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OCR for page 179
D
Predicting the joint Risk of a Mixture
in Terms of He Component Risks
One type of complexmixture problem is illustrated by the following para
digm. One has a complex mixture of known chemical composition, the toxici
ties of whose individual components can be estimated, and the overall toxicity
of the mixture needs to be estimated. At times, information on the toxicity of
the mixture has been obtained from a rodent experiment in which the mixture
has been tested at high dosages, or epidemiologic data have been gathered on
the mixture or on similar mixtures. However, usually the only available toxic
ity information on a mixture is related to its individual components, and one
needs to estimate the toxicity of the mixture.
This appendix considers the problem of estimating the joint risk of a mixture
in terms of the component risks that is, in terms of the risks of the individual
chemicals in the mixture. It begins with a class of models in which the joint risk
can be expressed in terms of the component risks. Thus, if one had reason to
believe in the validity of a particular model within this class and had informa
tion on the component toxicities, the joint risk could be estimated. It then
discusses an approach when there is no preferred model, but there is some
confidence that the true model may be within the class of models being consid
ered. The approach is analogous, in some respects, to one sometimes used to
assess the carcinogenic risk associated with a chemical on the basis of labora
toty data—that is, the use of several models (e.g., onehit, multistage, multi
hit, Weibull, and probit models) to predict the risks associated with small expo
sures. The resulting range of risks is presented as a "plausible" range. In the
present situation, we adapt that idea to estimate the risk associated with a
mixture. Specifically, we examine a variety of generalized additive models for
the joint risk associated with the mixture and express the risk for each model in
terms of the risks associated with the individual components.
179
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180
APPENDIX D
Of course, one usually cannot discriminate between various models for com
plex mixtures at low doses, because adequate response data are generally not
available. However, we hope that the range of predicted risks generated from
several of the commonly used models will provide a plausible range of risks.
The plausible range might be used in several ways, including application in a
sensitivity analysis of estimates based on a specific model.
Before the idea is described in greater detail, several cautions are in order.
First, the results given below are preliminary and are intended mainly to illus
trate the idea; full development and evaluation of the approach would require
additional research, which, although perhaps not extensive, is nevertheless
lacking. Second, the approach is not intended to be able to identify the tree risk
associated with a mixture at low doses that is unverifiable. Rather, we want to
produce an estimate of the lowdose risk that is consistent with a class of
models of the joint effects of the mixture.
EXPRESSING JOINT RISKS
IN TERMS OF COMPONENT RISKS
Consider the multistage model for two compounds that is developed in Ap
pendix E, in which it is assumed that no two components affect the same
transition. Let P(x~, x2) denote the risk (i.e., probability of developing a tumor)
associated with exposure to doses x~ and x2 of the two compounds. Then, for
this form of multistage model,
P(x~,x21= 1—expE—A(1 +B~x~1 +B2x211. (D1)
The background risk is P(0,0) = 1—expE—A], and the risks associated with
the individual components are P(x~ ,0) = 1—exp ~—A(1 + Boxy] and P(0,x2) =
1—exp ~—A(1 + B2X21]  We will demonstrate that the combined risk P(x~, x2)
can be determined by knowing P(0,0), P(x,,0), and P(0,x2~. Consider the
function
gyps = In ~—ln(1—pal.
By algebraic manipulation, it can be verified that Equation D 1 can be equiva
lently expressed as:
P(x~, x2) = 1—exp ~—exp [g(P[x~,03) + g[P(0,x21]—g[P(0,0~.
With Equation D 1, the joint risk, P(x~, X2), can be completely specified from
the component and background risks.
The above result can be extended to any class of generalized additive models
that have the form
F[P(x~, x2~] = Bo + BiZi~x~) + B2Z2(X2), (D2)
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APPENDIX D
181
where z1 and z2 are any transformations of x1 and x2 and F is any invertible
function. Withoutlossofgenerality, wecanassumethatcl(O) = Z2~0) = 0. The
background and component risks must then satisfy
F[P(O,O)] = Bo,
F[P(x1,01] = Bo + B1 Z1 EXIT,
and
F[P(O,x21] = Bo + B2z2(x21.
Thus, we can express Equation D2 as
F[P(xl, x21] = F[P(O,O)] + {F[P(xl,O)]—F[P(O,O)~)
+ {F[P(O,x21]—F[P(O,O)] 3
= F[P(x,,O)] + F[P(O,x2~]—F[P(O,O)],
and can write
P(xl, X2) = F—{F[P(xl, 04] + F[P(O,x21]—F[P(O,O)~} .
Hence, P(xl, x2) is determinable from the background and component risks.
The class of models that satisfy Equation D2 is broad and includes the
linear, multiplicative, logistic, and multistage models as subclasses. We illus
trate those in the following table, where we use x~ to denote zi~xi).
We also note that, for a multiplicative model, the relative risk of the mix
ture say, RR(xl, x2) can be expressed as the product of the component rela
tive risks; that is,
RR(xl, x2) = t~(xl,O)] [~0,X211.
