A Novel Flexible Approach for Evaluating Fixed Ratio Mixtures of Full and Partial Agonists

Chris Gennings*,{dagger}, W. Hans Carter, Jr.*,{dagger}, Edward W. Carney{ddagger}, Grantley D. Charles{ddagger}, B. Bhaskar Gollapudi{ddagger} and Richard A. Carchman{dagger}

* Department of Biostatistics, Virginia Commonwealth University, Richmond, Virginia; {dagger} Solveritas, LLC, Richmond, Virginia; and {ddagger} The Dow Chemical Company, Midland, Michigan

Received January 6, 2004; accepted March 29, 2004


    ABSTRACT
 TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
Assessing for interactions among chemicals in a mixture involves the comparison of actual mixture responses to those predicted under the assumption of zero interaction (additivity), based on individual chemical dose-response data. However, current statistical methods do not adequately account for differences in the shapes of the dose-response curves of the individual mixture components, as occurs with mixtures of full and partial receptor agonists. We present here a novel extension of current methods, which overcomes some of these limitations. Flexible single chemical concentration-effect curves combined with a common background parameter are used to describe an additivity surface along each axis. The predicted mixture response under the assumption of additivity is based on the constraint of Berenbaum's definition of additivity. Iterative algorithms are used to estimate mean responses at observed mixture combinations using only single chemical parameters. A full model allowing for different maximum response levels, different thresholds, and different slope parameters for each mixture component is compared to a reduced model under the assumption of additivity. A likelihood-ratio test is used to test the hypothesis of additivity by utilizing the full and reduced model predictions. This approach is useful for mixtures of chemicals with threshold regions and whose component chemicals exhibit differing response maxima (e.g., mixtures of full and partial agonists). The methods are illustrated with a combination of six chemicals in an estrogen receptor-alpha (ER-{alpha}) reporter gene assay.

Key Words: ray design; synergy; antagonism; interaction index.


    INTRODUCTION
 TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
Environmental and occupational human exposures to chemicals through the diet, drinking water, and air are rarely limited to a single chemical. While this fact has been recognized for many years, the subject has received an unprecedented amount of attention in recent years. Public and regulatory concern about chemical mixtures is reflected in amendments to the Food Quality Protection Act (FQPA, 1996Go), which required that the EPA consider the "cumulative effects" of substances having a "common mechanism of toxicity" when establishing food tolerances and when conducting risk assessments for pesticides.

Because of the unlimited number of potential mixture combinations, testing mixtures separately is only practical for a limited number of high priority mixtures. An alternative approach is to focus on representative compounds that are known to exert toxicity via a common mechanism of action (e.g., a common receptor) or a common target organ, and conduct studies to better understand the nature of chemical interactions and their underlying mechanisms. These studies would then form the basis for making inferences about related mixtures. The most common regulatory approach used to date for compounds thought to act through a common mechanism are based on the dose-addition approach as the default assumption (EPA, 2001Go; Wilkinson et al., 2000Go). Dioxins and furans exemplify one group of compounds regulated under the principle of dose-addition, which is manifest as the Toxic Equivalency Quotient (TEQ) approach. Each of the many dioxin and furan congeners is assigned a "Toxic Equivalency Factor" (TEF) based on its potency relative to the prototype compound, 2,3,7,8-tetrachlorodibenzo-p-dioxin (TCDD). The TEFs and dose of each congener are then multiplied, and the resulting values for each congener summed to derive the total mixture TEQ (Safe, 1998Go).

While dose-additivity approaches such as TEQs have practical advantages, they also have a number of fundamental limitations, particularly with their ability to predict mixture toxicity at low doses. First, individual TEF values are typically based on responses at relatively high doses (e.g., EC50 values), with the assumption that relative potency and mechanism of action are uniform across the dose-response range. For mixtures of chemicals in the environment, exposures typically are at relatively low levels, usually below individual chemical thresholds. This situation is distinctly different from drug interactions, which deal with pharmacologically active doses. Mechanisms of individual chemical action, as well as chemical interaction, are rarely uniform across the entire dose-response range and instead tend to be highly dose-dependent (Gennings et al., 2002Go). At very low exposures, dose-addition assumptions ignore the effectiveness of detoxification systems which often are overwhelmed at high doses, leading to overestimation of toxicity at low doses (Feron et al., 1995Go; Groten, 2000Go). Another problem with simple dose-addition approaches is that many receptor-based systems exhibit a high degree of tissue/organ specificity. For example, tamoxifen is an estrogen receptor (ER) agonist in the uterus, but an ER antagonist in the breast. For this and other reasons, Safe (1998)Go has cautioned against the use of TEQ approaches for hormonally active chemicals. Finally, even chemicals acting through a common receptor exhibit different thresholds, slopes, and maximal effect levels, such that simplistic predictions of combined effects based on individual TEFs can be grossly in error (Scaramellini et al., 1997Go). Using estrogenic chemicals as an example, many of the chemicals of concern are several orders of magnitude less potent than "natural estrogen" (i.e., estradiol 17ß or E2) and have a much lower maximal effect level as compared to E2. Chemicals exhibiting the latter property are referred to as partial agonists.

