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Parameterizing Dose-Response Models to Estimate Relative Potency
Functions Directly
Gregg E. Dinse1 and David M. Umbach
Biostatistics Branch, National Institute of Environmental Health Sciences (NIEHS), Research Triangle Park, North Carolina 27709
1To whom correspondence should be addressed at Biostatistics Branch, NIEHS, Bldg. 101, Room A-349, Mail Drop A3-03, 111 Alexander Drive, Research
Triangle Park, NC 27709-2233. Fax: (919) 541-4498. E-mail: dinse@niehs.nih.gov.
Received June 1, 2012; accepted June 5, 2012
Many comparative analyses of toxicity assume that the potency
of a test chemical relative to a reference chemical is constant, but
employing such a restrictive assumption uncritically may generate
misleading conclusions. Recent efforts to characterize non-constant
relative potency rely on relative potency functions and estimate
them secondarily after fitting dose-response models for the test
and reference chemicals. We study an alternative approach of spe-
cifying a relative potency model a priori and estimating it directly
using the dose-response data from both chemicals. We consider a
power function in dose as a relative potency model and find that it
keeps the two chemicals’ dose-response functions within the same
family of models for families typically used in toxicology. When dif-
ferences in the response limits for the test and reference chemicals
are attributable to the chemicals themselves, the older two-stage
approach is the more convenient. When differences in response
limits are attributable to other features of the experimental proto-
col or when response limits do not differ, the direct approach is
straightforward to apply with nonlinear regression methods and
simplifies calculation of simultaneous confidence bands. We illus-
trate the proposed approach using Hill models with dose-response
data from U.S. National Toxicology Program bioassays. Though not
universally applicable, this method of estimating relative potency
functions directly can be profitably applied to a broad family of
dose-response models commonly used in toxicology.
Key Words: dose response; Hill model; PeCDF; relative potency;
TCDD.
Toxicologists often compare the potency of one chemical to
another and use relative potency values for ranking chemicals
or for converting doses of a test chemical into their equivalents
for a reference chemical. Relative potency is also used when
evaluating mixtures of chemicals for dose additivity or when
calculating toxic equivalency factors.
Usually, relative potency is taken as constant, an assumption
that arises naturally in the context of dilution assays (Finney,
1965). When two chemicals exhibit constant relative potency,
their dose-response curves plotted on a log-dose axis are identi-
cal up to a constant horizontal shift; relative potency depends on
the magnitude and direction of that shift. Comparative assays
often, however, involve dose-response curves whose horizon-
tal separation is non-constant (Cornfield, 1964; De Lean et al.,
1978; DeVito et al., 2000; Guardabasso et al., 1988). In those
situations, methods that accommodate non-constant relative
potency are desirable to avoid misleading conclusions.
Recent work focuses on relative potency functions as gen-
eral descriptors of non-constant relative potency (Dinse and
Umbach, 2011; Ritz et al., 2006). The fundamental idea
is that possibly varying horizontal distances between two
dose-response curves may be indexed by any one of several
quantities: dose of either the test or reference chemical, mean
response, or response quantile. Thus, a relative potency func-
tion expresses local relative potency across the dose or response
range as a function of one of these quantities, thereby providing
a global description of the relative potency relationship.
In earlier work, we postulated a dose-response model for
each chemical and expressed relative potency functions in
terms of the parameters from those dose-response models
(Dinse and Umbach, 2011). For example, if the dose-response
relationship for each chemical obeyed a four-parameter model,
the relative potency function would be expressed using all eight
parameters. Of course, fewer parameters suffice if some param-
eters are constrained as equal when fitting the models. This
original approach may be characterized as a two-stage process
that initially estimates parameters for a pair of dose-response
functions and then plugs those estimates into an appropriate
expression to estimate the relative potency function. The rela-
tive potency function contains no fundamentally new informa-
tion beyond that contained in the dose-response functions—it
simply describes a relationship between them.
Here, we propose an alternative approach for estimating relative
potency functions. This alternative specifies a relative potency
function a priori and estimates it directly using standard nonlinear
regression methods. This approach requires specification of a
dose-response model for the reference chemical and a relative
potency model (these two determine the dose-response model
for the test chemical). When the relative potency function is the
toxicological sciences 129(2), 447–455 (2012)
doi:10.1093/toxsci/kfs209
Advance Access publication June 14, 2012
Published by Oxford University Press 2012.
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principal object of study, specifying a model for it explicitly has
intuitive appeal. This proposed approach is motivated by a desire
to simplify statistical inference on relative potency functions;
estimation and testing should be more straightforward when a
relative potency model is specified directly.
Selecting a model for relative potency requires care, however.
Comparative assays generally employ monotone dose-response
models for the test and reference chemicals that are within the
same family (i.e., have the same functional form but possibly
different parameter values). Matching an arbitrary relative
potency model to a given dose-response model for the refer-
ence chemical might induce a dose-response function for the
test chemical that is in a different family of models or even
one that is non-monotone. Consequently, we focus on relative
potency models that we call “compatible” in that they keep the
dose-response models for both reference and test chemicals
within the same family.
Our purpose is to develop this alternative approach and explore
its strengths and limitations. We consider monotone dose-response
models parameterized by lower and upper response limits, plus a
vector of additional parameters. We consider two situations: one
where any observed differences in response limits are attributable
to the chemicals themselves (intrinsic differences) and another
where such differences arise from other aspects of the experi-
mental procedure (extrinsic differences). We find that, when dif-
ferences in response limits are intrinsic, specifying a compatible
relative potency function is difficult so the two-stage approach is
more convenient. Alternatively, if the chemicals have the same
response limits or if any observed differences are seen as extrin-
sic, compatible relative potency functions are easy to specify for
commonly used dose-response models. Thus, the explicit speci-
fication of relative potency functions is widely, though not uni-
versally, applicable. We use data from U.S. National Toxicology
Program (NTP) bioassays to illustrate the direct estimation of
relative potency functions.
