
128.47
.09
.04
.85
.•4
.49
.40
.11
.44
.24
.95
.SB
.18
.89
.84
.14
.00
.09
.05
.44
.SI
.49
.88
.19
.•1
.98
.70
.44
.00
.72
.45
.15
.74
.11
.11
.07
.19
.78
.09
.99
.81
.70
.48
.•9
.08
.10
.14
.14
.44
127.20 1.15
14.14 1.99
A-36
401
402
401
404
408
404
407
408
409
410
411
412
415
414
418
414
417
410
419
420
421
422
425
424
42S
424
427
428
429
450
451
412
415
484
411
414
417
418
419
440
441
441
441
444
-445
444
447
440
449
450
107.24
100.44
472.08
100.11
•41.48
149.52
197.84
257.77
•11.45
•44.95
•51.72
119.59
155.74
101.77
•44.18
488.20
•99.44
104.04
•44.18
114.15
192.40
195.7*
112.18
•12.28
184.02
184.09
149.22
•12.82
•41.22
408.24
110.92
•12.13
•75.43
411.08
440.18
178.93
•74.17
424.88
•07.01
•19.82
•09. 45
480.02
71.85
•84.12
94.49
•02.44
•99.70
94.19
103.70
141.10 ,
.05
.95
.90
.44
.43
.04
.03
.99
.39
.44
.27
.41
.20
.51
.74
.49
.40
.41
.03
.39
.13
.SS
.73
.54
.11
.00
.73
.00
.33
.84
.55
.90
.15
.•5
.54
.54
.43
.11
.01
.44
.09
.45
.04
.47
.19
.80
.78
.94
l:ii
-------
ATTACHMENT A.li IXPOIUtE DUIATION (ED) AND IN6E5TION BATE fit) FOt
A HYPOTHETICAL SIMPLE BANDON SAMPLE OF 500 PEOPLE
MOM THE TAI6ET POPULATION
ID ID II
4SI
412
411
414
4S5
4S4
417
456
419
4«0
4*1
442
4ft)
444
4ft 5
4««
447
448
4ft9
470
471
472
473
474
475
474
477
478
479
410
461
462
46J
484
485
414
487
488
489
4*0
491
492
49 >
4*4
4*5
4*4
4*7
4*8
4*9
180
143.40
1)1.44
448.71
442.81
151.98
106.11
1)0.8)
147.5)
409.58
157.48
169.04
4*6. JO
•74.41
404.49
140.25
256.04
18). 19
48). 11
454.15
152.49
10*. 58
144.74
•41.))
85). 89
116.21
170.24
111.00
145.42
146.70
422.17
24). 22
140.45
4)7.45
101.48
144.43
140.**
171.78
220.78
177.90
443.75
1)9.90
4J2.94
144.82
48.84
12*. 15
1)7.94
.30
.24
.81
.85
.8)
.))
.85
.48
.99
.17
.97
.62
.78
.14
.22
.25
.27
.45
.21
.44
.99
.29
.87
.22
.57
.40
.78
.52
.44
.74
.12
.8)
.4)
.18
.87
.27
.28
.24
.21
.47
.61
.1*
.12
.84
.72
.18
814.77 1.21
4)4.08 4.47
illl', III
A-37
-------
ATTACHMENT A.It DOSE (V), EXPOSURE DUIATION ICO). AND INCEST10N
•ATI (II) FOR A HYPOTHETICAL S1HPIC IANDOH
OP seo PEOPLE rion THE TARCET POPULATION
IB ID
1R
ID
XR
10
11
12
19
14
IS
It
17
18
1*
20
21
22
13
24
25
2*
27
26
29
10
11
12
IS
14
IS
It
17
SB
19
40
41
4»
43
44
4S
44
47
48
i!
