Ds-optimal designs for studying combinations of chemicals using multiple fixed-ratio ray experiments

ENVIRONMETRICS

Environmetrics 2005; 16: 129–147

Published online 13 December 2004 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/env.666

Ds-optimal designs for studying combinations of chemicalsusing multiple fixed-ratio ray experiments

Michelle Casey1, Chris Gennings2*,y, W. Hans Carter Jr.2,Virginia C. Moser3 and Jane Ellen Simmons3

1GlaxoSmithKline, Collegeville, PA, U.S.A.2Department of Biostatistics, Virginia Commonwealth University, Medical College of Virginia, Richmond, VA, U.S.A.

3U.S. EPA National Health and Environmental Effects Laboratory/Office of Research and Development,Research Triangle Park, NC, U.S.A.

SUMMARY

Detecting and characterizing interactions among chemicals is an important environmental issue. Traditionalfactorial designs become infeasible as the number of compounds under study increases. Ray designs, whichreduce the amount of experimental effort, can be considered when interest is restricted to relevant mixing ratios.Simultaneous tests for departure from additivity across multiple fixed-ratio rays in the presence and absence ofsingle chemical data have been developed. Tests for characterizing interactions among subsets of chemicals atrelevant mixing ratios have also been developed. Of primary importance are precise estimates for the parametersassociated with these hypotheses. Since the hypotheses of interest are stated in terms of subsets of parameters, wehave developed a methodology for finding Ds-optimal designs, which are associated with the minimumgeneralized variance of subsets of the parameter vector, along fixed-ratio rays. We illustrate these methods bycharacterizing the interactions of five organophosphorus pesticides (full-ray) as well as a subset of pesticides(reduced-ray) on a measure of motor activity. Copyright # 2004 John Wiley & Sons, Ltd.

key words: additivity; efficiency; interaction

1. INTRODUCTION

Understanding the potential health hazards of chemical mixtures presents many challenges for

toxicology researchers and regulators alike. Proposed goals for mixtures research in toxicology and

applications to risk assessment were described recently in a consensus paper published by the Society

of Toxicology Expert Panel on Mixtures (Teuschler et al., 2002). A scientifically based approach to the

risk assessment of mixtures involves biologically based hypotheses and experimental approaches.

Recent advances in the statistical modeling of mixture data have moved the field from the analysis of

binary combinations to analysis of multiple chemicals using response surfaces or ray designs.

Received 17 June 2003

Copyright # 2004 John Wiley & Sons, Ltd. Accepted 21 December 2003

*Correspondence to: C. Gennings, Department of Biostatistics, Virginia Commonwealth University, 1101 E. Marshall St,#B1-039-A, Richmond, VA 23298-0032, U.S.A.yE-mail: [email protected]

Contract/grant sponsor: U.S. Environmental Protection Agency; contract/grant numbers: CR-827208-01; R82811401-0.

Ray designs have been proposed as an alternative to factorial designs in the study of chemical

mixtures. Brunden and Vidmar (1989) and others suggest the use of ray designs to support the

estimation of a response surface. Alternatively, Gennings et al. (2002) and Meadows et al. (2002)

propose restricting inference to relevant fixed-ratio ray(s). Focusing inference to fixed-ratio rays

reduces the dimensionality and amount of experimental effort associated with the study of chemical

mixtures. Furthermore, the chemical ratio within the mixture allows the evaluation of environmentally

relevant mixtures. Gennings et al. (2002) and Meadows et al. (2002) developed tests for departures

from additivity using these fixed-ratio ray designs. Casey (2003) extended this work to include

hypotheses that test for interactions involving subsets of chemicals. Such hypotheses, described in

Section 2, require the collection of data along a full fixed-ratio ray as well as a reduced ray where a

chemical or subset of chemicals is removed from the mixture.

Studies of mixtures toxicity can be simplified by grouping chemicals with a common mode of

action. This approach was codified in the 1996 Food Quality Protection Act, which directed the U.S.

Environmental Protection Agency (U.S. EPA) to base regulatory decisions on mixtures of pesticides

which act through a common mode of action, rather than on individual pesticides. Acetylcholinester-

ase-inhibiting organophosphorus (OP) pesticides were the first class to undergo a cumulative risk

assessment (U.S. EPA, 2002). With numerous agricultural, household, pet and garden uses, OPs have

been a widely used class of pesticides and thus there is high potential for cumulative exposures to OPs.

To apply the fixed-ratio designs developed by our group, we tested the additivity hypothesis using a

mixture of five OP pesticides: acephate, diazinon, chlorpyrifos, dimethoate and malathion. These

pesticides were chosen for study based on usage patterns (i.e. pesticides used on the same or similar

crops: USDA, 2000; FDA, 2000) and market share (i.e. highest volume pesticides: U.S. EPA, 1999).

The ratio of pesticides in the test mixture was determined by comparing the relative dietary exposure

to humans as projected by the U.S. EPA Dietary Exposure Evaluation Model (DEEM: U.S. EPA,

2001). Neurobehavioral and neurochemical endpoints were evaluated following acute exposure to the

mixture. The design and methodology described here utilize motor activity data.

Given an outcome of non-additivity among the five pesticides, a second study was designed to

determine the influence of malathion. The literature reports greater-than-additive interactions in binary

combinations of malathion and some other OP pesticides, presumably due to a toxicokinetic

interaction which slows pesticide detoxification (e.g. Murphy and DuBois, 1957; Cohen and Murphy,

1971). Traditionally, interactions between malathion and the other four pesticides would be assessed

with factorial designs studying binary or tertiary chemical combinations. However, the use of factorial

designs to consider the effects of exposure to all possible combinations of five chemicals at multiple

dose levels is not practical.

