A METHOD FOR ASSESSING THE HEALTH 3ISKS ASSOCIATED
WITH ALTERNATIVE AIR QUALITY STANDAKDS
FOR OZONE
Strategies and Air Standards Division
Office of Air Quality Panning anc Standards
U.S.. Environmental Protection Agency
"Triangle. Park, N.C. 27711
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Table of Contents
1.0 Introduction T-l
2.0 Underlying! Principles of the Method 2-1
2.1 Risk and Margins.of Safety 2-1
2.1..1 Legislative Guidance 2-1
2.1.2 Health Effect Threshold 2-4
2..1.3 Idealized Risk Surfaces 2-6
2.1.4 "Risk" Nomenclature ! 2-8
2..1.5 Basic Model 2-9
2.2 Uncertainty Concerning Health Effect Threshold-
Concentrations 2-12
2.2.1 Subjective Probability 2-12
2.2.2 Independence of Health Effects 2-16
2.2.3 Responses of Concern 2-18
2.2.4 Sensitive Population 2-18
2.2.5 Seriousness of Effect 2-19
2.2.6 Uncertainty about Causality 2-19
2.2.7 Defining, the Encoding Variable 2-23'
2.3 Uncertainty in Peak Air Quality Levels 2-30
2.4 Secondary Uncertainties and Public Probability" 2-40
3.0 General Description of the. Method 3-1
3.1 Mathematical Description of the Method 3-1
3.2 Obtaining the P-(C) Distributions 3-6
4.0 Application of the. Risk Assessment to Ozone. 4-T
4.1 Introduction 4-1
4.2 The Judgments of Health Experts 4-1"
4,3 Determination of P Functions for Ozone 4-17
4.4 Risk Tables and Risk Ribbons 4-27
References
Appendix A: The Meed for a Risk. Assessment A-l
Appendix B: Derivation of Basic Equations for Assessing
Health Risks Associated with Alternative Air
Quality Standards B-l
Aopendix Ot Derivations Related to the P Function C-l
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1.0 Introduction
The National Ambient Air Quality Standard (NAAQS) for photochemical
oxidants is being reviewed in conjunction with the reissuance of the
criten"a document. \ ' This review affords an opportunity not only to examine the __
standard in the light of a more extensive data base, but also to develop a
general method for setting NAAQS's.
Although the scientific data base is now more extensive than it was when
the original criteria document was published, many uncertainties relevant to
standard setting remain. These include uncertainties about the concentrations,
exposure times, and patterns of exposure which contribute to each category of
health effect associated with oxidants in general and ozone in particular. Due
in large part to the unpredictable nature of meteorological conditions, there
are also uncertainties about the maximum ambient ozone concentrations that will
occur in a given period of time, whatever the precursor emission situation. Dealing
with these uncertainties requires setting a standard with an adequate margin of
safety. The method described in this report provides a framework and suggests a
quantitative approach to accomplish this end.
The National Academy of Sciences has recommended that EPA make use of some
of the principles and techniques developed in the discipline of decision analysis
(2 3)
which are helpful to rational decision-making under uncertainity...__. The method
discussed below incorporates some of these principles and techniques. For example,
the technique of "probability encoding", which enables optimal use of the quantitative
judgments of health experts, plays an important role. The decision analysis principle
of reducing complex judgments to smaller,more manageable subjudgments whose logical
implications can be determined mathematically is employed.
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The kernel of the suggested approach is a systematic assessment of the risks
associated with alternative standards in a carefully defined sense* Unfortunately,
there are secondary uncertainties about how to represent the primary uncertainties
which give rise to the risks. Yet, an ambient air quality standard specifies a
precise averaging time, level, and expected number of exceedances of the level.
Another principle of decision analysis and the spirit of yet another are applied in
dealing with this situation. First, the method attempts to define terms precisely
so that uncertainty about what a quantity means is not added to the inherent uncertainty
about its value. Second, the output of the method clearly displays how a calculated
risk varies with the particular choice made from a reasonably comprehensive set of
representations of the primary uncertainties which give1 rise to the risk. In other
words the "softness" of the risk calculations which results from the secondary
uncertainties is dealt with directly in such a way as to give the decision-maker(s)
a conception of its degree.
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2.0 Underlying Principles of the Method
2.1 Risk and Margins of Safety
2.1.1 Legislative Guidance
Guidance for setting a primary NAAQS is given, by the Clean Air Act in
the following passage:
National primary ambient air quality standards ... shall be
ambient air quality standards the attainment and maintenance
of which in the judgment of the Administrator, based on [air
quality] criteria and allowing an adequate margin of safety,
are requisite to protect the public health. (4)
A fundamental motivation for the development of a method for assessing
the health risks associated with possible primary air quality standards is
the premise that in order for EPA to make the most meaningful judgment on
whether a possible standard provides an adequate margin of safety it needs as
clear a conception of the risks associated with the possible standard as it
is feasible to get at the given time. This important premise is supported in
Appendix A. The point that in general safety can only meaningfully be interpreted
in terms of risk is made in reference (5).
The provision in the Act for an adequate margin of safety and the following
clarification of the intent of the Act given in its Legislative History make
clear the sense in which the term 'risk1 must be interpreted to capture the
intent of the Act:
... In setting such air quality standards the Secretary
should consider and incorporate not only the results of
research summarized in air quality criteria documents,
but also the need for margins of safety. Margins of
safety are essential to any health-related environmental
standards if a reasonable degree of protection is to be
provided against hazards which research has not yet
identified.
... the Committee emphasizes that included among those
persons whose health should be protected by the ambient
standard are particularly sensitive citizens such as
bronchial asthmatics and emphysematics who in the normal
course of daily activity are exposed to the ambient
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environment. In establishing an ambient standard
necessary to protect the health of these persons,
reference should be made to a representative sample
of persons comprising the sensitive group rather than
a single person in such a group.
Ambient air quality is sufficient to protect the
health of such persons whenever there is an absence of
adverse effect on the health of a statistically related
sample of persons in sensitive groups from exposure to
the ambient air. An ambient air quality standard,
therefore, should be the maximum permissible ambient
air level of an air pollution agent or class of such
agents (related to a period of time) which will protect
the health of any group of the population.
For purposes of this description a statistically
related sample is the number of persons necessary to
test in order to detect a deviation in the health of
any person within such sensitive group which is
attributable to the condition of the ambient air. (5)
The passage explicitly states that the intent of the Act is to protect the
most susceptible group in the general population, implicitly assumes there is
an adverse health effects threshold concentration of any NAAQS pollutant, and
implicitly acknowledges that the threshold concentration will be unknown. If
there were no uncertainty the objective would be to set standards at the maximum
level (related to a period of time) for which peak pollutant concentrations
would not exceed the health effect threshold when the standard is met- Since
there is uncertainty, standards must provide a margin of safety. Even if the
assumption that there is an adverse health effects threshold concentration is
true, there is no positive concentration which the threshold is known to
be above. Hence, standards cannot be set so that there is no risk that
peak pollutant concentrations will exceed the health effect threshold
when the standard is met.
In order to make a meaningful judgment on whether a possible standard
provides an adequate margin of safety, a conception is needed of the threshold
risk associated with the possible standard. The threshold risk associated with
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a possible standard is the risk.that ambient concentrations of the pollutant will
exceed the health effects threshold concentration for the most sensitive group
in the general population when air quality just achieves, that standard. If
the threshold risk associated with the possible standard is deemed to be
acceptable in view of the circumstances then that standard is judged in a
meaningful way to allow an adequate margin of safety.
The part of the passage quoted above from the Legislative History of the Act
which says, "Margins of Safety are essential ... if a reasonable degree of
protection is to be provided against hazards which research has not yet identified,"
is ambiguous. 'Hazards which research has not yet identified1, could refer to:
(1) types of health effects of the pollutant that research has not yet identified which
are caused by lower concentrations than cause known or suspected health effects
of the pollutant; or (2) lower concentrations of the pollutant than have been
shown by research to date to contribute to health effects that research has shown
are effects of the pollutant at higher•_conc_entrat1ons_;_or_(3)_ both (1) and (2)
There is a way to assess the threshold risk associated with a standard for
health effects on which there is evidence, but there is no way to assess the
risk for health effects on which there is no evidence. Hence, the approach taken
here is to assess the risk for effects on which there is evidence. It is suggested
that the primary concern in standard setting should be with effects for which
there is evidence. Truisms are the most that can be said about effects for
which there is no evidence: there may not be any; if there are any their threshold
concentration may be either higher or lower than the threshold concentration of
effects for which there is evidence; and, the lower the risk of effects for which
there is evidence, the lower the risk of effects for which there is no evidence.
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Decision-makers may wish to keep these truisms in mind when deciding what
constitutes an acceptable risk for those effects for which there is evidence.
2.1.2 Health Effect Threshold
It is. obviously important, to give the health effect threshold concept
a precise definition. Uncertainty about what the concept means in general
would hinder attempts to deal with the uncertainty about particular health
effect threshold concentrations. Defining the health effect threshold concept
precisely must be distinguished from measuring particular health effect thres-
hold concentrations accurately. The latter cannot be done, but the former
can. The health effect threshold concept is a very useful one, despite the
inherent limitations in the accuracy with which particular threshold concentrations
can be determined.
One task involved in giving the concept of a health effects threshold a
precise definition is defining precisely what is to be regarded as a health
effect. A health effect threshold concentration must be distinguished from
a physiological response threshold concentration. It is well known that one
molecule of a pollutant can react biochemically witir:the human body- However, not
every physiological effect need be classified as a health effect. Effects
such as a disease or increased susceptibility to a disease are clearly health
effects, but there are other effects whose classification is not clear-
There is'nVpreclse and general technical definition of the health effect
concept which is in accord with common usage where common usage is clear, but
also guides classification of the unclear cases as health effects or non-health
effects. This is because there are inherent difficulties in trying to draw
the line in a non-arbitrary and general way between physiological changes which
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are health effects and those which are not. Many physiological responses are
on a continuum from trivial to obviously adverse effects with no obvious point
at which'the one leaves off and the other begins. In such cases, pragmatic
judgments have to be made about how much of the given response is to be
considered a health effect; i.e., how much of the given response to protect against.
A second aspect of defining theliealth effects threshold "concept precisely
requires distinguishing the health effects threshold concentration for a group
from the health effects threshold concentration for a person. People within
a group, even a sensitive group, will vary in their sensitivity. One person
can be said to be more sensitive than another if he has a lower health effect
threshold concentration. From a strictly theoretical point of view a natural
definition of the health effects threshold concentration for a group would be
that concentration which is the health effect threshold concentration for the
most sensitive member of the group. For practical reasons addressed in
section 2.2, a different definition has been adopted in the initial application
of this method, even though one reading of the above passage from the Legislative
History of the Clean Air Act is that this natural definition is the one intended.
The definition used is: the health effect threshold for the most sensitive
group is that concentration which is the health effect threshold concentration
for the least sensitive member of the most sensitive 1?» of the most sensitive
group.
A third aspect to making the concept of the health effect threshold
for the most sensitive group precise is the stipulation of the conditions
of exposure, such as: the concentrations of co-pollutants with which the
given pollutant may be additive in causing an effect; the concentration -
time patterns the exposure may have;, and, the state of the subjects with
regard to possible stresses such as exercise. This aspect will also be
addressed in Section 2.2~
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/kh/
2.1.3 IdealizedlRisk. Su.rf.asks //^
Suppose the conditions of exposure are uniform throughout a hypothetical
geographical region of concern, and constant over time; suppose for all persons
in the region under the conditions of exposure which obtain, the amount of
physiological response contributed to by pollutant X is a non-decreasing
function of its maximum concentration; suppose all necessary judgments have
been made on what is to count as a health effect; and, suppose a most
sensitive group has been identified, all of whose members will suffer a health
effect due to the presence of X if its concentration is high enough. Then, if
pollutant X reaches a certain concentration it will contribute to a health
effect in 1 percent of the people in the most sensitive group; if it reaches a
certain higher concentration it will contribute to a health effect in 10 percent
of the people in the most sensitive group; and so on.
Suppose that this somewhat idealized situation is realistic in the sense
that the relationship between the peak concentration and the percentage of the
most susceptible group affected is unknown. Suppose that the risk surface
depicted in Figure 2-1 represents this uncertainty. Then, if the health effect
threshold for the most sensitive group is defined to be that concentration
which is the health effect threshold concentration for the least sensitive member
of the most sensitive 1 percent of the most sensitive group, the intersection of
the plane which is perpendicular to the XZ - plane and to the Z-axis at the 1
percent point with the risk surface gives a probability density distribution
which represents the uncertainty as to what concentration is the health effect
threshold for the most sensitive group.
In Figure 2-f, line lib irfthe'XZ-plane is the set of medians of
the probability distributions obtained by intersecting the risk surface with
planes perpendicular to the XZ-plane and to the Z-axis at the full range of
I0/
points on the Z-axis. Concentration C0 is the expected concentration at
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Figure 2-1. Dose-Response Probability Surface
Percent
of
Hea/tW
Effect
H ^n.-__p-
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1 on/
which 1 percent of the group would be affected> C is t
tration at which 10 percent of the group would be affected, and so on.
2.1.4 "Risk" Nomenclature
In the public health literature where dose-response relationships have
been estimated without constructing a risk surface, concentrations corresponding
to C ]% have been said to have a risk of 1 percent. "Risk" in this sense
is an estimate of the percentage of a group which will suffer a health effect;
this percentage is called risk because it is interpreted to be the probability
that a generic member of the group will suffer a health effect. In the context
at hand, it is better terminology to refer to the risk that n percent of the
members of the group will suffer a health effect at various concentrations. These
risks can be determined from the r\% probability distribution, where the
n% probability distribution is the distribution which is the intersection
of the risk surface with the plane that is perpendicular to thexZ-plane
and to thej-axis at the n* point.
Obviously, a risk surface such as the one shown in Figure 2-1- would be
a very valuable input to a real-world attempt to determine the expected
concentration at which a certain percentage of a group would suffer health
effects, the expected percentage of people suffering health effects at a
certain concentration (slice the surface perpendicular to the x-axis), the
risk that a certain percentage of the group would suffer health effects at
a certain concentration, and so on. However, the idealized nature of the
situation described above must be kept in mind. Conditions of exposure
are not uniform throughout an area and are not constant over time: pollutant
mixes vary, concentration-time patterns vary, some people are sitting,
some people are walking, some people are jogging, etc. Different peoole
react differently to different concentration-time patterns. How many
people are being exposed varies over time. Within a given control area
concentrations of a pollutant vary at any given time.
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In view of the above complexities, the initial application of this
method (to ozone) concentrates on determining the risk that the health
effects threshold for the most sensitive group would be exceeded.
This is a risk that some sensitive peoplewould suffer adverse effects.
This approach, as well as seeming to capture the sense of the Clean Air
Act, avoids many formidable problems the above complexities pose for
attempts to make various types of "head count" estimates such as those
mentioned in the preceding paragraph. Of course, these estimates give a more
complete risk picture, and so the capability to make meaningful estimates
of this type is being developed.
2.1.5 Basic Model
In estimating the risk that a health effects threshold will be exceeded it
is necessary to take into account, in addition to the uncertainties concerning the
location of health effect thresholds, another major source of uncertainty. This
is the uncertainty concerning the maximum concentration of the pollutant which will
occur in the time, period over which the risk is to be estimated and when the
general air quality just meets the NAAQS for the pollutant. This uncertainty comes
about because the ambient concentrations of a pollutant are subject to the random
changes in meteorological conditions and in the emissions of the pollutant or its
precursors into the atmosphere. As a.result, even though pollutant or precursor
levels in an area have been brought to a general level at which the overall ambient
air quality is meeting the NAAQS, the highest concentration levels observed over
the area in a given time period (e.g. 1, 2,. or 5 years) will vary over a succession
of time periods. The uncertainty in the maximum concentration can be accounted
for by a probability distribution which can be estimated from aerometric data.
Figure 2-2 broadly represents~the"situatTon that must be dealt with. ~It
shows two probability distributions in the form of probability density
functions. The curve drawn in the upward direction is the probability density
function for a health effects threshold. The curve drawn in the downward
direction is the probability density function for the maximum hourly average ozone
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(C)
(C)
health effect threshold distribution
Concentration, C
max. CL concentration distribution
'STD
Figure 2-2, Probability Distributions on the Two
Major Sources of Uncertainty
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concentration observed over some time period. This distribution is drawn in
the downward direction so that both distributions can be conveniently
displayed on the same axis. The location on the concentration axis of the
distribution representing the uncertainty about the maximum ozone value
depends on which potential standard is under consideration.. The more
stringent the potential standard, the farther it is to the left, that is, shifted
toward lower concentrations. The farther it is to the left, the lower the risk that the
maximum ozone value will exceed the overall health effect threshold concentration.
To better take into account the statistical behavior of ambient pollutant
concentrations, the proposed standard for ozone has been given the following
statistical form.,
C<-yD ppm hourly average concentration with an expected number of exceedances
per year less than or equal to E.
Through the use of an expected exceedance rate an area is allowed to occasionally
experience a measured exceedance rate of greater than E so long as the expected
rate, estimated by averaging the rate over a number of years, is not exceeded.
The lower the numbers C<--TQ and E the more stringent the standard and the
more the lower curve in Figure 2-2 will be displaced to the left.. Raising one" of~~
these two numbers and" lowering the other sets up a tradeoff in which the particular
numbers and the particular context determine whether the standard is more or less
stringent. For any C^JQ or E a. standard of any desired stringency can be obtained
by making the other number high or Tow enough. However, for reasons which will be
discussed more fully in a Tatar section it is desirable to maintain the value of E
in the vicinity of one or Tower.
It should be clear that the standard level and the health effect threshold
concentration are not (in general) the same. Hence, a standard which allows one
exceedance of a standard level may have a very low threshold risk associated with it.
The standard, the standard level, and the health effect threshold are all distinct
concepts.
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The shape of the maximum ozone concentration distribution is a function
of the time period considered. For example, a one-year distribution and
a three-year distribution for the same standard will have different
shapes. Hence, the risk is a function of the time period; the longer
the period the greater the risk. It is required by law that the scientific
criteria for NAAQS's be reviewed and reissued every five years. Hence,
NAAQS decisions apply to at least a five-year period. At the end of five
years when the new criteria are reissued the risks can be reassessed.