12C.II
441. 44
•4S.18
411. IS
219.71
US. 12
149.04
181. IS
105.4*
12*. 10
•44.94
4S4.19
246.44
189.97
197.93
470.92
211.12
4S4.74
170.77
124.59
2S7.SO
418.19
217.44
198.72
112.49
104. 9S
194. 49
141.83
141.84
288. OS
284.45
141.74
411.44
14.22
141. 10
124.04
170.87
970. f 8
810.09
447.41
••9.81
•S4.I7
141.44
•18.81
144.47
485.18
104.71
258.42
.48
.17
.47
.79
.91
.42
.41
.45
.70
.70
.09
.04
.97
.85
.45
.S9
.08
.14
.08
.SO
.42
.40
.24
.85
.90
.97
.44
.98
.11
.52
.11
.04
.29
.24
.12
.87
.16
.10
.62
.16
.11
.81
.22
.11
.S4
.64
.11
.15
123.82 |. 81
487.17 S.84
2410.44
194.44
1894.55
2548.79
2912. S4
948.04
113.20
S792.94
61.74
1714.82
1474.04
14.18
465.50
29SS.47
4S9.32
444.81
1411.28
1944.45
620.63
841.49
1442.03
1847.24
1145.74
683.99
1388.27
1349.73
1874.05
454.27
1181.76
681.44
1123.49
24.91
164.03
• .44
1711.20
144.64
1724.16
1169.80
1235.82
1124.42
•616.07
1982.29
•41.17 •
215.78
1298.17
4011.81
1710.27
1101.41
1915.77
717.14
81
•2
S3
14
S3
84
87
86
89
40
41
42
43
64
45
44
47
46
• 9
70
71
72
73
74
75
74
77
76
79
60
61
82
• 3
•4
• 5
• 4
•7
•8
•9
to
tl
92
$3
94
95
94
97
98
100
114.01
•41.64
433.42
171.17
192.13
404.05
824.02
128.47
270.65
271.41
228.39
157.70
174.44
227.77
422.48
448.77
133.88
192.01
146. S3
145.54 •
209.23
147.06
209.99
123.47
68 .-84
239.64
292.97
112.22
284.70
150.41
424.11
118.80
419.42
176.92
417.76
119.18
440.80
•24.12
•7.28
•41.14
•42.49
212.68
471.13
110.14
484.61
474.11
121.16
404.13
.82 .
.01
.87
.42
.87
.97
.05
.11
.37
.74
.37
.65
.64
.84
.85
.01
.02
.16
.70
.23
.22
.14
.42
.61
.79
.45
.76
.27
.01
.26
.64
.20
.11
.27
••2
.40
.44
.94
.12
.18
.26
.19
.89
.11
.12
.04
.64
.SI
157.80 4.84
178.42 1.95
1029.58
1189.81
1230.45
1509.21
454.77
2426.0*
1727.41
2198.37
280.19
1119.49
94.13
1396.74
1123.25
1182.99
1227.11
20.33
•77.82
1099.40
748.43
97.43
1144.43
117.24
481.88
8641.90
68. 9*
257.77
2211.77
1889.22
134.45
154.51
932.71
1824.03
•291. 45
T02.49
1261.00
241.06
•619.47
1241.80
119.20
144.14
1444.80
41.14
1730.96
119.11
117.12
74.91
447.97
440.42
1470.48
1179.37
A-3B
-------
ATTACHMENT A.2i DOSE IV), IXPOSURE DURATION CED), AND INGCST10N
•ATE (ID FOR A HYPOTHETICAL SIMPLE RANDOM SAMPLE
OF SOO PEOPLE FROM THE TARCET POPULATION
ID
ID
IR
ID
ID
IR
101
101
103
104
10S
10*
107
108
109
110
111
lit
111
114
US
11*
117
118
119
120
111
122
123
124
its
12*
127
120
129
ISO
1S1
1S2
113
114
135
11*
117
ISO
1S9
140
1*1
142
14)
144
148
• • •
*•»•
147
140
149
ISO
191.10
l«*.17
127. 42
117.70
100.04
414.07
US. 91
10S.28
270.1*
144.02
471.19
192. 98
1*9.41
2*2.44
