The objective of this appendix is to provide details on an approach to the evaluation of “low-dose” mixture effects (see discussion in Chapter 5) by using data on a mixture of phthalates. There are many ways of conceptualizing a critical dose of each chemical in a mixture associated with “no observable effect,” such as no-observed-adverse-effect levels (NOAELs) or benchmark doses (BMDs). For illustration purposes, a BMD associated with a benchmark response (BMR) of 5% is estimated for each chemical in a mixture of phthalates and is used to determine a “mixture BMD” for a specified mixing ratio, assuming dose addition. The choice of a 5% BMR is for illustration only; other values may be selected. The mixture BMD depends on the mixing ratio of the components, and a tiered analysis strategy is described to determine critical doses of the chemicals in the mixture.

Howdeshell et al. (2008) reported on the effect that a mixture of five phthalate esters (BBP, DBP, DEHP, DIBP, and DPP) had on fetal testicular testosterone production. The mixture was selected so that the dose of each phthalate was proportional to a dose that yielded about equal reduction in testosterone when the components were given alone; that is, they used BBP, DBP, DEHP, and DIBP each at one dose and DPP at one-third that dose. Single-chemical data were used to predict the effect of the mixture at the specified ratio assuming dose addition; the observed fixed-ratio mixture dose-response data were compared with the dose-response predicted under dose addition. However, Howdeshell et al. did not use the dose-addition formula given in Chapter 4 (Equation 1) but rather an approximate approach to dose addition that used the average of the Hill slopes for the individual chemicals. The analytic method used in this appendix is based on a more general dose-additive model than and a somewhat different dose-response model from that used by Howdeshell et al. (2008). Here, dose addition is performed by using the formula from Chapter 4 (Equation 1) with the different slopes of the dose-response curves of the mixture components, and equality of the slopes is tested. Specifically, a nonlinear logistic dose-

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Appendix C
Analysis of a Mixture of Five
Phthalates: A Case Study
The objective of this appendix is to provide details on an approach to the
evaluation of “low-dose” mixture effects (see discussion in Chapter 5) by using
data on a mixture of phthalates. There are many ways of conceptualizing a criti-
cal dose of each chemical in a mixture associated with “no observable effect,”
such as no-observed-adverse-effect levels (NOAELs) or benchmark doses
(BMDs). For illustration purposes, a BMD associated with a benchmark re-
sponse (BMR) of 5% is estimated for each chemical in a mixture of phthalates
and is used to determine a “mixture BMD” for a specified mixing ratio, assum-
ing dose addition. The choice of a 5% BMR is for illustration only; other values
may be selected. The mixture BMD depends on the mixing ratio of the compo-
nents, and a tiered analysis strategy is described to determine critical doses of
the chemicals in the mixture.
Howdeshell et al. (2008) reported on the effect that a mixture of five
phthalate esters (BBP, DBP, DEHP, DIBP, and DPP) had on fetal testicular tes-
tosterone production. The mixture was selected so that the dose of each phthal-
ate was proportional to a dose that yielded about equal reduction in testosterone
when the components were given alone; that is, they used BBP, DBP, DEHP,
and DIBP each at one dose and DPP at one-third that dose. Single-chemical data
were used to predict the effect of the mixture at the specified ratio assuming
dose addition; the observed fixed-ratio mixture dose-response data were com-
pared with the dose-response predicted under dose addition. However, Howde-
shell et al. did not use the dose-addition formula given in Chapter 4 (Equation 1)
but rather an approximate approach to dose addition that used the average of the
Hill slopes for the individual chemicals. The analytic method used in this ap-
pendix is based on a more general dose-additive model than and a somewhat
different dose-response model from that used by Howdeshell et al. (2008). Here,
dose addition is performed by using the formula from Chapter 4 (Equation 1)
with the different slopes of the dose-response curves of the mixture components,
and equality of the slopes is tested. Specifically, a nonlinear logistic dose-
147