Recently an expert working group was convened by the Society of Toxicology to suggest future directions in chemical mixture research. The group highlighted the need for a specific focus on the key attributes of "real world" exposures, which include the examination of low dose exposures and a better understanding and incorporation of dose-response into mixture toxicity predictions (Teuschler et al., 2002Go). With this aim in mind, this article describes the development of a flexible statistical approach that seeks to address mixtures of compounds with differing response thresholds, slopes and maximal effect levels (i.e., full and partial agonists). Recently, statistical methods have been developed that permit the analysis of mixtures of many chemicals while maintaining feasible experimental designs (e.g., Casey et al., in pressGo; Gennings et al., 2002Go; Meadows et al., 2002Go) without the required constraint of parallel dose-response curves and while adequately accounting for biological variability in testing hypotheses of additivity. In short, single chemical concentration-effect data and mixture data at fixed-ratios are experimentally generated. An additivity model is estimated from the single chemical data, which is used for comparison to a model fit to the mixture data in terms of total concentration. The statistical comparison of the two models based on prediction along the specified fixed-ratio ray(s) is a test of additivity for the fixed ratio of the mixture of chemicals. This general strategy has been termed the "single chemicals required" (SCR) approach (Casey et al., in pressGo).

The definition of additivity (i.e., zero interaction) we use is given by Berenbaum (1985)Go and is based on the classical isobologram for the combination of two chemicals (e.g., Loewe, 1953Go; Loewe and Muischnek, 1926). That is, in a combination of c chemicals, let Ei represent the concentration/dose of the ith component alone that yields a fixed response, y0, and let xi represent the concentration/dose of the ith component in combination with the c agents that yields the same response. According to this definition of additivity if the substances combine with zero interaction, then

(1)
If the left-hand side of Equation 1, termed the interaction index, is less than 1, then a synergism can be claimed at the combination of interest. If the left-hand side of Equation 1 is greater than 1, then an antagonism can be claimed at the combination. The additivity models used in the SCR approach are algebraically equivalent to Equation 1 (Carter et al., 1988Go; Gennings, 2002Go). This definition of additivity is a general form for dose-addition. It should be pointed out that use of the TEF approach assumes common slopes across the chemicals under study; the dose-addition definition of Equation 1 does not require such an assumption.

An important feature of the SCR approach is that the additivity model estimated from single chemical data can be used to predict the "zero interaction" or additivity response for any specified mixture. It is necessary for the single chemical data to be adequately represented by the additivity model. Such a model often makes assumptions about the overall shape and characteristics of the single chemical concentration effectiveness. For example, a common maximum effect level is often assumed for all of the chemicals in the mixture. As mentioned earlier, this is not a valid assumption for mixtures containing partial agonists. When fitting threshold additivity models, the common parameter that defines each chemical's threshold concentration may not be sufficiently flexible. Extending the logic of Dawson et al. (2000)Go, who used Equation 1 with estimated single chemical dose-response curves to determine zero interaction, we have developed a new method of analysis useful for ray designs that permits more flexibility in modeling each single chemical concentration-effect curve. Prediction of the response along the fixed-ratio rays under additivity is determined indirectly using the interaction index with only single chemical parameters. A likelihood-ratio test of the hypothesis of additivity is then conducted by comparing a fully parameterized model of all of the data to the reduced model using only single chemical parameters.

Here we describe the general SCR approach using a sigmoid-shaped nonlinear threshold model, followed by a description of the more flexible approach, based on the interaction index for defining additivity. Both methods are demonstrated in an analysis of a mixture of six chemicals, which exhibit in vitro estrogenic activity, while possessing differing response thresholds and maximal effect levels.


