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Environmental
Health Perspectives Supplements Volume 110, Number 6, December 2002
Experimental Designs for Mixtures of Chemicals along Fixed Ratio Rays
Stephanie L. Meadows,1 Chris Gennings,2 W.
Hans Carter Jr.,2 and Dong-Soon Bae3
1Pfizer Inc., Groton, Connecticut, USA; 2Department
of Biostatistics, Virginia Commonwealth University, Medical College
of Virginia, Richmond, Virginia, USA; 3Center for Environmental
Toxicology and Technology, Department of Environmental and Radiological
Sciences, Colorado State University, Fort Collins, Colorado, USA
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Abstract
Experimental design is important when studying mixtures/combinations of
chemicals. The traditional approach for studying mixtures/combinations
of multiple chemicals involves response surface methodology, often supported
by factorial designs. Although such an approach permits the investigation
of both the effects of individual chemicals and their interactions, the
number of design points needed to study the chemical mixtures becomes
prohibitive when the number of compounds increases. Fixed ratio ray designs
have been developed to reduce the amount of experimental effort when interest
can be restricted to a specific ray. We focus on the design and analysis
issues involved in studying mixtures/combinations of compounds along fixed
ratio rays of the compounds. To obtain the inference regarding the interactions
among the compounds, we show that the only data required are those along
the fixed ratio ray. Key words: additivity, interaction index,
isobologram. Environ Health Perspect 110(suppl 6):979-983
(2002).
http://ehpnet1.niehs.nih.gov/docs/2002/suppl-6/979-983meadows/abstract.html
This article is part of the monograph Application
of Technology to Chemical Mixture Research.
Address correspondence to C. Gennings, Dept. of Biostatistics,
Virginia Commonwealth University, 1101 East Marshall St., B1-039-A,
Richmond, VA 23298-0032 USA. Telephone: (804) 827-2058. Fax: (804) 828-8900.
E-mail: gennings@hsc.vcu.edu
This research was supported by the U.S. Environmental
Protection Agency National Center for Environmental Assessment (Cincinnati,
OH) cooperative agreement CR-827208.
Received 18 December 2001; accepted 3 September 2002.
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Determining and characterizing the nature of interactions among components
of a combination of c drugs or chemicals is a problem of current interest
(where c is the number of drugs/chemicals in a mixture). Although assessments
based on single-drug/chemical exposure enable us to acquire fundamental knowledge
about individual drugs or chemicals under carefully controlled conditions, they
do not reflect real-world exposures. Thus, it is often of interest to study
the effects of exposure to multiple drugs/chemicals. Of ultimate interest in
such studies is the determination and characterization of interactions among
the components in a mixture. For example, Gennings et al. (1) report
on a study of the nature of the interaction involving the mixture of four metals.
The four metals chosen for the study were arsenic (As), cadmium (Cd), chromium
(Cr), and lead (Pb), which are among the top contaminants in site frequency
count by the Agency for Toxic Substances and Disease Registry (ATSDR) Completed
Exposure Pathway Site Count Report (2). In addition, human health risk
assessment associated with exposure to disinfection byproducts in drinking water
is of concern because of the widespread exposure of persons who receive disinfected
water. Other examples of human exposure to combinations of agents can be found
in the treatment of numerous diseases including cancer, AIDS, diabetes, and
asthma. These examples illustrate the importance of studying mixtures/combinations
of drugs or chemicals. Determining departures from additivity for a combination
of drugs or chemicals is a problem that has been considered by many authors
(3-7).
Classic Methodology for Detecting and Characterizing Departures
from Additivity
Isobolograms. The classic method for detecting and characterizing
departures from additivity between combinations of drugs or chemicals is the
isobologram. The isobologram, introduced as a graphical tool by Fraser (8,9),
is a plot of a contour of constant response of the dose-response surface
associated with the combination superimposed on a plot of the same contour under
the assumption of additivity. Its use was extended by Loewe and Muischnek (10),
Loewe (11), and Berenbaum (12), and reviewed by Gessner (13),
Wessinger (14), and Berenbaum (15). For a two-component mixture,
the analysis of an isobologram compares the observed isobol (e.g., combination
dose/concentration associated with 50% response [ED50]) to the line
of additivity. The line of additivity is formed by joining the ED50
associated with each of the individual components calculated from the dose-response
data for the individual components. Figure 1 presents illustrations of possible
isobolograms for a combination of two drugs/chemicals. As indicated, if the
isobol is below the line of additivity, a synergism is claimed. On the other
hand, if the isobol is above the line of additivity, an antagonism is claimed.
However, there are shortcomings associated with the use of isobolograms. For
instance, the method used in the construction of an isobologram typically does
not take data variability into account. Additionally, because it is a graphical
method, isobolograms effectively are limited to the study of combinations of
two or three drugs or chemicals.