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148 Phthalates and Cumulative Risk Assessment: The Tasks Ahead
response model is used to facilitate a point estimate of a BMD—corresponding
to a BMR of 5%—for each chemical alone. A mixture BMD is estimated from
the dose-additive model and compared with that estimated from the observed
mixture data at the specified mixing ratio. Furthermore, the dose-additive model
is used to demonstrate that the mixture BMD is not constant across mixing ra-
tios. That is, the point estimate of the mixture BMD predicted under dose addi-
tion is shown to be numerically different if observed and hypothetical mixing
ratios of the five chemicals are used. The illustration is concluded with a de-
scription of a tiered analytic strategy for mixtures.
METHODS
Data were kindly provided by Earl Gray, Jr., in the Reproductive Toxicol-
ogy Division, National Health and Environmental Effects Research Laboratory,
Office of Research and Development, Environmental Protection Agency, Re-
search Triangle Park, NC.
Experimental Data. Pregnant Sprague-Dawley rats were dosed by gavage on
gestation day (GD) 8-18 with either vehicle control (dose, 0), a dose of one of
the chemicals, or a dose of the mixture of five phthalates (BBP, DBP, DEHP,
DIBP, and DPP) in a mixing ratio of 3:3:3:3:1. DEP was also evaluated in the
single-chemical studies but showed no effect; the DEP data have been retained
because they provide additional information on variability. Both single-chemical
and mixture studies were conducted in blocks (incomplete block design) with
one or two dams per treatment per block with two to four blocks per chemical
for a total of 166 litters across chemicals and doses. Testosterone was extracted
on GD 18 from the testes of the first three males in each litter and measured with
radioimmunoassay. Details are given in Howdeshell et al. (2008). The average
of the two measurements (one per testis) for each fetus was used in the analysis
herein.
Initial Statistical Analysis. A mixed-effects analysis of variance was used to test
for differences in control-group means while adjusting for intralitter correlated
data. There was a significant difference in the control-group means of testoster-
one (in nanograms per milliliter of medium) between studies and a significant
block effect, so the data from all studies were adjusted by the average control-
group value per block (giving percent of control).
Construction of an Additivity Model. The general strategy for the analysis of
the data was to use the single-chemical data to fit a nonlinear logistic model of
the mean (µ) testosterone concentration (percent of control) for the five single
chemicals and for the fixed-ratio mixture (in terms of total dose), that is,

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Appendix C 149
(1 − α i )[1 + exp(− β 0i )]
µi = α i + ,
1 + exp[−( β 0i + β i x )]
where x is the dose, αi is the parameter associated with the maximum effect for
the ith chemical or mixture, βi is the (negative) parameter associated with the
slope for the ith chemical or mixture, and β0i is the parameter associated with the
shape of the curve. The term [1 + exp(− β 0i )] was included in the numerator to
force the mean to equal 1 for the control group (x = 0). It was assumed that the
observed relative testosterone concentration differed from the model mean, µ, by
additive independent zero-mean normally distributed random terms representing
between-pup (within-litter) and between-litter variations (that is, a nonlinear
mixed-effects model was used with a linear random-effect, adjusting for intrali-
ter correlations). Preliminary analyses demonstrated that the sample variances
among chemicals, doses, and litters increased with the sample means; this sug-
gested that the within-litter variation is proportional to the mean. When the
within-litter values were adjusted for the dose-group mean, the variation was
relatively similar and suggested a common interlitter variance. The model
adopted therefore set the within-litter variance to be proportional to the pre-
dicted mean and set the between-litter standard deviation to be constant. The
model was estimated with all three parameters per chemical and mixture (18
mean parameters and two parameters for the standard deviations).
When the model dose-response curve is inverted, the dose, EDi(µ0), of the
ith chemical that is required to produce a given mean, µ0, is
( µ0 − α i )
⎡ ⎤
⎥ − β 0i
ln ⎢
1 − µ0 + (1 − α i ) exp(− β 0i ) ⎦
⎣
EDi ( µ0 ) = .
βi
Therefore, if component doses of a mixture are given by ai tadd , where the ai are
fixed proportions and tadd is the total mixture dose, then the general dose-
additive model (see Altenburger et al. 2000 and Gennings et al. 2004) gives the
dose-response curve for this fixed-proportion mixture as
−1
⎡5 ⎤
ai
tadd = ⎢ ∑ ⎥
⎣ i =1 EDi ( µ0 ) ⎦
−1
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢5 ⎥
ai
= ⎢∑ ⎥.
( µ0 − α i )
⎡ ⎤
⎢ i =1 ⎥
⎢ ln ⎢1 − µ + (1 − α ) exp(− β ) ⎥ − β 0i ⎥
⎣ 0i ⎦
⎢ ⎥
0 i
βi
⎢ ⎥
⎣ ⎦

