A METHOD FOR ASSESSING THE HEALTH RISKS ASSOCIATED

     WITH ALTERNATIVE AIR QUALITY STANDARDS


                    FOR  OZONE
      Strategies and Air Standards  Division
   Office of Air Quality Planning and  Standards
       U.S. Environmental Protection Agency
       Research Triangle Park,  N.C.  27711
                 DRAFT
                  July  1978

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A METHOD FOR ASSESSING THE HEALTH RISKS ASSOCIATED

     WITH ALTERNATIVE  AIR QUALITY STANDARDS


                    FOR  OZONE
      Strategies  and Air Standards Division
   Office of Air  Quality Planning and Standards
       U.S.  Environmental Protection Agency
       Research Triangle Park, N.C.  27711
                  DRAFT
                  July   1978

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                         Table of Contents


                                                                Page


1.0   Introduction                                              1-1

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  "Risk11 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 PC(C) Distributions                    3-6

4.0   Application of the Risk Assessment to Ozone               4-1
      4.1  Introduction                                         4-1
      4.2  The Judgments of Health Experts                      4-1
      4.3  Determination of Pr Functions for Ozone              4-17
      4.4  Risk Tables and Ri5k Ribbons                         4-27

References

Appendix A:  The Need for a Risk Assessment                     A-l

Appendix B:  Derivation of Basic Equations for Assessing
             Health Risks Associated with Alternative Air
             Quality Standards                                  B-l

Aooendix C:  Derivations Related to the PC 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
criteria 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 NAAQS1s. I
     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)
wnich  are helpful to rational decision-making under uncertain! ty/  '  ' 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.
                                        1-1

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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 give 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.
                                           1-2

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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  'risk' 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
                                     2-1

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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. (6)

     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  concentrations;  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
risi< for health effects on which there is no evidence.  Hence, the approach taken
r;i.-e is to assess the risk for effects on which there is evidence.  It  is suggested
tiiat 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
.-vi-.icn tnere is no evidence:  there may not be any; if there are any their threshold
concentration may be either nigher or lower than the threshold concentration of
-rivets for which there is evidence; and, the lower the risk of effects for which
ci-.ere is evidence, the lower the risk of effects for ^hich there is no evidence.
                                    2-3

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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 concentratioi
 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 with 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 no  precise 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
                                     2-4

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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 the health 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.
                                     2-5

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2.1.3  Idealized Risk Surfaces
      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 2-1, line ab in the 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
                                       1%
points on the Z-axis.  Concentration Cg   is the expected concentration at

                                   2-6

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                                 Figure 2-1.  ' Risk Surface
 y
I
          erc
Effect
        501

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                                                  10^
which 1 percent of the group would be affected, Ce  * is the expected concen-
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  1X  have been said to have a risk of 1 percent.   "Risk"  in  this sense
     e                            ~~^~
 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 n% 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 thez-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 people would  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 situation 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
                                      2-9

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   (C)
                                          health  effect  threshold  distribution
                                                        Concentration, C
                              max. 0«  concentration  distribution
                         'STD
PC
  (C)
             Figure 2-2,  Probability Distributions  on  the  Two
                          Major Sources of Uncertainty
                                       2-10

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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 CSTD  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 oarticular
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 low enough.   However,  for  reasons which  will  be
discussed  more fully in a later  section  it  is  desirable  to maintain the value  of  E
 in the  vicinity of one  or  lower.
      It should be clear chat  the standard  level  ana  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.
                                       2-11

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      The shape of the maximum ozone concentration  distribution  1s 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),
      i .*.•	  i..«. cm  i.^i,*«^ r>,iM-tcat-innc rioerrihp thp tschnioue and the

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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)
                                    2-13

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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 well-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  (or 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.  3ut, 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 desired 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.
                                   2-15

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     Of course, either the Bayesian or the classical approach to statistical  infen
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 neede
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, by the 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

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effect threshold concentration for the other, then for that expert the two
health effects are threshold dependent; otherwise 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 in 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.
                                       2-17

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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.  But, since 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, character!zable in general terms, is the most sensitive group
 for that  category.  The most sensitive group for a category  is the  most

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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 2.1.2, 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  seme  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
t.o/dcological 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 rf 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 B); 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 2O  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
                        Concentrations, 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 B, 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

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relationship between the pollutant and it is uncertain,  figure 2-4 illustrates this
point.   Figure 2-4(b) shows three examples of composite functions derived fnom
two individual health effect threshold probability density functions (Figure 2-4(a>jT
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  be 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.
                                  2-23

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10 1


 8


 6


 4


 2


 0
                                                0.20 ppm

                                                0.04 ppm

                                                1
                                                                = 0.3 pom

                                                                = 0.08 ppm

                                                                    1
                            0.2            0.3
                                   Concentration, ppm
                                                               0.4
                                      0.5
        (a) Individual Health Effect Threshold Probability Density Functions
                                           e1 = 1, e2 = 1
                                                            = 0.15, e  -  1
             0.1
                                 0.2
        0.3

Concentration, ppm
0.4
   (b) Comoosite Probability Density Functions for; Various  Probabilities  of
        Existence are Assigned to Function with T = 0.2  opm
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 interest;
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 probaoility
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,

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and possibly some contradict each other; each of the three types of studies,
clinical, epidemiological, 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

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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 PC
function.  As is discussed more fully  in Section  3.2,  the P- function for
"m" or more exceedances is the  distribution of  the  mth highest concentration
observad  in the time oeriod.  The distribution  in this case  is referred to
as  r(~].  i'sing this notation,  the  distribution Pr  of the highest concentra-
    C                                             L
tion, wouid be designated as Pf
                              C
                                      2-31

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Figure 2-5.   .Probability Density Functions and  Cumulative  Distribution

                (Pr) of Annual  Highest Hourly Average Concentration
                  w
         0.06
0.08
                                                       Most probable
                                                    Wax. concentration
                                                          0.100

                                                     _    Expected
                                                     max. concentration
                                                          0.109
           0.10       0.12        0.14

                 Concentration, ppm
                             (a) Density Function
           0.16
       1.0 -
     £0.8
     .o
     2
     £0.6
     a.
     £0.4
      
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     To use the P- function in estimating risk it is necessary to be able
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 PC 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 Pp 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  CSTQ  is  the concentration level of the  standard, then  the PC  function
is placed so that its  mean  concentration value  corresponds to CSTD
 [Refer to Figure 2-5).  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  PC  function would be set at
                                    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 PC function.
In other forms the connection may be less direct.  For example, the form
of the ozone standard which has been proposed is:

        (B)  C-J- ppro 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 CSTD is that concentration
which corresponds to the 1 - 1/8760 = 0.999886  (where 3760  is  the number
of hours in one year) fractile on the distribution function for the hourly
averages.
     The Pr distribution and the hourly average concentration distribution
          w
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
                                 2-34

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•e 2-6.  Hypothetical Distribution of Hourly Average Concentrations and the

       Corresponding Distribution of the Annual Maximum Hourly Average Concentration
                                  I     I
                                                        Standard
                                                         level
                   0.2   0.4   0.6   0.8    1.0   1.2   1,4   1.6
                                 Concentration,
                                                   .-™
              (a) Distribution of Hourly Average Concentration
           1.0
           0.8
           0.6
           0.4
            0.2
1
                                  I
                0    0.2   0.4    0.6   0.8     1.0    1.2    1.4    1.6