So, the time period chosen over which to assess the risks is five years.
2.2 Uncertainty Concerning Health Effect Threshold Concentrations
An important part of assessing threshold risk is representing the uncertainty
about the health effect threshold concentration for the most sensitive group.
Representing this uncertainty requires subjective judgments that go beyond
strict scientific interpretation. However, these judgments are best made
by members of the set of health scientists who are most familiar with the
health science information reviewed and assessed in the criteria document.
For, although some of the judgments required go beyond strict scientific
interpretation, they can best be made on the basis of the scientific information
presented in the criteria document, and by those with the expertise to understand
the scientific implications of that information.
2.2.1 Subjective Probability
Since an integral step in arriving at the desired representation of the
uncertainty about the health effect threshold concentration for the most
sensitive group is the elicitation of subjective probability distributions for
individual categories of health effects, the required subjective judgments are
made in probability encoding sessions. "Probability encoding" is a term used in
the management science and decision analysis literature to refer to the elicita-
tion of subjective probability distributions from experts. It is an explicit,
precise, and formal technique for quantifying the probability judgments of
experts. The technique has been pioneered by Stanford Research Institute (SRI),
amongst others; two SRI-related publications describe the technique and the
history of its use. (7) (8) 2-12
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Several distinctions regarding "subjective" and "objective" probability
need to be understood. The first distinction is between:
(a) the method by which a probability distribution is derived.;., and
(b) the decision to accept a probability distribution as the representation
of uncertainty about the value of a quantity for some practical
purpose.
In the sense of (a), probability distributions are either objective or subjective;
that is, either the procedure by which a probability distribution is derived
from a body of information is completely determinate, so that two different
people following the procedure exactly will arrive at the same distribution,
or probability judgments are required in going from the information to the
distribution, so that in general different probability distributions will result
when different people make the judgments.
In the sense of (b), all probability distributions are subjective. The
decision to use a particular probability distribution to represent the uncertainty
about the value of a quantity is a subjective decision,, no matter whether
the distribution is derived objectively or subjectively.* The uncertainty
about the health effect threshold concentration for a pollutant could be
represented by any number of objectively derived distributions about which
any health scientist well informed on the topic would presumably make the
(subjective) judgment that he could contribute a subjectively derived
probability distribution which would represent the uncertainty better. In some
situations, a health expert might feel that a particular objectively derived
distribution represents the uncertainty best. However, as will be seen, the
situation is generally so complex that this will generally not be the case.
Certainly the situation is too complex for the type of objectively
derived probability distributions which have historically played a large
role in statistical inferences in the empirical sciences. Another
distinction needed here is between subjective and objective interpretations
*Hence, Quinn and Matheson suggest the term "judgmental probability" be used
rather than the term "subjective probability". "(8)
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of probability. The mathematical relationships between various probability
statements, distributions, etc. can be treated as part of an abstract
axiomatic system in which the terms remain uninterpreted. But an important
question concerns what these statements mean. Two basic interpretations
of probability statements dominate the history of the probability
concept. (9) One is the subjective (or epistemic) interpretation in which
a probability is interpreted as a measure of degree of belief, or as the
quantified judgment of a particular individual. Degree of belief consists
of a disposition to make certain specific kinds of choices in we11-defined choice
situations (10). The second interpretation is "the frequentist (or aleatory)
interpretation according to which the probability of an event is the relative
frequency of occurrence of that event to be expected in the long run (11).
The relative frequency interpretation has been the foundation of
sampling - theory for classical) statistical inference in the empirical sciences.
Often in the empirical sciences experiments can be designed such that random
samples can be taken from a well-defined population or process. If, by
agreement of those in a position to know, such an experiment satisfies certain
conditions, a probability calculated on the basis of a sample relative frequency
can be the only rational probability to assign to an event. Such a probability
is not only objective in its derivation, but also gains general acceptance because
of the compelling nature of its empirical content. Such general agreement
brought about by empirical content is the trademark of good empirical science.
In such situations, subjectivity has been reduced to a minimum. But,
there is still the subjective element of agreeing that the relevant conditions
obtain, and the subjective interpretation of what the probability means applies.
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Since the sample relative frequency is the only rational probability to
assign the event, it is the one assigned. Within the confines of such
contexts whether the probability is given the relative frequency or the
subjective interpretation makes little practical difference. However, the
concept of probability has always applied to a much wider range of uncertain
situations than those that can be investigated directly and empirically through
random sampling. Many of these situations are like the situation at hand; a
probability or probability distribution based on the available information is needed
for the most rational approach to decision-making, but it cannot be determined
on the basis of scientific data alone what probability or probability distribution
best represents the available knowledge and the remaining uncertainty. The
subjective interpretation of probability can be extended to these situations
and the relative frequency interpretation cannot. Hence, the subjective
interpretation of probability is the one that is used in decision analysis.
The subjective interpretation of probability has come to be identified
in some minds with the Bayesian approach to statistical inference, since in
that approach sample information in the form of a likelihood ratio" is often
combined with prior information in the form of subjective probability distri-
butions. But, in the Bayesian approach, which is an extension of the classical
approach, enough sample information will "swamp" the subjective prior distribution
in formally arriving at the posterior distribution. So, the only constraint to
"objectifying" the;"posterior "distribution "to any degree desired1 is a cost
constraint on the amount of sample information it is rational to obtain under the
circumstances. In the situation at hand it is not a resource constraint that
leads to the use of subjective probability distributions, but rather an inherent
inability to get sample information which is a direct input to the distribution
in question.
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Of course, either the Bayesian or the classical approach to statistical inference
can be used in fitting models to data outside the range of interest in order to
extrapolate from the range in which there is data to the range of interest
(12) (13). Such models can be used by an expert as an aid in arriving at the needed
distribution. But, unless there is wide agreement that the model holds within
the range of interest, it should not be felt that a distribution arrived at
using such a model is necessarily a better representation of the uncertainty
than a distribution arrived at without using the model.
To sum up, the representation of the uncertainty about the health
effect threshold concentration cannot be determined by solely scientific
means. The goals of empirical science and the goals of practical
decision-making must be distinguished. For excellent reasons, whereof
empirical science cannot speak it remains silent. The goal of practical
decision-making is to use mathematics and science as far as they go, but
if there remains a gap to adopt whatever means seem at the time to be best
for bridging the gap. The question is not whether subjective judgments will
be made, but rather how and by whom. It is argued here that these judgments
are best made explicitly, under well-defined choice situations, byrthe best-
informed health scientists; rather than implicitly, under ill-defined circum-
stances, and/or by non-experts.
2.2.2 Independence of Health Effects
The criteria document for the typical pollutant discusses several health
effects contributed to by the pollutant. A risk surface similar to the one
in Figure 2-1 of section 2.1 could be constructed for each of these effects.
However, if the risk of exceeding a health effect threshold is made the focus
of the risk assessment, several simplifications are possible. First, the
health effects can be grouped into independent categories according to the
following definition of independence: If for two health effects the judgments of
an expert, about the probability that the threshold for either effect is
below various concentrations, would be affected by knowing the true health
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effect threshold concentration-for the other, then for that expert the two
health effects are threshold dependent; other/vise the two health effects are
threshold independent. Threshold-dependent health effects are grouped into
the same health effect category.
One point of grouping the health effects into threshold-independent
categories is to enable the health experts to concentrate on one category
of health effects at a time in making their judgments.* Once subjective
probability distributions are obtained for each individual health effect
category, a composite health effect threshold distribution can be determined
mathematically from the individual distributions (see section 3.0). The
mathematical laws of probability used to obtain the composite distribution
can only be applied to independent distributions.
Although the grouping of the health effects into threshold-independent
categories reduces the number of effects that must be addressed by the health
expert at one time, he still may have to address several effects within a given
category. Just as in making his judgments about threshold independence the
expert focuses on whether knowledge about the threshold concentration for one ef-
fect would affect his judgments about the threshold concentration for another,
and does not concern himself with whether he believes there is any relationship
between the two effects at higher concentrations, so he focuses on the
effect within the category that he believes has the lowest threshold concen-
tration. One reason two given effects_may_ be jin the_ same category is that
the expert is sure or thinks it likely that one of the. effects occurs at a lower
concentration than the other.
*See subsection 2.2.6 for two other reasons.
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2.2.3 Responses of Concern
Once it has been determined how many independent health effect categories
an expert judges there to be for.which much or most of the evidence is from
his field of expertise (such as epidemiology), there are several types of
important judgments he makes for each category. First, he describes what
he considers the health effect to be that has the lowest threshold concen-
tration for the category. In those cases where the effect is a continuum
of physiological response, he judges how much response should be regarded
as a health effect; EPA representatives involved in the probability encoding
session clarify as much as possible for the expert the sense of the Act on
what is to be regarded as a serious enough response to be considered an
adverse health effect. 3ut, sines neither the Act nor its Legislative History
is explicit on this point, much is left to the expert's judgment. Since
experts naturally differ some in their judgments in cases where drawing
the line involves some inevitable arbitrariness, the definition- of a health
effect in such cases usually varies some among experts. So, differences
in subjective probability distributions on the health effect threshold con-
centration for such categories can be due both to differences in judgment
about what concentration (for the averaging time) will cause what response
and how much response should be considered an adverse effect on health.
Once one or more applications of the method have been made in which various
health experts have addressed the question of how much response of a given
type is an adverse health effect, the option of designating a given amount
of response as the amount EPA regards as a health effect and wants to
protect against may become more attractive.
2.2.4 Sensitive Population
Another judgment the expert makes for each category he addresses is
what group, characterizable in general terms, is the most sensitive group
for that category. The most sensitive group for a category is the most
2-18
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sensitive group for the health effect of that category which is judged to
have the lowest threshold concentration. Since experts may differ on
what group they judge to be the most sensitive group, the number of
people in the most sensitive group may differ for different experts. This
fact means that in specifying the health effect threshold concentration
for a group it may be that either the "target individual", whose threshold
concentration is used to define the threshold concentration for the
group (above in section 2 J.2, the least sensitive member of the most sensi-
tive 1 percent of the most sensitive group), will be different for the two experts,
or the number of people more sensitive than the target individual will be different,
2.2.5- Seriousness of Effect ~
There are two aspects to the risk of exceeding a health effect
threshold: the probability of exceeding the threshold and the seriousness
of the adverse effect. Determining the probability is the more complex
task, but giving EPA decision-makers and interested parties as good a
conception as feasible of the seriousness of the effect is also very
important. Everything else being equal, less risk is acceptable for serious
effects than for effects which are not serious. Hence, each expert
is asked to describe how serious he believes the given effect to be and
to say anything he feels would help clarify for a decision-maker how:
much concern there should be about the effect.
2.2.6 Uncertainty About Causality
For some categories of health effects there is no uncertainty about
the existence of a causal relationship between the pollutant and the
category of health effect. For others, the existence of a causal relation-
ship is uncertain. This uncertainty often arises in cases where there is
toxicological evidence from animal studies that the pollutant causes the
2-19
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effect in one or more species of animals, but there is no conclusive
evidence from clinical and/or epidemiological studies that a causal
relationship exists between the pollutant and the effect in humans. In
those cases where the existence of a causal relationship for a category
is in doubt, experts addressing that category are asked to judge the proba-
bility that a causal relationship exists between the pollutant and the
effect in humans.
Distributions which represent the expert's uncertainty about the health
effect threshold concentration _if_ there exists a causal relationship
between the pollutant and the effect in humans are encoded for such cate-
gories in the same way as the* distributions for categories about which
there is no doubt about the existence of such a causal relationship. The
existence probability is used to apportion the encoded distribution
between the range of concentrations the expert believes contains the
threshold concentration if a causal relationship exists and a range of
concentrations much higher than would, ever be found in the ambient atmosphere.
The latter portion must be maintained somewhere on the concentration
axis for mathematical reasons (see Appendix 8); its particular shape does
not affect the risk calculations. The fraction of the distribution
in the lower concentration range is equal to the existence probability.
The part of the distribution in this range has the same shape as the
encoded distribution. Figure 2-3 depicts a probability density function
for a health effect whose threshold is thought to be in the vicinity
of 0.20 ppm, but which has only a 20 percent probability of being real.
2-20
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,20
C»100,000
Corrcentrations, ppm
Figure 2-3 Hypothetical Threshold Probability Density Function
for a Health Effect with a 20% Chance of Being Real
2-21
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Having the expert address the question of the probability of existence
separate from the question of the most probable concentration of the health
effect threshold if a causal relationship exists is a second simplification
of the judgments which he has to make,* There can be a tendency when such a
breakdown is not made to either assume a causal relationship exists in risk
calculations if the existence probability is thought to be high, or neglect
the possible existence of a causal relationship in risk calculations if the
existence probability is not thought to be high. Unless the existence pro-
bability is negligibly different from 1 or 0, this tendency amounts to the
mistake of using outcomes rather than probabilities of outcomes in calcu-
lating probabilities (risks). The formal incorporation of an existence pro-
bability into the mathematics used to make the risk calculations, as is done
in Appendix 3, avoids this mistake.
The third reason for grouping the health effects into threshold-
independent categories is that the probability that at least one of several
\
unknown and independently distributed health effect thresholds is less than a
given concentration is greater than the probability that any selected one of
the thresholds is less than the given concentration. To put the point another
way, the composite health effect threshold distribution calculated from the
threshold distributions for single health effect categories will always be
displaced toward lower concentrations from all of the single category
distributions. For any given concentration the total area under the composite
density curve to the left of that concentration will be greater than the total
area under any one of the individual density curves to the left of that
concentration.
The leftmost individual health effect distribution will in general
have the most influence on the composite distribution. The one exception
to this generalization can be a case in which the leftmost distribution is a
distribution for a health effect category about which the existence of a causal
*See p. 2-1/.2~22
-------
relationship between the pollutant and it is uncertain. Figure 2-4" illustrates this
po.int. Figure 2-4JbJ^ shows three examples of composite functions derived fnom
two individual health effect threshold probability density functions (Figure 2-4(aU.
The two hypothetical individual functions used in the example are both normal
distributions. As can be seen in section 4.0, real subjective probability ._
distributions will tend to be neither symmetrical nor a well-known distribution.
2.2.7 Defining the Encoding Variable
The probability encoding technique is designed to minimize the moti-
vational and cognitive biases which psychologists have found can arise in
the elicitation of subjective probability distributions. Care can be taken
to guard against two possible "motivational" biases: the building of "safety
margins" into judgments, and the favoring of what has been scientifically
demonstrated over what is viewed to be most likely in view of the
current evidence. Judgments about the degree to which a health effect threshold
concentration has been scientifically demonstrated to be below certain
concentrations could also be elicited in the probability encodina sessions.
In fact, these would be useful judgments that would best be elicited in such
a session where the nature of the judgments could be carefully defined; belief
functions (14) on degrees of confirmation (15) could be1 elicited. But,
whether this is done or not, such judgments should not be confused with the
probability judgments made in the encoding of a subjective probability distri-
bution. (16) It is the latter that must be used to calculate risk.
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10
6 -
= 0.20 ppTi
=0.04 ppm
= 1
0.3 pom
0.08 ppm
= 1
0.3
Concentration, ppm
0.4
0.5
(a) Individual Health Effect Threshold Probability Density Functions
= 1
0.3
Concentration, ppm
0.4
0.5
(b) Composite Probability Density Functions for; Various Probabilities of
Existence are Assigned to Function with T = 0.2 ppm
Figure 2-4 Variation in Composite Health Effect Threshold Probability
Density Functions of Two Independent Density Functions as Different
Probabilities of Existence are Assigned to the Lower Threshold Function
2-24
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Defining well the uncertain quantity about which probability judgments are
elicited is a normal part of a probability encoding session. As was discussed
in section 2.1, there are three aspects to defining the health effects threshold
for the most sensitive group precisely:
(1) definition of what is to be regarded as a health effect;
(2) stipulation by his place in the distribution of sensitivities within
the most sensitive group the member of the group whose threshold
concentration is to serve as the threshold concentration for the
group;
(3) stipulation of the conditions of exposure.
In the initial application of the method (to ozone) the definition of a
health effect for each expert was based on his own best judgment of how much
response should be regarded as a health effect. The expert's judgments concerned
the concentration of ozone that would cause the effect in 1 percent of the group
he judged to be most sensitive for that effect if_ the whole group were exposed
under the stipulated conditions of exposure. The selection of the 1 percent
figure unavoidably involves some arbitrariness, but was selected for several
reasons. To use the health effect threshold of the most sensitive group
rather than the threshold of the least sensitive member of the most sensitive
1 percent of the most sensitive group would be impractical for at least one
reason. For the types of effects contributed to by ozone, the most sensitive
member of the most sensitive group is an unknown type of person who would
be extremely difficult for the health expert to make judgments about. Sub-
jective judgments grade from well-informed judgments to sheer guesses;
judgments about the most sensitive member of the most sensitive group would be
2-25
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at the latter end of the scale. The health expert is applying his own
expertise to make judgments on the basis of the scientific information in
the criteria document. While judgments about what dose of a pollutant
would affect a small percentage of the people in the most sensitive group
under certain conditions on the basis of the data in a criteria document are
very difficult and extrapolative, they are not as uninformed as would be
judgments about the most sensitive person.
As stated previously, the legislative history of the act indicates
that EPA is to protect the most sensitive group as a whole, rather than
the most sensitive person. The approach chosen satisfies this intent.
for the threshold risk determined on the basis of this definition of the
health effect threshold can reasonably be interpreted as the risk that the
pollutant will affect the group as a whole. It would be incorrect to
interpret this definition as an indication of a utilitarian judgment
to trade off the interests of 1 percent of a sensitive group against the interests
of society as a whole. Threshold risk so defined is not the risk that 1 percent
of the most sensitive group will be affected by the pollutant. How many people
would be affected if the ambient air level of a pollutant exceeds the health
effects threshold concentration as it is defined here is unknown; the point
is that it would affect some people.