This article describes the development of experimental design methodology for detecting and

characterizing interactions among these pesticides by use of fixed-ratio rays. It is important to

determine appropriate test doses (dose locations) and sample size allocation that provide precise

parameter estimates. We extend the work of Meadows (2001) and Meadows et al. (2004) to develop

designs for testing the effect of subsets of chemicals on the mixture (i.e. to test for non-additive

interactions involving subsets of chemicals) and for detecting departure from additivity across

multiple fixed-ratio rays simultaneously. The criterion we consider aims to minimize the generalized

variance of the parameters associated with the hypothesis of interest. While this criterion cannot claim

to maximize the power of the test of the hypothesis of interest, it should be related to an increase in

power since it reduces a measure of the variance of the estimates involved. To determine experimental

designs, appropriate models and hypotheses must be defined. The designs illustrated in this article are

based on the methods for detecting departure from additivity presented by Gennings et al. (2002),

130 M. CASEY ET AL.

Copyright # 2004 John Wiley & Sons, Ltd. Environmetrics 2005; 16: 129–147

Meadows et al. (2002) and Casey (2003) and are summarized in Section 2. Thus, the objective of this

article is to provide methodology to determine at what dose levels and in what proportion of the total

sample size observations should be taken for studies of mixture rays.

2. MODEL AND HYPOTHESES DEVELOPMENT

2.1. Additivity model

A generalized linear model is used to describe the relationship among the pesticides along the

two fixed-ratio rays. In the quasi-likelihood framework, proposed by Wedderburn (1974), the

generalized linear model relates the doses of the chemicals to the mean through a link function,

gð�Þ. Further, it assumes that the variance of the response is of the form �V(�), where V(�) is a knownfunction of �.

Let K be the number of fixed-ratio rays under consideration for a mixture of c chemicals and

aðkÞ ¼ ½a1ðkÞ; a2ðkÞ; . . . ; acðkÞ�, define the mixing ratio along the kth fixed-ratio ray, such thatPci¼1 aiðkÞ ¼ 1. Let xðkÞ ¼ ½x1ðkÞ; x2ðkÞ; x3ðkÞ; . . . ; xcðkÞ� define a vector of doses at a given mixture

point along the kth ray and t ¼ Pci¼1 xiðkÞ define total dose. When ray designs are considered, total

dose is the independent variable along the mixing rays and the amount of the ith compound in the

mixture along the kth ray is given by aiðkÞt. Following Meadows et al. (2002), the additivity (i.e. no

interaction) model along the kth fixed-ratio ray can be expressed as

gð�addðkÞÞ ¼ �0 þ �1a1ðkÞt þ �2a2ðkÞt þ � � � þ �cacðkÞt

¼ �0 þ ð�1a1ðkÞ þ �2a2ðkÞ þ � � � þ �cacðkÞÞt¼ �0 þ ��1ðkÞt

ð1Þ

where:

gð�addðkÞÞ is the specified link function (see McCullagh and Nelder, 1989)

�0 is the unknown parameter associated with the intercept

��1ðkÞ ¼Pc

i¼1 �iaiðkÞ is the unknown parameter associated with the slope along the kth ray.

2.2. Single chemical not required (SCNR) method of analysis

Meadows et al. (2002) and Casey (2003) describe a method of analysis that does not require

single chemical data where assumptions are made about the form of the underlying response

surface. The significance of higher-order terms in the polynomial approximation of the dose–

response relationship, expressed as a function of the total dose, indicates departure from additivity.

Meadows et al. (2002) showed that with this approach, only mixture data along a fixed-ratio ray are

necessary for detecting departure from additivity. Casey (2003) extended these results to detect

departure from additivity across multiple fixed-ratio rays in the presence or absence of single chemical

data.

Using the quasi-likelihood framework, the interaction model along the kth ray can be expressed

as

gð�mixðkÞÞ ¼ �0 þ ��1ðkÞt þ ��2ðkÞt2 þ ��3ðkÞt

3 þ � � � þ ��ckðkÞtck ð2Þ

OPTIMAL DESIGNS FOR CHEMICAL MIXTURES ON FIXED-RATIO RAYS 131


where:

��1ðkÞ ¼Xcki¼1

�iaiðkÞ; ��2ðkÞ ¼Xcki¼1

Xckj¼1

�ijaiðkÞajðkÞ;

i< j

��3ðkÞ ¼Xcki¼1

Xckj¼1

Xckl¼1

�ijlaiðkÞajðkÞalðkÞ

i<j<l

; etc:

gð�mixðkÞÞ is the specified link function (see McCullagh and Nelder, 1989)

��1ðkÞ is the unknown parameter associated with the first-order term

��iðkÞ for i ¼ 2; . . . ; ck is the unknown parameter associated with the ith-way interaction ck is the

number of chemicals under study along the kth fixed-ratio ray.

When single chemical data are available, the slope associated with each of the single chemicals is

estimable. Thus, the model fitted along the kth fixed-ratio ray is given by

gð�mixðkÞÞ ¼ �0 þ �1a1ðkÞt þ � � � þ �ckackðkÞt þ ��2ðkÞt2 þ ��3ðkÞt

3 þ � � � þ ��ckðkÞtck ð3Þ

The K fixed-ratio rays are generally fitted simultaneously assuming a common intercept. If multiple

control groups are experimentally evaluated, the assumption of a common intercept should be verified

by comparing the means of the control groups. When single chemical data are available they are used

in combination with mixture data to estimate the intercept and first-order terms and the higher-order

terms are estimated using only the mixture data. In the absence of single chemical data, all of the

model parameters are estimated using the mixture data. Parameter estimates are found by maximizing

the quasi-likelihood (e.g. McCullagh and Nelder, 1989).