                           Concentration,  (C/C^JQ)


              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
                                                   c
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 P- 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 PC 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 PC 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 are 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 PC functions
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
                                 2-36

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                      Figure  2-7.     Change  in  P«  Function  With  Change  1n  Standard  Concentration  Level
                                     'STD  -0.06
          *0.08
«0.12        =0.14
ro
i
OJ
               l.OL
                       0.05
0.10                        0.15


       Concentration G* ppn
              0.20

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a health effects threshold Is not the sane 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 PC 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
                                                       V*
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 2-8 shows the possible variations in the spread of a
hypothetical PC function.  Each of the P- functions in the figure passes
through P  a 0.368 at the level of the standard.  Section 3.2 treats PC
functions in more detail and describes methods for deriving them from the
distribution of hourly average concentrations.
      One further difficulty in estimating the appropriate PC functions for
a pollutant is that available air quality data may be at a significantly
                                    2-38

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                          C . Fimrt ton  I'or One-Year Period for Difl'ervnt Valuer  u.  UdbuVl  htidpc Factor, k
i
CO
ID
                     0.08
0.09        0.10
0.11        0.12         0.13

Concentration, ppm
0.14        0.15

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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
                                        c
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

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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 1f 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  PC distribution.
     Subjective probability distributions on threshold concentrations for
individual categories tend to be subjectively derived and hence different
for each health expert;  likewise  probabllity-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,  1t 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 1nad~v1sabil1ty "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 am
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 viewpoint:.  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" prior!, this  crlter 1 a cou 1 d elther~erifeF fn"seTecfThg~ th~e~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  1s 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) systematlcness, (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

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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
.oal 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
che 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-
cf-existence judgment:  suppose that three subjective probaoility 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
c'-fact distributions.  Letting 'd^1 represent the jth distribution for
the ith category,  'e^  ' represent the jth existence judgment  for the 1th
•rat^gory,  '!' represent the compositing function, and D. represent the  ith
composite  distribution, the situation can be  represented formally as follows:
                       I  (dj  , ej  , d^) = D1
                       I  (d*,  e^  , d2  ) - D2
                                 3  .3 »
                              >  e, ,  Oo ;  -  t

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     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 approxi-
mation  of the true PC distribution (Pc J, and that has a scale parameter which
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  * for that standard level  is unknown,  but 5 equiprobable
alternate values of k. (namely,  *&• wnere h 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 kn's.  Letting 'V represent the convolution function and 'ry represent
the jth risk number:
          V [I  (dj.  e],  dj),  k,] = V (Dr kj
          V [I  (df.  e],  dj).  kj]- V (Dr kj)
          V [I (df. ef, d^), kg] = V (D27, k5) - r135
Or, alternatively:
          R (d], e], dj, kj) = R (w.,)    = r,
          R (df, e], dj, k.j) = R (w2)    = rg
          R (df, ef, d|. ks) = R (w135)  - r135
where w1 = (d], e]. dg, 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:
                       1 rl i r2 i   • • *  = "135 ± ]

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Then, the 135 ordered pairs
      T35
                    
                                         I"
                                          I3F
                      r        ,5
                            •2  '
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

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     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
     than 0.5 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 PC
     distribution and  an  upper-bound P  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

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          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,
tne 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 ^hat 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 he 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.
   ;2)  it has a uniqueness that has importance for public regulatory
       decision-making.
Each of these will now oe discussed in turn:
   '!)   It can be checked that  the Kolmogorov" Axioms are  satisfied for  the
probability  space  ( n ,3, P),  where the  sample space  Z =  (w], w?,  .  .  ., w]35),
::ir  r-riald *3 - 2" (all of  the subsets of ft), and  the  probability
          -.,  of a suoset of P. is the number of elements  in the subset divided
                                   2-47

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     (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 PC 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 PC 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, pA to be the limiting frequency distri-
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;
                                        D
consider Std (Ln, 1 hr, 1 hr/yr).  Let 0  be the most rational distribution
                                                                R
representing the evidence on the health effect threshold; let P  (L ) be the
                                                               c   n
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^Lj |  = Gi(Ln).
Then the r- (L ) which gives the minimum G^(L ) is the Truest Risk Estimate (TRE)
for Std (Ln, 1 hr, 1 hr/yr).
     We have from (1) that the KoTmdgorov  Axioms are satisfied for a
probability space which has  & = (w-j, w^, . . . w,35) as its sample space.  If each
W-. =(d-, e.,d-,  k.)  is considered to be equally likely to give the TRE, then a
  I     I    '  «J
curve of the type depicted in Figure 2-9 is a cumulative probability'distribution
on the TRE for the given standard level.  Assuming arbitrariness has been
minimized  in the selection of the experts and each fy is equally likely to give
the best approximation of PC  (Ln), from the public point of view each wi is
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

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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"tlkaT^iut 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 rathei^.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 tfian or equal"16 the given ~rtslc~vaTu«*v  There-is  **so 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 w^  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,  RT(Ln)?   That depends  on whether RT(Ln)  exists.
*An endpoint adjustment  is  made in  the  way the curve is plotted; the points
 are *ri» LJ>  ™ther than (r.,  i	},  under the strong interpretation.   This
          135                 '  "T3J
 adjustment follows  the  recommendation  of Gumbel (22).

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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

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     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.
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 those 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 = 310 risk estimates are plotted.
     (3)  The. public probability distribution fora given standard in a given
location would include the judgments of every member  of "the set of best-
informed experts."  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" rather than just "Public Probability", is that on the air
quality side the public probability distribution for a given standard and location
                                         2-52

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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 case
(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).

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accepting risk one jj. 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 point 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.J  General Description of the Method
     The previous section dlscusssed the underlying principles involved in
assessing the risk to the most sensitive members of the population from a
j-fvsn air pollutant in an area just meeting a specified level of air quality.
This section describes the wethod  in wore 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
irt the calculations are shown.  Their derivations are given in Appendices
3 and C.  An understanding of the  underlying mathematics is not necessary
tc an understanding of the method  or its application.

     3.1  Mathematical Description of the Method
     In tha preceeding section it was pointed out that the risk of exceeding
i crua health effects threshold in a given period of time is determined by
i:.i uncertainty in the location of the health effects threshold (or in the
                                       3-1

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case of multiple thresholds,the uncertainty 1n the location of the lowest
threshold) and the uncertainty in the maxima* oxidant concentration over
the given tine period.  It can be shown, by application of the theory of
probability* that the follairing equation gives the relationship between
the risks and the above uncertainties:
          R  -I-/" Pc(C)pT(C)dC                                            (1)
where:
          R  » Probability (Risk) that a true health effect threshold,
               or the lowest of a multiple number of thresholds, 1s
               exceeded one or more tines 1n a given tine period
               (e.g. one year, five years, etc.)*

      PC(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 tine
               period.

      PT(C)  « The probability density function for the health effect
               threshold or 1n the case of multiple health effects
               the function for the lowest effect (the composite den-
               sity function).

      The derivation of Equation (1) 1s given 1n Appendix B.