By redefining the threshold in terms of 0.005 or 0.05 of the most sensitive
group we can say that roughly half as many or five times as many people
will be affected. So the somewhat arbitrary choice of 1 percent is not
unimportant. But this kind of choice must be made. The threshold risk can be
calculated for more than one definition to give the decision-maker an idea of
how much it varies with the definition chosen.
2-26
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The decision-makers can also keep in mind that the threshold risk
for individuals more sensitive than the least sensitive member of the
most sensitive 1 percent of the most sensitive group, including
the most sensitive member of the most sensitive group, is greater than
the threshold risk for the most sensitive group. The question of whether
the health effect threshold concentration for the most sensitive
member of the most sensitive group is greater than zero is left unanswered and
unaddressed. In general, for any low concentration the lower the probability
that the health effect threshold concentration of the most sensitive group
is less than the given concentration, the lower the (higher) probability that the
health effect threshold for the most sensitive member of the most sensitive
group is less than the given concentration. A relationship could be developed
to extrapolate from the most sensitive group threshold probability (which
is itself extrapolated from the available scientific data) to a probability'
for the most sensitive member of the most sensitive group.
The approach taken in stipulating the conditions of exposure is to have
the expert 'make his judgments for an idealized situation in which the con-
ditions of exposure are assumed to be the same for the whole sensitive group
and the whole group is assumed to be exposed. One condition of exposure is that
it is in the ambient air of an "average" United States city. If the
expert's best judgment is that there are additive effects from other
pollutants in the ambient air he is instructed to take them into account.
He is to assume, however, that the ambient concentration of other NAAQS pollu-
2-27
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tants is their standard level. This stipulation is to avoid double-counting,
yet deal with the fact that some 'NAAQS pollutants may be additive with one
another in causing some health effects. Non-NAAQS pollutants are to be
assumed to be at levels found in an average U.S. city.
Another condition of exposure for short-term effects concerns the
concentration-time patterns the exposure may have. If one exposure can
cause a health effect, the expert is instructed to make his judgments for
the following exposure pattern: the peak concentration lasts the length of
the averaging time with a "typical" build-up and drop-off in concentration
before and after the peak concentration. If possible, the averaging time is
selected before the elicitation of subjective probability distributions from
experts.
The experts are instructed to consider the members of the most sensitive
group to be under any normally occurring stress such as exercise; mild exercise
such as tennis or jogging is interpreted to be normally occurring, but heroic
exercise such as marathon running is not. To the extent that adaptation is
considered to be an important factor in making the judgments, the history of the
group is that they have all lived in that environment in which the expert
feels it most likely the pollutant would cause exactly 1 percent of the group
to suffer adverse health effects if they were all exposed on the worse days.
The point of having the health expert make judgments for an idealized
situation is to simplify the judgments. Even taking one health effect category
at a time, the expert is faced with portions of the criteria document which
review, critique, and interpret studies of different types, ages, and scientific
validity; some of these studies report positive results, some negative results,
2-28
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and possibly some contradict each other; each of the three types of studies,
clinical, ep.idemiological, and toxicological, has inherent limitations,
although they often complement each other. To take this scientific work
and the interpretation of it given in the criteria document and determine
its implications for even an idealized real world is a task only the
flexibility of the human mind can deal with, but a very complex task.
The degree of idealization which makes the judgments simplest
is an open question. If the evidence is mainly epidemiological the
degree of idealization suggested here is probably too great. When the evidence
is mixed among the three types of studies, as it usually is, it is hard to judge
what degree of idealization would be best. When risk surfaces are estimated
for the purpose of making the various kinds of head-count risk estimates which
would fill out the risk picture, there is another consideration. This con-
sideration is the type of risk surface which would be most convenient for the
second step from the risk surface to the real world— in that case a
quantitative step. The problem becomes that of designing the risk surface
optimally for serving as a pier between the information in the criteria
document and a full risk picture; the first span being the health expert
judgments and the second span being analysis, exposure pattern data, air
quality data, etc.
2-29
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2.3 Uncertainty in Peak Air Quality Levels
As indicated in Section 2.1.5 the second of the two primary uncertainties
which must be accounted for in assessing the risk of exceeding a health
effects threshold is the uncertainty as to the peak levels of oxidant con-
centration which may be encountered in the time period over which the risk
is assessed.
There exists a certain amount of confusion concerning peak pollutant
levels in that there is a tendency to think that a geographic area at a
given level of precursor emissions will experience a characteristic highest
concentration of pollutant in any calendar year or period of several calen-
dar years. This tendency is more evident in the belief that there is a charac-
teristic second-highest concentration which may be related to the level
of precursor emissions. In reality the highest concentration or second
highest (or third or fourth, etc.) are statistical variables which can
change significantly from one calendar year to the next even though the
average level of precursor emissions remains constant. These changes are
largely of a random nature and come about because of random fluctuations in
weather and in emission levels. It is precisely because of this random
behavior that the uncertainty in the highest concentrations experienced in
a given period of time needs to be accounted for in the risk assessment.
If the peak concentration for a given time period was a constant for a given
area at a given emission level, only the uncertainty in the position of the
health effect threshold would need to be of concern.
The uncertainty in the value of the highest hourly average concentra-
tion is represented in Fig"2^5(a)"by a probability density function. In the
2-30
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actual calculation of risk, as discussed in Section 3, it is more conven-
ient to use the cumulative distribution function which is just an alternate
way of expressing the same uncertainty. Figure 2-5 shows a hypothetical
probability density function and the corresponding cumulative distribution of
the annual maximum hourly average ozone concentrations. For the hypotheti-
cal situation depicted in the figure, the most probable value of the maximum
concentration in a calendar year is 0.10 ppm. The cumulative distribution,
customarily called the distribution, is such that the value of a point on
the distribution read from the vertical axis is the probability that the
observed maximum hourly average concentration for a given time period is
less than the concentration read off the horizontal axis. For example,
Figure 2-5(b) shows that there is a 50 percent chance that the maximum ozone
concentration in any year is less than or equal to 0.105 ppm. This con-
centration is usually referred to as the median value. The expected or
mean value of the annual maximum concentration is 0.109 ppm. This dis-
tribution is referred to in the following sections as the P~ function.
It may be desired to assess the risk of exceeding a health effect
threshold "m" or more times in a given period, where m is an integral
number such as 1, 2, 3,---* etc. In this case there is no change in the
threshold probability density functions, but there is a change in the Pr
L*
function. As is discussed more fully in Section 3.2, the Pr function for
L
"m" or more exceedances is the distribution of the mth highest concentration
observed in the time period. The distribution in this case is referred to
as P^. Using this notation, the distribution PC of the highest concentra-
C (1)
tion, would be designated as P .
2-31
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Figure 2-5. Probability Density Functions and Cumulative DIstriDimon
(P ) of Annual Highest Hourly Average Concentration
w
0.3 -
SO .2
o
0.06
0.08
Most probable
inax. concentration
0.100
Expected
rax. concentration
0.109
0.10 0.12 0.14
Concentration, ppm
(a) Density Function
0 .16
0.06
M Median value of
;x. concentration
= 0.105
I
_L
1
0.08
0.10 0.12
Concentration, ppm
0.14
0.16
(b) Cumulative Distribution Function, PC
-------
To use the Pr function in estimating risk it is necessary to be able
L»
to estimate the PC functions corresponding to potential or alternative
levels of the ambient air quality standard. The averaging time, the con-
centration level and the overall form of the standard are all factors in
making this connection. The averaging time of the concentrations to which
the PQ function is applied is the same as the averaging time specified in
the standard. Theoretically, there is no limitation in the averaging time.
It could be one hour, one day, one month or even one year. In the case of
ozone the one-hour average is most appropriate because protection is desired
against short-term peak concentrations. The remaining discussion will
assume hourly average concentrations, but it should be kept in mind that
the approach is not limited to this time period.
The concentration level of the standard determines the general position
of the PC function along the concentration axis. The manner in which it
does is dependent upon the form of the standard. For example, if the stan-
dard were to have the following form:
(A) CSTD ppm, expected maximum hourly average
concentration in one year,
where CSTD is the concentration level of the standard, then the. P- function
is placed so that its mean concentration value corresponds to CSTD
(Refer to" Figure~2-5)r~If the" standard level "had"corresponded to the median (0.5
fractile) of the maximum concentration instead of the expected
maximum value,then the 50% point of the Pr function would be set at C-Tn.
« STD
2-33
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In both of the above-mentioned-forms,the standard deals directly with the
behavior of the annual maximum hourly average concentration and, therefore,
there is a direct connection between the standard level and the Pr function.
L»
In other forms the connection may be less direct. For example, the form
of the ozone standard which has been proposed is:
(B) CSTn pprn hourly average concentration with an expected
number of exceedances per year less than or
equal to one.
In this case the standard is not based on the annual maximum hourly
average concentration but on the concentration which is expected to be
exceeded once a year on the average. That is, the proposed form is
directly connected to the distribution of hourly average concentrations
rather than to the distribution of the annual maximum hourly average con-
centrations. In fact, it can be shown that C~TD is that concentration
which corresponds to the 1 - 1/3760 = 0.999886 (where 8760 is the number
of hours in one year) fractile on the distribution function for the hourly
averages.
The P- distribution and the hourly average concentration distribution
are distinctly different, as shown in Figure 2-6. The hypothetical distribution
for the hourly averages shown is a Weibull distribution which, has been
demonstrated to provide a good fit to ambient hourly average ozone concen-
trations (17). The PC function in the figure corresponds to a time period of
one year. The concentration axis is in units relative to the standard level
as defined by (B) above; that is, the relative concentration 1.0 corresponds
to the level of the standard. As would be expected, the effective range
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Figure 2-6. Hypothetical Distribution of Hourly Average Concentrations and the
Corresponding Distribution of the Annual Maximum Hourly Average Concentration
I
I
_L
Standard
level
I
L
O.Z 0.4 0.6 0.8 l.a 1.2 K4 1.6
Concentration, C/C-—
Ca) Distribution of Hourly Average Concentration
1.0
0.8
0.6
0.4
0.2
JL
0 0.2 0.4 0.6 0.8 l.o 1.2 1.4 1.6
Concentration* (C/CSTO)
tb) Distribution of Annual Maximum Hourly Average Concentration
2-35
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(values sensibly above zero and below one) of the P function is much
L»
narrower than that of the hourly average distribution and encompases the
standard level. On the other hand, the hourly average distribution is
very close to its limiting value of 1 at the standard level. Under the
>
circumstances depicted in the figure, the value of the PC function at the
standard level is 0.368 and will be this value at all levels of the stan-
dard. Figure 2-7 shows a series of P~ functions for various alternative
levels of the standard. It is seen that the chief effect of the standard
level is to establish the position of the P- function along the concentra-
tion axis. Note, also, there is a tendency for the PC function to spread
as the standard level is increased.
To determine the corresponding PC function it is necessary to
know the distribution function for the hourly average concentrations. How-
ever, the exact connection is influenced by the degree to which the hourly
average concentrations;, in neighboring hours ara correlated and/or are
affected by the time of day, of the week and of the year. The concentrations
of pollutants tend to show such dependencies and ozone concentrations in
particular show strong autocorrelation and time dependence. The Pr functions
L
shown in the figures of this section are all based on the assumption of no
correlation and no time dependence. As discussed in Section 3.2 this case
is important in the development of approaches to taking these interactions
into account.
National Ambient Air Quality Standards are, of course, set for the
entire U.S. While the level of the standard tends to locate the position
of the corresponding P- function, the degree of spread or effective range
of the function can vary from one geographic area to the next. The shape of
the function can also vary. This, of course, means that the risk of exceeding
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Figure 2-7. Change in PC Function With Change in Standard Concentration Level
'STD =0.06
S0.08 =0,10
=0.12 =0.14
OJ
0.05
0.10 0.15
Cbncentration C» ppm
0.20
-------
a health effects threshold is not the same with a given standard level for
all parts of the country. This is not a limitation of the proposed risk
assessment method but rather a limitation in the procedure of setting a
single standard level applicable to all of the U.S. Therefore, in setting
a single national ambient air quality standard it is important to determine
the range of risks associated with each level of the standard. As indicated,
the standard level sets the position of the P- function and tends to be the
major factor determining the risk. The variation from one geographic area
to the next primarily affects the spread of the P- function and this effect
has less impact on the risk than the placement of the P function. Variations
U
in shape of the function as distinct from changes in spread are expected to
have a distinctly lower effect on risk. There may also be some influence of
geographic area on the placement of the P~ function if the nature of the
correlation between hourly average concentrations and time dependence were
to vary markedly from one area to the next. This possibility has not been
investigated sufficiently as yet but is presently judged to introduce less
variation than the change in spread since the broad aspects of the correlation
and time dependence probably do not vary substantially with location, at least
for urban areas. Figure 2-3 shows the possible variations in the spread of a
hypothetical Pr function. Each of the Pr functions in the figure passes
L \*
through P = 0.368 at the level of the standard. Section 3.2 treats P£
functions in more detail and describes methods for deriving them from the
distribution of hourly average concentrations.
One further difficulty in estimating the appropriate P- functions for
a pollutant is that available air quality data may be at a significantly
2-38
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Figure 2-8. Pc Funct1°" for One-Year Period for Different Values of Weibull Shape Factor, k
to
ID
0
0.08
0.09
0.10
0.11 0.12
Concentration, ppm
0.13
0.14
0.15
-------
higher level than would exist if the alternate levels of the standard
under consideration were being attained. In this case it will be necessary
to make an extrapolation to the more stringent air quality levels, making
the most of whatever information is available.
2.4 Secondary Uncertainties and Public Probability
In section 2.1, it has been pointed out that the two primary uncertain-
ties which give rise to the threshold risk associated with NAAQS's are the
uncertainty about the health effect threshold concentration and the uncertainty
about the maximum pollutant concentration which will be reached in a given
period of time if a given standard is just attained. Section 2.2 discussed
the factors involved in obtaining a representation of the first primary
uncertainty, a composite health effects threshold probability distribution.
Section 2.3 discussed the factors involved in obtaining a representation of
the second primary uncertainty, a P probability distribution. Once a
composite threshold distribution and a P.. probability distribution have been
determined, the mathematical formulas presented in section 3.1 can be used to
calculate a threshold risk estimate.
If the only uncertainties that needed to be addressed were primary
uncertainties, it would only be necessary to calculate one threshold risk
estimate for each alternative standard under consideration. However, there
are also secondary uncertainties about how to best represent the primary
uncertainties; these secondary uncertainties cannot be ignored because they
can greatly affect the calculated risk values, which in turn can affect the
decision on which alternative standard to adopt.
In saying that there is secondary uncertainty about how to best represent
the primary uncertainty about the health effect threshold concentration, the
possibility is left open that there may be no best representation of this
2-40
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uncertainty in one sense; that is, in the sense of there being a probability
distribution which most rationally represents the available evidence. If
for each health effect category, there exists a threshold distribution
which most rationally represents the available evidence and if there
exist probabilities which most rationally represent the available evidence
on the existence of a causal relationship between the pollutant and each
category of effect about which the existence of such a causal relationship
is in doubt, then the composite distribution obtained from these probabili-
ty distributions and these probabilities is the most rational representation
of the available evidence; otherwise, no such most rational representation
exists. It is an open philosophical question whether there exist most
rational representations of evidence in the two required senses. The
situation is similar with regard to the P distribution.
Subjective probability distributions on threshold concentrations for
individual categories tend to be subjectively derived and hence different
for each health expert; likewise probability-of-existence-judgments. One \
of the primary aims in standard setting is to minimize arbitrariness; to
minimize arbitrariness in standard-setting decisions, arbitrariness must be
minimized in the generation of the information on which the decision is made.
2-41
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From the public's point of view as a whole, it would be arbitrary to let
one health expert make the required.subjective judgments. Ideally, all
of the fully qualified health scientists who are best informed about the
information in the criteria document would contribute judgments. But
resource constraints and the inadyisabi_]1_t^_ of attempting to determine
the exact membership of the set of best-informed scientists dictate that only a
representative sample of the best-informed experts contribute judgments. Selection
of the experts is obviously an important juncture that deserves close attention,
both in terms of the particular experts to be selected for a particular assessment and
the criteria and process by which experts are selected in general.
Presumably, one criteria which deserves to play a role in the selection
of experts is diversity in viewpoTnt_._ The whole idea of having more than
one expert make the necessary judgments is to avoid arbitrarily basing a
standard on one point of view when more than one rational point of view
may exist about the most probable implications of the same evidence. To
the extent that diverse, well-informed viewpoints are recognizable
a''priori, this criteria could either enter in selecting the sample of
experts or in selecting a subset of the best-informed scientists from which
a truly random sample could be selected.
Perhaps the most widely known technique for the use of expert judgment
in decision-making is the Delphi Technique.(18) (19) Diverse approaches
have been included under the Delphi rubric, but at least three themes seem
to be common to most: (1) systematicness, (2) sharing of information and
perspective among experts, (3) convergence of judgment among experts.
The method being suggested here subscribes to (1) and (2). It also
subscribes to (3) to the extent that convergence of judgment is brought
about by the sharing of information and perspective. However, to the
2-42
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extent that convergence of expert opinion is made a goal in itself in the
Delphi approach, the philosophy of the.approach being suggested is different.
In some formal work, distinct from the Delphi approach, convergence of judgment
for the sake of convergence is a formal goal. (20) The method described here does
not accept the idea that convergence of judgment in representing the
implications of inconclusive evidence for the sake of convergence is a
legitimate goal; at least not for public decision-making. Rather, the
goal is to represent to the decision-maker, as much as is feasible, the
implications of the actual diversity of well-informed judgments that
can remain after information and perspective have been shared; ideally
the diverse judgments would be representative of the diversity of judgments
that would be made by the whole set of well-informed scientists were it
feasible to have them all participate.
Consider as an example a case in which there are two threshold-
independent health effect categories, one of which requires a probability-
of-existence judgment: suppose that three subjective probability distri-
butions are elicited on the health effect threshold concentration of
each category and three probability-of-existence judgments are made for
the category about which the existence of a causal relationship is uncertain.