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150 Phthalates and Cumulative Risk Assessment: The Tasks Ahead
The mixture data were also fitted to a nonlinear model of the same form, in
terms of total dose, as used for the components. The mixture BMD with a 5%
BMR was estimated from the mixture model and from the dose-additive model.1
For comparison, an independent-action model based on percentage of re-
sponse to individual chemicals (πi) was estimated, where2
µi − α i 1 + exp(− β 0i )
πi = = .
1 + exp ⎡ − ( β 0i + β i x ) ⎤
1 − αi ⎣ ⎦
If π measures the fraction of the maximum response, then
5
∏π .
π ind = i
i =1
It is important to note that the independent-action model as used here is
not a probabilistic model; it makes the conceptual leap of substituting fractional
effect (the fraction of the maximum response) for probability of occurrence (see
Chapter 4). It is not based on the assumption of statistical independence. More-
over, there is no way to estimate the maximum effect by using independent ac-
tion; here, for illustration, it is assumed that the maximum effect is 100% sup-
pression of testosterone because the maximum likelihood estimate for DEHP
alone has a maximum effect of 100% suppression.
RESULTS
Preliminary analyses indicated significant differences in mean testosterone
concentrations among the vehicle control groups and a significant difference in
the means among the blocks within control groups. Therefore, the later analyses
were based on percent control values, which were calculated by dividing the
average testosterone concentration per pup by the corresponding intrablock av-
erage control mean.
The nonlinear logistic mixed-effects model was fitted to the dose-response
data from each single chemical and to the mixture data in terms of total dose; the
model allowed intralitter correlated data. All five slope parameters were nega-
tive and significant, indicating that as the dose increases, there is a significant
decrease in testosterone concentration. The five slope parameters were statisti-
1
The model parameters were estimated by maximizing the likelihood of the observa-
tions, and confidence limits were estimated with the profile likelihood method. All calcu-
lations were performed in an Excel spreadsheet with components coded in Visual Basic
for Applications.
2
Recall that response in this case is the reduction in testosterone concentration and
that there is a maximum response.

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Appendix C 151
cally inhomogeneous (p = 0.03, likelihood ratio test) with point estimates rang-
ing from −0.002 to −0.040 per milligram per kilogram per day. Figure C-1 pro-
vides the observations and model-predicted dose-response curves (for the mean
response, at the maximum likelihood) for the five phthalates. In general, the data
are adequately represented by the nonlinear logistic model. Figure C-2 presents
the observed mixture data in terms of total dose. The solid curve is the model fit
based on the nonlinear logistic model, which adequately represents the observed
data. The dashed curve (Figure C-2A) is the dose-response model for the mix-
ture under the assumption of additivity. For comparison, the predicted independ-
ent-action dose-response curve is provided in Figure C-2B. In this case, the ex-
perimentally observed mixture data are adequately approximated by both the
dose-additive model and the independent-action model. In most cases, mixture
data are not available to make such a comparison, and single-chemical data are
used to approximate the mixture through an additivity model; in this case, dose
addition is a reasonable default to use when mixture data are not available.
It is of interest to determine a critical dose for the mixture of five phthal-
ates and compare the adjusted critical doses of the individual components with
their unadjusted critical doses. When the mixing ratio of the chemicals is speci-
fied, a BMD can be estimated for the mixture by using dose addition. BMD es-
timates for each of the five chemicals are provided in Table C-1 with lower one-
sided 95% confidence limits. BMDs for the mixture (with a specified mixing
ratio) and as predicted under additivity for the same mixing ratio (with the pro-
portion of the ith chemical denoted by ai) were estimated with the single-
chemical and mixture models (Table C-1). Specifically, the BMD for the mix-
ture (with fixed mixing ratio) under additivity is estimated as
−1 −1
⎛5 ai ⎞ ⎛ 0.23 0.23 0.23 0.08 0.23 ⎞
= ⎜∑ ⎟ =⎜ + + + + ⎟ = 52 mg/kg-d.
tadd
BMDi ⎠ ⎝ 116 30 49 25 126 ⎠
⎝ i =1
The mixture BMD as predicted by dose addition depends on the mixing
ratios of the chemicals. To illustrate that point, consider three mixing ratios of
the five phthalates for which single-chemical data are available (from the study
by Howdeshell et al. 2008). Table C-2 includes the mixing proportions for each
case and the corresponding concentrations of each chemical in such mixtures at
the mixture BMD (assuming dose addition). Such mixture BMDs depend on the
mixing ratio of the chemicals. A tiered analytic strategy is suggested by consid-
eration of the following and other cases.
● Case 1 is based on a mixture in which the mixing ratio for each single-
chemical component concentration is proportional to the BMD for each single
chemical. The single-chemical component concentrations in the BMD mixture
correspond to dividing each BMD by the number of active chemicals in the mix-
ture—here, five. The single-chemical component concentrations in the BMD
mixture can be considered as adjusted critical values—any mixture that contains