119.14
24*. 12
tfl.tO
ft. 97
140.09
4*2.41
107. «0
ISO. 09
101.05
49S.70
111.19
199.81
lit. 10
114.82
192.1*
120.91
S1J.SS
180.19
202.97
444.00
229.2*
197.17
00.40
401.8*
104.72
142.12
400.10
411.11
IOS.20
•42.82
141. 1*
8)1.77
111.10
424.10
.29
.10
.7*
.ts
.08
.8*
.17
.00
.71
.43
.«7
.03
.00
.00
.22
.21
.1*
.70
.02
.82
.11
.11
.72
.19
.24
.7*
.40
.40
.S3
.88
.0*
.87
.St
.23
.88
.14
.9*
.10
.27
.74
.04
.01
.4S
.00
.18
..JS
.10
.*s
!!!:»! US
1470.4*
•71.82
708.03
1841.93
tfO*. 12
til. OS
irs.08
127. «1
1*4.04
1002.71
819.77
It. 40
129. *4
1101.7*
1247.1*
141.49
140*. 70
•39.93
1140.40
2489.98
104.21
1009.10
21*7.90
839.04
1*91.01
1409.3*
101.17
90.02
•«0. 09
1282.8*
4053.12
1242.44
070.81
211.13
930.17
SO*. 00
400.1*
•92.01
111*. 07
•071.2*
1411.08
•297.01
1)78.77
•78.33
1044.41
049.42
1054.70
•179.19
i$i:ti
181
182
183
184
118
18*
187
180
1S9
1*0
1*1
1*2
1*1
1*4
1*8
l««
1*7
1*0
1*9
170
171
172
171
174
17S
17*
177
170-
179
100
101
182
103
104
US
10*
117
100
109
100
191
192
101
194
108
19*
197
190
199
00
100.14
•81. tO
1*9.94
374.75
149.11
414.42
t*0. 88
108. 98
179.84
402.82
4**. 00
77.2*
•10.11
•20.49
422.00
170.13
•72.01
8*4. 2S
144.42
•44.74
•19.79
147.19
•94.81
••4.71
204. -42.
•34.07
07.03
•89.10
•21.41
183.22
•92.80
•74.12
•41.18
144.23
•72.01
•44.14
•4*. It
00*. 04
•74.08
92.19
•19.78
104.82
•09.00
•09.01
1*0.44
410.18
124.07
•8«.tl
.01
.10
.17
.80
.11
.01
.to
.09
.11
.00
.20
.02
.1*
.10
.02
.1*
.04
.02
.87
.1*
.01
.11
.81
.17
.21
.•7
.01
.•2
.13
.09
.01
.4*
.29
.•4
.00
.10
.94
.08
.*7
.09
.10
.t4
.78
.09
.•1
.•9
.09
.08
U9.91 {.74
1.02 1.44
1905.**
1744.85
1008.72
1809.79
1121.10
1227.70
1*04.09
4092.04
• US. 47
702.43
1979.11
405.46
179S.I4
1404.14
071.49
219.80
1100.74
1174.80
011.79
•92.70
1804.82
408.4*
14J1.S8
. 794.14
1027.42
4447.10
1 129.9*
11*3.04
2*10.90
2402.15
400.12
180.9*
077.04
897.84
1944. 95
•209.2*
•221.72
•70. 45
1119.80
94.82
100*.**
192.47
400.01
•100.17
442.47
•It.lO
1101.49
1149.19
K0.74
7.22
A-39
-------
ATTACHMENT A.2t DOSE (V). EXPOSURE DURATION (ED). AND
•ATE (II) FOR A HYPOTHETICAL SIMPLE RANDOM SAMPLE
OP 100 PEOPLE PROM THE TAR6ET POPULATION
IB
ID
II
CD
IR
101
•02
10)
104
tos
•04
107
toe
109
tio
III
112
til
214
21S
Il«
117
210
lit
220
221
222
223
224
22S
224
227
220
229
210
2S1
212
211
214
2SS
214
217
110
219
240
241
242
141
244
243
24*
247
240
J«9
SO
1*2.44
277. 45
91.10
442.02
110.27
104.09
211.00
111.11
40S.20
•20. 84
209.17
144.49
227.78
401.44
44.19
S97.4S
199.70
400.10
491.40
244.10
117.18
277.11
407.47
07.71
S9S.42
477.10
125.41