     Equation (1) also holds for a more general case in which R 1s defined
as the risk of m or wore exceedances of a threshold in a given time period,
where m may have the Integral values 1, 2, 3, etc.  In this case PC is
redefined as the probability that the mth highest tine averaged pollutant
concentration does not exceed the concentration C in the given time period.

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In other words, P  is the cumulative distribution of the mth highest time
                 c
averaged pollutant concentration in the tine period of interest.
     It was pointed out in the last section that specifying the national
ambient air quality standard for a pollutant Units the range of Pr(C)
                                                                  L
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, 1t 1s only convenient to use Eq. (1) directly to calculate
risk when a single health effect with an existence probability of one 1s
Involved.  When the risk that lowest of n health effects will be exceeded
Is to be calculated and when there 1s uncertainty as to whether one or
more of the effects actually occur 1n the sensitive population at any
attainable pollutant concentration, the following expanded version of
Eq. (1) 1s used.  (See Appendix B):
        R » 1 - /*Pc(C)j4 dC - (l-tjKl-tg)-•••(!-€„)                      (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.
       a., *  The probability that the 1th health effect actually occurs
             in the most sensitive population in the possible range of
             concentrations of the pollutant.
                                   3-3

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The function p. 1s calculated from:

where
       P1  »  The probability density function for the threshold of
              the 1th health effect assuming Its e^ * 1.  (That 1s,

                 /"p* dC - 1).
                 o   l

     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 Q^ and Q in Eq. (3) are calculated from:

       Q1  -  1 - ei  /  p* dC                                             (4)
                                                                           (5)
                                        3-4

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The Qj are seen to be functions of the cumulative distribution of the
                               •
probability density functions p^.
     Thus, given the Pr(C) functions corresponding to different levels
                                                    •
of the standard, the probability density functions P| for n independent
health effects and their corresponding e^, Eqs. (2) through (5) 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
f:r 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 their seriousness.
     The calculations involved in Eqs, (2) through (5) can be most con-
                                                      «
/eniently carried out with a computer.  The function Py can also be
calculated by differentiating its cumulative distribution function.  The
       ive distribution is a function of the existence probabilities
                                                                      ,
;.id the cumulative distribution functions of the pt.  (See Appendix 3,)
L-2SS computational labor Is invol/ed with this method if thara is no
                                           0
;ncarsst in knowing the density functions p..

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3.2  Obtaining the PC(() Distributions
     As Indicated in earlier sections the PC(C) function 1s the emulative
distribution function for the highest time-averaged pollutant concentration
for a specified period of time.  If the risk 1s calculated for • or *ore
exceedances, It 1s the cumulative distribution for the nth highest concen-
tration.  To simplify the following discussion It will be assumed that the
concentration averaging time Is one hour.
     The PC(C) function for a pollutant 1n the air over a given region ts
a neasure 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, 1f 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-fi

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type of standard it Is 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 PC functions.  The approach
to taking 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 tfcen It can
easily be shown (See Appendix C) that:

        Pc  -  (1 - S(C)}n                                                 (6)

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
G(C) is defined by:

     S(C)   -  Pr [Cobs > C]                                               (7)
                                    3-7

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That,1s,G(C) 1s the probability that an observed hourly average concentra-
tion 1s greater than C.  It 1s an alternative way of expressing the emula-
tive distribution of hourly average concentrations depicted 1n Fig. 2-6(a) in
Section 2.3.  If the distribution function 1n Fig. 2-6(a} "is"labeled"F(C)
then:

       F(C)  -  Pr[Cobs  iC]                                             (8)

fro* which 1t follows that:

       6(C)  «  1 - F(C)                                                   (9)

If the broader definition of  PC 1s used, the expression  Is aore coaplex.
                v-0
 where pi1'  1s  the distribution of  the ath highest  hourly average concentra-
        c
 tlon for n  hours.  (See Appendix C for  derivation.)
      Thus,  1f  the distribution function 8(C)  Is  known,  the  desired PC
 function can be obtained fro* application of  Eq.  (6)  or (10).  Studies
 have found  that the distributions  of short-tern  time  averaged concentrations
 of air pollutants can usually be represented  by  lognormal,  Helbull, or g
 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 sinca its G(C) function can be stated explicitly.
       fi(C)  -  e vwo'                                                     (11)

7'-^ 3-ira.i>**er s is referred to as the scale factor.   It  is the concen-
r.-aCion corresponding to 6(C) - C.368.   It establishes the approximate
petition of the mid-concentration values of the distribution.  The  para-
.neter k Is called the shape factor.   It tends  to  be a aeasure 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
ir°s ~ht» corresponding P,. functions can  then be obtained by  use  of  Eqs.
{5} and (11) (or Eqs. (10) and (11)) at the given  level of air quality.
     FJT tr.e risk assessment it is necessary to conn^ict  alternative levels
:' tne ambient air quality standard with the correspcncnnn P. function.
7-..-S is aasily done through the Vfeibull  distribution, Eq. (11).   The  pro-
X.-^d form for the ozone standard is:

        C    Ppm hourly average concentration  with an axpected
         iTO
        ••:-nfaer of exceedancas per year less than  or equal to £.

    s - : *n  >, Appendix C that for any region  to  ^fhich the Welbull  function

-------
where
           •  level  of ambient air standard.
        E  *  expected number of exceedances in n_ hours.
The tem n£ Is customarily the number of hours in one year or 8760 hours.
The expected exceedance rate would normally be one for an afr standard.
In this case Eq.  (12) becomes:

     S(C)  -  e'  9'078 (C/CSTD)k                                           C13)

It should be pointed out that from the point of view of the risk assessment
nethod developed  in this report, the designation of the expected number of
exceedances 1s a  relatively arbitrary natter and could be set at any value
that gives a convenient level for CSTQ so long as the risk 1s the same
(the value of m in P^   of Eq. (11) has a. more direct Impact on health since
It directly bears on the number of exceedances of a true threshold}..  For
example, if It 1s decided that it Is undesirable to have any exceedances of
a true threshold  over a given time period then the PC used fn the calcula-
tion 1s PC    (see Eq. (10)) and once an acceptable level of risk 1s chosen
any combination of E and C^Q values which yields this risk value In a
given area give the same level of protection.
     The combination C^ and E determine the general location of P~ while
Ic determines Its spread.  Figure 2-7  (Sec. 2.3) shows the PQ functions for a
one-year period for a series of alternate levels of C«*Q with E « 1  and
for the Wei bull shape factor k • 1. Figure 2-8  (Sec. 2.3) shows the effect
                                      3-10

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of changing the shape factor at CSTD » 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 CSTD displaces the PC function over
a wide range while having a relatively snail 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 PC
function 1n 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 G^ than to changes
in k over the usual ranges of these parameters.
     From the proceeding discussion 1t 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
pi") functions.  The We1bull can be used with little loss in accuracy
even where other distributions such as the lognormal 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
« and k in the Ueibull 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 Meibull distribution
to hourly average concentration data obtained from air monitoring-sites.
                                   3-11

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Figure 3-1. P~ Function  for 5-Year Period for Different Values
               of Wei boll  Shape Factor, k
i.o -
3.6
9,4
t,2
  0.09
0.10
0.11        0.12         QJ3
      Concentration, ppnt
0.14
0.15
                                  3-12

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-------
also likely to contain the nth highest  hourly  average concentration, P-
could be obtained using Eq. (10).  The  tern  nP 1n  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.
                                  3-14

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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
         C
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
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 4-1.  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  i.s mainly from animal toxicological studies, so the three toxicologists
contributed judgments for that category.