Then, there are 3 x 3 x 3 = 27 different combinations of distributions
and existence judgments which give rise to 27 different composite health
effect distributions-—Letting 'd-j' represent the jth distribution for
the 1th category, 'e^ ' represent the jth existence judgment for the ith
category, T represent the compositing function, and D.. represent the ith
composite distribution, the situation can be represented formally as follows:
I (d] , e{ , d2) = D1
I (d*, B , d2 } * D
, e|, d| ) = D27
2-43
-------
On the air quality side assume there exists a two-parameter distribution
that best fits available air quality data, that can be used to derive an ap'proxi-
-i.
mation of the true P., distribution '(?„*}, and that has a scale parameter which
C ~ " , w
varies with the standard level and a shape parameter, k. Suppose that for
any given alternative standard level the value of k which gives the best
approximation to P J- for that standard level is unknown, but 5 equiprobable
alternate values of k.(namely, ^ wnere & ranges from 1 to 5} exhaust the
possibilities. Then 135 risk estimates can be generated by combining the 27 D-j's
with the 5 k^'s. Letting 'V represent the convolution function and 'r.1 represent
the . th risk number:
V [I (d], e], d2), k,] = V (Dr k,) = r1
V [I (d2, e], d2), k,> V (D2, k.,) =. r
V [I (dj, e^, d^), kgj = V (D27, kg)
Or, alternatively:
R (d], e], dj, kj). = R (w^ = r]
R (d2, e], d2, k^ - R (w2) = r.
R (df, ef, d^, ks) = R (w]35) = r135
where w-, = (d], e], dj, k^, etc,
Now, with no loss of generality the subscripts can be rearranged on the risk
estimates so that they are ordered from smallest to largest:
o
2-44
-------
Then, the 135 ordered pairs
C-
134,
can be plotted and a relatively smooth^uryej:an_bjej^r^
points; this is done for a hypothetical example in Figure 2-9.
Figure 2-9. Relative Frequency P.lot
1
35 .5 --
Even without its being given any further interpretation, the type of curve
shown in Figure 2-9 is worthy of the attention of decision-makers and
other interested parties:
(a) It gives a measure of the degree of consensus among the
experts on the risk implications for the standard under
consideration of the information in the criteria document;
a curve that stands up relatively straight indicates
that the data base reviewed in the criteria document
2-45
-------
is strong enough in its implications for the standard under
consideration to have lead to a high degree of consensus; a curve
that stretches from corner to corner indicates a low degree of
consensus for that standard.
(b) Let Std (L, 1 hr, 1 hr/yr) be a formal way of representing a
standard where "L" represents the concentration level, the
averaging time is one hour, and the expected number of ex-
ceedances per year is one. Then, a series of curves for different
standard levels L serve to indicate the range of standard levels
for which the risk is fuzzy; that is, if n is low enough the whole
curve is far to the left, which means the risk is low no matter
what combination of judgments is used; if n is large enough the
whole curve is far to the right,, which means the risk is high
no matter what combination of judgments is used. A risk greater
than1 Q.5 would be hard to interpret as providing a margin of
safety, so if r, > 0.5 for a standard,the standard level can be
regarded to be above the fuzzy risk range.
(c) The type of curve depicted in Figure 2-9 incorporates the secondary
uncertainty about how to best represent the primary uncertainty on
the air quality side as well; it simultaneously exhibits the
consequences of the two secondary uncertainties. Also, in
cases where geographical variation in the P distribution affects
the resulting calculated risk significantly, a lower-bound P
distribution and an upper-bound PC distribution can be used to
generate an upper-bound curve and a lower-bound curve for the same
standard; the resulting risk ribbon indicates clearly that the risk
2-46
-------
varies geographically throughout the United States for the
same standard.
(d) Curves or ribbons of the type depicted in Figure 2-9 can be
plotted for different combinations of health effect categories,
for different definitions of the health effect threshold
concentration, and for different numbers of exceedances of the
threshold concentration.
Despite its usefulness even without being given further interpretation,
the type of curve shown in Figure 2-9 needs to be given further interpretation
and can be . It needs to be given further interpretation so .that distinctions
of degree can be made within the fuzzy risk range. It can be given further
interpretation because:
(1) it has the formal properties of a cumulative probability distribution;
(2) it can be interpreted as a cumulative probability distribution on the
threshold risk, although it may not be a cumulative probability
distribution on the true threshold risk since there may be no such
thing.
(3) it has a uniqueness that has importance for public regulatory
decision-making.
Each of these will now be discussed in turn:
(1) It can be checked that the Kolmogorov Axioms are satisfied for the
probability space ( ft ,3, P), where the sample space ft = (w,, w~, . . ., Wl )
— l c, ^ 135 *
the ff-field O1 = 2 (all of the subsets of ft), and the probability
measure, P, of a subset of ft is the number of elements in the subset divided
by 135. (21)
2-47
-------
(2) The 135 r^'s are risk estimates, but they may not be estimates
of some "true risk" in the same sense that an estimate of the length
of a stick is an estimate of the actual length of the stick. There may be
no such thing as "the true risk" associated with a standard in the fuzzy risk
range. If there were a most rational P distribution and a most rational threshold
distribution then the risk value obtained by convoluting the two most rational
distributions could be considered the true risk; then the r^'s could be said
to be estimates of the true risk. But since whether such most rational distri-
butions exist is shrouded in philosophical obscurity, it cannot be assumed
they do.
Unlike the P distribution, the threshold distribution does not fit the
relative frequency mold, so the alternative available on the air quality side
of defining the true P distribution, P , to be the limiting frequency distri-
w C
bution of the maximum pollutant concentration over the relevant period as the
number of periods increases without limit when the given standard is just met,
is not available on the threshold side. Hence, the alternative possibility of
considering the convolution of the true P distribution with the true threshold
distribution to be the true risk is not available.
Since the question of whether there exists a true risk for the r^'s to
estimate is open, and is likely to remain open for the foreseeable future,
the further interpretation of the curve in Figure 2-9 must be broken down
into the two possible cases. The interpretations given for each case must
be such that for practical purposes it will make no difference which case
proves to be the actual case in the philosophical millennium.
2-48
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Case 1: For a given body of evidence relevant to the value of a quantity,
there exists (conceptually) a probability distribution which is the
most rational representation of the uncertainty about the value
of the quantity.
Let Std (L.j, 1 hr, 1 hr/yr) be a series of alternative standards;
p
consider Std (L , 1 hr, 1 hr/yr). Let 0 be the most rational, distribution
P
representing the evidence on the health effect threshold; let PC (Ln) be the
most rational distribution representing the evidence on the maximum pollutant
concentration in the relevant period if Std (L , 1 hr, 1 hr/yr) is just
attained. Call RT(Ln),the convolution of DR with PcR(Ln), the True Threshold
Risk associated with Std (Ln, 1 hr, Ihr/yr). Let) RT(Ln) - r.(Ln) ) = S.(Ln).
Then the r^ (Ln) which gives the minimum G-j(Ln) is the Truest Risk Estimate (TRE)
for Std (Ln, 1 hr, 1 hr/yr).
We have from (1) that the Kolmogorov Axioms are satisfied for a
probability space which has 8 = (w-j, W£, . . . w-j3e) as its sample space.. If each
wi =(d.., e..,d., kh) is considered to be equally likely to give the TRE, then a
curve of the type depicted in Figure 2-9. is a cumulative probability "distrTbufion
on the TRE for the given standard level. Assuming arbitrariness has been
minimized in the selection of the experts and each k^ is equally likely to give
the best approximation of P (L ), from the public point of view each wn- is
c n i
equally likely to give the TRE.
These circumstances suggest a public probability interpretation of the type of
curve depicted in Figure 2-9. "Public probability" for two reasons: First, the
particular experts contributing judgments were selected and encoded for the
express purpose of assessing the risk associated with alternative
standards by the public regulatory agency that has the responsibility of setting
2-49
-------
the standards. Second, it is the'public regulatory agency that would make the
decision to regard each w^ as being equally likely to give the TRE.
This interpretation of probability is like, but not exactly the same as, both
the subjective interpretation and the classical interpretation of probability.
The classical interpretation is of a relative frequency type, but relative
frequency in the sample space, not relative frequency in a random sample.
Games-of-chance examples in textbooks often for the sake of simplicity use the
classical interpretation. The 135 w's have the symmetric appearance of games-
of-chance examples, but, just as in such games, someone has to decide to regard the
elements of the sample space to be equally likely. In this case that someone
is not a private individual, but rather a public regulatory agency; hence the
interpretation is not a subjective interpretation.
This particular interpretation is the weak interpretation of public pro-
bability. The label for the vertical axis of the curve in Figure 2-9 under the
weak interpretation would read "Estimated Public Probability that the Truest Risk
Estimate is less than or equal to the given risk value". There is also a strong
interpretation of public probability. The label for the vertical axis of the curve
in Figure2-9 under the strong interpretation would read "Estimated Public
Probability", period. Under the strong interpretation a feature of the
\
weak interpretation is dropped. The probability distribution is no longer on a
*
point of the sample space. The probability distribution is on threshold risk, not
on the TRE. The public agency decision is to give each wi equal weight in a
cumulative probability distribution on threshold risk; that is, the decision
is to regard the whole curve as a probability distribution on threshold risk.
On the true threshold risk, R (LnJ? That depends on whether RT(Ln) exists.
*An endpoint adjustment is made in the way the curve is platted; the points
are (r., i ), rather than (r., i }, under the strong interpretation. This
1 T3T 1 --
adjustment follows the recommendation of Gumbel (22).
-------
Case 2; For a given body of evidence relevant to the value of a quantity,
there does not exist (even conceptually) a probability distribution
which is the most rational representation of the uncertainty about the
value of the quantity.
In this case, R (Ln), the true threshold risk, does not exist. So, under
the strong interpretation of public probability the curve is a public
probability distribution on threshold risk, but not on the true threshold
risk since no such thing exists. The weak interpretation of public probability
is not available in this case, because the TRE is not well-defined; none of the
135 r-'s can be nearest the true threshold risk if there is no true threshold
risk.
Since the weak interpretation of public probability is not available
in one of the two possible cases and the strong interpretation of public
probability is available in both cases, it is best to adopt the strong
interpretation^ The curve can be thought of as a probability distribution
on the true threshold risk if there is a true threshold risk*-just on threshold
risk otherwise. Threshold risk definitely exists, whether there is a true
threshold risk or not.
The practical import of the probability distribution is the same in either
case. The distribution indicates the public probability that threshold risk is less
than or equal to various values. Thus, distinctions of degree can be made in a
meaningful way within the fuzzy risk range. Each alternative standard has a public
probability distribution (actually, a ribbon) associated with it and these
distributions (ribbons) can be compared. The public has a meaningful way
of comparing the alternative standard selected to the alternative not selected.
2-51
-------
In generating public probability curves it is useful to recognize
their classical structure in dealing with a third type of secondary uncertainty
that can arise, disagreement among experts in their judgments on the
number of threshold-independent categories. For example, suppose that half of
the experts judge there to be three threshold-independent categories (only one
of which requires a probability-of-existence judgment) and half judge there to be two
in a situation similar in every other respect to the situation under consideration.
Assuming there are two independent categories has given 3x5= 135 risk estimates.
4
Assuming there are three independent categories gives 3 x 5 = 405 estimates.
Plotting 135 + 405 = 540 estimates would give undue weight to the judgments
of thase experts who judge there to be three categories; the resulting curve
would not be a probability distribution on the TRE under the weak interpretation
if the true risk exists. So, under the strong interpretation making no assumptions
about the existence of a true risk, the correct approach should give the two
groups of experts equal representation in the sample space; this is accomplished
by counting three times each of the 135 r^'s calculated while working under
the assumption that there are two independent' categories. Then, 3 x 135 + 405
- 405 + 405 = 810 risk estimates are plotted.
(3) The_ public probability distribution for a given standard in a given
location would include the judgments of every member of "the set of best-
informed experts.'1 Since the membership of this set could in principle be
specified by having the public agency make (in some cases arbitrary) decisions on
who is and who is not a member, the fact that it is unwise for the Agency to do so
does not affect the conceptual existence of the public probability distribution.
Another reason the label on the ribbon diagrams (see section 4.4) is "Estimated
Public Probability'1 rather than just "Public Probability", is that on the air
quality side the public probability distribution for a given standard and location
2-52
-------
is the limit of successive approximations which can be made by increasing
without limit the number of shape-factor values used (see table 4-3).
Obviously, the probability distribution which is used to estimate the.
public probability would be different if the subset of experts who contributed
probability judgments were different. So, why should any one distribution
be considered to give the estimated public probability? For the same sort of
reason any one jury's decision should be considered the decision in a court ease
(assuming there was due process), even though another group of twelve people might
have decided differently. In pragmatic decision-making, where subjective judgment
inevitably plays a vital role and resource constraints are inevitably a factor,
arbitrariness should be minimized, but it can't be avoided. The method suggested
here attempts to identify and isolate those junctures at which there is
inevitably some arbitrariness, so that how to proceed at such junctures in
general can be carefully considered,, and so that how the Agency does proceed
in particular cases can be scrutinized by the public.
The jury analogy cannot be pressed too far, of course. The regulatory
agency decision-makers, not the health experts,make the decision on what ribbon
*
to accept. The fact that in selecting a standard they are also accepting an
associated risk ribbon separates the normative judgment of what risk ribbon to
accept from the assessment of risk itself. These two types of judgment are
best kept clearly separated.
Can the arbitrariness which is unavoidable in selecting a particular
subset of health experts to make the judgments which help give rise to risk
ribbons be avoided by returning to simpler times when risks, not risk ribbons,
were accepted? An assumption of this rhetorical question is false. In
*The topic of risk acceptance is not addressed in this report; see Rowe (23),
2-53
-------
accepting risk one js. always accepting a risk ribbon, since a particular risk
number is a special case of a ribbon— the special case in which one judgment
is accepted at each poi.nt where there is a secondary uncertainty. Even if
these judgments are made by highly qualified experts, there is more, not less,
arbitrariness in ignoring the secondary uncertainty problem in assessing risks,
2-54
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3.0 General Description of the Method
Tie previous sectfon dtscusssed the underlying principles involved in
assessing the risk to the most sensitive members of the population from a
given air pollutant in an area just meeting a specified level of air quality.
This section describes the method in more detail giving the basic tools for
estimating the risks. The mathematical framework is presented showing how
the uncertainties concerning the health effects thresholds and the maximum
pollutant concentrations observed in a given time period are combined quantitatively
to determine the risk. Next, the methods for obtaining the probability functions
for maximum concentrations are discussed.
In the following subsections only the actual working equations used
in the calculations are shown. Their derivations are given in Appendices
B and C. An understanding of the underlying mathematics is not necessary
to an understanding of the method or its application.
3.1 Mathematical Description of the Method
In the proceeding section it was pointed out that the risk of exceeding
a true health effects threshold in a given period of time is determined by
the uncertainty in the location of the health effects threshold (or in the
3-1
-------
case of multiple thresholds,the uncertainty in the location of the lowest
threshold) and the uncertainty in the maximum oxidant concentration over
the given time period. It can be shown, by application of the theory of
probability* that the following equation gives the relationship between
the risks and the above uncertainties:
R -I-/" P (C)p_(C)dC 0)
Q '
where:
R = Probability (Risk) that a true health effect threshold,
or the lowest of a multiple number of thresholds, is
exceeded one or more times in a given time period
(e.g. one year, five years, etc.).
Pr(C) = Probability that the highest observed time-averaged
\*
(e.g. one hour, two hour, etc.) pollutant concentration
does not exceed the concentration C in the given time
period.
P_(C) = The probability density function for the health effect
threshold or in the case of multiple health effects
the function for the lowest effect (the composite den-
sity function).
The derivation of Equation (1) is given in Appendix B.
Equation (T) also holds for a more general case in which R is defined
as the risk, of rn or more exceedances of a threshold in a given time period,
where ra may have the integral values 1, 2, 3, etc. In this case PC is
redefined as the probability that the mth highest time averaged pollutant
concentration does not exceed the concentration C in the given time period.
3-2
-------
In other words, P is the cumulative distribution of the mth highest time
C
averaged pollutant concentration in the time period of interest.
It was pointed out in the last section that specifying the national
ambient air quality standard for a pollutant limits the range of PC(C)
functions which will satisfy the air quality requirements of the standard.
If this limitation can be expressed quantitatively, and the health effect
threshold density functions have been determined, then for any given
specification of the standard, a range of risks associated with that speci-
fication can be calculated from Eq. (1).
In practice, it is only convenient to use Eq. (1) directly to calculate
risk, when a single health effect with an existence probability of one is
involved. When the risk that lowest of n health effects will be exceeded
is to be calculated and when there is uncertainty as to whether one or
more of the effects actually occur in the sensitive population at any
attainable pollutant concentration, the following expanded version of
Eq. (1) is used. (See Appendix B):
R - 1 - / Pc(C)pT dC - (l-e^a-e^ — d-^) (2)
where:
P = The probability density function for the location of the
lowest of n thresholds over the possible range of concen-
trations of the pollutant.
e^ * The probability that the ith health effect actually occurs
in the most sensitive population in the possible range of
concentrations of the pollutant.
3-3
-------
o
The function p_ is calculated from:
e,p,
where
a
pi 3 The probability density function fop the threshold of
the ith health effect assuming its e^ s 1. (That is,
J~p* dC = 1).
o
/
In practice p. is obtained by asking the subject health expert in
an encoding session to first give his best judgment of the value of e.
and then encoding him as to the location of the threshold assuming that
the effect actually occurs in the sensitive population at an attainable
pollutant concentration. As was discussed in Section 2.2.7, the encoding
procedure gives the cumulative distributions for the health effects thres
holds. These functions are differentiated by numerical methods to obtain
the p*.
The terms Qj and Q in Eq. (3) are calculated from:
C
1
.
-e, / P" dC (4)
• i
Q - Q1Q2Q3-«--Qn C5)
5-4
-------
The Q> are seen to be functions of the cumulative distribution of the
o
probability density functions p^.