Under the hypothesis of additivity, the parameters associated with interaction along the K fixed-

ratio rays, namely (��2ð1Þ; ��3ð1Þ; . . . ; �

�c1ð1Þ; �

�2ð2Þ; �

�3ð2Þ; . . . ; �

�c2ð2Þ; . . . ; �

�2ðKÞ; �

�3ðKÞ; . . . ; �

�cKðKÞÞ, are zero.

Define the p� 1 vector c ¼ ½�0; ��1ð1Þ; ��2ð1Þ; . . . ; ��c1ð1Þ; . . . ; ��1ðKÞ; ��2ðKÞ; . . . ; ��cK ðKÞ�0as a vector of model

parameters and define badd as a matrix of contrasts (zeros and ones) such that

badd c ¼ ��2ð1Þ; ��3ð1Þ; . . . ; �

�c1ð1Þ; . . . ; �

�2ðKÞ; . . . ; �

�ckðKÞ

h i0

An overall hypothesis of additivity, based on the significance of higher-order terms, is given by

H0 : baddc ¼ 0 ð4Þ

In addition to the hypothesis described above, Casey (2003) developed hypotheses for detecting and

characterizing interactions due to subsets of chemicals. The hypothesis that a chemical or subset of

chemicals is not involved in interactions with the remaining components of the mixture is given by

H0 : binteractc ¼

��2ðfullÞða1ðfullÞÞ2

� ��2ðreducedÞða1ðreducedÞÞ2


� ��3ðreducedÞða1ðreducedÞÞ3

..

.

��mðfullÞða1ðfullÞÞm

� ��mðreducedÞða1ðreducedÞÞm

266666666664

377777777775¼ 0 ð5Þ

132 M. CASEY ET AL.


where the reduced ray (parameters denoted with subscript ‘reduced’) represents the mixture study

where the chemical or subset of chemicals of interest is removed from the mixture and the remaining

components are studied at the same relative ratios as given in the full ray. Parameters along the full ray,

where all of the chemicals are considered in the mixture, are denoted with subscript ‘full’. This

hypothesis compares interaction parameters along the reduced ray (e.g. the no-malathion ray) to

interaction parameters along the full ray.

For example, consider a mixture study with three chemicals. Let (a1(full): a2(full): a3(full)) denote the

mixing ratio. The model along the fixed-ratio ray is given by

gð�ðfullÞÞ ¼ �0 þ ��1ðfullÞt þ ��2ðfullÞt2 þ ��3ðfullÞt

3

where:

��1ðfullÞ ¼ �1a1ðfullÞ þ �2a2ðfullÞ þ �3a3ðfullÞ

��2ðfullÞ ¼ �12a1ðfullÞa2ðfullÞ þ �13a1ðfullÞa3ðfullÞ þ �23a2ðfullÞa3ðfullÞ

��3ðfullÞ ¼ �123a1ðfullÞa2ðfullÞa3ðfullÞ

Suppose the third-order term is not significant and it is of interest to determine the effect of the third

chemical on the mixture. Thus, data are collected along a second fixed-ratio ray given by (a1(reduced):

a2(reduced)), where the third chemical is removed from the mixture and

aiðfullÞajðfullÞ

¼ aiðreducedÞajðreducedÞ

The model fitted along this reduced ray is given by

gð�ðreducedÞÞ ¼ �0 þ ��1ðreducedÞt þ ��2ðreducedÞt2

where:

��1ðreducedÞ ¼ �1a1ðreducedÞ þ �2a2ðreducedÞ

��2ðreducedÞ ¼ �12a1ðreducedÞa2ðreducedÞ

Since the reduced experiment is performed at the same relative ratios as those in the original or full ray,

�12a1ðfullÞa2ðfullÞða1ðfullÞÞ2

¼ �12a1ðreducedÞa2ðreducedÞ

ða1ðreducedÞÞ2¼

��2ðreducedÞða1ðreducedÞÞ2

and��2ðfullÞ

ða1ðfullÞÞ2becomes


¼��2ðreducedÞ

ða21ðreducedÞÞ2

þ �13a1ðfullÞa3ðfullÞða1ðfullÞÞ2


)��2ðfullÞ

ða1ðfullÞÞ2�


¼ �13a1ðfullÞa3ðfullÞða1ðfullÞÞ2




The hypothesis, given by

H0 :��2ðfullÞ

ða1ðfullÞÞ2�


" #¼ 0

tests for the effect of the third chemical on the mixture (i.e. the significance of �13 and/or �23). If thereis sufficient evidence to reject this hypothesis, we conclude that the third chemical is involved in at

least one two-way interaction. Further experimentation would be necessary to determine specifically

which chemicals are involved in these interactions. If we fail to reject the hypothesis and the study is

reasonably powered, we can conclude that the third chemical does not interact with the remaining

components of the mixture.

AWald-type test for testing the hypotheses given in (4) and (5) is given by

W ¼ ðbcÞ0½bXb0��1ðbcÞM�

ð6Þ

where �X is the variance–covariance matrix of cc and b is any contrast matrix (e.g. badd; binteract).

Since cc is distributed asymptotically normal (McCullagh and Nelder, 1989) with mean c and variance�X, it follows that W is approximately distributed chi-square with M degrees of freedom (M is the

number of tests considered simultaneously or the number of rows in b). The moment estimate for � isexpressed as

�� ¼ 1

ðN � pÞXi;j

ðyij � ��iÞ2Vð��iÞ ¼ X2

ðN � pÞ ð7Þ

where X2 is the generalized Pearson statistic (McCullagh and Nelder, 1989), which is asymptotically

distributed chi-square with N� p degrees of freedom. In the quasi-likelihood framework, McCullagh

(1983) defines the large sample variance–covariance matrix for cc as � ½IðcÞ��1, where IðcÞ is the

expected quasi-information matrix. Let N be the number of observations. McCullagh (1983) expressed

the expected quasi-information matrix as

IðcÞ ¼ D0ðVð�ÞÞ�1D ð8Þ

where

D ¼ @�i@�0j

� �i¼1;2;...;Nj¼1;2;...;p

and Vð�Þ ¼Vð�1Þ 0 � � � 0

0 Vð�2Þ � � � 0

..