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                            TABLE 4-1
           HEALTH EXPERTS PARTICIPATING  IN THE ANALYSIS
Or. David Bates
Dr. Robert Carroll
Dr. Robert Chapman
Dr. Timothy Crocker
Dr. Richard Erlich
Dr. Bernard Goldstein
Dr. Jack Hackney
    fyt&/
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     The Advisory Panel on Health Effects of Photochemical Qxidants,
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
pro-.=ct against.  Therefore, the health effect categories considered
represent short-tern effects only.  Most of the evidence for the re-
gaining short-terra affects 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 ara 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 *re:
reduction in pulrranary 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
r%at he feels emphysema and chronic bronchitis belong in  the same
car^cry with caucn, chest discomfort, ate.  One expert judges that
zr.err- 3ra two additional threshold-independent categories; he would
.;":••;? the effects similar to the way the first expert who judges there
 •.. ':.-. *our categories does, except he groups reduction  in pulmonary
 r< •,-,-." :n .:-.d £.3-5,--/:;:-on of jsthna, emphysema, and chronic bronchitis
 '•  ::.  v- same ca~8crry.  Three experts feel that all of the remaining
 '.;ii:h  in'fec'c ara threshold interdeoendent.  In ::unmation, 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 membrances 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

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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 levels to assume for other NAAQS pollutants were
not specified.  Second, tht way 1n which adaptivity should be incor-
porated into their judgments was not specified precisely.
     Although care was taken to emphasize that their best judgment, not
the easiest-to-rationalize 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 B 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.
                              4-5

-------
     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-Q.QS 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 epidemiclogical
 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 aboutjo.2  PPi> ™ 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

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                                                                                                 D/YI t
0-1
0
o.n
                                                                                                                       u o
                  .lib             0.1             0.15           0.2             0.2!i
                       figure 4-1.   Reduction  in f'u Imonary Function (Experts  A.B.&C)
0.35
 1.0
:,ppm
   VAillAIll f!

   COMMLN i _

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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

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   i.jTf:MVlEW£R-.
                                               bUBJECT.
1.0 rr
   VAHIABLE.

   COMMtNT.
                0.1             0.2            0.3            0-4

                 Figure 4-2.  Cough, Chest Discomfort, and Irritation of Mucous Membranes  of
               	Nose. Throat, and Trachea  (Experts A.D.&C)	
C.ppm

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     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 epidemiological 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 bxidants, 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 B defined the health effect to be an increased incidence
of bacterial infections in humans oir 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 bom, 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 young children stop exercising before the dose necessary to
cause the effect ever reached the lower lung.  In support of this
                             4-11

-------
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.t 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 B 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  (0.1 ppm) which has  been found  to cause  the effect in rodents,
                                  4-12

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I.

— J
U)
            INTERVIEWER
SUBJECT.
 SR|_PROBABILITY ENCODING FORM V—JANUARY  i. .3


	DATE	
                         0.1           0.2             03            0.4           0.5

                            Figure 4-3.  Reduced Resistance to Bacterial Infection   (Experts A.B.&C)
                                                         C,ppm
            VAHIADLE	


            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 themselves.
(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 B 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

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 :-:j i-iild aithrra~ic  attacks  which are very easily triggered  in  sensitive
 -•rc.-rnatics are clearly distinguishable from the severe  attacks  which
-re serious.  He rr.ade  his  judgments on the latter.
       Expert A judged the most sensitive group to be individuals  who
uva asthma or emphysema.  Experts  B and C judged asthmatics to be the
-csc sensitive group.
       Li figura 4-4 it can  be seen that all three experts think it very
.i:1kaly this effect occurs  below O.C5 ppm; axpart A mentioned  the
.isgative result at that concentration of Rokaw and i-1assey  (1962).  Expert A
-ancioned studies oy Molley, et. al. (1959), Reirsners and Balcham (1965),
arc Schcettlin and Landau   (1S61) in coming to his probability  judgments; he
alsc mentioned thai  there  are problems in interpreting  them, especially
:;--2 study by Schcettlin and  landau.  Expert C also rrentioned tnat  the
Jon,:^""! i,-; -~nd Lanc^i.  stuay  was a basis for h:s juag-ants, -jassita rns
rrsarvations abo-j"  :ne study, because in his opinion, there  15  vary little
•••.-•idence otr,-;:- tnar  that study to base juacnients on.
       The median for  axperts A and 3 is about 0.14 ppm; this may  reflect
:ra fact that Rermers  and  aalcham found a benef^'cia'i effect  of  air filtri-
:  - i«- jti-dies of  "our exercising oaiienti *~. in -;;^ne concentration of
   ..  £.,pert C's cedian is  ^uch hign-sr it 0.25 opm.  Experts A ar-j C
-3.-i;-ced that the^s  is a -jreat deal of uncertainty aocut where  the
     ;:••:•": io •:-• --:'.±  ifficc. snd thair ^p,^:;-.:jt i iatributicns reflect
 •  , .-.=:•;.  r-,-':- z<-;'-:iZ~~:ir.y juagment-: ~^.:~ :;-.s tnrashold is  raac chan
                                                     Expert 3, wncsa ci-.
                                                   is a'rosi  sure tne

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      INTERVIEWER
SUBJECT
                                                                                                DATE
I
o,
                                                                                                               ——  0.6
   0.0
                  . 0.05
                                                                                                                    0.8
                                                                                                                    0.9
                                                              C,ppm
                          Figure 4-4.  Aggravation of Asthma, Emphysema, and Chronic Bronchitis (Experts A,B,&C)
       VARIABLE.

       COMMENT.

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4.3  Determination of PC Functions for Ozone

     As suggested In Sections 2 and 3 estimating the appropriate PC
function involves:  1. determining the mathematical function or functions
which best describe the distribution of tike-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 tine 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 perforated under contract for EPA involving 14 sites scattered
around the United States and involving 22 site-years of data showed that
the Weibull Distribution (Eq. Q.1}, Section 3.2) provides an excellent fit
to hourly ozone concentrations^]/).  In only two cases did a lognormal
distribution give a superior fit.
     Ozone hourly average concentrations exhibit strong correlation and
strong dependence on time of day and ti«e 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 PC function; however, the tine dependence does.
     The method of dealing with the time dependence discussed in Section 3
was to find a tine 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
                                     4-17

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complete independence of the tine-averaged concentrations.  For ozone this
period fs daring the midday hours of the Mm Months of the year.
     In a continuation of the above-mentioned study"(177 it was found that
Heibull distributions could be fit to data obtained between 11 AM and 6 PM
both from May through September and July through August.  It MS further shown
that the maximum ozone concentration had a significantly greater chance of occurring
during the longer period.  Therefore, this tine period, which contains 1071
hours, MS used in the derivation of the PC function.
     The fora of the National Ambient A1r Quality  Standard proposed for ozone
1s-3n€STD PP» hourly average conetntratientirith  an  expected number of exceedances
per year less than or equal to one.  For a region  whose air quality just
meets this standard, the Ueibull distribution of hourly averages for the
hours 11 AM to 6 PM, May through September, would  be according to Eq, 02)
In Section 3.3:
          6CC)»  e-(ln 1071)(C/CSTD)k
(12')
And the Pr function for a period of nw years would be from Eq.  (6)  In
         u                           y
Section 3.3.