Thus, given the Pr(C) functions corresponding to different levels
L o
of the standard, the probability density functions p^ for n independent
health effects and their corresponding e^, Eqs. (2) through C5} can be
used to calculate the range of risks associated with alternate specifica-
tions of the ambient air quality standard. The risks can be calculated
for the individual health effects and for composites of two or more of the
effects in any combination. As will be discussed in later sections, cal-
culating risks for individual health effects and various combinations can
be of value where the effects differ significantly in thefr seriousness.
The calculations involved in Eqs. (2) through (5) can be most con-
a
veniently carried out with a computer. The function pj can also be
calculated by differentiating its cumulative distribution function. The
cumulative distribution is a function of the existence probabilities e.
and the cumulative distribution functions of the pt. (See Appendix B.}
Less computational labor is involved with this method if there is no
•
interest in knowing the density functions p..
3-5
-------
3.2 Obtaining the PC(Q Distributions
As indicated in earlier sections the P/.CC) function is the cumulative
distribution function for the highest time-averaged pollutant concentration
for a specified period of time. If the risk is calculated for m or more
exceedances, it is the cumulative distribution for the mth highest concen-
tration. To simplify the following discussion it will be assumed that the
concentration averaging time is one hour.
The P/-(C) function for a pollutant in the air over a given region is
a measure of the air quality for that region with, respect to the pollutant.
Specifying a National Ambient Air Quality Standard places a limitation on
the range of PC(C) functions corresponding to air quality just meeting the
standard. For example, if the ambient air quality standard specified an
expected (average) value of maximum hourly average concentration for a one-
year period was not to exceed a given level, this would immediately locate
the mean value of the distribution of maximum values and thus define the
concentration region in which the preponderance of maximum values must
occur. However, depending upon the area and the control methods used to
meet the standard, the distribution of maximum values about the mean could
be relatively narrow or spread out. It is expected, however, that there
would be practical limitations on the degree of spread of the distribution.
Therefore, specifying the expected maximum concentration limits the PC(C)
distributions just satisfying the standard.
In applying the risk assessment method it is necessary to determine
the range of PC(C) functions just meeting each alternate specification of
the standard. While, in principle, this should be possible for almost any
3-6
-------
type of standard it fs more readily done for standards with statistical
forms than for standards with deterministic forms.
The PC distribution function is related to the distribution of
hourly average concentrations. However, the presence of correlation
between hourly average concentrations observed in different hours and the
dependence of concentrations on time of day or period of the year can
strongly affect this relationship. Air pollutants commonly show this
correlation and time dependence and these effects must, therefore, be
taken into account in developing suitable P/. functions. The approach
to talcing these effects into account that will be discussed here makes
use of the case in which independence of hours and no time dependence of
hours are assumed. This case will be discussed first.
If no correlation or time dependence of hours: exists the» it can
easily be shown (See Appendix C). that:
Pc - (1 - GCC}}'1 C6)
where PC is the distribution of the highest concentration for n hours.
(n = 8760 hrs. for one year or 43,800 hrs, for 5 years,) The function
e(C) is defined by:
GCC} * Pr [Cofas > CJ (7)
3-7
-------
That is,6(C) is the probability that an observed hourly average concentra-
tion is greater than C. It is an alternative way of expressing the cumula-
tive distribution of hourly average concentrations depicted in Fig. 2-6(a) in
Section 2.3. If the distribution function in Fig. 2-6(a) is labeled F(C)
then:
F(C) • Pr [Cobs <.C] (8)
from which, it follows that:
S{C) = 1 - F(C) (9)
If the broader definition of P- is used, the expression is more complex.
^rao.a do,
v=0
(m)
where P^ is the distribution of the rath highest hourly average concentra-
V*
tion for n hours. (See Appendix C for derivation.)
Thus, if the distribution function S(C) is known, the desired PC
function can be obtained from application of Eq. (6) or (10). Studies
have found that the distributions of snort-terra time averaged concentrations
of air pollutants can usually be represented by lognormal, Weifaull, or gamma
distribution functions (24). Of these the Weibull function provides a good
3-8
-------
fit to photochemical oxidant air monitoring data and is convenient to
use since its G(C) function can be stated explicitly.
6(C) - e'CC/5) (11)
The parameter 5 is referred to as the scale factor. It is the concen-
tration corresponding to 6(C) = 0.368. It establishes the approximate
position of the mid-concentration values of the distribution. The para-
meter k is called the shape factor. It tends to be a measure of the
spread of the distribution. The larger k the more compact the distribu-
tion. If the values of k and 5 have been determined for a given geographic
area the corresponding PC functions can then be obtained fay use of Eqs.
(6) and (11) Cor Eqs. (10) and Ol))at the given level of air quality.
For the risk assessment it is necessary to connect alternative levels
of the ambient air quality standard with the corresponding P- function.
This is easily done through the Weibull distribution, Eq. (11). The pro-
posed form for the ozone standard is:
CSTD Ppm hourly average concentration with an expected
number of exceedances per year less than or equal to E.
It is shown in Appendix C that for any region to which the Weibull function
applies and just meets the standard;
8(C) - e- E STo 02)
3-9
-------
where
C-_- s level of ambient air standard.
E = expected number of exceedances in n^. hours.
The term n_ is customarily the number of hours in one year or 8760 hours.
The expected exceedance rate would normally be one for an air standard.
In this case Eq. (12) becomes:
S(C) = a' 9'073
It should be pointed out that from the point of view of the risk assessment
method developed in this report, the designation of the expected number of
exceedances is a relatively arbitrary matter and could be set at any value
that gives a convenient level for C-~ so long as the risk Is the same
(thevalue of m in PC of Eq. 01) has a.more direct impact on health since
it directly bears on the number of exceedances of a true threshold}* For
example, if it is decided that it is undesirable to have any exceedances of
a true threshold over a given time period then the P^ used in the calcula-
tion is P^ ' (see Eq. (10)} and once an acceptable level of risk is chosen
any combination of E and C^yg values which yields this risk value in a
given area give the same level of protection.
The combination C~,Q and E determine the general location of P. while
k determines its spread. Figure 2-7 (Sec. 2.3) shows the P- functions for a
one-year period for a series of alternate levels of C^Q with E = 1 and
for the Weifaull shape factor k = 1. Figure 2-8 (Sec. 2.3) shows the effect
3-10
-------
of changing the shape factor at CSJD « 0.1 ppm and E * 1. Figure 3-1 shows
the effect of changing k for a PC covering a 5-year period. It is seen
from the three figures that changing C^ displaces the PC function over
a wide range while having a relatively small effect on its shape. Chang-
ing k causes little actual displacement of the PC for a one-year period
but has a large effect on its shape. The changing k causes the P-
function in Figure 2-8 to pivot about the point (0.10, 0.368). The effect
of k on the PC for a five-year period (Fig. 3-1) is still largely in the
shape of the function, but there also seems to be more displacement. This
results from the fact that the pivot point for the distribution is now
very close to the concentration axis (0.10, 0.007). In general, calculated
risk values will be more sensitive to changing values C-— than to changes
in k over the usual ranges of these parameters.
From the preceeding discussion it is seen that the assumption of
independence of hours and the use of the Weibull function to represent
the distribution of hourly average concentrations readily yield PC and
p£m' functions. The Heibull can be used with, little loss in accuracy
even where other distributions such as the lognorroal distribution provides
a better fit to the concentration data. The primary concern in estimating
the appropriate P_ function is to have it placed properly along the con-
centration axis and have the correct degree of spread. The parameters
s and k in the Weibull function provide wide flexibility in this regard.
As shown above, the standard level essentially places the PC function.
The appropriate values of k can be obtained by fitting a Weibull distribution
to hourly average concentration data obtained from air monitoring sites.
3-T1
-------
Figure 3-1. PC Function for 5-Year Period for Different Values
of Wei bull Shape Factor, k
1.0 h
0.8
>, 0.6
o
Q.
0.4
0.2
k«2.0
0.09 0.10 0.11
CSTT> = °«10
1
8760
0.14 0.15
Concentration, ppm
3-12
-------
The range of applicable k values for standard-setting purposes should be
determined by examining aerometric data in areas that are close to the
concentration ranges of the alternate levels of the standard under con-
sideration. Where this is not possible, the k values can be determined
at existing levels of air quality and the results extrapolated to potential
standard levels.
While using a Weibull distribution where a lognormal or gamma distri-
bution might be more appropriate does not appear to lead to serious errors
in the risk estimates, ignoring the possible dependence of hourly average
concentrations can lead to significant error. An internal EPA study (25)
showed that dependence of one hourly concentration on the value of another
did not lead to serious errors if independence of hours was assumed. How-
ever, it also showed that dependence of concentrations on time of day or
year can lead to P- functions which were placed lower on the concentration
axis than would be obtained assuming no time dependence. When this time
dependence can be modeled it should be possible to generate the corres-
ponding P- tables of functions and express them mathematically. This was
done in the EPA study for the daily maximum hourly average ozone concentra-
tion.
Another approach can be taken if the time dependence is such that the
maximum concentration tends to occur only within some determinate period
and the probability distribution of concentrations is approximately the
same for all hours within the time period. In this case all hours outside
the time period can be excluded and independence assumed for the hours within
the period. A Weibull distribution then could be fit to the hours within
the time period to determine the appropriate k values and Eqs. (6) and 02)
used to calculate PC. To the extent that the period under consideration is
3-13
-------
also likely to contain the mth highest hourly average concentration, PC
could be obtained using Eq. (10). The term n^ in Equation (12) would be
set equal to the number of hours per calendar year of the time period. The term
n in Equation (6) would be n~ times the number of years for which the risk
is to be estimated. As will be discussed in Section 4, this procedure was
applied to ozone.
-------
4.0 Application of the Risk Assessment Method to Ozone
4.1 Introduction
The risk assessment method described in the previous sections has
been developed during EPA's review of the Photochemical Oxidant NAAQS, which
EPA proposes to rename the Ozone NAAQS. The initial application of the
method has been to ozone. In order that the public be made aware of the method
and some preliminary results of its application, a preliminary report was
issued on January 6, 1978. Since that time two more health experts have
contributed judgments, the question of the number of threshold-independent
categories has been investigated further, and more suitable shape parameters
for the P distributions have been chosen. All of these changes affect some
or all of the risk estimates. The risk estimates are now presented in risk
ribbon diagrams, as well as in tables as averages.
This section presents the final results of the ozone, risk assessment;.
Subsection 4.2 presents the judgments of the health experts, including their
subjective probability distributions. Subsection 4.3 presents the method
used to derive the P probability distributions for ozone. Subsection 4.4
c
presents the results of the risk assessment.
4.2 The Judgments of Health Experts
Judgments were elicited from the nine health scientists listed in
Table 4-U There is a. consensus that there are. at least two threshold-
independent health effect categories for ozone. The category of reduced
resistance to bacteria infection was judged to be threshold independent of
the remaining health effects of ozone; The evidence on this category of
effect is mainly from animal toxicological studies, so the three toxicologists
contributed .JadgmeBts for tfeat category.
-------
TABLE 4-1
HEALTH EXPERTS'PARTICIPATING IN-THE"ANALYSIS
Dr. David Bates
Dr. Robert Carroll
Dr. Robert Chapman
Dr.. Timothy Crocker
Dr. Richard Erlich
Dr. Bernard Goldstein
Dr. Jack Hackney
Dr. Steven Horvath
Dr. .Carl Shy
Clinical Investigator
Epidemiologist
Epidemiologist
Toxicologist
Toxicologist
Toxicologist
Clinical Investigator
Clinical Investigator
Epidemiologist
4-2
-------
The Advisory Panel on Health Effects of Photochemical Oxidants,
several of whose members supplied distributions for this assessment,
has advised that the information in the revised criteria document
indicates a one-hour averaging time represents a satisfactory estimate
of the exposure duration which a primary ozone or oxidant NAAQS should
protect against. Therefore, the health effect categories considered
represent short-term effects only. Most of the evidence for the re-
maining short-term effects of ozone is from clinical and epidemiological
studies. Hence, the six clinical investigators and epidemiologists
contributed the judgments on these effects.
There is not a consensus on how many threshold-independent cate-
gories there are of the remaining short-term effects. Two experts
felt that there are three additional threshold-independent categories.
For one of these two experts the three additional categories are:
reduction in pulmonary function; cough, chest discomfort, and irritation
of mucous membranes of nose, throat, and trachea; and aggravation of
asthma, emphysema, and chronic bronchitis. The other agrees, except
that he feels emphysema and chronic bronchitis belong in the same
category with cough, chest discomfort, etc. One expert judges that
there are two additional" threshold-independent categories; he would
group the effects similar to the way the first expert who judges there
to be four categories does, except he groups reduction in pulmonary
function and aggravation of asthma, emphysema^ and chronic bronchitis
into the same category. Three experts feel that all of the remaining
health effects are threshold interdependent. In summation, it is
-------
uncertain whether there are two, three, or four threshold-independent
categories for ozone.
Judgments were elicited from'three experts for each of the following
four categories: (1) reduction in pulmonary function; (2) cough, chest
discomfort, and irritation of mucous raembrances of nose, throat, and
trachea; (3) reduced resistance to bacterial infection; and (4) aggra-
vation of asthma, emphysema, and chronic bronchitis. The secondary
uncertainty about how many of these categories are threshold independent
is taken into account in the risk estimates. Of the nine sets of judgments
elicited for categories (1), (2), and (4), two each were contributed by Drs.
Bates, Chapman, and Shy, one each by Drs. Carroll, Hackney, and Horvath.
In this initial application of the method several subjective pro-
bability distributions were elicited from Drs. Shy and Goldstein before
the final scheme for matching special fields of expertise to categories
was determined. Several of those distributions did not fit the scheme
arrived at later, and thus have not been used. It is contrary to the
Spirit of the method, of course, to encode more distributions than are
intended to be used, and then select the ones to include in the
assessment. When those distributions which do not fit the scheme
arrived at later were encoded, they were intended to be used. In
future applications the scheme will have been determined before
judgments are elicited.
The definition of conditions of exposure will improve with time.
They were not as precise as described in section 2.2.7 in two
respects for the application to ozone. First, although the experts
were asked to make their judgments for the ambient air of an average
4-4
-------
United States city, to take additive effects from other pollutants
in the air into account and to not double-count for other NAAQS
pollutants, ambient Tevels to assume for other NAAQS pollutants were
not specified. Second, the way in which adaptivity should be incor-
porated into their judgments wa:s not specified precisely./
Although care was taken to emphasize that their best judgment, not
the easiest-to-rationa.lize judgment, was desired, the health experts were
encouraged to verbalize their thoughts about how they came to their
probability judgments.: These, comments were noted and are used, to sketch a
very rough picture of the reasoning behind each expert's distribution-
(1) Reduction in Pulmonary Function
Three clinical investigators contributed subjective probability
distributions for health effect category 1, reduction in pulmonary
function. The amount of decrement in pulmonary function considered a
health effect(as measured by percentage reduction in forced expiratory
volume (FEV) response)was described by Expert 8 as 2-3% above the noise
level, which will be different for different groups, and by Expert A
as just above noise level. The percentage changes in mind when judgments were
made were roughly 5-10% for Expert A, 15% for Expert B, and 10% for
expert C. By consensus, asthma and infrequent exposure to high ozone
levels were two characteristics of the most sensitive group. Expert A
further characterized the most sensitive group as exercising, particularly
those exercising at high altitudes. Expert B further characterized the
most sensitive group for whom data was available to him, as a working
population of asthmatic subjects. Expert C further characterized the most
sensitive group as young (children) or elderly.
-------
None of the three experts felt that one occurrence of the effect is
serious; one occurrence is reversible, even for the" most susceptible
group. However, Expert A expressed concern about any impairment of
functioning, and Expert C stressed that the seriousness goes up rapidly with
the 'frequency -
The three subjective probability distributions elicited for
reduction in pulmonary function are given in Figure 4-1. Although
at most concentrations experts B and C differ some in their assessment
of the probability that the health effect threshold is less than the
given concentration, the trend of their judgments is close compared to
those of expert A. The three medians (0.5 probability) are approximately
0.075 ppm for expert A, 0.175 ppm for expert B, and 0.18 ppm for
expert C. Experts B and C assign a probability.in the 0.03-0.05 range to
the proposition that the threshold is less than 0.1 ppm; expert A assigns
a probability of about 0.97 to the 0.18 ppm concentration which is roughly
the median for both B and C.
In making his judgments, Expert A mentioned the results of
DeLucia and Adams (1977) on the effects of ozone on exercising
individuals. Experts B and C did not give the epidemiological
studies reviewed in the criteria document and the von Nieding, et. al.
(1976) clinical study very much weight. In light of the questions
that have been raised, they will be skeptical about the von Nieding results
until they are replicated. Expert B mentioned that he has noticed hints of an
effect at about! 0.2 Ppro in his investigations of asthmatic subjects; taking
variation in susceptibility into account, he estimated a median of 0.175 ppm.
Expert C estimated that it would take about a two-hour exposure to 0.37 ppm
to cause a health effect in normal people, about a one-hour exposure to 0.37 ppm
4-6
-------
INTERVIEWER.
SUBJECT.
DATE.
0.1 !
0
b.o
0.05 0.1 0.15 0.2 0,25
Figure 4-1. Reduction in Pulmonary Function (Experts A,B,&C)
0.3
0.35
0.8
0.9
1.0
C,ppm
VARIABLE.
COMMENT.
-------
to cause a health effect in very sensitive asthmatics such as those
for whom the threshold is defined in this assessment.
(2) Cough, Chest Discomfort, and Irritation of Mucous Membranes of
Nose, Throat, and Trachea
One clinical investigator (Expert A) and two epidemiologists
(Experts B and C) contributed subjective probability distributions
for health effect category 2. A health effect for this category
was defined to be a coughing spell, sore throat, etc. sufficient to cause
discomfort. By consensus of all three experts, the most susceptible
group is exercising children. Expert C added that healthy children are
probably the most sensitive since their nervous system would be the
most acute. By consensus, category 2 is the least serious of the four
categories of health effects; one occurrence is reversible.