. ... . .

.0

0 0 � � � Vð�NÞ

26664

37775

Replacing c with cc in (8), XX ¼ ½IðccÞ��1is a consistent estimate forX (McCullagh, 1983). Replacing �

with �� and X with XX in (6), W is approximately distributed F with M numerator degrees of freedom

and N� p denominator degrees of freedom, where p is the number of parameters.

134 M. CASEY ET AL.


3. DATA DESCRIPTION

The methods in the previous section permit applying experimental design techniques to the testing of

complex statistical hypotheses. To demonstrate these methods, we consider the situation where after

an interaction in a mixture of chemicals is detected along a full ray, it is of interest to design a study

(i.e. to determine the total dose locations and allocations of experimental subjects) for a reduced ray to

determine whether a subset of the mixture components interacts with the remaining chemicals. The

data for this illustration are the five OP pesticides described in the Introduction.

Motor activity was the endpoint used. In brief, adult male Long-Evans rats were administered the

test chemical(s) via oral gavage. Motor activity was measured as photobeam interruptions in a

chamber shaped like a figure-8 (see Ruppert et al., 1985). Counts were tabulated electronically over a

30-min test session, which began 4 h after dosing. Dose–response data were collected for each

pesticide, followed by the mixture dose–response (summary statistics are provided in the Appendix).

The proportions of the five OPs in the mixture were: acephate, 0.040; diazinon, 0.002; chlorpyrifos,

0.031; dimethoate, 0.102; malathion, 0.825. Preliminary analysis of single chemical data indicated

that malathion was ineffective in altering motor activity, even at relatively high doses (�500mg/kg).

Since motor activity is a count variable with a positive mean, a ln link function (McCullagh and

Nelder, 1989) was used to fit the generalized linear additivity model given in (1) and the variance was

assumed to be of the form ��. A plot of this additivity model, seen in Figure 1, was useful in specifying

the active dose range up to 450mg/kg along the ray. An equally spaced design with equal allocation

of experimental subjects and extra dose points in the low dose region was used for the full ray

(see Appendix). This design was one of convenience and did not satisfy any statistical optimality

criteria.

Figure 1. Predicted generalized linear interaction model (dotted line) fit with the loge link function using single chemical data

and mixture data along the full five-pesticide fixed-ratio ray (0.040: 0.002: 0.031: 0.102: 0.825) and the additivity (first-order)

model (solid line) where the parameter estimates are provided in Table 1. The asterisks represent the sample mean motor activity

responses



Since the mixture contained five chemicals, the single chemical and mixture data were fitted to the

fifth-order generalized linear model given in (3). Maximum quasi-likelihood estimates were found

using the Gauss–Newton algorithm (PROC GENMOD in SAS1 (version 8.2)) and are provided in

Table 1. The hypothesis given by

H0 : baddc ¼��2ðfullÞ��3ðfullÞ��4ðfullÞ��5ðfullÞ

2664

3775 ¼ 0 ð9Þ

where c ¼ ½�0; �1; �2; �3; �4; �5; ��2ðfullÞ; ��3ðfullÞ; ��4ðfullÞ; ��5ðfullÞ�0 and

badd ¼0 0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 0 1

2664

3775

was used to test the hypothesis of additivity. The resulting p-value was 0.007; thus, there was sufficient

evidence to reject this hypothesis and conclude that departure from additivity exists along the full five

pesticide fixed-ratio ray. The fifth-order interaction term was not significant (p-value, 0.22); thus, it was

removed from the model. In addition, the parameter associated with malathion was removed from the

model due to lack of significance (p-value, 0.85) which indicates the lack of activity of malathion. The

fourth-order interaction term was marginally significant with a p-value of 0.07. Although marginal, we

may conclude that up to fourth-order interactions exist along the full ray; however, without further

experimentation, we cannot determine which chemicals are involved in these interactions.

The existence of fourth-order interactions along the full ray provided motivation for studying the

influence of malathion on interactions. That is, it was of interest to determine if malathion interacts

with the other four pesticides in the mixture. The methodology developed in Section 2 using a reduced

ray can be used to test such a hypothesis. The reduced ray (i.e. the other four pesticides) uses the same

relative ratios as those in the full ray (i.e. acephate: 0.229; diazinon, 0.011; chlorpyrifos, 0.177;

dimethoate, 0.583). For example, the ratio of acephate to diazinon in the full ray (0.040:0.002) was

Table 1. Estimated model parameters where the single chemical and mixture data along the full fixed-ratio rayare fitted to the model given in (3) with the ln link (allowing Vð�Þ ¼ �Þ

Parameter Estimate SE p-value

�0 5.326 0.020 < 0.0001�1 (acephate) �0.018 0.001 < 0.0001�2 (diazinon) �0.004 0.0004 < 0.0001�3 (chlorpyrifos) �0.025 0.002 < 0.0001�4 (dimethoate) �0.0123 0.001 < 0.0001��2 (2nd-order interactions) �3� 10�5 1� 10�5 0.0097��3 (3rd-order interactions) 1.5� 10�7 7.2� 10�8 0.0345��4 (4th-order interactions) �1.9� 10�10 1� 10�10 0.0714� 3.503

Note:P5

i¼1 �iaiðfullÞ ¼ ��1ðfullÞ ¼ �0:0028. The fifth-order interaction term was removed from the model due to lack ofsignificance (p-value, 0.2203). Similarly, the slope for malathion was removed from the model (p-value, 0.8485).

