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G Empincal Modeling of Me Toxicity of Mixtures This appendix reviews some statistical methods for empirical modeling of toxicity at experimental doses. Suppose the goal is to study the toxicity of various mixtures of m chemical components. A major difficulty is the appropri- ate measurement of a toxic end point. Chemicals act on various organs and systems, so a proper measure must integrate several response factors. Assum- ing that an appropriate measure has been selected, the toxicity of a chemical is defined as the expected excess response in a population of organisms exposed to the chemical. If response is recorded on a binary scale, such as 0-1 oralive- dead, then toxicity is defined as the difference in the probability of, say, death between the presence and the absence of exposure to the chemical: tox = P(death with chemical)P(death without chemical). Other response events could be substituted for death. A mixture containing m chemicals will be represented by a vector z = (at, Z2, ~ Zm), meaning that the mixture contains zoo dose units of chemical 1, Z2 dose units of chemical 2, and so on. The goal of empirical modeling is to obtain a parsimonious description of the toxicity surface, toxfz), over the experimen- tal range. An empirical model will summarize important patterns in the data; this will allow some inferences to be drawn about the toxicity of the mixture and suggest avenues for further experimentation. If the available data consist of direct joint measurements on the toxicity surface, then standard regression-analysis procedures can be used to model toxicity. In this setting, there is a broad class of powerful statistical tools ac- companied by excellent computation software for exploratory investigation of the dose-response surface, for model fitting and diagnostic checking, and for experimental design. For toxicity data in the above format, the job of empirical 209
210 APPENDIX G modeling is relatively easy. A good reference text is Mosteller and Tukey (1977). Many toxicity measurements, however, are quantal for example, the pro- portion of animals that die prematurely when exposed to a chemical might be recorded. Such data do not follow a standard regression format (i.e., the under- lying assumptions for regression theory are not satisfied). A useful approach to such data is based on extensions of the standard linear model developed over the last 20-30 years. For quantal response data, there are logistic and probit regression methods, which, in turn, are special cases of the more extensive collection of generalized linear models (GLMs) to be found in the book by McCullagh and Nelder (19831. For survival data, there are the Cox propor- tional hazards model and its extensions; relevant references are Carter et al., (1983), Cox and Oakes (1984), and Kalbfleisch and Prentice (19801. An important aspect of the GLM and Cox model methods is that they can be implemented easily, given an adequate understanding of the application and theory of standard linear models (McCullagh and Nelder, 1983; Carter et al., 19831. In particular, the common exploratory and confirmatory data-analysis techniques developed for the standard regression setup have counterparts in the new framework. There are analogues of graphic exploration, variable selec- tion, analysis of variance, residuals, and diagnostic checking. An exception occurs in experimental design, because most of the standard optimal experi- mental design literature on the linear model focuses on construction of designs that minimize the variance of some regression parameters. In the standard linear model, variance expressions are independent ofthe value ofthe response surface, so the optimal design is relatively easy to define. That does not happen with quantal response data, and the optimal design here depends on the un- known characteristics of the toxicity surface. As a result, the optimal design can be difficult to specify. It could be argued that, because the optimality crite- ria used to define designs are often very narrow, efforts spent on developing an optimal design might not always be justified. An appealing approach to experimental design is that of Carter et al. (1983~. Their work is concerned with finding favorable doses for chemotherapy. The typical measurements are either quantal response or survival times. The tech- niques that they propose involve sequential designs taken from response-sur- face methods (see, for example, Box et al., 19781. In several compelling exam- ples given by Carter et al. (1983), the approach is shown to work very well. In the light of that experience, there is much to be said for using standard frac- tional-factorial, screening, and other familiar designs to explore toxicity sur- faces, at least unless it becomes apparent that the designs are grossly ineff~- cient. It is clear that further research needs to be done. The purpose of this section is to show how the GLM strategy works. A complete account of the method, with many more examples, can be found in McCullagh and Nelder's (1983) book. The illustration of the use of GLM below demonstrates how the method preserves many of the most important
