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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
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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
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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).
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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.
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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.
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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,
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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) = 1—Qfz) = probability of death,
given dose z of the mixture, and
Qitzi) = probability of survival,
given dose zi of component i.
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216
The model assumes
APPENDIX G
m
Q(z)= II Qi(zi)
i = 1
If g(x) = log(1—x) 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 = 1—exp—[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 E—log(1—x)], 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
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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.
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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
OCR for page 219
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.
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220
APPENDIX G
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Representative terms from entire chapter:
generalized additive