136 M. CASEY ET AL.


20:1, as was the ratio in the reduced ray (0.229:0.011) except for roundoff error. This hypothesis is

given by

H0 : binteractc ¼


��2ðreducedÞ

ða1ðreducedÞÞ2��3ðfullÞ

ða1ðfullÞÞ3�



��4ðreducedÞ

ða1ðreducedÞÞ4

2666666664

3777777775¼ 0 ð10Þ

where c ¼ ½�0; �1; �2; �3; �4; ��2ðfullÞ; ��3ðfullÞ; ��4ðfullÞ; ��2ðreducedÞ; ��3ðreducedÞ; ��4ðreducedÞ�0 and

binteract¼

0 0 0 0 01

ð0:04Þ2 0 0 � 1

ð0:2286Þ2 0 0

0 0 0 0 0 01

ð0:04Þ3 0 0 � 1

ð0:2286Þ3 0

0 0 0 0 0 0 01

ð0:04Þ4 0 0 � 1

ð0:2286Þ4

26666664

37777775

The objective of this article is to provide methodology to determine at what dose levels and in what

proportion of the total sample size observations should be taken.

4. EXPERIMENTAL DESIGNS

Ds-optimality is a special case of D-optimality, defined by Kiefer and Wolfowitz (1959). D-optimal

designs minimize the generalized variance of the model parameters, where the generalized variance

is defined as the determinant of the variance–covariance matrix. While this criterion is not directly

related to optimal hypothesis test power, use of such designs should result in increased power since

the parameters of interest are estimated more precisely. Ds-optimality minimizes the generalized

variance for a subset of model parameters. We have seen in (4) and (5) that certain hypotheses of

interest in chemical mixture problems can be expressed in terms of subsets of the parameter vector.

Ds-optimality is appropriate when the precision of a subset of parameters is of interest (Atkinson and

Donev, 1992).

In the quasi-likelihood framework, the determinant of the variance–covariance matrix is given by

�X�1�� , where X is the expected quasi-information matrix given in (8). When the hypothesis of

interest is given by a subset of parameters (e.g. the hypothesis of additivity given in (4)) or is based on

linear combinations of model parameters (e.g. the hypothesis given in (5)), the variance–covariance

matrix is given by �ðbXb0Þ�1, where b is an appropriate matrix of contrasts (e.g. badd; binteract). The

expected quasi-information matrix depends on parameter values, total dose values, and the allocation

of observations to the dose groups. Given expected parameter values, the total sample size, and the

form of the hypothesis of interest, the Nelder–Mead algorithm (Nelder and Mead, 1965) can be used to

determine the total dose levels and the allocation of observations that minimize the generalized

variance, �ðbXb0Þ�1�� , for the model parameters of interest.



For clarity, consider a ray with S distinct design points (here, in terms of total dose, t). Using these

design points the mixture model gð�iðc; tiÞÞ with the specific form given in (3) can be estimated,

i ¼ 1; . . . ; S. The basis of the Ds-optimality criterion is that the generalized variance is a function of

the S total dose values and the number of replications at each point. These are introduced through theD

matrix in (8), which can be expressed as

D ¼

@�11ð�; t1Þ=@�0...

@�1n1ð�; t1Þ=@�0

9>>=>>; n1 � p

@�21ð�; t2Þ=@�0...

@�2n2ð�; t2Þ=@�0

9>>=>>; n2 � p

..

.

@�S1ð�; tSÞ=@�0...

@�SnSð�; tSÞ=@�0

9>>=>>; nS � p

26666666666666666666664

37777777777777777777775

where ni; i ¼ 1; . . . ; S, represents the number of replications at the ith total dose group such that

N ¼XSi¼1

ni

Using the methods described in Section 2, it is of interest to determine the design along the reduced

ray which minimizes the generalized variance associated with the parameters involved in the

hypothesis, given in (10), that malathion does not interact with the four active pesticides. The

Ds-optimal design methods condition on the total sample size, the model parameters, and the number

of dose groups of interest. Thus, these values must be specified prior to implementing the methods

developed here. Since the use of polynomial models can be justified as a basis for mixture models on

fixed-ratio rays, it is difficult to specify parameter values that may be associated with a biologically

meaningful alternative hypothesis.

In what follows we describe three scenarios that lead to reasonable alternative hypotheses. Values

for the intercept ð�0Þ, single chemical slopes ð�1; �2; �3; �4Þ, interaction parameters along the full ray

(��2ðfullÞ; ��3ðfullÞ; �

�4ðfullÞ), and the variation parameter (�) are provided from the analysis performed on the

full ray (Table 1). Preliminary mixture data along the reduced fixed-ratio ray would be ideal for

specifying the additional higher-order interaction terms (��2ðreducedÞ; ��3ðreducedÞ; �

�4ðreducedÞ). In the

absence of such information, the higher-order interaction parameters along the full ray are used as

a guide to specify similar parameters along the reduced ray under the alterative hypothesis that

malathion is involved in interactions with the four active pesticides. Since ��2ðfullÞ; ��3ðfullÞ; �

�4ðfullÞ,

a1(full) and a1(reduced) have been specified we can determine parameter values along the reduced ray

under the null hypothesis (i.e. under the assumption that malathion does not interact with the

138 M. CASEY ET AL.


remaining pesticides). For example, consider the case where malathion is not involved in any three-

way interactions. Under this assumption:


��3ðreducedÞ

ða1ðreducedÞÞ3¼ 0

)��3ðfullÞ

ða1ðfullÞÞ3¼


)��3ðfullÞ

ða1ðfullÞÞ3� ða1ðreducedÞÞ3 ¼ ��3ðreducedÞ

ð11Þ

Parameter values along the reduced ray under the assumption that malathion does not interact with the

active pesticides are provided in Table 2 (the ‘no malathion interactions’ column).

