Control group means ranged from 41.7 to 55.6 gµ/dL, with an average coefﬁcient of vari- ation of 15.3. All data values were standardized to percentage of control for each chemical [(experimental value/control mean) × 100]. Statistical modeling. The definition of additivity (i.e., zero interaction) used is given by Berenbaum (1985) and can be related to the isobologram for a combination of chemi- cals (Loewe 1953; Loewe and Muischnek 1926) through the interaction index. That is, in a combination of c (here, c = 18) chemicals, let Ei represent the concentration or dose of the ith component alone that yields a fixed response (i.e., ED30), y0, and let xi represent the concentration/dose of the i th component in combination with the c agents that yields the same response. According to this defini- tion, if the substances interact in an additive fashion, then c x ∑ i = 1. i =1 Ei If the left side of Equation 1, termed the interaction index, is < 1, then a greater-than- additive interaction (e.g., synergism) can be claimed at the combination of interest. If the left side of Equation 1 is > 1, then a less-than- additive interaction (e.g., antagonism) can be claimed at the combination. The 18 chemicals were combined accord- ing to a speciﬁed mixing ratio (Table 2) and Additivity model 140 120 )SD100 ± n a e m ( lro 80 t n o c % , T4 60 40 20 [1] i = 1, . . . ,19 evaluated experimentally. The mixing ratio was selected based on the ratios found in breast milk, ﬁsh, and other sources of human expo- sure, as described above. The mixing ratio is denoted in terms of the proportion, ai, of each chemical in the mixture (Table 2) such that 18 i ∑=1ai = 1. [2] The FSCR approach of Gennings et al. (2004) allows for different threshold parame- ters and range parameters for each chemical and ﬁxed-ratio mixture. The empirical mixture data were modeled (termed the empirical model) using a nonlinear exponential model of the form ⎧⎪ αi + γ i x < δi ⎫⎪ µi = ⎨ ⎬, [3] ⎩⎪αi + γ i exp[βi (x − δi )] x ≥ δi ⎪⎭ where αi + γi = 100, x is the dose of the ith chemical (i = 1, . . . ,18) or the mixture dose for the mixture ray (i = 19), αi is an unknown parameter deﬁned by the maximum effect for the i th chemical or mixture, βi is an unknown parameter deﬁned by the slope for the i th chemical or mixture, and δi is an unknown parameter deﬁned by the threshold along the i th ray for single chemicals (in terms of dose) or mixtures (in terms of total 140 ) SD120 ± an100 e m l( 80 o r t on 60 c ,%40 4 T 20 Empirical model Control 10 100 1,000 Total mixture dose (µg/kg/day) 0 300 600 900 1,200 Total mixture dose (µg/kg/day) 1,500 1,800 2,100 Figure 1. Predicted effects on serum total T4 from the single-chemical data in an additivity model and empirical effects of the PHAH mixture. Results demonstrate a significant deviation from additivity at the three highest mixture doses. The effects of the lower mixture doses were not signiﬁcantly different than that predicted by additivity. The inset illustrates the same data plotted as log-dose. Environmental Health Perspectives • VOLUME 113 | NUMBER 11 | November 2005 Cumulative risk of thyroid disruptors dose of the mixture). Preliminary inspection of the data provided evidence that the variance of the observed response increased with the mean [i.e., Var(Y ) = τµ for unknown dispersion para- meter τ]. Unknown parameters were estimated using the method of maximum quasi likelihood (McCullagh and Nelder 1989). The model in Equation 3 was used to estimate the ED30 (i.e., the dose associated with a 30% decrease from control) for each single chemical. The delta method (Agresti 2002) was used to esti- mate large sample 95% conﬁdence intervals on the ED30 estimates. We estimated the mixture dose–response curve (Figure 1, empirical model) using the model in Equation 3 for the speciﬁed mixture. The dose–response curve for the mixture assuming additivity (Figure 1, additivity model) was estimated using only the single- chemical dose–response model parameters and the single-chemical data (Equation 3), then predicting along the mixture ray with the con- straint of additivity given in Equation 1. To determine whether there was a statisti- cally signiﬁcant deviation from additivity, we used a quasi-likelihood ratio test to compare the empirical mixture model to the restricted additivity model based on an F-distribution. The restricted additivity model (Gennings et al. 2004) included only the single-chemical dose–response model parameters but used both the single-chemical and mixture data. We used this restricted model to predict the mean responses for the mixture data using the con- straint of additivity given in Equation 1. In addition we compared the predicted responses from the mixture data under the hypothesis of additivity (following the methods in Gennings et al. 2002) to the observed sample means using an F-test (df = 6, 1,305). Results We noted no visible signs of toxicity after the short-term PHAH treatments. There were no treatment-related effects on body weight gain. Summary statistics for the single-chemical curve ﬁts, including ED30 estimates and 95% conﬁ- dence limits, are shown in Table 3. Summary statistics for the individual chemical dose groups can be found in Supplemental Data Table 1 (http://ehp.niehs.nih.gov/docs/2005/ 8195/supplemental.pdf). Data modeling with the FSCR method provided maximum effect parameters (asymp- totes) and dose threshold parameters for each chemical and for the mixture. Note that the dose–response model for OCDF was reduced to background (100%) because the slope parameter (β) was not significant (p = 0.84) and the maximum effect parameter was not different from 100%. The estimates for the maximum effect parameters (α) for the single chemicals clustered into three groups, with maximums at 14, 31, and 50% of control 1551