S72.SO
022.04
170.08
171.01
170.02
208.47
111.10
120.14
402.04
114.47
170. SI
2)4.41
140.08
411.74
•24.14
144. IS
140.47
248.24
404.12
71.41
IIS. 71
140.90 :
400.48 1
.11
.44
.00
.47
.10
.14
.14
.17
.10
.14
.29
.22
.12
.24
.40
.10
.99
.70
.41
.95
.02
.42
.48
.21
.44
.24
.44
.14
.44
.88
.14
.04
.44
.21
.77
.49
.01
.70
.24
.79
.to
.44
.12
.11
.'« 1 -
.01
.74
.19
1.07
.00
1104.71
1404.04
414.19
1090.77
1101.10
448.04
1427.40
411.49
108.79
144.90
020.47
728. SO
489.24
1181.95
S04.0B
1801.49
089.10
2007. SI
S00.24
1021.07
274.90
047.11
080.07
I9S.29
1102.14
I40S.01
1894.11
2412.4$
181.17
701.44
1002.00
10.10
.989.09
8447.12
1410.00
194.04
1023.19
1149.70
721.43
494.02
1188.97
1209.17
114.71
194.10
•40.04
1109.41
211.00
1499.04
008.07
00.19
281
282
281
114
288
284
287
280
289
240
241
242
243
244
248
244
247
240
249
270
271
272
271
274
278
274
277
270
279
200
201
202
201
104
208
•04
207
100
209
•90
291
•92
•93
•94
198.-
•94
297
290
III
S90.74
438.27
194.00
•91.10
133.20
494.73
280.01
417.81
442.78
78.90
281.78
449.49
97.12
240.94
210.94
•82.74
247.38
412.39
•38.33
210.44
•24.34
•41.31
348.3*
107.04
•on 44
100.10
194.49
•92.27
•40.71
210.91
140.74
110.92
210.40
194.00
•11.10
•40.71
•44.10
•47.29
•29.01
•10. 10
412.21
•41.47
414.94
141.14
129.09
•40.40
114.91
•12.17
255:15 <
.28
.08
.42
.14
.12
.19
.89
.04
.40
.74
.41
.20
.74
.70
.11
.98
.05
.01
.91
.01
.24
.17
.97
.00
.87
.40
.82
.94
.94
.71
.74
.71
.01
.11
.40
.11
.07
.78
.04
.47
.11
.00
.04
.41
.07
.17
.48
.44
1.94
.04
711.04
428.48
138.41
1084.04
3401.92
1040.05
1402.27
1702.38
3411.15
149.44
1224.78
4105.41
240.21
1145.52
1184.71
1177.04
814.11
3949.05
440.11
748.41
1444.91
22.92
1745.54
20.10
191.04
091.20
2014.14
908.87
2414.12
2401.81
1739.48
474.19
898.40
142.72
101.19
1282.01
1099.21
1412.49
1712.29
117.07
2411.30
172.21
1009.79
981.92
2182.31
' 1177.11
421.28
704.74
JS50.11
1490.40
A-40
-------
ATTACHMENT A.2t DOSE (V). IXPOSUIE DURATION (CD). AN8 1N6ESTION
•ATE (IR) FOR * HYPOTHETICAL SIMPLE RANDOM SAMPLE
OP 100 PEOPLE PIOH THE TAISET POPULATION
20
ID
IR
ID
ID
IR
101
102
10)
10*
SOS
101
107
108
109
110
111
112
IIS
114
SIS
11*
117
lie
11*
120
121
122
12S
*24
12S
524
127
128
129
ISO
111
112
1SS
SI*
IIS
SS4
117
•11
•If
1*0
•41
1*2
Ml
144
•41
•44
•47
}"
ISO
S72.12
24*. 97
141. SS
218.9*
1«S. IB
411.49
IIS. 48
1S7.S8
454.84
1*8.75
11.17
tis.ts
1«7.«0
84. 41
110. U
128.89
181.22
SOS. 24
1*8.09
112.40
S94.*7
445. «S *
1*7.41
187.41
1SS.4S
287. «S
1*9.87
2)1.39
1*1.41
28). 82
114.22
117. 7S
241. «4
17*. 79
1*1.28
170.1*
428.*!