Using the no-malathion-interaction case as a guide, we can specify values of model parameters

under the alternative hypothesis based on cases that are of interest to investigators. In the first scenario,

we consider the case that malathion is involved in only two-way interactions. In this case the third- and

fourth-order interaction terms are determined following the process defined in (11) and the second-

order term is defined based on changes in the mean values that researchers define as biologically

meaningful. In the second scenario, we consider the case where malathion is involved in two- and

three-way interactions and the third scenario is the case where malathion is involved in two-, three-

and four-way interactions. The interaction model parameters along the reduced ray for these three

cases are provided in Table 2.

In addition to using the parameter values under the null hypothesis of the no-malathion-interaction

case, plots associated with each of the three interaction scenarios are provided in Figure 2(a–c). The

additivity model along the reduced ray is also provided to aid in the interpretation of the interaction

model specified under the alternative hypothesis. These plots permit the elucidation of the effect due to

malathion under the alternative hypothesis, which is represented by the difference between the solid

and dashed curves. The design problem addressed here is the location and allocation of experimental

subjects to minimize the generalized variance associated with the test statistic given in (6).

Table 2. Model parameters along the reduced fixed-ratio ray under the null hypothesis (i.e. malathion does notinteract with the remaining pesticides) and the alternative hypotheses that (a) malathion is involved in two-wayinteractions, (b) malathion is involved in two- and three-way interactions, and (c) malathion is involved in two-,

three- and four-way interactions

Parameter No malathion Two-way only Two- and Two-, three-interactions three-way and four-way

�0 5.326 5.326 5.326 5.326�1 (acephate) �0.018 �0.018 �0.018 �0.018�2 (diazinon) �0.004 �0.004 �0.004 �0.004�3 (chlorpyrifos) �0.025 �0.025 �0.025 �0.025�4 (dimethoate) �0.012 �0.012 �0.012 �0.012��2 (2nd-order interactions) �9.8� 10�4 �9.5� 10�4 �7.84� 10�4 �6.664� 10�4

��3 (3rd-order interactions) 2.8� 10�5 2.8� 10�5 2.52� 10�5 1.96� 10�5

��4 (4th-order interactions) �2.0� 10�7 �2.0� 10�7 �2.0� 10�7 �1.4� 10�7

� 3.503

Note:P4

i¼1 �iaiðreducedÞ ¼ ��1ðreducedÞ ¼ �0:0157.



Tables of total dose values where a given percent change in the mean value occurs between the no-

malathion-interaction case and the specified alternative cases may also be useful in defining

biologically meaningful interactions. Table 3 provides the doses where a given percentage change

in the mean response is observed between the no-malathion-interaction case and the model specified

under the alternative hypothesis for the three scenarios (Table 2, Figure 2). Since the generalized linear

model is non-linear, the differences in the curves are not constant. For example, if we are interested in

detecting at least a 5 per cent change in the mean for the two-way interaction case, the minimum total

dose value that is associated with such a change in the mean is 40.3mg/kg.

Thus far, we have used available information provided by the analysis on single chemical and

mixture data along the full ray, in combination with specified alternative hypotheses that are

considered to be biologically interesting. Now we must specify the size of the design and the available

sample size. Suppose researchers indicate that a maximum sample size of 80 animals is possible for

the mixture study along the reduced ray. For a mixture of c chemicals, cþ 1 is the minimum number of

total dose groups necessary to detect departure from additivity along a fixed-ratio ray. Thus, the

minimum design along the reduced ray is a five-point design.

The Nelder–Mead algorithm (Nelder and Mead, 1965) is used to determine the optimal allocation

of the 80 available animals to five total dose groups that minimizes the generalized variance associated

with the parameter values related to the hypothesis that malathion is not involved in interactions with

the remaining four pesticides. The Nelder–Mead algorithm requires starting values for the dose levels

and the sample size allocations. For each of the three cases considered, we force the design to include a

control group, and a range of starting values (0 to 80 by 5) for the remaining total dose values were

considered. Initially, equal allocation across the dose groups was assumed. The Ds-optimal designs are

provided in Table 4. Notice that the design for the case where it is assumed that malathion is involved

in two-way interactions is the same as the design for the case where malathion is assumed to be

involved in two-, three- and four-way interactions. Also, notice that in each of the three cases the

allocation of the total sample is equal across the dose groups. Finally, if we look at the plots of the

models under the alternative hypotheses, provided in Figures 2(a–c), we see that the total dose levels

determined by the Ds-optimal designs are located where the curvature exists along the model

Table 3. Lowest total dose values along the reduced fixed-ratio ray that result in 5 per cent,10 per cent, 15 per cent and 20 per cent changes in mean responses between the model underthe null hypothesis and the model under the alternative hypotheses that (a) malathion isinvolved in 2-way interactions, (b) malathion is involved in 2- and 3-way interactions, and (c)

malathion is involved in 2-, 3- and 4-way interactions

Change in mean Two-way Two- and three-way Two-, three- and four-wayinteractions interactions interactions

5% 40.33 18.37 15.7810% 56.36 28.71 27.0415% 68.25 42.98 77.6920% 77.96 81.88 81.46

—————————————————————————————————"Figure 2. Generalized linear model along the reduced ray (0.2286: 0.0114: 0.1767: 0.5833) under the null hypothesis (i.e. no

interactions due to malathion) (solid line), under the assumption of additivity (dotted line), and under the alternative hypotheses

(dashed line) that (a) malathion is involved in second-order interactions, (b) malathion is involved in second- and third-order

interactions, and (c) malathion is involved in second-, third- and fourth-order interactions. Model parameters are provided in

Table 2

140 M. CASEY ET AL.




considered under the alternative hypothesis and where the largest differences occur between the no-

malathion-interaction model and the model under the alternative hypothesis.