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Figure 1. Illustrations
of isobolograms for a combination of two drugs/chemicals. The dashed
line is the line of additivity. When the isobol bows below the line
of additivity, a synergism is claimed; when the isobol bows above the
line of additivity, an antagonism is demonstrated.
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Interaction index. The interaction index, introduced by Berenbaum
(12), provides a convenient method to determine and characterize departures
from additivity for a combination of c > 2 or 3 components. The interaction
index, II, is defined by
[1]
where c is the number of components, X1, X2,...,Xc
are the doses in combination associated with a desired effect, and ED100µ
(CHEMi), i = 1,...,c is the dose of the ith
component that, when administered alone, produces the same effect. When the
interaction index, defined in Equation 1, is equal to 1, the c components
interact additively. When II is greater than 1, the components interact antagonistically;
when II is less than 1, the components interact synergistically. Again, it should
be noted that the individual component dose-response information is required
to calculate the interaction index. As described by Berenbaum (12), the
interaction index is directly related to the isobologram, i.e., when II = 1,
the isobol is coincident with the line of additivity; when II > 1, the isobol
bows above the line of additivity; and when II < 1, the isobol bows below
the line of additivity. An advantage of using the interaction index over the
isobologram is that the interaction index is not limited to combinations/mixtures
of just two or three components. However, as developed by Berenbaum (12),
the biological variability associated with the data is not taken into account
by the interaction index.
Statistical models. Statisticians frequently use models of the
form
to approximate the relationship between a response of interest, Y,
and concentrations of c chemicals (x1,x2,...,xc).
Carter et al. (4) showed that a relationship exists between the interaction
index proposed by Berenbaum (12) and the parameter in a statistical model
associated with the interaction of the components of the combination. Without
loss of generality, consider that the combination/mixture of interest involves
two chemicals and that the response is continuous. Therefore, following the
logic of Carter et al. (4), for the linear models case, the relationship
between the response and the doses or concentrations of the components in combination
can be expressed as
[2]
where
µ is the mean response, E(Y),
ß0 is the unknown intercept,
ß1 is the unknown slope parameter associated with the first
component,
ß2 is the unknown slope parameter associated with the second
component,
ß12 is the unknown parameter associated with the interaction
of the two components, and
x1 and x2 are the doses of the respective
chemicals.
From the model defined in Equation 2, the ED100µ (CHEMi)
for the respective components can be derived to be
Thus, after algebraic manipulation, the model defined in Equation 2 becomes
or
Therefore, it follows that when ß12 = 0, the combination
of components 1 and 2 is additive, i.e., the isobologram is coincident with
the line of additivity and the interaction index equals 1. Similarly, when ß12
> 0, the combination of components 1 and 2 is synergistic, i.e., the isobologram
bows below the line of additivity and the interaction index is < 1; and when
ß12 < 0, an antagonism is present, i.e., the isobologram
bows above the line of additivity and the interaction index is > 1. This
demonstrates the algebraic equivalence between the statistical model and the
interaction index. Gennings et al (16) demonstrated the experimental
convergence of the statistical modeling approach and the interaction index.
The number of components that can be considered in the statistical model can
be generalized to c, and data variability is appropriately accounted
for in the resulting inference.
The various methods used to determine and classify departures from additivity
described above use both single-compound data as well as combination data. Consider
the situation in which the single-compound dose-response data are not available.
The approach discussed in the following sections of this article allows one
to test the null hypothesis of additivity using only combination/mixture data
collected along a fixed ratio ray. Thus, single-compound dose-response
data are not needed.
New Methodology
Problems with statistical modeling are associated with the size of the experiment
required to generate data to support the model. Factorial experiments, e.g.,
2c or 3c, are often considered. When c
is large such experiments may not be feasible, so in a sense, this approach
is limited to combinations of relatively few drugs or chemicals. An alternative
to the traditional factorial design for studying interactions, and the design
to be considered here, is the ray design. Ray designs, described by Martin (17),
Mantel (18), Finney (19), Bruden and Vidmar (20), and others,
are used to study mixtures of c drugs or chemicals
at a fixed mixing ratio, [a1:a2:...:ac],
where (sum)i = 1c
with the total dose, t, varying. The fraction of the total dose represented
by the ith drug or chemical is ai, and the amount of
the ith drug or chemical in the mixture is ait. This
approach is appealing because the dimensionality of the study is reduced along
each ray, i.e., each ray can be considered as an individual drug or chemical,
with only the total dose varying. For example, in a study involving c
drugs or chemicals, the fitted model based on a response surface approach is
a (c + 1)-dimensional surface. In contrast, the fitted model based on
a ray design defines a set of two-dimensional dose-response curves.