The three subjective probability distributions elicited for category
2 are given in Figure 4-2. The three medians are about 0.13 ppm (Expert C),
0.15 ppm (Expert B), and 0.18 ppm (Expert A). Expert A assigns a pro-
bability of about 0.97 to the threshold being in the 0.1 ppm to 0.25 ppm
concentration range; Expert B assigns a probability of about 0.95 to the
threshold being in the 0.09 ppm to 0.3 ppm concentration range; and Expert
C assigns a probability of only about 0.90 to the threshold being in the
0.05 ppm to 0.4 ppm concentration range. Hence, the three experts not
only differ in the relative weight they put on various relevant studies,
but also in the absolute weight they put on the body of evidence that is
available.
4-8
-------
INTERVIEWER..
.SUBJECT.
DATE.
C,ppm
VAHIABLE.
COMMENT.
Figure 4-2. Cough, Chest Discomfort, and Irritation of Mucous Membranes of
Nose, Throat, and Trachea (Experts A.B.&C)
-------
Different relative weight was put on the experimental findings of
Bates and Hazucha (1973), Hammer's study of student nurses (1974),
and the results of several Japanese epideraiological studies (Shimizu,
1975; Makino and Mizoguchi, 1975; Japan Environment Agency, 1976).
Possible additive effects had to be factored into the weighting of
these studies because the Hammer study and most of the Japanese results
were for oxidants, whereas the Bates and Hazucha findings were for ozone.
Also, the subjects of the Hammer and the Bates and Hazucha studies
were young adults, whereas the subjects of the Japanese studies were
school children.
All three experts considered the Japanese studies significant;
they also found their implications hard to assess since total oxidants
were measured, the relative contribution of other pollutants is not
clear, and in some cases group dynamics may have been a factor. In
making their judgments, Expert C,. an epidemiologist, gave the Japanese
studies the most weight and Expert A, a clinical investigator, gave them
the least.
(3) Reduced Resistance to Bacterial Infection
The subjective probability distribution of three toxicologists
was used for category 3, reduced resistance to bacterial infection.
Most of the scientific basis for the health effect is a set of toxicolo-
gical studies on animals. Experts A and C define the health effect for
this category to be an increased incidence of bacterial infections in
humans. Expert 8 defined the health effect to be an increased incidence
of bacterial infections in humans or an increase in the severity of already
occurring infections.
4-10
-------
By consensus, the most susceptible group for category 3 is young
children. Young children do not have fully developed lungs and immuno-
logical protection. Expert A characterized the most susceptible group
more finely as: (a) prematurely born, since the lung development of
prematurely born children lags behind that of the normal child until
the age of about five; (b) not asthmatic; hence, airways are in good
condition, and they will breathe deeply; (c) exercising vigorously.
Reduced resistance to bacterial infection itself was not described
as being serious, but concern was expressed about: (a) increased severity
of bacterial infections; (b) the obvious consequence to some of the people
who have their resistance to bacteria infections reduced, namely, a
bacterial infection they would not have had otherwise; and (c) the
possibility of an increase in the risk of other, even more serious,
health consequences due to the existence of the resulting bacterial
infection. Expert B felt that the very old would belong to the group
that is most susceptible in this sense.
Expert A's subjective probability of the health effect existing in
humans was the lowest, namely 0.3. He believes that concentrations of
ozone high enough to injure macrophages (which are a vital part of the
body's defense mechanism against bacterial infection) may never reach
the alveoli (air cells of the lung) of humans in commonly occurring ambient
situations. First, he observed that there are anatomic differences
between human and rodent lungs such that the exposure required to achieve
the same dose to the alveoli is larger for humans. Second, he conjectured
that irritation to the upper respiratory tract might be great enough to
make most yoimg children stop exercising before the dose necessary to
cause the effect ewer readied t&e lower lung,. In sapport :sf this
-4-T1
-------
conjecture, he referred to "a piece of work done at Riverside, California,
in the 1960's" in which mice stopped running after a certain amount of
exposure to ozone.
Because of the anatomical difference in the lungs of rodents and
humans, and because of the difference between the conditions of actual
human exposure in the ambient atmosphere and the conditions of exposure
in the Gardner, et. al., infectivity model, Expert A felt it would take
much higher concentrations than 0.1 ppm to cause the effect in humans,
if it indeed caused the effect at all.
Expert B expressed a very different point of view on the probability
of existence of the effect in humans. His subjective probability of
existence in humans is 0.95. He expressed confidence that the lower
lung of humans would receive the dose necessary to cause the effect if
the exposure concentrations is high enough* This confidence was partially
based on the results of the mathematical modeling approach to estimating
ozone uptake in the deep lung done by Miller (Ph.D. Thesis). Some of
the differences between Experts A and 8 on this question may be due to
the fact that Expert B explicitly introduced an increase in severity
of an already occurring bacterial infection into his definition of the
health effect, whereas Expert A did not.
As can be seen in Figure 4-3, Expert B also has a very different
view of the probability that the threshold is below various concentra-
tions. His median is 0.11 ppm. He feels that the threshold for very
susceptible humans is most likely about the same concentration as the
concentration (Q.I ppm) which has been found to cause the effect in rodents,
4-12
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,-*
co
INTERVIEWER.
.SUBJECT.
SRI—PROBABILITY ENCODING FORM V—JANUARY 1975
DATE
o.i 0.2 a3 0.4 0.5 0.6
Figure 4-3. Reduced Resistance to Bacterial Infection (Experts A.B.&C)
C.ppm
VAUIABLE
COMMENT
-------
Expert C's subjective probability that ozone contributes to
the effect in humans is 0.5. He feels that if the effect does occur
in humans, the threshold for its occurrence in very susceptible indi-
viduals is most likely to be even less than the 0.1 ppm concentration
which apparently will cause the effect in rodents. His median is about
0.07 ppm. He thinks there is a significant possibility that the
threshold is as low as about 0.04 ppm. But, he acknowledges the
possibility that the threshold might be as high as 0,35 ppm, either
because humans are different than rodents or because, despite
their apparent validity, the results at the 0.1 ppm level are
misleading for rodents thamselves.
(4) Aggravation of Asthma, Emphysema, and Chronic Bronchitis
Three epidemiologists have contributed distributions for health
effect category 4, aggravation of asthma, emphysema, and chronic bronchitis.
For expert 3 aggravation of asthma is threshold independent of aggravation
of emphysema and chronic bronchitis. He groups aggravation of emphysema
and chronic bronchitis with the effects of category (2). Expert C feels
the evidence for this category is sparse, and that what little evidence
there is applies mainly to aggravation of asthma.
A health effect for this category was defined to be an aggravation
of one of the three lung diseases. It was observed that such aggravation
not only increases discomfort, but also affects the individual's functioning,
which is already restricted, and can have more serious irreversible conse-
quences. The effect was described as serious. Expert C remarked that
4-14
-------
very mild asthmatic attacks which are very easily triggered in sensitive
asthmatics are clearly distinguishable from the severe attacks which
are serious. He made his judgments on the latter.
Expert A judged the most sensitive group to be individuals who
have asthma or emphysema. Experts B and C judged asthmatics to be the
most sensitive group.
In figure 4-4 it can be seen that all three experts think it very
unlikely this effect occurs below 0.06 ppm; expert A mentioned the
negative result at that concentration of Rokaw and Massey (1962). Expert A
mentioned studies by Molley, et. al. (1959), Remmers and Balcham (1965),
and Schoettlin and Landau (1961) in coming to his probability judgments; he
also mentioned that there are problems in interpreting them, especially
the study by Schoettlin and Landau. Expert C also mentioned that the
Schoettlin and Landau study was a basis for his judgments, despite his
reservations about the study, because in his opinion, there is very little
evidence other than that study to base judgments on.
The median for experts A and B is about 0.14 ppm; this may reflect
the fact that Remmers and Balcham found a beneficial effect of air filtra-
tion in studies of four exercising patients at an ozone concentration of
0-.13. Expert C's median is much higher at 0.25 ppm. Experts A and C
remarked that there is a great deal of uncertainty about where the
threshold is for this, effect, and their spread-out distributions reflect
this view. Their probability judgments that the threshold is less than
0.30 ppm are only 0.80 and 0.72,^:^ respectively. Expert 8, whose distri-
bution is similar to Expert A's up to the median, is almost sure the
less than 0^30 .ppm.
4-K
-------
INTERVIEWER
SUBJECT
DATE
0.2 hi1 T
0.1
00
. 0.05
Figure 4-4. Aggravation of Asthma, Emphysema, and Chronic Bronchitis (Experts A.B.&C)
VARIABLE.
COMMENT.
-------
4.3 Determination of P^ Functions for Ozone
As suggested in Sections Z and 3 estimating the appropriate PC
function involves: 1. determining the mathematical function or functions
which best describe the distribution of time-averaged ambient concentrations
of the pollutant; 2. given the distribution of time-averaged concentrations,
the standard, and the nature of the correlation between concentrations in
neighboring time periods and dependence upon time of day and year, deriving
suitable PC functions; and 3. estimating the range of values of parameters
appearing in the P~ function.
A study performed under contract for EPA involving 14 sites scattered
around the United States and involving 22 site-years of data showed that
the Weihull Distribution (Eq. OU> Section 3.2) provides an excellent fit
to hourly ozone concentrations^17). In only two cases did a lognorraal
distribution give a superior fit.
Ozone hourly average concentrations exhibit strong correlation and
strong-dependence on time of day and time of year. The day of the week.
also has some effect. As indicated in Section 3, the correlation between
neighboring concentrations does not appear to have an important effect on
the Pr function; however, the time dependence does,
w
The method of dealing with the time dependence discussed in Section 3
was to find a time period in which the time dependence was relatively con-
stant and which was highly likely to contain the maximum hourly average
concentration for the time period. If such a time period existed and a
distribution could be fit to the time-averaged concentrations within this
period, then the appropriate PC function could be derived by assuming
4-T7
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complete independence of the time-averaged concentrations. For ozone this
period is during the midday hours of the warm months of the year.
In a continuation of the above-mentioned study (17) it was found that
Weibull distributions could fae fit to data obtained between 11 AM and 6 PM
both from May through September and July through August. It was further shown
that the maximum ozone concantration had a significantly greater chance of occurring
during the longer period. Therefore, this time period, which contains 1071
hours, was used in the derivation of the P- function.
The form of the National Ambient Air Quality Standard proposed for ozone
is: C5T« ppm hourly average concentration with an expected' number of exceedances
per year less than or equal to one. For a region whose air quality just
meets this standard, the Weibull distribution of hourly averages for the
hours 11 AM to 6 PM, May through September5 would be according to Eq. 02}
in Section 3.3:
S(C) = e Vl" '""^"-STD* (12«
And the Pr function for a period of nu years would be from Eq. (5) in
*•* y
Section 3.3.
fin 1071UC/C )k 1071 nv
Pc = fl-«n HWUgmJ } y (gl)
By substituting alternate levels, C-,-, of the standard into Eq, (61)
the Pr function for n years needed for the risk assessment can be obtained.
»* y
However, before this can be done it is necessary to estimate the range of
values of the parameter k (Weibtill shape factor) over different regions in
4-18
-------
the U.S. Table 4-2 shows measured k values for the 1071-hour time period
as well as other time periods. The range for the values based on 1071
hours is 1.31 to 2.04.
For the risk assessment best estimates were made of the lower-bound
and upper-bound values of k.. This was done by probability encoding two
researchers involved in the development of the Weibull distributions and
the PC functions. The information base was the data in Table'4-2, plus the
data developed during the Weibull distribution studies. The data shown in
Table 4-2 are, by and large, for geographic regions above the range of
alternative ozone standards considered. The Weibull studies suggest that
the k factors at the standard levels would be somewhat higher than those
shown in Table 4-2. This factor was taken into account in the encoding.
The median values for the location of the lower- and upper-bound shape.
factors were 1.36 and 2.54 respectively. The distributions obtained in
the encoding sessions are shown in Table 4-3. Since the range of k values
varies somewhat with the standard level, the range, strictly speaking,
should be estimated for each alternative level of the standard. In the
case of ozone the difference is not likely to be large enough- to seriously
affect the risk estimates.
Given the above range of values for the Weibull shape factors, Eq (61)
can be used to calculate the PC functions. Figure. 4-5 shows the PC functions used.
in succeeding sections to calculate risk estimates for a standard level of
0.10 ppm and an expected exceedance rate of once per year. The time period
is five years. That is, the functions in Figure' 4-5 can be used to calculate
estimated risks of exceeding an ozone health effect threshold one or more times
in five years when the standard level is at 0,10 p.pm. The functions were calculated
4-T9
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Table 4-2 Change in Weifaull Distribution Shape Factors With Changing Time
Segment in Which Hourly Average Concentrations are Collected3
Shape Factor, k
Site
Kansas City, Kansas
Des Hoines, Iowa
Louisville, Kentucky
Memphis, Tennessee
Mamaroneck, New York
Racine, Wisconsin
Year
1975
1975
1974
1974
1975
1974
Full Yearb
1.24
1.92
0.80
1.32
0.80
1.49
May-Sept. c
11 AM-6PM
—
2.04
1.56
2.28
1.31
__
July-Aug.
11 AM-6PN
4.34e
2.40
2.02
2.34
1.57
1.99s
a Reference 17.
b 8760 hours.
c 1071 hours
d 434 hours.
e Poor Heibull fit.
4-20
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Table 4-3 Subjective Probability Distributions for Upper- and Lower-Bound Weibull
Shape Factors (k) for Distributions of Hourly Average Ozone Concentrations
Probability
that k is
below specified
value
0.10
0.30
0.50
0.60
0.90
Lower-
Bound
k
1.15
1.30
1.36
1.41
1.47
Upper-
Bound
k
2.41
2.50
2.54
2.66
2.98
-------
Figure 4-5 PC Function for Estimating Risks of Exceeding an Ozone
Health Effects Threshold One or More Times in Five Years
1.0 -
0.8
0.6
0.4
0.2
0.0
Standard Level S0.10 ppnr
Expected Excsedances
per Year = 1
Effective Hours
per Year = 1071
0.08
0.09
0.10 0.11 0.12
Concentration, (ppm)
0.13
0.14
4-22
-------
using the median lower-bound and median upper-bound shape factors. The
two curves, in the figure therefore bound the large majority of PC functions
(for the above parameters) that would be encountered in different areas of the
U.S.
Because the standard level enters into Eq (61) as a divisor of the concen-
tration, Figure 4-5 can also be used to indicate the PC function for other
standard levels. This is done by relabeling the concentration axis. For
example, if the function is desired for a new level, C^yp, each number on the
concentration axis is multiplied by CSTQ/0.10.
Estimates were also made of the risk of exceeding an ozone health effect
threshold five or more times in five years. The corresponding PC functions
for the median upper- and lower-bound shape factors are shown in Figure 4-6. It
is seen that these functions are much closer together than those in Fig. 4-5. As
a result, the range of risk estimates for five or more exceedances of a health
effects threshold in a five-year period will be much smaller than for one or
more exceedances in the same period.
In dealing with risks of multiple exceedances of a health effect threshold
it should be noted that a risk of five or more exceedances in five years is
not the same as the risk of one or more exceedances in one year since the
respective P functions are different. This point is illustrated in Fig.4-7
which shows a series of P.;1 ' functions where m varies from one to five years.
w
It is seen that the "one or more in one" risk will be greater than the "five
or more in five" risk.
It is tempting to think of the risk of five or more exceedances of a.
threshold in five years as equivalent to the risk of an average of one or more ex-
ceedances of a threshold per year. This is not the case. An average of one or
more exceedance per year would correspond to the limiting PC for m years when m
increases without limit. For the situation depicted tn Figure 4-7, this limiting
?,. is a -step ftmctiori which is zero for concentrations below 0.10 .ppra (the
c
$ta»diS0"d "bevel) and sse for csRcefrtratioas of Q,K) ppm and above. Depending
whT-r-h "S^ho D IvRl) -i i ^
wn.r.cn sne r^ > TS.correl-ated, -the
-------
Figure 4-6 PC Functions for Estimating Risks of Exceeding an Ozone
Health Effects Threshold Five or More Times in Five Years
1.0
0.8
0.6
p(5)
C
0.4
0.2
0.0
k » 2.54
Standard Level
Expected Exceedances
per Year
Effective Hours per
Year
0.10 ppm
0.08
0.09
0,10 0.11
Concentration, (ppm)
0.13
4-24
-------
Figure 4-7 p Functions for Estimating Risks of Exceeding an Ozone
Health Effects Threshold "m" or More Times in "m" Years
0.08
Standard Level
Expected Exceedances
per Year
Effective Hours per Year
Shape Factor
O.TOppm
1 ~.-
1071
1.36
I I I I I 1 I I I [
0.09
0.10
0.11 0.12 0.13
Concentration, (ppm)
a.
0.15
4-25
-------
average risk can be significantly different from the risks for
low values of m. In general, it would give a lower risk than
that obtained using the PC for lower values of m.
4-26
-------
4.4 Risk Tables and Risk Ribbons
This section presents the .quantitative results of the ozone risk
assessment. Table 4-4 presents average threshold risk estimates for several
alternative standards. Table 4-4(a) presents estimates of the risk that the
health effect threshold will be exceeded one or more times in five years
if the given standard is just met. Table 4-4(b) presents estimates of the risk
associated with the given standard that the health effect threshold will be
exceeded five or more times in five years if the given standard is just met.
The average risk estimates, are obtained by averaging the risk estimates
which result when the single most likely value of the shape parameter for the P
c
distribution is used»
Figures 4-8" through 4-17 are a series of risk ribbons. Section 2.4 explains
how these ribbons are derived. The plots are ribbons rather than simply
curves because, as is explained in section 2.3, the risk varies over the
United States. The lower-bound and upper-bound curve for each alternative
standard are obtained by estimating the extremes for the shape parameter of
the P distribution, as is explained in section 4.3.
Figure 4-8 presents the ribbons which are obtained for the two most
serious health effect categories when the health effect threshold for
the most sensitive group is defined to be the health effect threshold for the
least sensitive member of the most sensitive 1 percent of the most sensitive
group; figure 4-9- similarly for the least sensitive member of the most
sensitive 5 percent of the most sensitive group; figure14-10 similarly for the
least sensitive member of the most sensitive 10 percent of the most sensitive
group. The risk ribbons shift toward lower risk values as the definition
changes from the 1 percent definition to the 5 percent definition to the 10
percent definition.