APPENDIX G 211 aspects of ordinary regression analysis. Nonparametnc extensions of GLMs and their use in estimating mixture toxicity are discussed further later. In the final section, the so-called biomathematical models introduced by Ashford and Cobby (1974) and Hewlett and Plackett (1979) are discussed. These models are developed from general quasimechanistic pnnciples. Their implementa- tion and interpretation, however, are more complicated than the GLM frame- work. It should be emphasized that the purpose of either model system (gener- alized linear or biomathematical) is to approximate the relationship between toxicity and dose over the observed dose range; extrapolation on the basis of these models without additional scientific insight might be inappropnate. Where extrapolation is required, the empirical model will, at best, suggest relationships that a scientist can use to postulate more refined theones. ANALYSIS OF AN INHALATION EXPERIMENT ON RATS To illustrate the use of the GLM method, we present a simple case study. In one of a series of inhalation experiments carried out by B. C. Levin et al. (in press) at the National Bureau of Standards, groups of rats were exposed to mixtures of CO and CO2 at high doses. Some of the experiments are described in Chapter 3 and Appendix C. For each mixture, the number of animals that survived was recorded. Preliminary data are listed in Table G-1 and plotted in TABLE G-1 Mortality of Rats Exposed to CO2 and Coa CO2, ppm CO, ppm No. Rats Dead No. Rats Exposed 34,300 3,000 1 5 37,500 3,600 5 5 36,600 3,400 3 5 47,500 2,400 0 4 43,400 3,200 3 6 44,800 2,600 3 6 49,700 2,700 1 4 51,100 2,700 3 5 103,600 1,600 0 4 118,800 1,500 0 4 104,700 2,400 0 4 100,300 2,900 4 6 137,600 2,900 1 3 146,900 2,900 0 4 147,600 3,100 1 5 142,400 3,800 6 6 55,600 3,000 1 4 73,000 2,200 0 5 74,300 2,800 2 5 52,500 2,500 4 4 177,100 3,200 0 4 173,200 3,600 1 4 a Data from inhalation experiment, Levin et al. (in press).
212 APPENDIX G Figure G-1. Note that the plot reveals some wide gaps in the design. Further experimentation might be directed toward filling the gaps, if estimation of the entire concentration surface were the objective of the expenment. A close examination of Figure G-1 suggests that higher concentrations of CO2 seem to decrease the toxicity of CO; hence, the ratio of CO to CO2 might be an important explanatory vanable. After some trial and error, a log-logistic model was found to give a reasonable fit to the data: log{p~z)/~1 - p~z)~} = 50 + BACON + 32(CO2) + 33(CO/CO2), where pro) represents the probability of death after exposure to mixture z and the hi are estimated under the model. The goodness of fit of this kind of model is measured by what is known as Me scaled deviance (i.e., the analogue of a residual sum of squares). Here, the scaled deviance is 24. 17 on 18 degrees of 1 80,000 1 60,000 1 40,000 1 20,000 ~ 1 00,000 Q o 80,000 60,000 40,000 o o o 4 13 0 2 20,000 1 1 1 1 1 1 1 o 0 1 2 3 2 2 5 0 3 5 1,500 2,000 2,500 3,000 3,500 4,000 CO, ppm FIGURE G-1 Data from inhalation experiments (Levin et al., in press). Points are labeled 0-5, with O indicating that all animals survived at that concentration combination and other number indicating number of deaths.
APPENDIX G 213 freedom. For a standard regression model, the deviance reduces to the residual sum of squares. The scaled deviance has an approximate chi-squared distribu- tion with a number of degrees of freedom equal to the sample size minus the number of parameters in the model. The deviance for our fit is larger than one would hope the expected value for a chi-squared variate is equal to its number of degrees of freedom, so, because 24.17 is over 30% larger then the expected 18, one might worry about the adequacy of the model. The standardized resid- uals from the fit are plotted against the model-predicted response rates, y (the number expected to die at each observed combination of CO and CO2), in Figure G-2. If the model were correct, the residuals would approximate standard normal deviance. Note that the largest residual is more than two units away from zero. Given the sample size, that residual is somewhat suspect. The point corres- ponding to it is labeled "4" in Figure G-1. One can see that the point seems high relative to its neighbors. All the animals in that group were cannulated for blood withdrawal during exposure, and that might explain why the group is an outlier. Refitting the model with the outlier removed results in a model with a scaled deviance of 17.87 on 17 degrees of freedom. The reduction in the devi- ance of 6.3 units for 1 degree of freedom is dramatic. Residuals from the fit are acceptable, and the model provides a plausible summary of the data. The iso- boles (curves of constant toxicity) for the fitted model are shown in Figure G-3. 3 2 1 a) - ._ In - 3 · · - · ~ · ~ - ~. · -1 _ ~ ~ -2 _ ~ 1 1 1 1 1 0 1 2 3 Predicted Values 4 s 6 FIGURE G-2 Residuals from model fitted to all data. The largest residual corresponds to the point labeled "4" in Figure G-1.