What can be stated about departures from additivity in the mixture? Meadows
(21) showed that when mixture data are collected along a fixed ratio
ray, the additivity model reduces to a simple linear regression model. In addition,
the interaction model reduces to a higher-order polynomial model. Thus, the
test for additivity is equivalent to the test of the adequacy of the simple
linear regression model. Consider that the combination/mixture of interest involves
c drugs or chemicals and that the response of interest is continuous.
The underlying additivity model, i.e., the model with no cross-product terms
is defined by
Y = ß0 + ß1x1 +
... + ßcxc , [3]
where
Y is the observed response,
xi is the dose of the ith drug or chemical,
ß0 is an unknown parameter associated with the intercept,
and
ßi is an unknown parameter associated with the slope
of the ith drug or chemical.
When the mixing ratios are invoked, the dose of the ith drug or chemical
is xi = ait, where ai
is the mixture fraction for the ith drug or chemical and t is
the total dose. As a result, the additivity model becomes
[4]
where
For convenience, we assume the experimental region along the fixed ratio ray
in terms of total dose is transformed to the region -1 ¾ t
¾ 1. Thus, under additivity, the dose-response relationship along
the ray can be described with a simple linear regression on total dose.
It also follows that when the slope of the regression line for total dose
is ß1a1 + ß2a2
+ ... + ßcac , the interaction index, defined
in Equation 1 equals 1. This would suggest that single-component dose-response
data would be required. In the absence of single-chemical data, the slope ßi
for each drug/chemical alone is unknown, so that the hypothesis
H0:ß1* = (sum)i=1c
ßiai cannot be directly tested.
However, consider the following model:
Notice that this model is the model that would be supported by a factorial
experiment, and the ßij,ßijk,...ß12...c
terms are coefficients associated with the various two-factor, three-factor,
and higher-order interactions. The ray design will not support this model; however,
invoking the mixing ratio associated with the ray design results in
[5]
It follows that interactions among pairs of chemical components are associated
with second-degree terms, interactions among three chemicals are associated
with third degree terms, and so on.
Of ultimate interest is the determination of departure from additivity among
a particular combination/mixture of drugs or chemicals. When comparing the model
under additivity to the interaction model, defined in Equations 4 and 5, respectively,
evidence of curvature indicates departure from additivity, i.e., there is interaction
among the compounds if at least one ß*i ‚
0, i = 1,...,c. Thus, any polynomial lack of fit associated with
the additive model Y = ß0 + ß1*t
would be associated with a lack of additivity. Meadows (21) showed
that the test statistic for the null hypothesis of additivity,
is given by
[6]
Experimental Design
Experimental design implications for studying a c component mixture
include the following:
- Place a minimum of c + 1 points on the ray of interest to maximize
the power of the test for lack of fit of the additivity model, i.e., Y
= ß0 + ß1*t.
- Replicate the experiment at these points to make the lack of fit test possible.
When the response variable is continuous and the method of least squares has
been used to estimate the model parameters, Meadows (21) showed that
we can incorporate the statistical results of Jones and Mitchell (22)
to determine values of total dose that maximize the design's ability to detect
lack of fit or departure from additivity.
The overall lack of fit answers the question of whether there is a departure
from additivity. Rejection of his hypothesis that simultaneously tests that
the interaction parameters are equal to zero, i.e., H0: ß2
= ß3 =...= ßc = 0, implies that interaction
is present among the chemicals globally. Thus, if the overall test for additivity
is rejected, tests of the form
using Hochberg's (23) correction for multiple testing, can be used
to answer the question of whether a j-factor interaction exists. If such
an interaction is detected, recall that
Here, interest will be focused on which of the j-factor interactions
are present. This can be determined by performing 1cj2
additional ray experiments at the same ratio as were present on the original
ray.
Illustration
The methodology introduced in this article is illustrated with cytotoxicity
data obtained from assessing interactions among As, Cd, Cr, and Pb in human
keratinocytes. The experimental data were obtained from R. Yang and colleagues
at Colorado State University (Fort Collins, Colorado). The end point of interest
is the percent viability of treated NHEK (normal human epidermal keratinocytes)
cells using the MTT (3-[4,5-dimethylthiazol-2-yl]-2,5-diphenyltetrazolium bromide)
assay. The mixture point of interest for As, Cr, Cd, and Pb contained the estimated
dose/concentrations associated with 50% lethality (LD50s) of 7.7
µM, 4.9 µM, 6.1 µM, and 100 µM, respectively. This 1X solution
was serially diluted at a 1:3 ratio to get 0.333, 0.111, 0.037, 0.0123, 0.004,
and 0.0014 dilution groups. Double deionized water was used as the vehicle control
in all cases.