               .  0.,-
-------
ch3 U.S.  Table 4-2 shows measured k values  for the 1071-hour time period
*s .veil as other tisie periods.  Tha range for the values based on 1071
 vurs  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
,-ssearchers involved in the development of the Vteibull distributions and
-.iii* ?r functions.  The information base was  the data in Table 4-2, plus the
^a^i developed during the Heibull distribution studies.  The data shown  in
."ii'ie  4-2 are, by and large,  for  geographic  regions above the range of
altamattve ozone standards considered.  The Hetbull studies suggest that
:ha k  factors at the standard levels would be somewhat higher than those
saotm  in Table 4-2.  This factor  was  taken into account in the encoding.
Tha aeaian values for the location of the lower- and upper-bound shape
-icr^r* 2are 1.36 and 2.34 respectively.  The distributions obtained in
;„« ^needing sessions are shown in Table 4-3.  Since the range of k values
•«\rias scsewnat with the standard level, the range, strictly speaking,
--.O'jl'3 be »sti3satad for aach  alternative level of tte standard.  In the
-..j.= -f iicre the diffar«nca  is net  likely to be large enough to seriously
_f~\j<:t t;*!i risk estisiatas.
     •!'..=-• -PS aDove range of valuss  for the Ueibull shape -actors, Eq  -.5')
.,  ;-. ,::J 'co calculate the  P., functions.  Figure 4-5 shov.s the PC functions  -jsec
    .-;.:= = J-:rg sections tc  calculate  risk estimatas for a standard level of
     -. -  ,.-.:  r.r>  =xpact5i  =xc32danc=  rate cf  cnca oar yaar.  The time period
  .  ..   ••-.;-•-'.   ~- = -  -,'..  : *2  ^^r.c'i^ni ir. r\~'^* 4-5 csn oe ussd to calculata
  •--'--:?'. -::.-  of  ex.se-'.r:c  an ozone hea'.tn affact tnresr.old one or more tirnes
       .. . 3-.r.-   ...   -  ...  .•--.->.-„'•: -=vel   is at  O.'O per..  The function wera calciil ited

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Table 4-2  Change 1n Weibull Distribution Shape Factors With  Changing Tine
          Segment In Which Hourly Average Concentrations are Collected8
                                                Shape Factor, k
. 	 Site
lBsas City, Kansas
*s Molnes, Iowa
L9«1svllie, Kentucky
**PMs, Tennessee
**wroneck, New York
kclne, Wisconsin
Year
1975
1975
1974
1974
1975
1974
Full Yearb
1.24
U9E
0.80
1.32
0.80
1.49
Hay-Sept.c
11 AM-6PH
—
2.04
1.66
2.28
1.31
—
July-Aug.
11 AM-6PN
4.34*
2.40
2.02
2.34
1.57
* 1.99*
 a  Reference 17.
 b  8760 hours.
 c  1071 hours
 d   434 hours.
 e  Poor Welbull fit.
                                 4-20

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Table 4-3 Subjective Probability Distributions for Upper- and Lower-Bound Welbull
           Shape Factors (k) for Distributions of Hourly Average Ozone Concentrations
                    Probability
                     that k 1s            Lower-         Upper-
                  below specified         Bound          Bound
                       value                k              k
                       0.10               1.15           2.41
                       0.30               1.30           2.50
                       0.50               1.36           2.54
                       0.60               1.41           2.66
                       0.90               1.47           2.98
                                          4-21

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     Figure4-5   PC Function for Estimating Risks of Exceeding an Ozone
                 Health Effects Threshold One or More tiroes in Five Years
1.0  -
0.8  -
0.6
0.4
0.2
0.0
                                     Standard Level  » 0.10 ppn
                                  Expected Exceedances
                                    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

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t,;o 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


• : r


     •Because  the  standard level enters  into  Eq  (61) as a divisor of  the  concen-


 .r.,.'or.. -igure 4-5  can also be used to  indicate the PC function for other


 standara levels.   This is done by relabeling the concentration axis.  For


 example, if the function is desired for  a new level, CSTD, each number on the


 MiiC^tratlcn  axis is multiplied by CSTD/OJO.


      £:. urrvates v/ere  also made of the risk of exceeding an ozone health effect
                                                                    / r \

 :•;•.-•;•,:', I  fi.a or more tirr.^s in five years.   The corrasponding ?_     functions


 r'ir  I'ne median upper-  and  lower-bound  shape  factors  are  shown  in  Figure 4-6.   It


 -is  jean that these functions are much  closer together than those  in  Fig. 4-5.   As


 •  -'is-i-,  "-fie range of risk estimates  for five or no re excaedances  of a health


 -••~-:ct:  -.nrrshold in  r- f^ve-year  period vvill oe much  -.^aith  effect threshold


    ;,-•;, id  ':e  note^  tiiat  \  ri-'<  of  five or more -3xc:-iJ>incis  ^n five je'u  i is


  . •; ',:_  san-i  -is the  r":S,-.  of one or more exca=a.ir!C=;  -r •>:/,.-  \-i-.r  3^102 tre

               • m '
  .;;a:^iv£ P,     functions  are different.  This point  is  illuitracec  in Fi:.4-7


        -• .•;<; a  scr-ie-i  cf  P  ';71''  functions  vnece m varies fr.v.n  one  vo  five years.
     -.•-••-               L


     ,> i«=e;i :ha: t.-,^  'or!=  or more  ^n one" ris< wili  :.i  greater :har  tns  ''five
     to  is  tempting to thinx of  the  risk  of five or mere exceedar.ces  ,: s


-•<•-*:.-.:;  -i  ::i ?:,/Q ;--i^i 5'j equivalenc  co  :iV2 risii of '-in average  of  one or mere ex


  :   •:-.-:  of a cnrssfioi: oer year.   Tiiiu  i-i -ot me cisa.   An  average ^f c;.e
                                                                                 or
                                         iticn  Je-fcra.j in :-igure 4-7,  this  limiting





                            ;•  ic.K'ni era;. >:.> cf  0.10 -cn~, ii'-j ice s.  Occencl'.ng

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     Figure  4-6   PC Functions for Estimating Risks of Exceeding an Ozone
                Health Effects Threshold Five or Hore Times 1n Five Years
(5)
                                         Standard  Level        • 0.10 ppm
                                         Expected  Exceedances
                                           per Year           « i
                                         Effective Hours per
                                           Year               -1.071
                                               I
                      0.09
 0.10       0.11
Concentration, (pp«)
0.12
0.13
                                      4-24

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         Figure *-7  PC Functions for Est1»at1ng Risks of Exceeding an Ozone

                     Health Effects Threshold "m" or More Tines 1n "M" Years
 1.0
 0.8
0.6
0.4
0.2
                                         Standard Level           « 0.10
                                         Expected Exceedances
                                             per Year            « 1

                                         Effective Hours per Year * 1071

                                         Shape Factor            * 1.36



                                            i      l     i     i      t
   0.08
0.09
0.10
O.V1        0.12        0.1?