    MATERIALS AND METHODS
 TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
Chemicals. The six chemicals selected for study included: Methoxychlor (MXC), o,p-DDT, beta-hexachlorocyclohexane(ß-HCH) obtained from Chemservice (West Chester, PA). Bisphenol A (BPA) and octylphenol (OCT) were purchased from Sigma Chemical Company (St. Louis, MO), and 2,3-bis(4-hydroxyphenyl)-propionitrile) [DPN] from Tocris (Ellisville, MO).

Cell culture. For use in the ER-{alpha} gene transcription assays, MCF-7 human breast cancer cells (obtained from Dr. Leigh Murphy, University of Manitoba, Canada) were maintained in phenol red-free DMEM supplemented with 10% FBS, 2 mM L-glutamine, 15 mM HEPES, penicillin/streptomycin (100 IU/ml/ 100 µg/ml), and amphotericin B (2.5 µg/ml). Cells were maintained at 37°C and 3–4% CO2.

Transfection/treatment. Cells were plated in triplicate in 96-well plates at a density of 6–8 x 103 cells/well in 5% FBS-DCC (Hyclone, Logan, UT). After attachment and growth for 6 h, the cells were transfected using LipofectinTM (Life Technologies). To each well was added 50 ng of the ß-galactosidase (ß-gal) expression vector pCH110 (Pharmacia, Piscataway, NJ), 150 ng of 17m5-G-Luc, the Gal4-regulated luciferase reporter vector, and 5 ng of Gal4-HEG0, an ER expression vector (both provided by P. Chambon, INSERM, France). The plasmids were transfected in serum-free, antibiotic-free DMEM supplemented with 2 mM L-glutamine. Cells were allowed to incubate overnight at 37°C in a humidified atmosphere of 3% CO2/air. Sixteen to eighteen h following transfection, cells were treated in triplicate with individual test compounds, mixtures, or the solvent (≤0.2% dimethylsulfoxide, DMSO, Sigma) in 5% FBS-DCC. The ER agonist, E2, was used at 10–8 M as a positive control. Twenty-four h following treatment, cells were lysed and aliquots from each well divided into two 96-well plates for luciferase and ß-gal activity determination using a Packard Topcount NXTTM luminescence counter (Packard Instrument Company, Meriden, CT). The reference plasmid pCH110 was cotransfected as an internal control in order to correct for variations in transfection efficiency. The data was assessed as units of luciferase activity normalized to the ß-gal activity from individual wells. Experiments were evaluated with fold induction being the endpoint of interest. Treatment regimens which resulted in a reduced ß-gal activity relative to that of transfected cultures exposed to E2 under unaltered media conditions were considered cytotoxic and were not used for further analysis.

Data Analysis
Two differing statistical methods for detecting and characterizing interaction in a mixture of six chemicals are illustrated in the Example/Results section.

SCR approach. This method has been described previously by Gennings et al. (2002)Go and Casey et al. (in press)Go. An important feature of the SCR method is that single chemical dose-response data are used to fit to an additivity model which is algebraically equivalent to the definition of additivity as given in Equation 1. This model is used to predict under additivity along one or more fixed-ratio mixing rays of the chemicals in the mixture in terms of total dose or concentration. A mixture model is fit to observed mixture response data along the fixed-ratio mixture ray(s) and compared to that predicted by the additivity model. Lack of coincidence of the results of both the mixture and additivity models using an appropriate statistical test is considered to be a demonstration of departure from additivity. Although the additivity and mixture models have parameters incorporated to allow for a flexible model fit, these models are constrained to have, for example, a common maximum response. In some cases, this assumption (common maximal response) and the subsequent resulting models do not adequately represent the observed data.

Flexible alternative approach. In contrast to SCR, a new and more flexible approach is developed and illustrated here. In general, single chemical dose-response data are used to estimate dose-response curves for each single chemical. The increased flexibility is due to the inclusion of additional parameters to describe the dose-response relationship for the individual chemicals. These curves are assumed to have a common background parameter and are then allowed sufficient flexibility (i.e., enough parameters) to adequately fit the single chemical data. This combination of single chemical dose response curves is made into an additivity model by imposing the definition of additivity in Equation 1, thus creating an indirect additivity model. A test of additivity for a fixed-ratio mixture in terms of total dose/concentration is based on statistical comparisons of a "full" and "reduced" model. The full model allows a simultaneous fit of the single chemical and mixture data using sufficiently flexible model assumptions. These can allow for different maximum effect plateaus, flexibility in estimating dose or concentration thresholds, and different slope parameters. The reduced model uses only the models from the single chemical data and the constraint in Equation 1 to predict responses along the fixed-ratio mixture ray(s). Details of the statistical methods are provided in the Appendix. (The equation numbers referenced in the following sections refer to equations in the Appendix.) Direct inquiries regarding available software to the first author.