4-27
-------
Table 4-4(a). Risk that Health Effect Threshold Will be Exceeded 1 or More Times
in 5 Years for Alternate Standard Levels*
Hourly average standard
level (1 expected ex-
ceedance per year)
0.06 ppm
0.08 ppm
0.10 ppm
0.12 ppm
0.14 ppm
(1 Deduction
in pulmonary
function
0.14-0.16
0.22-0.26
0.31-0.36
0.41-0.47
0.52-0.60
(2)Chest discomfort
and irritation
of the respira-
tory tract
0.03-0.05
0.09-0.14
0.21-0.27
0.34-0.42
0.47-0.56
(3 Deduced resistance to
bacterial infection (animal
studies)
0.03-0.12
0.17-0.20
0.24-0.26
0.28-0.29
0.31-0.32
(4)Aggravalion of asthma
emphysema, and chronic
bronchitis
0.03-0.04
0.10-0.15
0.22-0.29
0.36-0.41
0.45-0.50
Risk of exceeding 1 or
more of the thresholds for
the individual categories
0,25-0.30
0.42-0.50
0.60-0.67
0.74-0.79
0.83-0.87
I
p*y
co
Table 4-4(b). Risk that Health Effect Threshold U111 be Exceeded 5 or More Times
in 5 Years for Alternate Standard Levels*
Hourly average standard
level (1 expected ex-
ceedance per year)
0.06 ppm
O.Ofl ppm
0.10 ppm
0.12 ppm
0.14 ppm
(l)Reduction
In pulmonary
function
. 0.11
0.19-0.20
0.27
.0.35..
0.44
(2)Chest discomfort
and irritation
of the respira-
tory tract
0.03
0.06
0.15
0.27
0.39
(3 Deduced resistance to
bacterial Infection (animal
studies)
o.oa
0.15
0.21
0.26
0.29
(4)Ag
-------
Figure 4-8. RISK OF EXCEEDING THE THRESHOLD* OF
" AT LEAST ONE OF THE FOLLOWING HEALTH EFFECTS:
O REDUCED RESISTANCE TO BACTERIAL INFECTION
O AGGRAVATION OF ASTHMA, EMPHYSEMA, AND
CHRONIC BRONCHITIS
Estimated
Public
Probability
1.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Risk of Exceeding a Health
Effect Threshold One or More
Times in a Five Year Period.
* HEALTH EFFECT THRESHOLD: Concentration at which 1%
of group most sensitive to the effect would suffer
the effect if the whole group were exposed under
specified conditions.
4-29
-------
Figure 4-9. RISK OF EXCEEDING THE THRESHOLD* OF
AT LEAST ONE OF THE FOLLOWING HEALTH EFFECTS:
O REDUCED RESISTANCE TO BACTERIAL INFECTION
O AGGRAVATION OF ASTHMA, EMPHYSEMA, AND
CHRONIC BRONCHITIS
Estimated
Public
Probability
0.1 '0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Risk of Exceeding a Health
Effect Threshold One or More
Times in a Five Year Period.
* HEALTH EFFECT THRESHOLD: Concentration at which 5%
of group most sensitive to the effect would suffer
the effect if the whole group were exposed under
specified conditions.
4-30
-------
Figure 4-10. RISK OF EXCEEDING THE THRESHOLD* OF
AT LEAST ONE OF THE FOLLOWING HEALTH EFFECTS:
O REDUCED RESISTANCE TO BACTERIAL INFECTION
O AGGRAVATION OF ASTHMA, EMPHYSEMA, AND
CHRONIC BRONCHITIS
Estimated
Public
Probability
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Risk of Exceeding a Health
Effect Threshold One or More
Times in a Five Year Period.
* HEALTH EFFECT THRESHOLD: Concentration at which 10%
of group most sensitive to the effect would suffer
the effect if the whole group were exposed under
specified conditions.
4-31
-------
Figure 4-11 presents the ribbons for the risk of exceeding the health
effect threshold for the two most serious categories five or more times
in a five year period when the threshold is given the 1 percent definition.
Figure 4-12 similarly for the 10 percent definition. The ribbons generally
collapse to curves when the risk values are rounded to two decimal places for
five or more exceedances because the shape of the P. distribution for the
W
fifth high hourly average ozone concentration is not as sensitive to
changes in its shape parameter k (see section 4.3)-.
Figure 4-13 presents the risk ribbons for all four health effect categories
when the health effect threshold is given the 1 percent definition. These
are the only risk ribbons presented which must deal with the fact that there
is secondary uncertainty about the number of threshold-independent health affect
categories for ozone. See section 2.4- for an explanation of how this type
of uncertainty is handled. The weights used were
3 - two independent categories
1 - three independent categories
2 - four independent categories
in view of the judgments related in section 4.2.
Figures 4-14 through 4-17 present the risk' ribbons for the individual
categories. They are not as smooth because there are only 15 (=3x5)
points being plotted, except for category 3. Category 3 requires a probability-
2
of-existence judgment so 45 (= 3 x 5) points are plotted for it.
4-32
-------
Estimated
Public
Probability
Figure 4-11.RISK OF EXCEEDING THE THRESHOLD* OF
AT LEAST ONE OF THE FOLLOWING HEALTH EFFECTS:
O REDUCED RESISTANCE TO BACTERIAL INFECTION
O AGGRAVATION OF ASTHMA, EMPHYSEMA, AND
CHRONIC BRONCHITIS
1.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Risk of Exceeding a Health
Effect Threshold Five or More
Times in a Five Year Period.
* HEALTH EFFECT THRESHOLD: Concentration at which 1%
of group most sensitive to the effect would suffer
the effect if the whole group were exposed under
specified conditions.
4-33
-------
Figure 4-12. RISK OF EXCEEDING THE THRESHOLD* OF
AT LEAST ONE OF THE FOLLOWING HEALTH EFFECTS:
O REDUCED RESISTANCE TO BACTERIAL INFECTION
O AGGRAVATION OF ASTHMA, EMPHYSEMA, AND
CHRONIC BRONCHITIS
Estimated
Public
Probability
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Risk of Exceeding a Health
Effect Threshold Five or More
Times in a Five Year Period.
* HEALTH EFFECT THRESHOLD: Concentration at which 10%
of group most sensitive to the effect would suffer
the effect if the whole group were exposed under
specified conditions.
4-34
-------
Figure 4-13 RISK OF EXCEEDING THE THRESHOLD* OF
AT LEAST ONE OF THE FOLLOWING HEALTH EFFECTS:
O REDUCTION IN PULMONARY FUNCTION
o COUGH, CHEST DISCOMFORT, AND IRRITATION
OF THE NOSE, THROAT, AND TRACHEA
O REDUCED RESISTANCE TO BACTERIAL INFECTION
O AGGRAVATION OF ASTHMA, EMPHYSEMA, AND
CHRONIC BRONCHITIS
Estimated
Public
Probability
1.0
0.9
0.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Risk of Exceeding a Health
Effect Threshold One or More
Times in a Five Year Period.
HEALTH EFFECT THRESHOLD: Concentration at which 1%
of group most sensitive to the effect would suffer
the effect if the whole group were exposed under
specified conditions.
4-35
-------
Figure 4-14. RISK OF 'EXCEEDING THE THRESHOLD*
OF THE FOLLOWING HEALTH EFFECT:
O REDUCTION IN PULMONARY FUNCTION
Estimated
Public
Probability
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Risk of Exceeding the Health
Effect Threshold One or More
Times in a Five Year Period.
* HEALTH EFFECT THRESHOLD: Concentration at which 1%
of group most sensitive to the effect would suffer
the effect if the whole group were exposed under
specified conditions.
4-36
-------
Figure 4-15. RISK OF' EXCEEDING THE THRESHOLD*
OF THE FOLLOWING HEALTH EFFECT:
O COUGH, CHEST DISCOMFORT, AND IRRITATION
OF MUCOUS MEMBRANES OF NOSE, THROAT, AND
TRACHEA
Estimated
Public
Probability
1.0
0.9
0.3
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Risk of Exceeding the Health
Effect Threshold One or More
Times in a Five Year Period.
* HEALTH EFFECT THRESHOLD: Concentration at which 1%
of group most sensitive to the effect would suffer
the effect if the whole group were exposed under
specified conditions.
4-37
-------
Figure 4-16. RISK OF EXCEEDING THE THRESHOLD*
OF THE FOLLOWING HEALTH EFFECT:
O REDUCED RESISTANCE TO BACTERIAL INFECTION
Estimated
Public
Probability
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 l.l
Risk of Exceeding the Healtl
Effect Threshold One or Mor$
Times in a Five Year Period
* HEALTH EFFECT THRESHOLD: Concentration at which 1%
of group most sensitive to the effect would suffer
the effect if the whole group were exposed under
specified conditions.
4-38
-------
Figure 4-17. RISK OF EXCEEDING THE THRESHOLD*
OF THE FOLLOWING HEALTH EFFECT:
O AGGRAVATION OF ASTHMA, EMPHYSEMA, AND
CHRONIC BRONCHITIS
Estimated
Public
Probability
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Risk of Exceeding the Health
Ef-fect Threshold One or More
Times in a Five Year Period.
* HEALTH EFFECT THRESHOLD: Concentration at which 1%
of group most sensitive to the effect would suffer
the effect if the whole group were exposed under
specified conditions.
4-39
-------
REFERENCES
1. Air Quality Criteria for Ozone and Other Photochemical Qxidants,
U.S. EPA, Office of Research and Development, EPA-600/8-78-004,
April, 1978.
2. National Academy of Sciences, Decision Making for Regulating
Chemicals in the Environment, Washington, D.C., 1975.
3. National Academy of Sciences, Decision Making in the Environmental
Protection Agency, Washington, D.C., 1977.
4. Clean Air Act, Section 109, 42 U.S.C.
5. William W. Lowrance, Of Acceptable Risk, William Kaufmann, Inc.,
Los Altos, California, 1976.'
6. Senate Committee on Public Works, A Legislative History of the
Clean Air Amendments of 1970, 1974.
7. Carl S. Spetzler and C.A.S. Stael von Holstein, "Probability
Encoding in Decision Analysis", Management Science, Vol. 22,
No. 3, November 1975.
8. Daniel J. Quinn and James E. Matheson, The Use of Judgmental
Probability in Decision Making, prepared for U.S. EPA, May 1978.
9. Ian Hacking, The Emergence of Probability, Cambridge University
Press, 1975.
10. Henry E. Kyburg, :Jr., and Howard E. Smokier (eds.), Studies
fn Subjective Probability; John Wiley and SOPS, Inc.; New York, 1964.
11. Robert L. Winkler and William L. Hays, Statistics,. 2nd edition,
Holt, Rinehart and Winston, New York,-1975.
12. Bernard Altshuler, "A Bayesian Approach to Assessing Population
Risks from Environmental Carcinogens," Proceedings of SIMS Research
Application Conference on Environmental Health. Alta, Utah;
July 5-9, 1976; Revised November 2, 1976.
13. V- Hasselblad, W.C. Nelson, and G.R. Lowrimore, "Analysis of
Health Effects Data: Some Results and Problems", in John W.
Pratt (ed.), Statistical and Mathematical Aspects of Pollution
Problems, Marcel Dekker, Inc., New York, 1974.
14. Glenn Shafer, A Mathematical Theory of Evidence, Princeton
University Press, Princeton, New Jersey, 1976.
-------
15. Rudolf Carnap, Logical Foundations of Probability, 2nd edition,
The University of Chicago Press, 1962^.
16. L. Jonathan Cohen, The Probable and the Provable, Oxford University
Press, 1977. ;
17. "The Validity of the Weibull Distribution as a Model for the
Analysis of Ambient Ozone Data." Draft Report to EPA by PEDCo
Environmental, November 17, 1977.
18. Harold A. Linstone and Murray Turoff (ed's.}» The Delphi Method.
Addison-Wesley Publishing Co., Inc., Reading, Massachusetts, 1975
19. Steve Leung, Elliot Goldstein, and Norman Dalkey, "Human Health
Damages from Mobile Source Air Pollution: A Delphi Study-Volume I,"
performed for California Air Resources Board, April, 1977.
20. Morris H. OeGroot, "Reaching a Consensus," Journal of the American
Statistical Association, March 1974, Volume 69, Number 345.
21. Terrence L. Fine, Theories of Probability, Academic Press, Inc.,
New York, 1973.
22. E. J. Gumbel, Statistics of Extremes, Columbia University Press,
New York, 1953^
23. William D. Rowe, An Anatomy of Risk, John Wiley and Sons, New
York, 1977.
24. T. C. Curran and N. H. Frank, "Assessing the Validity of the
Lognonnal Model When Predicting Maximum Air Pollution Concen-
trations" Presented at 63th Annual Meeting of the Air Pollution
Control Association, Boston, June 15-20, 1975.
25. Joel Horowitz and Sam Barakatz, "Statistical Analysis of the
Maximum Concentration of an Air Pollutant: Effects of Autocorrelation
and Nonstationarity", submitted for publication to Atmospheric
Environment.
26. "Summary Statement From the EPA Advisory Panel on Health Effects
of Photochemical Oxidants," Prepared for EPA under the supervision
of the Institute for Environmental Studies of the University of
North Carolina at Chapel Hill, January, 1973.
-------
APPENDIX A
THE NEED FOR A RISK ASSESSMENT
How is it to be determined which alternative standards would provide
an adequate margin of safety? Assume for the present that short term'
(one hour) exposure to pollutant X results in only one adverse human
health effect, that the suitably defined threshold concentration for this
effect is L, and that the maximum hourly concentration of X in a given period
of time can be set at any concentration desired by adoption of a suitable
control program. One approach would be to consider various margins of
concentration, as measured in the direction of lower levels from some
nonarbitrary point of reference, and try to determine the smallest margin
that provides an adequate margin of safety. The point of reference could
be either l_d, the lowest concentration for which the effect is judged to
be demonstrated to a given (high) degree, or L ,- the concentration which
is judged to be the most probable concentration of the threshold.
Even in the simplified context this approach will not suffice to put
EPA in position to make the most meaningful judgment as to which alternative
standards provide an adequate margin of safety. For consider Figure 1 where
m-j and ITU, are any two candidates for the "margin of safety,"~Lj',""is the point
of reference (the argument is similar for L ), and LI and L~ are the two
potential levels of the standard corresponding to m, and ITU. Obviously,
the greater the margin the more protection provided by the standard, so
nu. provides more protection than mr "But without further information nothing
can be said in absolute terms about the safety or degree of protection
provided by either mj or ITU.
A-T
-------
L. Concentration
Figure A-1. , "Margins of Safety"
The approach just outlined falls short because it does not
provide for an estimation of the risk associated with each possible
standard. The risk associated with each possible standard is the risk to
the most susceptible group in the general population of suffering adverse
health effects when air quality just achieves that standard. In order to
make a meaningful judgment on whether a possible standard provides an
adequate margin of safety,, a conception is needed of the risk associated
with the possible standard. If the risk associated with the possible
standard is deemed to be acceptable in view of the circumstances, then
that standard is judged in a meaningful way to allow an adequate margin
of safety.
To see clearly why the approach of comparing margins outlined above has
inherent problems, consider the two hypothetical probability density distribu-
tions shown in Figure 2. If Ld is the lowest concentration at which human
health effects are judged to be scientifically demonstrated to a certain
'(high) degree, the probability density distribution representing uncertainty
about the health effect threshold concentration could be flat, as in (a), or
relatively spiked,, as in (b). That is, the criteria document could either
have a sparse scientific data base which leaves a wide range of uncertainty
A-?
-------
m, >
Figure A-2. J.arger Margin with Higher Risk
A-3
-------
about the threshold concentration, or it could have a more complete
data base which strongly suggests the threshold concentration lies
within a relatively narrow range. The margin m, is greater than the
margin nu. But, if (a) is the case m-, gives a standard level, U, that
has more risk associated with it than the standard level nu gives, L^.
if (b) is the case.
The point made by the example does not assume that there would ever
be agreement among experts as to the exact shape of a probability distribution •
only that in the one case they would agree that the range of uncertainty is
relatively large and the probability distribution relatively flat, whereas
in the other case they would agree that the range of uncertainty is relatively
small and the probability distribution relatively spiked. The point reinforced
by the example is that in setting a iNAAQS one should do more than just
identify a nonarbitrary point of reference and then choose the length
of a margin to be called "the margin of safety." The Clean Air Act
requires that the standard provide an adequate margin of safety, not
identify some margin as "the margin of safety". Hence, setting a NAAQS
is fundamentally a matter of choosing the least stringent standard which
has an acceptable level of risk associated with it.. Only then can any
margin be meaningfully identified as an adequate margin of safety.
To be sure, one of the circumstances EPA decision-makers may want to
consider in determining what alternative standard to set are the concentrations
at wh-ich it is scientifically demonstrated to a given degree in the judgment
of health experts that the pollutant in question contributes to various
A-4
-------
•adverse human health effects. As mentioned above, the difference between
the lowest such concentration and the standard level could be called "the
margin of safety". Everything else being equal a decision-maker may prefer
to accept more risk the larger "the margin of safety" in this sense.
But if this were the case it would not affect the fact that the so-called
"margin of safety" is not a measure of the margin of safety provided by
the standard; the measure of the margin of safety provided by the standard
is the risk associated with the standard.
The above arguments can be reformulated so that they show that the
ratio m/T. (or m/T ), the ratio of "the margin of safety" to the demonstrated
effects level, is not a measure of the margin of safety provided by a
standard. However, this point can be made more succinctly by considering
more realistic situations than the oversimplified one we have considered
so far. Dropped first is the unrealistic assumption that the maximum hourly
concentration of the pollutant in a. given period of time can be set at any
concentration desired by adoption of a suitable control program.