  Concentration, (pom)
0.14
0.15
                                                4-25

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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 P  'm'  for lower values of m.
                                4-26

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     .,.4   Risk  Tables  and Risk Ribbons
     This  section presents the quantitative results  of the ozone risk
. :  •5^,.-i(-:-:.   Tab]a 4-4 presents average thrssivolc  risK estimates for several
d.r.prnacive  standards.  Table 4-4(a)  presents estimates of the risk that the
health  effect threshold will be exceeded one or acre times in five years
•f  rne  riven standard  is just met.   Table 4-4(b) presents estimates of the  risk
 •.--,;-.'irs-j ,vuh  tiie  r/en stanaard  that the health effect threshold will be
 •;-':ece'   : A'2 or r-,ore  times  in five years if tie given standard is just wet.
 *-.;.-  a*. ;."..•; Ms' 2stliiiat?s  are ocv;i.":£d by averse: ir.; trc rii'-i estirr.atii
 -.--i.---!  -e;;,~- when  the  sif.sle most likely ^';ue  zf  t,-,e Enapa parameter  for the  P__
     *                       -'                                                        L.
o'stritution is  used.
     figures 4-8 througn  4-17  are a series cf  risk tycoons.  Section 2.4 explains
 :•:,••! :;-.as3 r-iboons  are  derived.   The plots are  r;Jbo.*s "at!:i" th.iii slnply
c..rv.j5  -acaLise,  as  is  explained  in  Srctlcn Z.3,  t.ia '-HX  .^rie: •:•.-.:• ~-'r-.
   •-;-  ;:^.-^.   ~'vi  lo-ver-cound  and ^.;re'"-:).:-jr,r curve f" -;.TC;:  -'   •-."•-"-. :
   -;vJ.i-':  ^ra retained  by  estimating the ^xtrairis  fo:* "re :r;a:'r ps/sfr^rer or
 -ri'3..  -"iistr-ioution, as  is  explained  in section 4.3.
      ~- •--• .  *-c  preseri.;i  t,ie ribocns '.vhicn irs  obc^i,;co for  •:!! -. iv-c  .••-i
     ;. :   M".tr  effect  C2~;;nciries  v.'f.S!"  the heii"r.  ^'""i':" rr.";iriO=^  ;'-;:•
 • >; -n'st  sensitive  group  is  .defined to be  che  health .affa^t  tnr~sho^  ;":"  -ire
 "•:;'3t  :-,'••'t'-'a n^oer oi' the  nost seni't'/s  i  ,:er":;--  o~" ".; :  •"...~ ;.rf'"."  -

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                           Table  4-4(a).  Risk  that Health Effect  Threshold Mill  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
O.S2-0.60
(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)Reduced 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)Aggravation 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
                            Table  4-4(b).  Risk  that Health Effect Threshold Will 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
0.08 ppm
0.10 ppm
0.12 ppm
0.14 ppm
(1 Deduction
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)Reduced resistance to
bacterial Infection (animal
studies)

0.08
0.15
0.21
0.26
0.29
(4)Aggravation of asthma
emphysema, and chronic
bronchitis

0.02
0.06
0.16
0.28-0.29
0.40
Risk of exceeding 1 or
more of the thresholds for
the individual categories

0.21
0.36
0.52
0.67
0.78-0.80
*R1sk values expressed as range  to reflect  estimated variation in risk throughout U.S.

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        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
                       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

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       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
-.imated
ilic
obability
                    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 5%
             of group most  sensitive to the effect would suffer
             the effect if  the whole group were exposed under
             specified conditions.
                                    4-30

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           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
TTstimated
Public
Prooability
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 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

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     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 PC distribution for the
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 effect
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  (=  3 x  5)
points  being  plotted,  except  for  category 3.  Category 3  requires  a  probability-
of-existence  judgment so 45 (=  32  x 5)  points  are  plotted  for  it.
                                  4-32

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            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
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 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

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        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
stimated
ublic
robability
                     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

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     Figure 4-13     RISK OF EXCEEDING THE THRESHOLD* OF
          .-.T  LEAST ONE OF THE FOLLOWING HEALTH EFFECTS:
           c    REDUCTION IN PULMONARY FUNCTION
               C3LGK,  CHEST DISCOMFORT, AND  IRRITATION
               OF THE MCSE, THROAT, AND TRACHEA
           o    REDUCED RESISTANCE TO BACTERIAL INFECTION
           o    AGGRAVATION OF ASTHMA, EMPHYSEMA,  AND
               CHRONIC BRONCHITIS
-
1.0


0.9


0.3


G. 7
             0.6  	
              .5
             0. 4
                                   -  §P3-       ,.;.  -r:.--.L

                                            ^-:^-
          '
        - -


                                                 , o
                                                            .a   0.3  i.o
                                           r.^j -  :f Zxcaeding  a Health
                                           Effect Thr35h3ld One or Mere
                                            i.T.es in a Fi-.e Year ??ricd.
              - -. r:
                            i
                           . . ^       ;_ »-/
                    •  i : the wh   2  jr^ ^p
                                        . - -   '-:
                                             rlor
                                   :-:!

-------
         Figure 4-14.  RISK OF EXCEEDING THE THRESHOLD*
           OF  THE  FOLLOWING HEALTH EFFECT:
             o   REDUCTION IN PULMONARY FUNCTION
imated
lie
•bability
                        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
                                            Tiroes 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

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            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
Pu; lie
Probability
                          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

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          "Fi'qure 4-16.   RISK OF EXCEEDING THE THRESHOLD*
              OF THE FOLLOWING HEALTH EFFECT:
               O   REDUCED RESISTANCE TO BACTERIAL INFECTION
stimated
tiblic
robability
1.0 r
                      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  iMore
                                               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

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           Figure 4-17.  RISK OF EXCEEDING THE THRESHOLD*
              OF THE FOLLOWING HEALTH EFFECT:
               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 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.

                                   4-39

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                              REFERENCES
 1.  Air Quality Criteria for Ozone and Other Photochemical  Oxidants,
     U.S. EPA, Office of Research and Development,  EPA-600/8-78-004,
     April, 1978.

 2.  National Academy of Sciences, Decision Making  for Regulating
a king f(
  jT~^
, U.j. >
     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
     In Subjective Probability; John Wiley  and  Sons,  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.

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15.  Rudolf Carnap, Logical Foundations of Probability, 2nd edition,
     The University of Chicago Press, 1962.	

16.  L. Jonathan Cohen, The Probable and the Pwable, 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. tinstone 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. DeGroot, "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, 19587	

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
     Lognormal Model When Predicting Maximum Air Pollution Concen-
     trations" Presented at 68th Annual Meeting of the Air PoTlution
     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, 1978.

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                             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



naalth effect, that the suitably defined  threshold concentration for this



5ff-:ct i; 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



'•-.na'.'^tr'sry point of reference, and  try  to determine the smallest margin



chat provides an adequate margin of  safety.  The point of reference could



be either Lr, the lowest concentration  for which the effect  is juoged  to



 -= carr.-rstrated tc a 31 van  (high) degree,  or L , the ccncer.i.'aticn which



i.-  :'•:•-^i'J to be the .Tout probable concentration of  che  tr.rsshoid.