    RESULTS
 TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
The objective of the analysis is to determine if there is evidence of interaction among a combination of six chemicals in an ER-{alpha} reporter gene assay. The concentration ranges for the single chemical studies were determined from preliminary range finding studies. The test concentration ranges were 0–10 µM for MXC, DDT, ß-HCH, and OCT; 0–1 µM for BPA; 0–0.1 µM for DPN. Summary statistics from the observed data are provided in Table 1. Based on a plot of sample variances versus sample means (Fig. 1), we assume the variance of fold induction increases with the mean (i.e., Var(Y) = {tau}µ. In the following analyses, a quasi-likelihood criterion was used to estimate model parameters.


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TABLE 1 Summary Statistics for Single Chemical and Mixture ER-{alpha} Reporter Gene Data Using Fold Induction

 


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FIG. 1. Observed sample variances versus sample means across all single chemical and mixture data. The variance is evidently proportional to the mean.

 
SCR Analysis
Using this approach the threshold additivity model given in Equation 2a was initially fit to the single chemical concentration-response data. However, the analysis resulted in the estimated threshold parameters being outside of the region of the experimental data. Thus, the corresponding nonlinear additivity model (Equation 2b) was used. The resulting parameter estimates and corresponding p values are provided in Table 2. All six chemicals had positive and significant slope parameters (ßi), indicating that as the concentration of the chemical increased, there was an increase in induction of the measured assay response relative to the solvent (DMSO) controls. Figure 2 provides plots of the observed and model predicted concentration-effect additivity model for each of the six chemicals. A goodness-of-fit test based on the comparison of the deviance between the SCR model and a more general model indicated lack of fit (11.87 with 7147 degrees of freedom; p < 0.001). The more general model allowed for different slope and threshold parameters for individual chemicals. Upon inspection of Figure 2, the lack of fit seems to be primarily due to lack of flexibility in the model, and is present throughout the concentration range.


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TABLE 2 Parameter Estimates for the Additivity Model in Equation 2b and the Mixture Model in Equation 4a after Model Reduction Due to Estimates of Thresholds Outside the Experimental Region

 





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FIG. 2. Observed and model predicted responses for the additivity model given in Equation 2b with parameter estimates given in Table 2. (A–F) represent the single chemical concentration-effect curves. (G) provides the observed data along the fixed-ratio xeno and mixture ray in terms of total concentration with the predicted responses for the mixture model given in Equation 4a. (H) includes the data and fitted mixture model from (G; solid curve) overlaid with the curve estimated under the additivity model (determined from the single chemical data with model given in Equation 3b; dashed curve) in terms of total concentration.

 
Continuing the SCR analysis, assuming the estimated additivity model adequately approximates the single chemical data, we can use the model to estimate the total concentration-effect curve along the specified mixture ray. Here the mixing ratios for the six chemicals were selected from no-observable-effect concentrations (NOECs) from preliminary concentration range-finding studies. Given the focus on low-level exposures, this approach was considered preferable to selecting ratios based on higher levels of response. The resulting ratio is approximately (0.4715:0.0047:0.4715:0.0471:0.0047:0.0005) for (MXC:DPN:DDT:ß-HCH:OCT:BPA), respectively. The associated slope parameter under the additivity model is given by , which was estimated as 0.132 (SE = 0.012). Assuming a common maximum effect, the mixture threshold model (Equation 4a) was fit simultaneously with the additivity model while allowing for a threshold model for the mixture data in terms of total concentration. The estimated slope parameter for the mixture data (Table 2) was 0.104 (SE = 0.023) with threshold parameter 0.166 (SE = 0.068). The hypothesis of additivity is that these two slope parameters are the same and that the threshold concentration is zero, i.e., that the mixture concentration-effect curve is equivalent to that under the additivity model. Here, the test of additivity is rejected (p < 0.001; Table 2) indicating that there is evidence of departure from additivity among the six chemicals. Both the predicted concentration-effect curves under additivity and that based on the mixture data are presented in Figure 2H. The solid curve is the fitted curve for the mixture data; the dashed curve is what was predicted under the additivity model using single chemical data along the specified fixed-ratio ray. Because the modeled concentration-effect curve is increasing and the experimental mixture data fall below the curve under additivity, it can be claimed that departure from additivity is associated with a less than additive (antagonistic) overall effect.