214 FIGURE G-3 Isoboles for fitted empirical model. APPENDIX G 1 80,000 r 160,000 1 40,000 120,000 O Q 100,000 ~ _ 80,000 60,000 40,000 20,000 1,500 2,000 2,500 3,000 3,500 4,000 ~ , / - CO (ppm) The 0.5 isobole corresponds to the set of combinations of CO and CO2 that would lead to a 50% death rate in the population. The isoboles are highly nonlinear for lower concentrations of CO2. The example gives the essential flavor of the GLM approach to data analy- sis. Because the method preserves the common steps in the analysis of regres- sion data, the analyst can learn to use it quickly. In the example, there were 22 data points and two explanatory variables; by modern standards, this is a mod- est size of data set. If the data set were smaller, it might have been more difficult to find an appropriate fit to the data; with more data, we would hope to get a more detailed description of the toxicity surface. The dangers of extrapo- lation are also shown by the example; clearly, animals cannot survive in a pure CO or pure CO2 environment, as would be predicted by the model. However, for zero concentration of CO2 combined with any concentration of CO (includ- ing concentrations above the Low, about 4,600 ppm), the model predicts zero mortality. Thus, extrapolation on the basis of this purely empirical model is inappropriate. As is often the case in data analysis, progress would have been very slow without graphics. A variety of modern statistical software packages could have been used to implement our analysis. We used the GLIM statistical package that is distributed by the Royal Statistical Society and the S statistical package,
APPENDIX G 215 which is an environment for data manipulation and graphics distributed by Bell Laboratories. The analysis here used only functions of the concentrations of mixture com- ponents in estimating the mixture's toxicity. There is no reason, however, why other variables, such as mixture properties, could not be considered in the analysis. A GENERALIZED CLASS OF NONINTERACTING MODELS SIMPLE ADDITIVE MODEL The simple model often used to describe the toxicity of a mixture is the additive model. The toxicity of the mixture is represented as toxkz) = tox~(z~) + tox2(z2) + . . . + toxm~zm), where toxfz) denotes the toxicity of the mixture with dose vector z and toxi~z`) describes the toxicity of the ith component of the mixture. If toxicities are measured as excess risks, the additive model says that the total excess risk is the sum of the excess risks associated with the components. This section shows that additive models constitute a large class of models, including many often used to address dose-response issues. GENERALIZED ADDITIVE MODEL An additive model might not be appropriate in the raw toxicity scale, but it might hold for some alternative scale. gEtoxkz)] = Dozy + ¢2(Z2) + · + ~m(Zm)' where g is a transformation of the toxicity scale and Lizzie describes the effect of the ith component on the overall toxicity of the mixture. If the responses are expressed as quantal data, then the transformation g may be a simple logit or probit, and the functions Hi will typically be linear in dose or log dose. Some examples of generalized additive models follow. In the additive-risk model, g is the identity function, toxfz) is the excess risk associated with the mixture, and ~i(Zi) is the excess risk associated with the ith component. The public-health literature notions of additivity and synergism are based on this model. In the multiplicative-risk model, let toxkz) = 1Qfz) = probability of death, given dose z of the mixture, and Qitzi) = probability of survival, given dose zi of component i.
216 The model assumes APPENDIX G m Q(z)= II Qi(zi) i = 1 If g(x) = log(1x) and ~i(zi) = log[Qi(zi)], the multiplicative model can be written as m g[tox(z)]= ~ (), i = 1 which is in the form of a generalized additive model. In standard probit/logit analysis, fox(z) is the probability of death, given dose z of the mixture; g is the probit or logit transformation; and the His are linear in Zi or log(z)). To show joint action with a multistage model where the exposures act at separate stages and only at one stage each, an example is taken from Appendix E. The model is probability of death, given dose z = 1exp[A(1 + BIZI)(1 + B2Z2)] where A, Bi, and B2 are parameters. To fit that into the format of the general- ized additive model, take g to be the complementary log-log transformation, g(x) = log Elog(1x)], and let ~i(zi) = log(1 + Bizi) and ~ = log A. Then gEtox(z)] = or + ¢~(z~) + ¢2(Z2). Additive models have served statisticians well for many years. In the case where the His are parametric, the generalized additive models are a special case of the generalized linear models of McCullagh and Nelder (1983). If a generalized additive model holds, the toxicity of a mixture can be as- sessed from knowledge of the effects of its individual components on the trans- formed toxicity scale. Interaction depends crucially on the scale in which toxicity is measured. This dependence has led to debate as to whether interaction ought to measure departure from additive or multiplicative models, and so on. To rationalize the various interaction measures, it would help if the underlying models were al- ways explicitly stated. In addition to removing some of the confusion, that approach would make it easier to plan experiments to detect interaction. In the context of generalized additive models, it seems reasonable that interaction ought to reflect the presence of first-order interaction (nonadditive) terms in the model. Hinkley (1984) has developed some analogues of Tukey's one-degree- of-freedom test that would aid in that purpose. If functions g and Hi have known parametric forms, such as g logit and ~ linear, there is a well-established inference system for the model, because the