     Zv^n  ir,  the simplified context this  approach  will  not  suffice to  put



 "•-.  i- :••-;; it ion to tnaki-  tne most meaningful  judgment  as  to  which alternative



 it:n~ards provide an adesuate  margin of safety.  Fcr  cans1a=r Figure  I where



-,  cic T- are iry two candidates for the  "margin of  ;«ifet./.''  L..,  is  the point
                                                                C3


 jf  -2f2-er;va  (tha argument  is  similar for L  }, and L-j and L,  are the two



 -, :-=r.rV.'  ieveis Of tn-2  standard corresponding to m- and ni7.   Obviously,



 '.-*.  -r-?.'--;-  tr«? "rargir the  more  prv^tecrion provided oy  the  standard, so



  -.  •r.;,:;.;;  icrs >rct^:t1'.'n Lhan ~,.   SUT: witno^t furrier  •'nforr-ation  nothing



 -c,,  ^e ia:-.-:  :r.  ib-c]jt5  terns  about  tha safety or degrae of  protection
                                       A-1

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                                  K-
                                                                »oncen tra.ti on
                  Figure A-l.   | "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-2

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Figure A-2.   Larger Margin with Higher Risk
                        A-3

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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-j  is greater than the
margin iru.  But, if (a) is the case m1 gives a standard  level, LI( that
has more risk associated with it than the  standard  level ^ gives, L2,
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 NAAQS 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 safety1*.  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 which it is scientifically demonstrated to a  given degree in the judgment
 of health experts that the  pollutant in question contributes to various
                                   A-4

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                             *
•:.:•'••=:rse human health effects.  As mentioned ioove, the differenca between
-n-2 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  niargin of safety" in this sense.
B'.r. If this were the case  it would not affect t!ie fact that the so-called
;'T.-arg>-~- of safety" is not  a measure  of the margin of safety provided  by
.-..- ...... JarJ; t.ie me^-fa  of the margin of safety provided by tne standard
is cne risk associated with the standard.
     The above arguments can be reformulated so that they show that the
ratio i7!/T. (or m/T ), the  ratio of "the margin of safety" to tne demonstrated
= ff~ct:s live], is not a  '-neasure of the margin of safety provided by d
.tandai-.i.  However, this point can be  made mors succinctly by considering
~.V2  'ialiitic situations  than tns oversirro] ifiau one we have consvJiraa
io far.  Drcpoed first  is  the unrealistic assumption tiiat me ^axinuiT! .-ourl;,-
ccrcantration of the- pollutant  in  a  given period of time ;a;- oi  sdi ac any
-C:icar::,-aticn desired by adoption  of a suiteDle cc.--.-rol progra.n.

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     Let Std(L, t, e/u) be a formal way of representing a standard,
where 'L' represents 'the concentration level,'  't' represents  'the averaging
time,1 'e1 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(Lj, 1 hr. 1 hr/yr) provides more protection, that is a greater margin
of safety, than Std(L1§ 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(Llf  1 hr, 1 hr/yr) and Std(L1,
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 Std(Llt t, e/u) with Std(L2,  t,  e/u),  where L2 < Lr   If  either
m or  m/Lj 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  m  or m/Ld
 as  a measure  of  the margin of safety provided by a standard.   Let  X  and  Y  be
 two  pollutants for  which Ld 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

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c«0
 Figure A-3.
Small Change In Risk vrs.  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/Ld.
     Suppose Z 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/Ld 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

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                             APPENDIX B-
       DERIVATION  OF  BASIC  EQUATIONS FOR ASSESSING HEALTH RISKS
         ASSOCIATED  WITH ALTERNATIVE AIR QUALITY STANDARDS
 -!:k  of  Exceeding  3  T.-ua Health  Effect Threshold or the Lowest of
 c'-e-a'!  Thresholds
:vrst  .K-ternir.e  probability  ? where:
    P  =  Probability  mat  rvo  hourly average concentration exceeds
        the  true health effect  threshold or the lowest of several
        thresholds in  i given period.
.at
    ?.(C) =  Probability that no time averaged concentration exceeds
            the  concentration C in the given period.
    ?-{C) =  Probability density function expressing tne uncertainty
            in  the location  of  the true threshold or the lowest cf
            several  thresholds.
    .-'°
      ?-.'C}dC  =  Probability tnat a trua threshold or the lowest of
    f   t
    3            several  thresholds is in the interval a,b.

    •~>'• 'cws  that:

      .'.iC}p,(C}dC = Probability that tne true threshold or the lowest
            t
                     of several  thr^s^clcis  is conts^ned :n the in^e^va"
                     uC and tnls /al-je is r=oc exceeded by any hourly
                     i/eraTe co'-.centrdtior  in the ai-^sn period.

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By integrating over all values of C the quantity P, defined above, is
determined:

      P  =  f 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
                 0»
      R  =  1 -f  Pc(C)pT(C)dC                                             (B-2)
                o

     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, PC is
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
V 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 averaged 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

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              «•

       P.  -/"  PC(C)PT> (C)dC                                              (B-3)
             A         '
   Let:
        P  a  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                0»               «•
     Pr(C)PT  (C)dC •   / P,  (x)dx  •   I PT  (x)dx	/ PT  (x)dx
      c    Tl          J   T2           J  r3          J  Tn
                       C                C              C

This term is the probability that:  1. no hourly average concentration
exceeds the value C;  2. the threshold T^ is in the interval dC; and
3. all the other thresholds are above this value.
   If:
                 w

         C>  '  /PT.
   Q<(C)  •  /PT  (x)dx
             c
And:
              n
      Q(C)   =   TT q
                     «

the above expression simplifies to:

             L.  dC
                                      B-3

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jtegrating  over  the  concentration range gives:
Ms Integral  Is  the probability that no hourly average concentration exceeds
(/of the n thresholds  in the time period and T-| is the lowest threshold.
i order to enumerate, all  cases, it is necessary to form the sum of integrals
rfttrfs type in which each threshold in turn is assumed to be the lowest.

lerefore:
      P  -   I- IPS-PT  dC
                        M
         - /PC CQ  Z  PJ
           A         I*]    1
      P  =    p  PT dC
Hs equation is,  as would be expected, identical to Eq. (B-l) except that
i is now a composite probability density function such that:
               n
     Pj  3  Q  H PT /Q1                                                  (B-4)
              i*l   1
tere:

  Q.(C)  =  /"pT  dC                                                      (B-5)
            f+   1
                                     B-4

-------
and

                 n
        Q  »     TT Q,
                i=l

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:



       »T  '  Pr lowest iC)


and:
                                                                            (B-6)
It follows from probability theory that:


     1-D-  =     IT d-D_ )                                                 (B-7)
        T       i=i     Ti


The composite probability density function is by definition:


               dD,
       PT  .  _±                                                          (a-8)
Differentiating (B-7) yields:



                               n         dDT

      dC~  ~    ..__   " T.
                                   3-5

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Since by definition:

     1-D_  =     p_  dC - Q,                                                 (B-10)
      -D_  =   /p_
        TI      c  TI
           *  dDT /dC
               Ti
                                                                            (B-ll)
Equation (B-9)  1s equivalent to Equation (B-4).