Flexible Alternative Analysis
For comparison, the second and more flexible approach was used to analyze the same data. The second method allows for more flexibility in fitting individual chemical concentration-effect curves. For example, the additivity model in Equation 2b assumes a common maximum effect parameter ({gamma}) across the six chemicals. The response data for DDT (Fig. 2C) and OCT (Figure 2E) suggest plateau (i.e., partial agonist) effects that are below that observed as defined by the response of the positive control E2 alone (14-fold induction). Thus, the model in Equation 5 was fit to each of the single chemical concentration-effect data. This model allows for different maximum-effect parameters ({gamma}i), slope parameters (ßi), and threshold parameters ({delta}i) for each chemical (i = 1, ..., 6). The model, however, assumes a common background response ({alpha} = 1.0) in the DMSO control group. After model reduction (i.e., dropping the concentration-threshold parameter for DDT, which was originally estimated to be less than zero, allowing plateau parameters for DDT and OCT, and fixing the maximum effect parameter for the other chemicals to a fold induction of 14 as observed for Equation 2), the resulting model parameter estimates, standard errors and p values are provided in Table 3 with the fitted responses plotted in Figure 3. From Figure 3, this more flexible model adequately fits the single chemical and mixture data and leads to visually improved fits compared to those presented in Figure 2 (from the additivity model for SCR). A goodness-of-fit test comparing the deviance for the model in Table 3 to a more general model with seven concentration threshold parameters did not indicate a lack of fit (p > 0.99). The maximum effect parameter ({gamma}) for MXC, DPN, ß-HCH, and BPA was set at 14.0 based on the E2 positive control data. From this model (Table 3), five of the six individual concentration-effect curves had positive and significant slope parameters (ß1, ß2, ß3, ß4, ß6) while the only exception OCT had a positive and borderline significant slope (p = 0.061). The positive and significant slope parameter ({theta}1) for the mixture indicated an increase in fold response as the total concentration of the chemical mixture increased along the fixed-ratio ray (p < 0.001). Further, there were positive point estimates for the threshold parameters for MXC, DPN, ß-HCH, OCT, and BPA (p = 0.374, p = 0.071, p = 0.835, p = 0.137, and p = 0.034, respectively). The estimate for the threshold for the mixture in terms of total concentration was estimated as 1.60 µM (Table 3, ) which was significantly different from zero (p < 0.001).


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TABLE 3 Parameter Estimates from the Model in Equation 5 after Model Reduction Due to Threshold Values within the Experimental Region

 





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FIG. 3 Observed and model predicted responses for the additivity model given in Equation 5 with parameter estimates given in Table 3. (A–F) represents the single chemical concentration-effect curves. (G) provides the observed data along the fixed-ration xeno mixture ray in terms of total concentration. (H) includes the data and fitted mixture model from (G; solid curve) overlaid with the curve estimated under the additivity model (determined from the single chemical data) in terms of total concentration.

 
In order to test the hypothesis of additivity, a likelihood ratio test was conducted. This test requires the likelihood value under the "full" model (without restrictions of additivity along the mixture ray) and a "reduced" model that restricts the prediction along the ray to be associated with additivity. The latter was accomplished by estimating the single chemical model parameters simultaneously with the mixture data while constraining the estimates to agree with the definition of additivity along the fixed-ratio mixture ray as given in Equation 6. Using an iterative algorithm with a quasi-likelihood estimation criterion, the restricted model parameter estimates under the hypothesis of additivity were determined (Table 4). Notice that this restricted model omits the parameters associated with the mixture ray. The predicted values along this ray are found implicitly from the restriction in Equation 6. The likelihood-ratio test of additivity was performed using quasi-likelihood values from both models. The hypothesis of additivity along the specified fixed-ratio ray was rejected (p < 0.001; Table 4).


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TABLE 4 Parameter Estimates from the Corresponding Additivity Model to Equation 5 with the Constraint of Additivity as Given in Equation 6

 
Figure 3H presents the observed mixture data, the unrestricted fit to these data from the full model (solid curve), and the predicted responses associated with additivity (dashed curve). Using this approach, the constraint in Equation 6 is limited to the data below the minimum plateau level across the single chemical concentration-response data. Since the OCT data had a range parameter ({gamma}) of 4.598 (Table 4), then the predicted response under additivity can be no greater than 5.598 (). In order to calculate the contribution to the quasi-likelihood test for observations with total concentration at and above 3.0 µM, the contribution to the restricted quasi-likelihood for these observations was conservatively assumed to be that fit using the full model (which essentially cancels the effect of these data from the test statistic). Thus, the test of additivity is based on the effect in the low range of total concentration.