(~3)
That result is useful for situations where only the relative risks of the compo
nent chemicals are available (and not their actual risks). Moreover, for suff~
ciently small xl and x2, the above result also applies to the multistage model
given above and the logistic model.
ILLUSTRATIVE EXAMPLES
Reif ( 1984) gave several examples of the collection of epidemiologic data on
both the individual and the joint effects of two "substances," one of which is
tobaccosmoking. We use the data to illustrate the multiplicative and logistic
models. For example, Reif examined the risk of lung cancer as a function of
smoking status and uranium exposure (see Table D21. Letting A = smoking
and B = uranium, he got:
P*
P(xl, O)
P(O, X2)
P(x1, X2)
1.5/26,392 = 0.57 x 104,
5.87x 104,
2.27 x 104, and
22.7 x 104.
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182
TABLE D1 Alternate Response Models
APPENDIX D
Fonn of
Model Formula for P(x,,x2) FO

Linear Bo + B,x,* + B2X2* P
Multiplicative exp [Bo + B,X,* + B2x2*] log p
Logistic exp [Bo + B,X,* + B2X2*]
1 + exp [Bo + B,X,* + B2X2*] log Lp/1—P)]
Multistage 1—exp [—Bo(l + B,X,*) (1 + B2X2*)] log [—log(1—p)]
The estimate of P(x~, x2) based on the multiplicative model (see Table D 1) is
23.4 x 104. The logistic and the multistage models give very similar esti
mates. Thus, in this example, all three models give very similar estimates of
P(x~, x2), and ones that closely agree with the value P(x~, x2) estimated by
mixture data. With each model, we can predict P(x~, x2) accurately with only
the information on individual cancer risks.
Reif gave five other examples (his Tables 261. We summarize the above
results and those for the other examples in Table E2.
The modelbased estimates of P(x~, x2) for the first three examples, in which
there are data on P(0,0), P(x~,0), and P(0,x2), are quite good. The three
models give similar predictions, all closely approximating the P(x~,x2) esti
mated directly. The last three examples are from casecontrol studies in which
only relative risks are available (and not estimates of P(0,0), P(x~ ,0), and P(0,
x21. Thus, we can predict only the relative risk of the mixture on the basis of the
multiplicative model. This estimate closely approximates RR(x~,x2) for the
fourth example, but underestimates it for the fifth and sixth examples. The
results of the multistage model might seem to be inconsistent with those de
scribed in Appendix E, where it is argued that the excess risk associated with a
mixture (at low exposures) can often be accurately approximated as the sum of
the individual excess risks. However, on the basis of Reif's data (which reflect
relatively high exposures), the excess risks associated with mixtures are con
sistently higher than the sums of the individual excess risks. Moreover, evalua
tion of the multistage model for the data produces a joint risk estimate that
exceeds the sum of the individual estimates. Thus, whereas the multistage
model gives good predictions of the joint risk in the examples, the exposures
were not so low that it approximates a linear model. These results verify the
critical dependence of the argument in Appendix E on magnitude of exposure.
The risk estimates derived from the various epidemiologic studies are also
subject to considerable uncertainty.
MIXTURES OF MORE THAN TWO CHEMICALS
The results extend in a straightforward way to the situation of mixtures of K
compounds. The analogue of Equation D2 is
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184
F[P(xl, x2, . . ., Xk)] = Be + Blzl(xl) +
For example, the logistic model (see Table D 1) generalizes to:
. .
P(xl, X2, . . . , xk) = PIP2 · · · Pkl~ ]
APPENDIX D
. + BkZk(Xk) (D4)
where Pj = P(0, 0, . . ., x;, 0, . . . ,0) forj = 1 to k the effect with exposure to
only substance j (at dose Xj).
ADDITIONAL RESEARCH
For a proper assessment of the potential of this approach, several problems
need to be investigated. First, a variety of models should be considered, and
the formula for P(x~ ,x2) for each (in terms of the individual effects) should be
derived for each model. Then, when information on individual and joint effects
is available, all the estimates should be computed with the models. Given the
results for several data sets, some models might be deemed more realistic,
because they yield results that are consistent with what the real risk is thought
lobe.
The range of plausible values of P(x~,x2) can also be applied to data sets in
which there is no direct information on P(x~,x2), as an aid in designing (or
deciding to undertake) an extensive experiment. For example, if the plausible
range is within achievable or acceptable limits, it might be decided that little
would be gained from carding out an extensive experiment.
SUMMARY
Generalized additive models are discussed more specifically in the context
of a situation in which risk estimates associated with individual components of
a mixture are available. Here, one goal is to be able to express the risk associ
ated with the mixture in terms of the component risks, so that, for a given
model, the former can be estimated in terms of the latter. Another use of the
result is to predict a plausible range of risks associated with the mixture.
REFERENCE
Reif, A. E. 1984. Synergism in carcinogenesis. J. Natl. Cancer Inst. 73:2539.