APPENDIX G 217 TABLE G-2 Influence of One Chemical on the Biologic Action of Another Site of Primary Action Interaction Similar Dissimilar No Simple Independent Yes Complex Dependent model is in the category of GLMs. For these models, parameter estimation, confidence intervals, and procedures for analysis of diagnostic residuals are well established. Experimental designs for GLMs are only in an early stage of development (Steinberg and Hunter, 1984~. Given a large number of doses (say, more than 20), transformations g and hi can be nonparametrically estimated (see Breiman and Friedman, 1985, and Hastie and Tibshirani, 19861. More elaborate methods allowing for general interaction patterns between variables have been described by O'Sullivan et al. (1986~. Although the reliability of the procedures might not be very high in small samples (fewer than 30 data points), the exploratory value of such meth- ods can be very useful. In particular, parametric forms for g and hi are often suggested by those methods. QUASIBIOLOGIC MODELS Hewlett and Plackett (1979) have described a more complicated system of empirical models (see also Hoel, 19831. Those models have been developed by Ashford and Cobby (1974) and Hewlett and Plackett (1959~. The latter models, although widely used, make assumptions about the underlying bio- logic factors that might not always be appropriate; moreover, it is still awkward to apply them to any mixture more complex than binary. The Hewlett and Plackett approach considers mixtures of two chemicals A and B whose action occurs at either of two sites So and S2. The joint action of the two chemicals is classified as similar or dissimilar, depending on whether the sites of action of the chemicals are the same or different. The chemicals interact if the presence of one affects the amount of the other that reaches its site of action. In the Hewlett and Plackett system, the joint action of the chemicals is classified as in Table G-2. In a Hewlett-Plackett model, a toxic event is said to occur if the administered dose exceeds some tolerance or threshold in the organism and a probability distribution for tolerances in the population is specified. The Hewlett and Plackett (1979) formulation is developed in some detail for noninteracting chemicals. The general model is as follows: Qfz) = probability of survival, given dose z of the mixture.
218 APPENDIX G The administered dose z = (Z~,Z2) gets translated to an effective dose w = Owl, w21. The tolerance of the organism w-scale is denoted cat = (c0~,c02~. A toxic event occurs (the organism fails) if the combined effect at either site of action exceeds some threshold. Formally, that is described as: For O c v c 1, either or both of the following will hold: w~/~2 + vw2/~2> 1; vow/ + W2lw2 > 1- (G-1) (G-2) Expression G-1 describes the combined effect of the mixture at site 1, and Expression G-2 the combined effect at site 2. Parameter ~ measures the degree of similarity in the joint action of the chemicals. If v = 1. the action is a simple similar action; if ~ = 0, it is independent. For mathematical simplicity, an alternative form of the model is more often used. Here, a toxic event occurs if (W~/~) + (W2/~2) > 1, (G-3) where O ~ ~ c 1. Parameter ~ is a measure of action similarity. If ~ = 1, the action is a simple similar action; if ~ = 0, it is independent. Given a specified distribution for 1 and a known relationship between w and z, Liz) can be computed and, in principle, any unknown parameters can be estimated by the method of maximum likelihood. It is not known whether there is a software package that will conveniently implement the analysis. Model discrimination is difficult in the Hewlett-Plackett framework. Alternative specifications for the population tolerance distributions and the type of interac- tion can lead to indistinguishable forms for the toxicity surface (Hewlett and Plackett, 1979, pp. 49-501. Extension of the Hewlett and Plackett models to mixtures of more than two components has not been worked out in any detail. For consideration of mixtures with many components, such models would probably be vein complicated. Hewlett and Plackett models are based on receptor theory. (Hewlett and Plackett use "sites" in much the same way as Ashford and Cobby use "recep- tors.") Chemicals are assumed to "tie up" receptors at the site of action, thereby diminishing the activity at the site. For the case of a mixture of m chemicals, the activity at the site of action that results from a dose z = (at, Z2, ~ Zm) iS expressed as m 1 + ~ (Xia ACT(z) m 1 + ~ aiZ i = 1 , (G4) where al are association constants, aizi corresponds to the effective dose of the ith component at the site of action, and hi measures the effect on the activity due