Upon closer inspection, since MXC and DDT cover 94% of the mixing ratio, their low concentration effects are influential in the shape of the predicted response under additivity. Since the fold induction increases so rapidly in the low concentration range of DDT, the curve under additivity also rises rapidly. In fact, the test concentrations of DDT were not low enough to identify the threshold (i.e., without a threshold parameter, the concentration of DDT associated with the background fold induction of 1.0 was 0.0). Hence, the threshold concentration under additivity was estimated to be essentially zero (Fig. 3H). Similar to the previous analysis, since the experimentally derived mixture data along the specified mixing ratio of the six chemicals fall below that predicted under additivity (Fig. 3H), there is evidence of departure from additivity associated with an overall less than additive (antagonistic) interaction.


    DISCUSSION
 TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
Following the SCR method from Gennings et al. (2002)Go, the single chemical data were used to estimate an additivity model. This model was algebraically manipulated to determine the concentration-effect curve along the mixture ray. The mixture data were fit to a model incorporating parameters similar to the additivity model, which assumed a common maximum response. The test of additivity corresponds to the test for the equivalence of the slope parameters for these models (i.e., the mixture model and the additivity model along the fixed-ratio ray) in terms of total mixture concentration. This hypothesis was rejected for these data. If we are interested in determining the impact of a subset of chemicals in this mixture, the approach taken by Casey et al. (in press)Go could be used. That is, if we want to determine if a subset of the chemicals interact with the remaining chemicals, then a reduced ray design (fewer components of the mixture) could be experimentally evaluated and proper comparisons made to test for the impact of the individual chemicals left out of the reduced ray experiment.

A novel approach for testing the hypothesis of additivity along a fixed-ratio ray of chemicals was described. It has the advantage of more flexible model fits for the single chemical data than imposed by the additivity model for the SCR method. It is an extension to the work of Dawson et al. (2000)Go using the interaction index to predict under additivity. The proposed method uses a direct search algorithm to estimate the parameters required under additivity with a bisection algorithm to predict the mean response for the mixture data. The approach concludes with a likelihood ratio test of additivity.

The flexibility of the second approach is important in allowing different maximum effect levels for individual chemicals, such as commonly found for receptor-mediated toxicants. This flexibility is of biological importance as it allows for mixtures of full and partial agonists to be simultaneously considered in combination (mixtures) toxicity assessments. Here, the concentration-effect curves for MXC, DPN, ßHCH, and BPA were apparently still increasing based on the observed data. These chemicals were assumed to plateau above the observed experimental region. By contrast, DDT and OCT apparently plateau at a fold-induction below that assumed for the other four chemicals, consistent with the notion that they are partial agonists.

Further, the flexibility of the second approach allows for more precise modeling of threshold regions for individual chemicals. This is particularly evident by comparison of the predicted models in Figures 2 versus that in Figure 3 for MXC, DPN, and OCT. The threshold region under additivity includes all concentrations below the plane that connects the estimated concentration thresholds for each single chemical and is associated with background response levels. Figure 4 is a schematic for this threshold additivity surface for three chemicals. When concentration thresholds are not detected for a subset of the chemicals in the mixture, then it implies that any increase in exposure to these chemicals results in an increase in response levels. These are important features of environmental mixtures which are accommodated by more flexibility in assessing additivity and departure from additivity as demonstrated by the second statistical approach to the analysis of multi-component mixture responses.



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FIG. 4. A schematic of a threshold additivity surface. When three chemicals are combined in an additivity model (e.g., as given in (5)) the additivity threshold surface is a plane which intersects at the concentration thresholds for each of the three chemicals. The response associated with any combination of the three chemicals below this plane (i.e., between the plane and the origin) is the same as the background response.