491.79
412.18
2)2.01
415. SI
111.00
ass. is
.4)
.5)
.71
.08
.45
.17
.24
.08
.9)
.5)
.5)
.45
.17
.74
.OS
.93
.48
.00
.52
.17
.7*
.18
.IS
.««
.10
.40
.89
.74
.94
.S4
.OS
.50
.47
.OS
.27
.19
.47
.41
.8*
.17
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.ts
.11
4SS.lt 1.4}
•S9.fl 0.1S
toi.st r.7*
•74.11 l.tl
171.09 l.SS
«i:» t:!J
1159. S*
209.72
•10. 2i
1*2.11
911.9*
1111.77
2144.92
10.25
1417.**
S130.ll
84.32
1083.00
1*34.0*
420.17
14.29
S105.25
8154.47
1978. S)
105.91
•55.72
.1117. *9
•S27.77
1125.09
204.45
S49.ll
119.51
520 ..96
10*1.80
S02.S8
1799.85
420.4*
8*9.48
872. *S
780.04
tsss.s*
•sst.ss
14)4. *7
1954.07
•ISO. 88
1475.41
4204.94
1157. 1)
1175.75
1772.88
141.91
1101.57
4*47.5)
1122.10
1441.90
850.11
151
152
55)
154
155
15*
157
158
159
1*0
1*1
1*2
1«S
1*4
1*5
S**
1*7
1*8
1*9
170
171
172
17)
174
175
17*
177
178
179
180
181
182
18 S
SS4
•S5
>•*
•87
•88
189
190
•91
•92
19)
•94
•95
•9*
197
•98
HI
105.52
599.24
101.34
152. SI
121. 85
18S.S5
179.44
420.29
SS4.19
190. 43
224.80
111.91
15*. 81
282.8*
S2S.S8
15*. 07
475.01
194.84
18*. 18
171.97
•14. 89
189.57
147.24
144.12
102.48
2S7.S4
444.45
229.14
154.47
419.15
144.95
144.70
S47.S4
480.74
•10.10
SS9.S9
•55.41
SS4.SS
145.94
212.11
425. SS
177.48
.5*
.55
.1)
.18
.45
.08
.20
• 20
.52
.04
.49
.19
.92
.17
.79
.25
.94
.84
.5*
.18
.18
.44
.04
.72
.5*
.44
.9*
.17
.79
.71
.97
.55
.10
.2*
.74
.48
.97
.25
.80
.18
.40
.89
1S7.71 0.42
119.01 1.47
197.41 1.11
458.45 a.tl
191.17 4.11
•42.79 S.I8
«S:H Ml
1SSS.BB
4048.5*
105.47
87). C6
910.51
1084.55
1154.70
1851.95
2S27.10
10)2.80
1575.7*
1259.1*
19*9.2*
483.41
188.7)
1189.54
2577.04
1554.59
I42S.10
1714.55
S1S.47
757.21
1215.70
2825.99
191.54
7*S.04
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-------
ATTACHMENT A.2i DOSE (V). EXPOSURE DURATION (ED). AND IN6ESTION
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OP 100 PEOPLE MOM THE TAI6ET POPULATION
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JU42
------- |