After exposure to individual metals or metal exposure, cells were re-fed with
fresh metal-free KGM medium and incubated for 3 days prior to viability analysis
by the MTT assay. Details of the experimental protocol and methods are described
elsewhere (1,24) and are not included here. The summary statistics for
the LD50 mixture data presented in Table 1 are linearized cytotoxicity
response data from Gennings et al. (1).
The nonlinear additivity model selected for fitting the single-compound data
by Gennings et al. (1) was based on a Gompertz function where the mean
viability was modeled as
From this model, 
is the parameter associated with the minimum mean response, and 
is the range of mean response values. Therefore, for this example, it is reasonable
to assume that a transformation on the response, conditioning on the values
of
= 8.76 and =
109 obtained by Gennings et al. (1), will induce linearity in the additivity
model. As a result, the additivity model becomes
As shown in "New Methodology," because the mixture data were collected along
a fixed ratio ray, the additivity model can be rewritten as
[7]
Additionally, the interaction model along the same fixed ratio ray becomes
[8]
Therefore, conditioning on the values of
= 8.76 and =
109, the transformation
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Figure 2. Observed responses
and fitted curve under the additivity model for the fit of the mixture
data using the NHEK cells.
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on the observed responses for the mixture data was performed. The additivity
model given in Equation 7 and the interaction model given in Equation 8 were
fit to the mixture data using the method of least squares. The GLM (general
linear model) procedure of SAS (25) was used to estimate the unknown
parameters in Equations 7 and 8. Parameter estimates and their p values
are provided in Table 2. Figure 2 presents the fitted concentration effect curve
under additivity for total concentrations of the four metals along the ray associated
with the LD50 mixing ratio. Asterisks (*) indicate the observed transformed
responses at the seven dilution points. From this figure, there is some question
as to whether the data fall along the line of additivity. In comparison, Figure
3 presents the observed mixture data and the fitted interaction (higher-order
polynomial) model. Dots (•) indicate the design locations of the total
dose values selected by the 
1-optimal
design, which are presented by total doses of 0, 16, 59.4, 102.7, and 118.7
µM. Notice that the values selected as the 1-optimal
design are symmetrically spread throughout the total dose region, whereas the
majority of the points used in the current study are directed toward the lower
total dose region. An enlarged version of the lower total dose region of the
plot of the fitted interaction model is presented in Figure 4. The test statistic
for the null hypothesis of additivity,
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Figure 3. Observed and
predicted responses for the higher-order polynomial model for the fit
of the mixture data using the NHEK cells. Dots (•) indicate the
design locations of the total dose values selected by the 1-optimal
design.
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Figure 4. Observed and
predicted responses for the higher-order polynomial model for the fit
of the mixture data using the NHEK cells (enlarged).
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is given by
Table 3 presents the overall test for departure from additivity given in Equation
6. Based on this test, we reject the null hypothesis of additivity (p
value < 0.001) and conclude that at least one of the j-factor interactions
exists, j = 2,...4. Because the overall test for additivity is rejected,
it is of interest to determine whether two-, three-, or four-factor interactions
are present. Therefore, we want to test the following hypotheses using Hochberg's
correction (23) for multiple testing:
Table 3 also presents the single-parameter tests associated with each of the
j-factor interactions, j = 2,...4. Using Hochberg's (23)
correction for multiple comparisons, all three parameters are significantly
different from zero when the overall significance level is set at 5%. However,
if we consider the case where an overall significance level of 1% is used, only
the two smallest p values are significant using Hochberg's correction
(23). Therefore, we conclude that a three-factor interaction exists,
implying that the two-factor interactions are not constant. Now it is of interest
to determine which three metals are interacting with one another. This can be
accomplished by performing 1432 = 4 additional
experiments at the same ratios of metals that were used in the original ray.
Table 4 gives the ratios of compounds along with the corresponding total dose
to be used for the four additional experiments. This approach limits the inferences
of the original experiment, as well as the four additional experiments, to be
made about the particular mixing ratio used in the experiment.
Conclusion
It was shown that the classic methodology used in evaluating an interaction
requires single-drug/chemical data. In "New Methodology" it was shown that the
evaluation of interactions could be accomplished with a ray design that did
not generate single-drug and single-chemical data. When a ray design is used,
departure from additivity is associated with higher-order polynomial terms in
a linear model. Additivity, or absence of interaction, is described by a simple
linear model in terms of total dose. As a result, we have shown that we can
obtain information about departures from additivity from data collected along
a fixed ratio ray. This result is important in that it permits a reduction in
the total experimental effort for studying a combination when compared with
that associated with a traditional factorial design. Additionally, by incorporating
the approach taken by Jones and Mitchell (22), we have presented methodology
for determining optimal levels along the fixed ratio ray (total dose) to be
considered in the experiment for detecting model inadequacy.
References and Notes
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Last Updated: January 17, 2002