 

    APPENDIX
 TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
Assume a study of c chemicals with concentration-effect data available for each chemical. In addition, concentration-effect data of r fixed-ratios of mixtures of the c chemicals are available for a total of R = c + r rays. Single chemical and mixture rays are defined by their fixed ratios denoted by [a1:a2: ...: ac] such that ; where, for the ith single chemical, ai = 1 and ab = 0 for b != i, for i = 1, ..., c. Let yijk be the response from the kth observation in the jth dose of the ith chemical; i = 1,...,c + r; j = 1,..., di; k = 1,..., nij with mean µij. The variance of Y is assumed to be a function of the mean, i.e., {tau}V(µ), and the method of maximum quasi-likelihood is used for parameter estimation (e.g., McCullagh and Nelder, 1989Go, Chapter 9). A special case of this general form for the variance is that of a common variance, which results in a least squares estimation approach to estimate model parameters. In the example, we assume the variance of Y increases with the mean, i.e., {tau}µ.

Testing for Interaction Using a Fixed-Ratio Ray Design: SCR Approach
It is of interest to determine whether the components in a specified mixing ratio of c chemicals interact. Interaction is detected in the SCR approach by comparison of mixture models to the corresponding additivity model. Carter et al. (1988)Go and Gennings (2002)Go demonstrated that generalized linear models and nonlinear models, respectively, which can be linearized (i.e., h(µ) = {Sigma}ißixi) are algebraically equivalent to the definition of additivity in Equation 1. Further, these models can be adjusted to be in the form of a threshold additivity model (Schwartz et al., 1995Go). Consideration of threshold additivity models permits inference regarding low concentration regions that are associated with background response. Although the SCR approach is applicable to general forms of nonlinear and threshold models (e.g., Gennings et al., 2002Go, 2003), we describe the method via the models used in the example in section 3.

Using single chemical data, a threshold additivity model (e.g., Gennings et al., 1997Go) for increasing concentration-effect curves is given by

(2a)
where

xi is the concentration of the ith chemical,
ßi are the unknown parameters associated with the slope of the ith chemical in the mixture of c components,
{alpha} is a minimum effect parameter,
{gamma} is the response range assumed to be constant across all c chemicals,
{alpha} + {gamma} is the maximum effect, and
{delta} is an unknown threshold parameter.
The concentration threshold for the ith chemical is given by , i = 1,...,c. Thus, the model assumes that the concentration-effect curves for each of the chemicals in the mixture ranges between {alpha} and {alpha} + {gamma}; the concentration-effect curves are allowed to have different slope parameters (ßi); and, the concentration thresholds, , are somewhat restricted to be jointly related to {delta}add.

If all estimates for the concentration thresholds are outside of the experimental region (where negative estimates indicate lack of evidence for a concentration threshold and estimates above the experimental region demonstrate no concentration-effect relationship), then the corresponding nonlinear model is considered, i.e.,

(2b)

We are interested in detecting and characterizing an interaction among the c chemicals in the mixture for the fixed mixing ratio of the chemicals. Following Gennings et al. (2002)Go define t as the total concentration of the xeno mixture (i.e., ). Along a given ray with total dose t, the amount of the ith chemical is xi = ait, where ai is the ratio of the ith chemical in the mixture for i = 1,...,c, and . The slope associated with this mixture under the assumption of additivity is given by

Thus, the concentration-effect curve of the mixture in terms of total concentration for the fixed mixing ratio under the hypothesis of additivity is given by

(3a)
which is associated with the threshold additivity model given in Equation 2a, or

(3b)
which is associated with the nonlinear additivity model given in Equation 2b.

The mixture data along the fixed mixture ray is fit to a similarly parameterized mixture model of the form

(4a)
for a threshold model, or to

(4b)
for the corresponding nonlinear model if there is no evidence a threshold exists.

The hypothesis of additivity along the specified ratio of the chemicals is an hypothesis of coincidence between the additivity model in either Equations 3a or 3b and the mixture model given in either Equations 4a or 4b, i.e., for the threshold models, H0: {{theta}mix = {theta}add and {delta}mix = {delta}add} versus H1: {{theta}mix != {theta}add or {delta}mix != {delta}add}. An F test is used to test this hypothesis of additivity (e.g., Casey et al., in pressGo).

Alternative Approach: Defining Additivity Using the Interaction Index in Equation 1 Set to One
Of concern in conducting the SCR method is that the additivity model is underparameterized so that it lacks the flexibility to accommodate various features of the individual curves (e.g., the existence of a threshold dose, or differing plateau levels). For this reason, we were motivated to develop a different strategy for testing for interaction on a ray. The general strategy is to use more flexible models for single chemical concentration curves combined with a common intercept parameter. In short, an additivity model is only implicitly defined by combining these model fits with the definition of additivity as given in Equation 1. In addition, the concentration-effect curve in terms of total concentration is fit using a mixture model along the mixture ray(s). A likelihood-ratio test is then constructed to test for interaction. The approach is subsequently defined in more detail.