For each of the three cases considered above, we found the Ds-optimal design with the minimum

number of design points, cþ 1. It is of interest to examine the effect of adding additional dose groups.

For example, consider the six-, seven- and ten-point Ds-optimal designs, provided in Table 5, for the

alternative hypothesis that malathion is involved in two-way interactions only. Recall that each of the

designs forces a control group. Notice that the five-point design places two dose groups in the middle

of the dose region at the first curve in the model and two doses in the high dose region where there is

additional curvature and the largest difference between the null and alternative models occurs. The six-

point design continues to follow this pattern. In fact the six-point design is not much different from the

five-point design as 79mg/kg is not much different from 80mg/kg. As more design points are added,

the doses are added to the middle and high dose regions and are relatively equally spaced in these dose

regions. Figure 3 illustrates the placement of dose points for the five-, six-, seven- and ten-point

designs given in Table 5.

Table 4. Five-point Ds-optimal designs for the hypothesis, given in (10), thatmalathion does not interact with the four active pesticides. The top row correspondsto the total dose locations and the bottom row of the design corresponds to thesample size allocation. Parameter estimates along the full fixed-ratio ray areprovided in Table 1. Assumed model parameters under the alternative hypothesisalong the reduced ray for each of the three cases considered are provided in Table 2

Ds-optimal design

Two-way interaction case W ¼ 0 33 35 65 80

16 16 16 16 16

� �

Two- and three-way interaction case W ¼ 0 30 35 61 80

16 16 16 16 16

� �

Two-, three- and four-way interaction case W ¼ 0 33 35 65 80

16 16 16 16 16

� �

Table 5. Six-, seven- and ten-point Ds-optimal designs for the hypothesis, given in (10), thatmalathion does not interact with the four active pesticides and the alternative hypothesisthat malathion is involved in two-way interactions. The top row of the design corresponds tothe total dose locations and the bottom row corresponds to the sample size allocation.Parameter estimates along the full fixed-ratio ray are provided in Table 1. Assumed modelparameters under the alternative hypothesis along the reduced ray are provided in Table 2

Two-way interaction case Ds-optimal design

Six-point design W ¼ 0 30 35 60 79 80

13 13 13 14 13 13

� �

Seven-point design W ¼ 0 25 31 36 56 66 80

10 12 12 12 12 12 12

� �

Ten-point design W ¼ 0 26 30 35 40 55 60 65 79 80

8 8 8 8 8 8 8 8 8 8

� �

142 M. CASEY ET AL.


5. EFFICIENCY

When it is of interest to detect and characterize interactions among chemicals in a mixture, multiple

hypotheses are often considered. Different hypotheses can result in different designs. In choosing the

appropriate design, consideration can be given to the primary hypothesis of interest. LetM(.) represent

a variance–covariance matrix associated with the (.) design. Given the design corresponding to the

primary hypothesis of interest (�) and the design corresponding to the secondary hypothesis ð��Þ,efficiency, described by Atkinson and Donev (1992) and given by

Eff ¼ Mð Þj jMð �Þj j

� �1=p

can be used to determine how efficient� is relative to�* for testing the secondary hypotheses (p is the

number of model parameters). For example, we are primarily interested in testing for interactions

between malathion and the four active pesticides. However, to ensure that appropriate conclusions are

made with respect to the effect of malathion, it is important to consider the hypothesis for additivity,

given in (4), along the reduced ray. If the design chosen to study interactions between malathion and

the remaining components of the mixture does not adequately allow for the detection of higher-order

terms along the reduced ray, conclusions about the effect of malathion may be misleading. Thus, it is

important to consider the efficiency of the designs given in Table 4 for testing the hypothesis of

additivity given in (4).

Let �interact represent the design associated with the hypothesis, given in (10), that malathion is not

involved in any interactions and let �add represent the design for the overall hypothesis of additivity

Figure 3. Results of 5-, 6-, 7- and 10-point Ds-optimal designs for the hypothesis given in (10), where the alternative hypothesis

assumes malathion is involved in two-way interactions. Parameter estimates are provided in Table 2. Generalized linear model

along the reduced ray (0.2286: 0.0114: 0.1767: 0.5833) under the null hypothesis (solid line), under the assumption of additivity

(dotted line) and under the alternative hypothesis (dashed line) are also provided



along the reduced ray, given in (4). Following Atkinson and Donev (1992), the Ds-efficiency of�interact

for testing the hypothesis of additivity along the reduced ray can be expressed as

Ds-eff ¼ Mð interactÞj jMð addÞj j

� �1=p

ð12Þ

where Mð interactÞj j is the determinant of the variance–covariance matrix associated with the

hypothesis of additivity when the �interact design is considered, Mð addÞj j is the determinant of the

variance–covariance matrix associated with the hypothesis of additivity when the �add design is

considered, and p is the number of parameters. If Ds-eff¼ 1, �interact is fully efficient for testing the

hypothesis of additivity. The smaller Ds-eff is the less efficient �interact is for testing the hypothesis of

additivity.

Consider the Ds-optimal design (�interact), given in Table 4, for the case where it is assumed that

malathion is involved in two-, three- and four-way interactions. We are interested in determining the

efficiency of this design for testing the overall hypothesis of additivity. The Ds-optimal design for

testing for departure from additivity along the reduced ray is given by

�add ¼ 0 30 35 65 80

16 16 16 16 16

� �

The efficiency of�interact for testing for the presence of the higher-order terms along the reduced ray is

given by

Ds-eff ¼ Mð interactÞj jMð addÞj j

� �1=11

¼ 0:99

Thus, the Ds-optimal design associated with testing interactions between malathion and the four active

pesticides under the assumption that malathion is involved in two-, three- and four-way interactions is

efficient for testing for the presence of higher-order terms along the reduced ray. Notice that these two

designs are similar to one another, and thus we would expect high efficiency.

