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D Predicting the joint Risk of a Mixture in Terms of He Component Risks One type of complex-mixture 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 datathat is, the use of several models (e.g., one-hit, 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 low-dose 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= 1expEA(1 +B~x~1 +B2x211. (D-1) The background risk is P(0,0) = 1expEA], and the risks associated with the individual components are P(x~ ,0) = 1exp ~A(1 + Boxy] and P(0,x2) = 1exp ~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(1pal. By algebraic manipulation, it can be verified that Equation D- 1 can be equiva- lently expressed as: P(x~, x2) = 1exp ~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), (D-2)

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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 D-2 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 D-2 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 tobacco-smoking. 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 D-21. 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 10-4, 5.87x 10-4, 2.27 x 10-4, and 22.7 x 10-4.

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182 TABLE D-1 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/1P)] Multistage 1exp [Bo(l + B,X,*) (1 + B2X2*)] log [log(1p)] The estimate of P(x~, x2) based on the multiplicative model (see Table D- 1) is 23.4 x 10-4. 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 2-61. We summarize the above results and those for the other examples in Table E-2. The model-based 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 case-control 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 D-2 is

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183 =_ ~ o . . o Cq C) ~ - _ . ~ X Phi to o .,- ~ .O Cal =' _ .~- . ~ ^ Cal ;^ o - . - o ._ Pi o U. Cal m Cal ad Cal x ._ cq - m x or o - * m ._ o ED ._ ~ ~ r~ 1 1 0 0 ~ _. x x C~ O 1 1 0 0 x x - 0 C ~ 1 1 0 0 X c-, X C~ . . . . ~ O ~ ~ ~r 1 o ~ O C~ o x 00 0 . . C~ 0 o - x ~ 00 0 U~ 1 1 0 0 _4 ~ X X ~ 0 ~ . . . O 0 ~ ._ ._ .s C) ~3 () r t_ ~ ~_ ._ _ O O Ct O ~ C D u: O ,,~ cq s~ au O D 3 ~: ._ o C~ C) C) t:: Ct C~ o X Ct ~0 ~ O _ ~ CD~i C) C) ~ Ce o X r CL =e =t .c .~ C) ~ ,~ ~ .c 0,;> 5 ct =0 0 ~ C) ~ X~ a~ X ;> C: ~ ^ Ct X t: ~O, O ~ O O ~ O U, ~ C~ D ~ C .~- ~ D O C~ Ct D ~ ~ ~ ~ ~ ~ G ~ C~ ~ ~ 00 o - ~ O ~ ~ . . . . O o X 00 ~ ~ 00 . . . . ~ ~ ~ o o - ~ o ~ o o o . ~ . ~ - ~ ~ ca ~ s~ o ~ .= ~ ~ ~ . - . - . - o o o v) v) s~ 5~ D c5 ~ ~ c: c~ ~ ~ ~ o ~ c) 'e c~ - o o c) o c~ ~: u' .c ~ - - s: o o - 'e r. . - o o c) - o c) D - o ~: Cq - Ct - cn C) C~

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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) (D-4) 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. 73:25-39.