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152 Phthalates and Cumulative Risk Assessment: The Tasks Ahead
single-chemical component concentrations that are each less than or equal to
their adjusted critical values3 will (under dose addition) invoke a response less
than the BMR. In case 1, the mixture BMD is 69 mg/kg-d under additivity, and
the adjusted critical values for the five chemicals range from 5 mg/kg-d for DPP
to 25 mg/kg-d for DIBP (Table C-2). This case is especially simple because the
adjusted critical values are just one-fifth of the single-chemical BMDs (Table C-
1). When the exposure concentration of each single chemical in some mixture is
below the adjusted critical value (for any mixing ratio), the response to the mix-
ture is associated with a lower BMR than that used to construct the adjusted
critical values (here, the BMR is 5%).
● Cases 2 and 3 are based on exposure data presented in Table 2-2; data
on DPP as a parent compound were not included, and it is omitted from these
two cases. Table 2-2 presents urinary concentrations of metabolites of the parent
compounds, but the fraction of the parent diester that ends up in the urine varies
widely among the phthalates. For example, 5-10% of DEHP is excreted as
TABLE C-1 Estimated BMDs Associated with 5% BMRa for Single Chemicals
and Mixture Data Based on Nonlinear Logistic Model and Estimated with
Mixed-Effects Model Accounting for Intralitter Correlated Datab
Lower One-Sided 95%
Chemical BMD (mg/kg-d) Confidence Limit (mg/kg-d)
BBP 116 66
DBP 30 20
DEHP 49 31
DPP 25 10
DIBP 126 47
Mixture 74 39
Mixture (additivity) 52 39
a
The response evaluated here is the fractional reduction of testosterone concentration
relative to the testosterone concentration at zero dose. Other definitions could be contem-
plated, such as the change relative to the maximum reduction achievable or, in view of
the variation observed in average testosterone concentrations at zero dose between differ-
ent groups of animals, some change related to a measure of the width of the distribution
of those zero-dose testosterone concentrations. The choice here has been arbitrarily se-
lected for demonstration purposes.
b
The mixture components are each at 23% except DPP, which is 8% of the mixture.
Study details are included in Howdeshell et al. (2008).
3
Any particular set of adjusted critical values have to be treated together as a set for a
particular mixing ratio of the components. There must be no mixing and matching of
adjusted critical values obtained from different mixtures.