6. DISCUSSION

The goal of this article was to develop a methodology for determining dose locations and sample size

allocation that provide precise parameter estimation and enough power to detect departure from

additivity and interactions due to subsets of chemicals along fixed-ratio rays. The precision of

parameter estimates is an important issue as it is associated with increased power. Consideration has

been given to Ds-optimal designs which are concerned with minimizing the generalized variance

associated with the hypothesis of interest.

The hypotheses developed in Section 2 require the fit of higher-order polynomial models. Working

with such models can cause numerical problems, particularly as the order of the polynomial increases.

For example, consider the model parameters that result from fitting the generalized linear model with

the log link to the single chemical data and to the data along the full fixed-ratio ray (estimates provided

in Table 1). Notice that the parameter associated with the fourth-order term is �1.9� 10�10 with a

standard error of 1.0� 10�10. We are considering a dose range up to 450mg/kg along the full ray and

up to 80mg/kg along the reduced ray. Raising such dose values to powers of 52 results in relatively

144 M. CASEY ET AL.


large numbers; thus, parameter estimates associated with higher-order terms become smaller as the

order of the polynomial term increases. In some situations such small numbers cause calculation

problems. It should be noted that the degree of the polynomial model chosen follows directly from the

number of components in the mixture (Meadows et al., 2002; Casey, 2003). Therefore, these higher-

order polynomial terms are important in detecting and characterizing interactions. One proposed

solution to ease numerical problems is to consider scaling the dose range. For example, we could

divide the doses by 10 or 100 and work with either centigrams or decigrams instead of milligrams.

Although the results are not presented here, we examined the effect of scaling the dose on the

Ds-optimal designs. For the examples reported here the designs were equivalent to the designs along

the scaled dose region.

Since the hypotheses of interest involve only a subset of the model parameters, Ds-optimal designs

were developed for fixed-ratio rays. The Ds-optimal examples presented here considered the design

along a single ray. These methods can be extended to simultaneously consider dose location and

sample size allocation for multiple rays. Although the examples presented here resulted in relatively

equal allocation, it is important to note that equal allocation is not always optimal. Ds-optimal designs

condition on model parameter values specified under the alternative hypothesis. In the case where

competing hypotheses are of interest, methods for determining the Ds-efficiency of one design relative

to another were also developed.

The designs presented here were developed for the generalized linear model. In evaluating the risk

associated with exposure to mixtures it is often of interest to detect a threshold (Schwartz et al., 1995).

The methods for detecting overall departure from additivity and for detecting interactions between

subsets of chemicals and the remaining components of a mixture are readily extended to the threshold

model (Casey, 2003). The methodology for determining Ds-optimal designs can readily be applied

when threshold models are of interest.

ACKNOWLEDGMENTS

The information in this document has been funded in part by the U.S. Environmental Protection Agency (#CR-827208-01, #R82811401-0). It has been subjected to review by the National Health and Environmental EffectsResearch Laboratory and approved for publication. Approval does not signify that the contents reflect the views ofthe Agency, nor does mention of trade names or commercial products constitute endorsement or recommendationfor use. The authors gratefully acknowledge the excellent technical assistance of Ms. P. M. Phillips and K. L.McDaniel in the collection of the experimental data and the thoughtful comments of Drs. L. Birnbaum, T. A.Henneman and Ms. L. Teuschler and anonymous referees on an earlier version of the manuscript.

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Meadows SL, Gennings C, Carter WH Jr. 2002. Experimental designs for mixtures of chemicals along fixed-ratio rays.Environmental Health Perspective 110: 979–983.

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APPENDIX

Summary statistics for motor activity response associated with the single chemical data

Chemical Dose Mean motor activity Standard Sample

(mg/kg) response deviation size

Acephate 0 217.88 35.77 8

3 200.13 34.13 8

10 165.88 25.02 8

30 108.75 62.51 8

60 58.25 24.23 8

120 33.25 27.41 8

Diazinon 0 206.69 34.77 16

5 190.88 28.49 16

25 215.56 24.02 16

50 183.13 24.44 8

75 165.69 33.05 16

125 152.00 38.65 8

150 76.5 35.4 8

250 61.25 47.07 8

Continues

146 M. CASEY ET AL.


APPENDIX (CONTINUED)

Chemical Dose Mean motor activity Standard Sample

(mg/kg) response deviation size

Chlorpyrifos 0 198.56 24.56 16

1 213.13 31.42 8

2 192.75 38.61 8

5 213.37 31.04 8

10 178.31 34.02 16

20 157.13 28.31 8

25 162.00 32.36 8

30 80.13 34.85 8

50 49.44 29.32 16

Malathion 0 195.86 19.06 7

100 201.5 28.38 8

500 203.75 28.34 8

Dimethoate 0 195.75 33.12 8

5 188.25 24.21 8

10 188.63 53.05 8

25 107.75 37.23 8

50 103.75 51.59 8

75 101.50 59.57 8

Summary statistics for motor activity response associated with mixture data along the full fixed-

ratio ray

Total concentration Mean motor Standard deviation Sample size

dose (mg/kg) activity response

0 199.43 20.77 14

10 200.92 27.10 12

55 167.92 37.55 12

100 117.08 47.34 12

200 95.17 34.03 12

300 72.25 40.93 12

450 60.08 46.36 12



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Ds-optimal designs for studying combinations of chemicals using multiple fixed-ratio ray experiments