A more general form of the additivity model in Equation 2a allows for different threshold parameters and range parameters for each chemical. In addition, we allow for different threshold parameters and range parameters for each fixed-ratio mixture. For convenience in notation, we define a model for each single chemical and for each fixed-ratio mixture ray with subscript i, i = 1,..., R(= c + r), i.e.,

(5)
where

µi is the mean response for the ith ray, i = 1,..., c for single chemicals; i = c + 1, ..., R for the mixture ray(s),
x is the concentration of the ith chemical (i = 1,...,c) or the total concentration for the mixture ray(s),
{gamma}i is an unknown parameter associated with the range for the ith chemical/mixture,
ßi is an unknown parameter associated with the slope along the ith ray for single chemicals (in terms of concentration) or mixtures (in terms of total concentration), and
{delta}i is an unknown parameter associated with the threshold along the ith ray for single chemicals (in terms of concentration) or mixtures (in terms of total concentration).

Thus, the model does not restrict the range of the concentration-effect curves for the chemicals in the mixture (i.e., range between {alpha} and {alpha} + {gamma}i); the concentration-effect curves are allowed to have different slope parameters (ßi); and, each chemical is allowed to have a concentration threshold if the data support the parameter and otherwise the {delta}i parameter is removed from the model.

The model in Equation 5 is fit simultaneously to all of the single chemical data and the mixture data along the fixed-ratio ray(s). In the example we assume the mean model in Equation 5 and that the variance is proportional to the mean, i.e., Var(Y) = {tau}µ, across concentration and chemicals. Under these assumptions, a quasi-likelihood criterion is used to estimate the model parameters using an iterative algorithm (e.g., the Gauss-Newton algorithm). Assuming the variance is proportional to the mean, the quasi-likelihood function of the "full model" given in Equation 5 to the single chemical and mixture data is denoted as Q(full;{tau}), i.e., with the scale parameter estimated using the moment estimate, , where N is the total sample size and p is the number of model parameters.

In order to test for interaction along the fixed-ratio ray of interest, a reduced additivity model is estimated for comparison. This is accomplished by using only the parameters necessary to estimate the single chemical data from Equation 5 with a constraint of additivity as given in Equation 1 to determine the predicted values along the mixture ray. That is, define the ECi(µ) as the concentration for the ith chemical alone that produces a response of µ. From the model in Equation 5,

From Equation 1, under additivity, for a mixture of c chemicals along a fixed-ratio ray defined by [a1: a2: ...: ac] such that ,

Thus, for a specified value of the mean µ0(add), the corresponding total concentration that yields that mean response under additivity is given by

(6)

Note that Equation 6 can be used to estimate the threshold along a fixed-ratio mixture ray under additivity by setting µ0(add) = {alpha}, i.e., . Notice that if the ith chemical in the mixture is not associated with a threshold concentration (i.e., {delta}i(add) = 0), its contribution to the summation dominates the sum as the denominator is zero making be zero.

Under the assumption that the variance is proportional to the mean, a quasi-likelihood estimation criterion is used for estimation along the R rays (=c single chemical rays + r mixture rays) using only single chemical model parameters and the constraint in Equation 6, i.e.,

The estimation of the additivity model is accomplished by imbedding a bisection algorithm into a Nelder-Mead direct search algorithm. Given candidate values for the model parameters ({alpha}(add), ßi(add), {gamma}i(add), and {delta}i(add), i = 1,...,c), the bisection algorithm is used to find the value of µ(add) that is associated with each mixture data point where the observed total concentration values, tobs, were such that

Then the Nelder-Mead algorithm is used to find the constrained quasi-likelihood estimates for the model parameters by maximizing Q(add).

A likelihood ratio test of additivity along the fixed-ratio mixture ray(s) of interest is constructed as

which for large samples follows an F(dffull dfadd, dffull).


    ACKNOWLEDGMENTS
 
The data presented are from a grant funded by the American Chemistry Council (ACC Project #1718).


    NOTES
 

1 To whom correspondence should be addressed. Chris Gennings, Dept of Biostatistics, Virginia Commonwealth University, 1101 E. Marshall Street, #B1-039-A, Richmond, VA 23298-0032. Fax: (804) 828-8900. E-mail: gennings{at}hsc.vcu.edu.


    REFERENCES
 TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
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