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Appendix C 153
TABLE C-2 Three Mixtures to Illustrate an Approach to Calculating Adjusted
Critical Doses for Single Chemicals in a Mixturea
Mixing Ratios That Sum to 1 Mixture BMD,
(Single-Chemical Dose [mg/kg] in Mixture BMD, Assuming Additivity
Assuming Additivity)b (mg/kg)
BBP DBP DEHP DPP DIBP
Case a1 a2 a3 a5 a6 tadd
0.336 0.086 0.143 0.072 0.363
1
(23.3) (6.0) (9.9) (5.0) (25.2) 69.3
0.13 0.19 0.66 0.02
2
(6.2) (9.0) (31.4) — (1.0) 47.6
0.02 0.38 0.48 0.12
3
(0.8) (16.1) (20.3) — (5.1) 42.4
a
Case 1 corresponds to dividing each single chemical BMD by 5 (the number of active
chemicals in the mixture). Case 2 is based on the relative proportion of the parent com-
pound from its metabolites at the 50th percentile as evaluated in the NHANES study for
the five chemicals (see Table 2-2). Case 3 is based on the relative proportion of the parent
compound from its metabolites at the 50th percentile as evaluated in the Wittassek et al.
(2007) study (see Table 2-2). The mixture BMD depends on the mixing ratio.
b
The single-chemical doses for the mixture BMD under additivity sum to the mixture
BMD in the last column.
MEHP, whereas more than 90% of DBP is excreted as MBP. For this example,
we assumed that the sum of MEHP, MECPP, MEOHP, and MEHHP (DEHP
metabolites) represents 50% of parent DEHP. Because less is known about the
excretion of BBP, DBP, and DIBP as measured by the listed metabolites, we
assumed that the excretion of the corresponding metabolites is roughly similar to
the exposure to the parent compound. So, for illustration only, the mixing pro-
portions of the four parent compounds were calculated on the basis of the pro-
portion of the sum across the metabolites (using the 50th percentile values) for
the four parent compounds, with the DEHP metabolites doubled. Case 2 corre-
sponds to the values from the National Health and Nutrition Examination Sur-
vey (NHANES); case 3 corresponds to the German study values (see Table 2-2).
For case 2, the mixture BMD is 48 mg/kg under dose addition, and the adjusted
critical values for the remaining four chemicals range from 1 mg/kg for DIBP to
31 mg/kg for DEHP (Table C-2). For case 3, the mixture BMD is 42 mg/kg un-
der additivity, and the adjusted critical values for the remaining four chemicals
range from 1 mg/kg for BBP to 20 mg/kg for DEHP (Table C-2).
In contrast with the evaluation of single chemicals, the critical dose (here,
69, 48, and 42 mg/kg for the three cases considered) of a mixture and the ad-
justed critical values for the components clearly depend on the mixing ratio.
How should adjusted critical doses be specified for individual chemicals in
a mixture when exposure information is not available (that is, the doses and mix-

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154 Phthalates and Cumulative Risk Assessment: The Tasks Ahead
ing ratios of the chemicals in the mixture are not known or are not constant)?
The committee suggests that a tiered approach be considered.
● First, determine whether the single-chemical doses in the exposure of
concern are all below the adjusted critical value specified by dividing the critical
values (here, the BMD associated with a fixed BMR) of the single chemicals by
the number of active chemicals in the mixture (case 1 in Table C-2). If so, the
response to the mixture is less than the BMR, assuming general dose addition.
There is no need to go any further.
● Second, if one or more of the single-chemical doses in the exposure of
concern exceeds the adjusted critical value specified for the mixture in step 1,
determine the mixing ratio of the exposure of concern and recalculate the critical
dose for the specific mixture ratio (for example, cases 2 and 3 in Table C-2). If
all single-chemical exposures are below the adjusted critical doses for the mix-
ture of concern, the response to the mixture is less than the BMR, assuming gen-
eral dose addition.
In Table C-2 for cases 2 and 3, assumptions would be made to determine
doses of a parent compound on the basis of metabolite concentrations. If, for
example, the calculated dose of DEHP exceeds 10 mg/kg (from case 1), a more
refined estimate of an adjusted critical dose could be based on the mixing ratios
obtained from exposure estimates (case 2 or 3). That is, the exposure to DEHP
may be increased if exposures to other chemicals are lower than considered in
case 1. If the exposure to each chemical is below the single-chemical adjusted
critical value for the specified mixture ratio (case 2 or 3), the response could be
claimed to be less than the selected BMR.
DISCUSSION
The additivity model described here was based on a nonlinear logistic
model with the potential for a maximum effect other than zero testosterone.
Howdeshell et al. (2008) used the nonlinear Hill model, assuming that the
maximum effect was complete suppression of testosterone, and approximated
the dose-addition procedure by using an average Hill slope for the mixture. The
analyses of each model included similar figures (Figure C-2 here; Figure 2B in
Howdeshell et al. 2008) that compared the mixture data, the nonlinear model fit
to mixture data, and the model predicted by dose addition. Both showed that the
dose-additive model fell below the mixture model. Howdeshell et al. did not
make a statistical comparison of the two models; they claimed that a dose-
additive relationship adequately represented the data. As seen in Figure C-2, the
dose-additive model used here is similar to the observed mixture model; a for-
mal statistical comparison of the two curves was not conducted.
The point of the analysis illustrated here was to determine a mixture BMD
by using dose addition and to show that its value depends on the mixing ratio.