     Equations  (B-7)  and (B-8)  provide an alternate route to obtaining the
composite density function, p_, which can then be used 1n 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 1n 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 1t has been shown that Eq. (B-2) gives the risk that the true
threshold T^ 1s exceeded one or more times in a given period.
     If there 1s uncertainty as to whether the 1th effect occurs In humans,
assign the probability e^ that the effect does exist. Then choose a value
of C • u such that u 1s 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 the 1 th effect if it does exist, a
lew overall  probability density function can be written:
     P  (C)  -  e.P   + (1-e   «(C-u)        1-1
                                 B-6

-------
where  6 (C-u) is the Dirac delta function.  It has the property:

             .         • 1  if a < u < b
           r                   ~~   ~
           I  6(C-u) dC                                                      (B-13)
           a           =0ifa>uorb
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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. (B-4) and rearranging gives:
              n   e.               JL 1-e,
                                            ] «(C-u)                       (B-15)
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. (B-5).
     Q1  -  61  /""p^dC  *
                C
For all C < u
     Q1
     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
°f Eq. (B-16) is zero by definition.
                                   B-8

-------
nu :er~ 'f £q.  (S-?-,:  gives:
                           ^ ••%
                                                    . -u

-------
 w that the derivative has the properties of the Dirac delta function,
 fete between the limits a and b.
C-u)dC
              -b
          -   /d*n -
            r*         n      r"
       « C  / «(C-u)dC ]  - [  /  «(C-u)dC ]                           (8-23)
           a                   b
   »  (B-13) to evaluate the quantities  in brackets for different values
   b it follows that:
                         * 1  if a    u  <  b
 /*{C-u)dC                                                           (B-24)
 4                       *0ifa>uorb a

'ore, *(C-u) is also a Dirac delta  function.
"» therefore, write:

         v^ei   o             n
 PT  "  "Y<57  PTI' + *(C'U)   ?   ^
 '1  •  e1
           I
            c
                             8-10

-------
or alternatively:
                .C
     Q1 " I-*!     PTdC                                                    (B'26)
and
Substituting  Eq.  (B-25) Into Eq.  (B-l)  gives:
     P  «
           o
and therefore:
            /*Pcp°dC  +   0   (1-e^                                         (B-28)
             - / v?
                      ,dC  +   H  (l-ej)                                     (B-29)
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 1n 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 1th effect
can be written:
                                rC
                         -C1)   /
     0T   - e1 Dj   +  (1-C)     «(C-u)dC                                  (B-30)
                                     B-ll

-------
  differentiating Eq.  (B-30)  1t can  be  shown to be the appropriate distri-
      function for the density function Eq. (B-12).
        dDT         dD"
         PT   *  ep!   +  (1-eJ  «(C-u)
          Ti        T1          1
Hves:
     1s Eq.  (B-12).
    Next,  it can be  shown that the use of Eq.  (B-30) and Eq.  (B-7) yields
   correct form of the composite,  Pr   Substituting Eq. (B-30) in Eq. (B-7)
                                          /•C
                                          /  «(C-u)dC]
    that for c < u:

         (1-0.)  »   0  d-e.  0*
             r       ^      i   »j
   c * u
    therefore:
                                      rc
          -0T)  •  II  [ 1^  -(1-e^  J  5(C-u)dC
                   I                  JB
                                  B-12

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For C > u
        1-D   =  0   or    DT  =  1
It is, therefore, possible to write:

                                                 C
                                                                            (B-32)
                                                 C
        1-DT  =   II (1-e.O^ ) - ( 0 (1-e.) ) y  6(C-u)du
                  111
where Eq. (B-32) gives the same result as Eq. (B-31) over the whole range
of the concentration, C.

If 0° is defined by:

        1-D;  =   II (1-e.o; 8)                                               (B-33)


then Eq. (B-32) can be written as:

                                     C
        DT  =  D^  +  ( riO-e^ )y «(C-u)dC                               (B-34)
                        1
which is the composite analog of Eq.  (B-30).

     The derivative of Eq.  (B-34) is  ta
from the definition of (   in Eq. (B-26)
     The derivative of Eq.  (B-34) is taken to obtain p_.   But first,  note
               1"eiDT.
and from Eq. (B-27)
        Q  =  II  Q,
              1   '
                                   3-13

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(aking  the  derivative of  (B-34) then:

       dDT           dQ              d
       AT"9   PT  a  V   *  n  0-e.) 3/r
       dC      T    dC     .     1  at,
                           1           o
or
However:
      I   -
        •
  or
                     o
                  eipT
                      1  *  5(c"u)
which 1s the desired composite density function and 1s identical to Eq. (B-25).
     Note that starting with the Dy.  and  their corresponding e^ it is possible
to first form the composite O  and   then by:
HT  '
obtain the function p^ for insertion into Eq.  (B-29)  to  estimate the risk
of exceeding the lowest of n thresholds.
                                                                           (B'36)
                                      B-14

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                                 APPENDIX  C
                   DERIVATIONS  RELATED  TO THE  PC  FUNCTION
1.0  Derivation of PQ Function Given Distribution of Time  Averaged
     Concentrations and Ho Interactions  or Time  Dependence
     The following basic relationship from probability theory may  be  used
to relate the PC function to the distribution function for the  time averac
concentrations G(C) which is defined by Eq.  (3)  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 8760 trials.  The event is the occurence of an observed
concentration above some specified value C.
     The collection of values P(1''>TP2> *" are referred to as the binomial
distribution.
     From the above definition of the event it follows that:

          P  s  S(C)                                                        (C-2)

Where G(C) is distribution of the averaged concentrations defined by
Eq. (ft) 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  =  (l-fi(C)  )n                                                 (C-3)

which is Eq. (10) of Section 3.2.
                                        C-l

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 tf  the P-  function  Is to be used to calculate the risk that the
                                                            /_\
 bid will  be exceeded m or more times, then it 1s labeled P-v  '  and
 iponds to  the  probability that a threshold will be exceeded not more
 i-l times  in the time period.
 is  case:
               m-1
           •   IX                                                    (C-4)
 Pu  is given by Eq.  (C-l).   If Eqs.  (C-l) and  (C-2) are  substituted
 Eq.  (C-4):
           m-1
                v!n-v!        '

 C-3)  is  the  special case of Eq.  (C-5) corresponding  to m - 1.
 Distribution of Time Averaged Concentrations When Represented by
 Mel bull  Distribution Just Meeting an Air Quality Standard.
 The  National Ambient Air Quality Standard is assumed to have the
Dwing form:

      CCTQ 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:

       6(C)  • e-(C/5)k                                                  (C-6)
                                C-2

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G(C) can be interpreted as a relative frequency.   As  such,  the  quantity
E/nE (where n£ « 8760 hrs in a year) is the expected  frequency  of occurence
of concentrations above the level  C
Therefore:

          E       -
          f-  •  «   ~                                                   (c-7)
          n
Solving this expression for the parameter 6 gives:
                             1
Substituting this value in Eq. (C-6) yields:
          G(C) - e~llntnE/EJJ IC/CSTDJ                                      (C-9)

which is Eq. (13) of Section 3.2.
                                      C-3

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