The appropriate health experts to make these judgments may be the authors
and reviewers of the criteria document
A-5
-------
Let Std(L, t, e/u) be a formal way of representing a standard,
where 'L' represents 'the concentration level,' 't1 represents 'the averaging
time,' 'e' represents 'the number of expected exceedenaces of the level,' and
'u1 represents a unit of time. Now, the analysis of how to estimate the risk
associated with a standard presented in. the main body of the report shows that
Std(U, 1 hr. 1 hr/yr) provides more protection, that is a greater margin
of safety, than Std(Lp 1 hr, 2 hrs/yr). Yet, the so-called "margin of
safety", whether defined in terms of m or m/Td, is the same for the two
standards. The same would be true for Std(t-,, 1 hr, 1 hr/yr) and Std(L-|,
1 hr, n hrs/yr.), no matter how large n. Yet, if n is large enough the degree
of protection or safety provided by the two standards can differ substantially.
Compare StdfL-j, t, e/u) with Std(l_2, t, e/u), where L^ < L^. If either
m or m/Lu were a good measure of the margin of safety provided by a standard,
the difference in the degree of protection provided by the two standards
would be the same in cases (a) and (b) of Figure 3. But in fact if the
evidence indicates something like- (a) is the case, there is a small difference
in the degree of protection or risk associated with the two standards, whereas
if the evidence indicates something like (b) is the case, there is a large
difference.
•' When the simplifying assumption that the pollutant only contributes to
«*
one health effect is dropped there are additional problems with using ra or m/1_d
as a measure of the margin of safety provided ty a standard. Let X and Y be
two pollutants for which L^ is the same concentration; suppose there is a thin
criteria document for X which indicates X only contributes to one health
effect; suppose there is a thick criteria document for Y which indicates that
(
there are h health effects which, by agreement of the experts, group into j
A-6
-------
Figure A-3. " "Small Change in Risk yrs.. Large Change in
Risk for Same Change in Standard Level
A-7
-------
independent categories; suppose also that the demonstrated effect levels
for each of the j-i categories that don't have the lowest demonstrated
effect level are at concentrations only slightly greater than L,. Then, as
the analysis in the main body of the report shows, the risks associated with
Std(L, t, e/u) are not the same for pollutants X and Y; yet, they have the
same so-called "margin of safety" as measured by m or m/L^.
Suppose I were a pollutant just like Y except that its criteria document
indicates there is evidence from animal studies that Z may contribute to a
j * 1st human health effect category; suppose the experts agree that if the
effect is an effect of Z in man as it is in animals, then its threshold
concentration is most likely less than the threshold concentrations of the
other j categories of health effects; suppose the expert judgments of the
probability that Z contributes to the effect in man are neither close to
0 or 1.0. Then, clearly the risk associates with Std(L, t, e/u) is even
greater for Z than for Y; yet m and m/l, remain the same.
Illustration has been given of several ways an approach to setting NAAQS's,
that does not involve risk assessment, can fail to put EPA in a position to make.
the most meaningful judgment as to which alternative standards provide an adequate
margin of safety. In most real world cases several of the logical difficulties
with alternate approaches illustrated by the above hypothetical situations
will obtain. Therefore, risk assessment should be an integral part of
setting NAAQS's.
A-8
-------
APPENDIX
DERIVATION OF BASIC EQUATIONS FOR ASSESSING HEALTH RISKS
ASSOCIATED WITH ALTERNATIVE AIR QUALITY STANDARDS
1.0 Risk of Exceeding a True Health Effect Threshold or the Lowest of
Several Thresholds
First determine probability P where:
P = Probability that np_ hourly average concentration exceeds
the true health effect threshold or the lowest of several
thresholds in a given period.
Let
P-(C) = Probability that no time averaged concentration exceeds
the concentration C in the given period.
P_(C} = Probability density function expressing the uncertainty
in the location of the true threshold or the lowest of
several thresholds.
Note
PT(C)dC = Probability that a true threshold or the lowest of
several thresholds is in the interval a,b.
It follows that:
P (C)pT(C)dC = Probability that the true threshold or the lowest
of several thresholds is contained in the interval
dC and this value is not exceeded by any hourly
average concentration in the given period.
B-l
-------
By integrating over all values of C the quantity P, defined above, is
determined:
aa
P = /" Pc(C)PT(C)dC (B-l
If R is the probability or risk of exceeding a health effects threshold or
the lowest of several thresholds one or more times in the given period, then:
R - 1 - P
1 -/ Pc(C)pT(C)dC (B-2
Note that if the function P- is defined as the probability that no
time averaged concentration exceeds the concentration C it must also be the
probability that the highest time averaged concentration observed in the
time period does not exceed C (Therefore, is <_C). In other words, Pr is
L*
the cumulative distribution of the highest time averaged concentration
occuring in the given time period.
It is further noted that if it is desired to estimate the risk of
exceeding a health effects threshold or the lowest of several threshold
"m" or more times in a given time period, then PC becomes the probability
that a threshold will not be exceeded more than m-1 times in the given time
period. In this case PC is the probability distribution of the mth highest
time averaoed concentration.
2.0 Determination of Probability Density Function for the Lowest of Several
Health Effect Threshold
Assume n different health effects and that a-probability density function
has been obtained for each. If the function PC(C) is known, Eq. (B-l) can
then be used to evaluate the individual probabilities for each health effect.
B-2
-------
ea
P =/" P (-C)P (C)dC (B-3)
1 /, C Tf
Let:
P = The probability that no hourly average concentration
exceed any of the n true thresholds in the given time period.
To evaluate this probability for all possible configurations it is assumed
that each threshold in turn is the lowest threshold.
Differential elements of the following type can be formed:
CO o» 0»
/ P_ (x)dx • / PT (x)dx — / PT
J TZ J T3 J Tn
PC(C)PT (C)dC * / PT (x)dx • / PT (x)dx — / P, (x)dx
C " C
This term is the probability that: 1„ no hourly average concentration
exceeds the value C; 2. the threshold T-j is in the interval dC; and
3. all the other thresholds are above this value. -
If:
= / PT (x)dx
J Ti
C
And:
Q(C) = TT
1-1
the above expression simplifies to:
8-3
-------
Integrating over the concentration range gives
This integral is the probability that no hourly average concentration exceeds
any of the n thresholds in the time period and T-j is the lowest threshold.
In order to enumerate all cases, it is necessary to form the sum of integrals
of this type in which each threshold in turn is assumed to be the lowest.
Therefore:
n
P - P §- PT
'
or:
f
Pr CQ L, PT /°^1 dr
C ,*~1 T. "iJ Ql-
1=1 T
or:
f?
- L
p dC
'
This equation is, as would be expected, identical to Eq. (8-1) except that
PT is now a composite probability density function such that:
n
PT = Q IT PT>/Qi (B-4)
i =1 i
where:
r"
» / PTt dC (B_5)
A 1
B-4
-------
and
n
TT Q, (B-6)
The quantity pT(C)dC is the probability that the lowest of the n thresholds
is in the interval dC.
An alternate method of deriving the composite probability density
function is to start with the composite cumulative distribution for the
health effects threshold. Let:
DT = Pr lowest
and:
DT » Pr (T. <.C)
'i
It follows from probability theory that:
n
1-D_ = TT (1-D_ ) (B-7)
T 1-1 Ti
The composite probability density function is by definition:
do
PT • -- (E-8)
Differentiating (B-7) yields:
dDT n n dDT.
TT (1-D_ ) ) V T4--HF
T- * J I ~U-r dL
i=n i 1=1 T.
Jl .
dC
B-5
-------
Since by definition:
1-0 - /PT_ dC - Q. (B-K
1 C 1
g = TT (1-0- )
i i
PT. = doT /dc (B-1]
Equation (B-9) is equivalent to Equation (B-4).
Equations (B-7) and (8-8) provide an alternate route to obtaining the
composite density function, p_, which can then be used in Equation (B-2) to
estimate risk.
3.0 Inclusion of Uncertainty That One or More Health Effects Exist
For some health effects there may be uncertainty that the effect
actually occurs in humans. It would be desirable to include this considera-
tion when considering the uncertainty in the location of the threshold on
the concentration axis. Given uncertainty only in the location of the
threshold it has been shown that Eq. (B-2) gives the risk that the true
threshold T. is exceeded one or more times in a given period.
If there is uncertainty as to whether the ith effect occurs in humans,
assign the probability e. that the effect does exist. Then choose a value
of C = u such that u is many times larger than any concentration likely to
be encountered. In other words, u is many times beyond the concentration
range of interest. In this case, if p!. (C) is defined as the probability
density function for the location of thi f th effect if it does exist, a
new overall probability density function can be written:
PT (C) = e.PZ + (i-e.) «(C-u) i - 1 , n (B-12
1 i 1 ' i '
8-6
-------
where 5 (C-u) is the Dirac delta function. It has the property:
= 1 if a < u <_ b
/b
S(C-u) dC (B-13)
a = 0 if a > u or b < u
In other words, if the interval of integration includes C = u the
value of the integral is unity. If it does not include C = u the integral
is zero. The Dirac delta function itself may be considered as zero for
all values of C except u. At u it has an infinite value. It is also
assumed that p° (C) is essentially zero in the vicinity of C = u and above.
Ti
Equation (B-12) is based on the premise that saying an effect does not
exist is mathematically equivalent to saying that the true threshold is at
a very high concentration which is above any concentration likely to be
encountered. Thus, if the probability e can be assigned to the certainty
that it does exist, the fraction e of the total area under the probability
density curve can be assigned to the concentration range in which the effect
is thought to be located if it does occur. The rest of the area, (1-e),
can be assigned to a range above any concentration likely to occur.
Note also, the use of the Dirac delta function is only a matter of
convenience since it leads to a simple form of Eq. (B-12). Any function
can be used in this outer region as long as it has the value zero in the
region of interest and has the area (1-e). The Dirac function is simply
a convenient form for including the desired property in the probability
density function.
Substituting Eq. (B-12) into (B-3) gives:
•< = e* / pr PT dc + 0-e*) /
1 1 J C Ti 1 J
£(C-u)dC
0
Since P~ will be essentially 1 at C = u, the above equation yields:
j
B-7
-------
To find the probability that no hourly average concentration exceeds any
of n thresholds, it is necessary to derive the appropriate form of the
composite threshold probability density function. Its general form is
given by Eq. (B-4).
Substituting Eq. (B-12) into Eq. (3-4) and rearranging gives:
e.
PT = Q Zl 7T PT- * t Q Zi ST"1 1 5(C~U) (B~15
i=l 1 i i=l 1
The first term on the right side of the Eq. (B-15) is evaluated in the
ambient concentration range of the pollutant well below the value C = u.
In this region the functions Q. have a simple interpretation. Substituting
Eq. (B-12) into Eq. (3-5).
0»
q. = e. /""py.dC + (1-e.) J «(C-u)dC (B-16
c 1 c
For all C < u
Q1 = e. y'py.dC H- (1-e.) (3-17
The second term on the right side of the Eq. (B-15) is evaluated in
the vicinity of C = n, far above the ambient concentration range. In this
region the behavior of the Q. functions needs to be more carefully consid-
ered. The problem is the behavior of the second term on the right side
of Eq. (B-16) as C passes through u.
Note that in the vicinity of C = u the first term on the right side
of Eq. (B-16) is zero by definition.
B-a
-------
Thus:
/•"
Q1 » (1-ei) / 5(C-u)dC for C % u (B-18)
C
Expanding the second term of Eq. (B-15) gives:
C Q2Q3-"Qn(l-e1) + Q1Q3---Qn(l-e2) + ••• + P^^Vl (1'en) 1
which on substituting Eq. (B-18) yields:
OB
|n «(C-u) [ J 6(C-u)dC l""1 | n (1-e.)
The portion of this term in large brackets can be shown to have the proper
ties of a Oirac delta function (Eq. (b-13)) as follows:
Let:
*(C-u) = n S(C-u) Cy*5(C-u)dC l""1 (B-19)
and
CO
$(C) = / 5(C-u)dC (B-20)
C
then
dC
Substituting into the bracketed terms:
dC
B-9
-------
or
(B-22;
To show that the derivative has the properties of the Dirac delta function,
integrate between the limits a and b.
/•b f b
/ t(C-u)dC = - /d«,n =
= [ / 6(C-u)dC ]" - [ / o(C-u)dC ]" (B-23]
a b
Using Eq. (3-13) to evaluate the quantities in brackets for different values
of a and b it follows that:
= 1 if a <_ u <__ b
a. =0ifa>uorb a
Therefore, 4>(C-u) is also a Dirac delta function.
We can, therefore, write:
i n\.—* io\/.,\I_,/-,\ , .
PT = (QZq7 PT.} + o(c-u) n (1-ei} ^B-25)
i ^
where:
Q. = e. C p° dC + (1-e.)
vi i / T. r
8-10
-------
or alternatively:
£
Q, = l-e. / pi dC (B-26)
i i j i •
o 1
and
Q = QiQ2*'"Qn ^B"27^
Substituting Eq. (B-25) into Eq. (B-l) gives:
aa
P = /"pCPTdC + n (T-eJ (B"28)
o
and therefore:
.
1 - / PrP°dC + II (l-e.) (B-29)
i \j V ^ *
y
The Eqs. (B-25) through (B-29) are the basic working equations for estimating
risk when a multiple number of health effects thresholds are involved, and
where one or more of the health effects may not occur in humans.
The above results can be derived in an alternate manner starting with
the cumulative distribution functions for the location of the health effects
thresholds. Note that Eq. (B-7) relating the composite cumulative distribu-
tion to the individual distributions continues to hold, although now the
composite and individual density functions are given by Eqs. (B-25) and
(B-12). The definition of the cumulative distribution for the ith effect
can be written:
rc
DT> * e1 Dy + (1-e^ / 5(C-u)dC (B-30)
o
8-11
-------
By differentiating Eq. (3-30) it can be shown to be the appropriate distri-
bution function for the density function Eq. (B-12).
dDT_ dD°
~dC~ = ei ~dT
or
PT = ep! H- (1-e.) s(C-u)
Ti ' i 1
which is Eq. (3-12).
Next, it can be shown that the use of Eq. (B-30) and Eq. (B-7) yields
the correct form of the composite, p_. Substituting Eq. (8-30) in Eq. (3-7)
gives:
!-D_) = n [1-e.Dl - (1-e.) /
i ,- i i.j j
(1-DT) = nn-e,DT - (1-e.) / 6(C-u)dC] (B-3'
o
Note that for c < u:
.o
(1-D_) = n O-e. DT )
1 i i
For C -^ u
and, therefore:
(1
r
-DT) = n [ l-e1 -0-e.) J 5(C-u)dC ]
B-12
-------
For C > u
I-DT = o or DT = i
It is, therefore, possible to write:
1-DT - n 0-e.D° ) - ( n 0-e.) ) f 6(C-u)du (B-32)
i i i -n
where Eq. (B-32) gives the same result as Eq. (B-31) over the whole range
of the concentration, C.
If P° is defined by:
1-Dj - n (1-e^) (B-33)
then.Eq. (B-32) can be written as:
/•C
(B-34)
o
which is the composite analog of Eq. (B-30).
The derivative of Eq. (B-34) is taken to obtain p,.. But first, note
from the definition of Qi in Eq. (8-26)
Q1 » 1-e^. (B-35)
and from Eq. (B-27)
Q - n Q1
B-13
-------
Taking the derivative of (B-34) then:
dDT _ _ _dQ_
dC T dC . i (
or
PT • - * «(C-u) n (l-e,)
However:
dC
or
o
eiPT.
•i i
which is the desired composite density function and is identical to Eq. (5-25).
Note that starting with the Oj. and their corresponding e- it is possible
to first form the composite O and then by:
dD
T
= PT (B-36)
obtain the function pZ for insertion into Eq. (B-29) to estimate the risk
of exceeding the lowest of n thresholds.
8-14
-------
APPENDIX C
DERIVATIONS RELATED TO THE PC FUNCTION
1.0 Derivation of Pc Function Given Distribution of Time Averaged
Concentrations and No Interactions or Time Dependence
The following basic relationship from probability theory may be used
to relate the P- function to the distribution function for the time averaged
concentrations G(C) which is defined by Eq. (8) of Section 3.2.
Where Pv is the probability that a certain event, whose probability of
occuring in a single trial p, will occur v times in n independent trial.
Here a trial is the period covered by one averaging time. For example,
for an averaging time of one hour a trial is a single hour. Thus, a
year will have 8750 trials. The event is the occurence of an observed
concentration above some specified value C.
The collection of values PQ>P-]'P2' "* are referred to as the binomial
distribution.
From the above definition of the event it follows that:
P = G(C) (C-2)
Where G(C) is distribution of the averaged concentrations defined by
Eq. (8) of Section 3.2.
If there are to be no exceedances of the concentration C, then m =0
and it follows from Eqs. (C-l) and C-2) that:
P0 - (1-6(0 )n (C-3)
which is Eq. (10) of Section 3.2...
C-l
-------
If the PC function is to be used to calculate the risk that the
threshold will be exceeded m or more times, then it is labeled P^ and
corresponds to the probability that a threshold will be exceeded not more
than m-1 times in the time period.
In this case:
m-1
Pv (C-4)
v=0
where Py is given by Eq. (C-l). If Eqs. (C-l ) and (C-2) are substituted
into Eq. (C-4):
m-1
n! v M_r\"-v ,- c*
(] Gj (C 3]
v!(n-v)<
Eq. (C-3) is the special case of Eq. (C-5) corresponding to m = 1 .
2.0 Distribution of Time Averaged Concentrations When Represented by
'/Jeibull Distribution Just Meeting an Air Quality Standard.
The National Ambient Air Quality Standard is assumed to have the
following form:
C<-Tn ppm hourly average concentration with an expected
number of exceedances per year less than or equal to E.
The Weibull distribution is defined by Eq. (12) of Section 3.2:
S(C) - e
C-2
-------
G(C) can be interpreted as a relative frequency. As such, the quantity
E/nc (where n_ = 8760 hrs in a year) is the expected frequency of occurence
t E
of concentrations above the level C =
Therefore:
.
JL = e" —p (C-7)
n
Solving this expression for the parameter 5 gives:
1
5=CSTD/(ln(nE/E))lr (C-S)
Substituting this value in Eq. (C-6) yields:
G(C) = e-On(oE/E» (C/CSTD)k CC-9)
which is Eq. (13) of Section 3.2.
C-3
------- |