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Appendix C 155
That required an estimation of a BMD for each component in the mixture; a
nonlinear logistic model was used here for illustration. A comparison of the re-
sults that would be obtained with other models was not conducted. The devel-
opment and dissemination of methods that result in inference that does not de-
pend on a specific model constitute an important field of research. Bayesian
approaches have been suggested in which the resulting inferences include the
uncertainty associated with model selection, as well as parameter uncertainty.
In accordance with the discussion in Chapter 5, the evaluation of critical
doses in this illustration was based on BMDs. Nyribihizi et al. (2008) compare
BMDs for experimentally observed mixture data with a fixed mixing ratio and
the corresponding BMD under additivity. Their approach is similar to that used
here. Other approaches, such as the use of NOAELs, are possible; the limitations
of the use of NOAELs have been discussed extensively (see, for example, EPA
2000).
The illustration in this appendix included the use of approximate mixing
ratios of the chemicals estimated from urinary concentrations. Such estimates
required many simplifying assumptions. The availability of the supporting data
relating urinary metabolites and parent compound exposure concentrations var-
ies among the chemicals. Exposures probably differ between infants, children,
and adults—a variation not considered in our calculations. However, the ap-
proach is generic and can be repeated for different mixing ratios to account for
observed exposures.
A BBP
1.6
1.4
Testosterone (fraction of control) .
1.2
1
0.8
0.6
0.4
0.2
0
0 100 200 300 400 500 600 700 800 900 1000
Dose (mg/kg-d)

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156 Phthalates and Cumulative Risk Assessment: The Tasks Ahead
B DBP
1.6
1.4
Testosterone (fraction of control) .
1.2
1
0.8
0.6
0.4
0.2
0
0 100 200 300 400 500 600 700
Dose (mg/kg-d)
C DEHP
1.6
1.4
Testosterone (fraction of control) .
1.2
1
0.8
0.6
0.4
0.2
0
0 100 200 300 400 500 600 700 800 900 1000
Dose (mg/kg-d)

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Appendix C 157
D DPP
1.6
1.4
Testosterone (fraction of control) .
1.2
1
0.8
0.6
0.4
0.2
0
0 50 100 150 200 250
Dose (mg/kg-d)
E DIBP
1.4
1.2
Testosterone (fraction of control) .
1
0.8
0.6
0.4
0.2
0
0 100 200 300 400 500 600 700 800 900 1000
Dose (mg/kg-d)
FIGURE C-1 Average testosterone concentration (as percent of control) per pup (*) vs
dose of five single chemicals with maximum likelihood dose-response curves used in
additivity model.

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158 Phthalates and Cumulative Risk Assessment: The Tasks Ahead
A MIXTURE
1.4
1.2
Testosterone (fraction of control) .
1
0.8
0.6
0.4
0.2
0
0 200 400 600 800 1000 1200 1400
Dose (mg/kg-d)
Model Data mean and 90% confidence interval
Data Dose addition
B MIXTURE
1.4
1.2
Testosterone (fraction of control) .
1
0.8
0.6
0.4
0.2
0
0 200 400 600 800 1000 1200 1400
Dose (mg/kg-d)
Model Data Independent action
FIGURE C-2 (A) Observed (*) and model-predicted dose-response curves for mixture of
five phthalates based on the nonlinear logistic model for the mixture data (solid curve)
and as predicted under additivity (dashed curve). The mixing ratio of the five phthalates
was 3:3:3:3:1 for BBP, DBP, DEHP, DIBP, and DPP, that is, 0.23, 0.23, 0.23, 0.23, and
0.08 of the mixture. (B) For comparison, the prediction using an independent-action
model based on percentage of response.

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Appendix C 159
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