APPENDIX G · AL 219 to the in component of the mixture. The activity at the site of action is mono- tonically related to toxicity of the chemical: toxkz) = fEACT(z)], (G-S) where fis a monotonic function. Thus, the greater the activity ofthe mixture at the site of action, the greater the risk of a toxic event. The biologic sophistication in both biomathematical modeling systems is relatively modest, so one would be hard pressed to choose one system over the other solely on the basis of biologic plausibility. In addition, the parameters in both models are vaguely defined. In the receptor-theo~y models, if f were of known parametric form, a system of inference for the models could tee obtained (see McCullagh and Nelder, 1983) by expressing a model in the format of a GLM [with a nonlinear predictor related to the activity function, Act. For that reason, it could be argued that receptor models have a practical advantage over the models of Hewlett and Plackett. SUMMARY Conventional toxicity data, typically represented by dose-response func- tions, offer possibilities for modeling and analysis by many different ap- proaches. A mixture by itself, administered as a unitary treatment, poses no novel problems in principle, and expressions of the function can be derived with standard methods. The standard method, however, is less versatile than the class of GLM for modeling mixture data. The models are congruent with, and extensions of, the more familiar methods of regression analysis and pro- vide other desirable features. For example, normality and constant variance are not required for estimates of the random-error component, and interactions of systematic effects can be specified to hold on an additive scale by appropri- ate transformations of the scale. Also, GLMs are accessible through an exten- sive collection of computer programs that permit various approaches to be explored and that can model joint excess risks without reliance on arbitrary biologic mechanisms. Those methods, however, require larger data sets than are ordinarily collected by toxicologists; the precision of dose-response surface estimates improves as the number of data points increases. Several quasibiologic models (e.g., Hewlett-Plackett and Ashford-Cobby) based on biologic mechanisms have been advanced and, in essence, are based on assumptions about the overlap and nature of interactions at common sites. Although some might be translated into GLM terms, their suppositions limit the ease with which they can be expanded into models for multicomponent mixtures.
220 APPENDIX G REFERENCES Ashford, J. R., and J. M. Cobby. 1974. A system of models for the action of drugs applied singly or jointly to biological organisms. Biometrics 30: 11-31. Box, G. E. P., W. G. Hunter, and J. S. Hunter. 1978. Statistics for Experimenters. John Wiley & Sons, New York. Breiman, L., and J. H. Friedman. 1985. Estimating optimal transformations for multiple regression and correlation. J. Am. Stat. Assoc. 80:580-619. Carter, W. H., G. L. Wampler, and D. M. Stablein. 1983. Experimental design, pp. 108-129. In Regression Analysis of Survival Data in Cancer Chemotherapy. Marcel Dekker, New York. Cox, D. R., and D. Oakes. 1984. Analysis of Survival Data. Chapman and Hall, New York. (201 pp.) Hastie, T. J., and R. J. Tibshirani. 1986. Generalized additive models. Stat. Sci. 1:297-318. Hewlett, P. S., and R. L. Plackett. 1959. A unified theory for quantal responses to mixtures of drugs: Non-interactive action. Biometrics 15:591-610. Hewlett, P. S., and R. L. Plackett. 1979. An Introduction to the Interpretation of Quantal Responses in Biology. University Park Press, Baltimore. (82 pp.) Hinkley, D. V. 1984. Diagnostics for Transformable Non-Additivity. Technical Report No. 6. Center for Statistical Sciences, University of Texas, Austin, Texas. Hoel, D. G. 1983. Statistical Aspects of Chemical Mixtures. Working Paper for SGOMSEC 3 Work- shop, Guilford, England, August 15- 19, 1983. National Institute of Environmental Health Sciences, National Institutes of Health, Research Triangle Park, N.C. [17 pp.] Kalbfleisch, J. D., and R. L. Prentice. 1980. The Statistical Analysis of Failure Time Data. John Wiley & Sons, New York. (321 pp.) Levin, B. C., M. Paabo, J. L. Gurman, and S. E. Hams. (in press) Effects of exposure to single or multiple combinations of the predominant toxic gases and low oxygen atmospheres produced in fires. Fundam. Appl. Toxicol. McCullagh, P., and J. A. Nelder. 1983. Generalized Linear Models. Chapman and Hall, New York. (261 pp.) Mosteller, F., and J. W. Tukey. 1977. Data Analysis and Regression: A Second Course in Statistics. Addison-Wesley, Reading, Mass. (588 pp.) O'Sullivan, F., B. S. Yandell, and W. J. Raynor, Jr. 1986. Automatic smoothing of regression functions ingeneralizedlinear models. J. Am. Stat. Assoc. 81:96-103. Steinberg, D. M., and W. G. Hunter. 1984. Experimental design: Review and comment. Techno- metrics 26:71-97.