MATERIALS AND METHODS
Dose-response models. Let f(d;θ) denote the mean response elicited by
non-negative dose d of a chemical of interest. Here, f, the dose-response model,
is a monotone function of d that depends on a vector θ of unknown parameters.
We focus on dose-response curves for which the mean response increases from
a lower limit or background level L at a dose of zero (d = 0) to an upper limit U
at an infinite dose (d = ∞). (If, instead, response decreases with dose, then U is
associated with d = 0 and L with d = ∞.) We express such dose-response curves
using the general formula:
f(d;θ) = L + (U − L) g(d;ϕ), (1)
where g is a monotone function that ranges from 0 to 1 as d ranges from 0 to ∞
and depends on a vector ϕ of unknown parameters. Here, θ = (L, U, ϕ). Various
functions that meet the requirements for g are commonly used in toxicology
(Table 1). For our illustrations, we will use the Hill (1910) model, a sigmoid
function often used for dose-response relationships, where
g(d;ϕ) = dS/(dS + MS) = {1 + exp[Slog(M) − Slog(d)]}−1. (2)
Here, ϕ = (S, M), where S regulates the shape of the dose-response curve and
M is the median effective dose (or ED50). For convenience, we use the notation
“log” for the natural logarithm.
Relative potency. Let subscripts 0 and 1 index the reference and test
chemicals, respectively. In dilution assays where relative potency (denoted ρ)
is constant, the dose-response functions (denoted f0 and f1, for the reference and
test chemicals, respectively) must have the same functional form (say, f) though
they each have distinct parameter vectors θ0 and θ1, respectively. When relative
potency is constant, then f(d1;θ1) = f(ρd1;θ0) for any dose d1 ≥ 0 (Finney, 1965).
This equation states that, at any specified dose d1 of the test chemical, a dose d0
of the reference chemical equal to a constant ρ times d1 yields the same mean
response as a dose d1 of the test chemical. Consequently, ρ = d0/d1, where d0
and d1 are any two doses of the respective chemicals that elicit the same mean
response. Equipotency corresponds to ρ = 1.
Recent efforts to accommodate non-constant relative potency are based on
relative potency functions (Dinse and Umbach, 2011; Ritz et al., 2006). When
plotted with dose on a logarithmic axis, the horizontal distance between two
dose-response curves at a given response level µ corresponds to the magnitude
of the log-relative potency at that value of µ, i.e., log(ρ) = log(d0) − log(d1).
When that horizontal distance changes along the curves, it (or, equivalently, the
relative potency itself) can be indexed by the mean response, the dose of the
reference chemical, or the dose of the test chemical. The resulting three relative
potency functions—denoted ρµ(µ), ρd0(d0), and ρd1(d1), respectively, by Dinse
and Umbach (2011)—describe the same fundamental relative potency informa-
tion but through different variables.
Let response quantile π be defined as response level µ expressed as a frac-
tional distance from the lower response limit L to the upper response limit U,
i.e., π = (µ − L)/(U − L). As µ ranges from L to U, π ranges from 0 to 1. We
call the function g(d;ϕ) in Equation 1 a dose-quantile model, a dose-response
model that gives the response quantile as a function of dose. Let ED100π denote
the dose that produces a response 100π% of the distance from L to U. When
the two dose-response functions have the same upper and the same lower
response limits, the relative potency at response level µ with corresponding
response quantile π is simply the ratio of the ED100π values, reference over test.
Consequently, for chemicals with identical response limits, relative potency
also can be indexed by π, leading to another relative potency function that we
denote ρπ*(π). When chemicals differ in their response limits, one can still index
the ratio of ED100π values by π and so construct a function ρπ*(π); however, with
unequal response limits, the ratio of ED100π values no longer corresponds to
the classical definition of relative potency but represents a modified definition
TABLE 1
Dose-Quantile Functions for Several Dose-Response
Models Whose Compatible Relative Potency Model Is a Power
Function in Dosea
Dose-quantile function
Dose-response model
g(d;ϕ)b
h(X)c
Logistic/Hill
Weibull
Probitd
Generalized logistic/Hill
Generalized Weibull
[1 + exp(α + βlog(d))]−1
1 − exp[−exp(α + βlog(d))]
Φ(α + βlog(d))
[1 + exp(α + βlog(d))]−γ
{1 − exp[−exp(α + βlog(d))]}γ{1 − exp[−exp(X)]}γ
[1 + exp(X)]−1
1 − exp[−exp(X)]
Φ(X)
[1 + exp(X)]−γ
aA power function in dose has form ρ°(d) = eηdψ or, equivalently, log[ρ°(d)] =
η + ψ log(d).
bThe vector ϕ contains parameters α and β and, optionally, additional param-
eters γ.
cThe dose-quantile function g(d;ϕ) is obtained by substituting X = α + βlog(d)
into h(X).
dΦ(X) = (2π)−1/2∫-∞
X exp(−t2/2) dt is the standard normal distribution function.
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ρ*(d1), however, will not degrade fit even when response limits
differ between chemicals. It incorporates a modified definition of
relative potency that is constructed to recover the classical con-
cept in the face of differences in response limits—but the con-
struction is meaningful only if those differences are not due to the
chemicals themselves.
Using EROD and A4H enzyme activity data from NTP bio-
assays, we illustrated the utility of directly estimating a relative
potency function. For liver EROD activity, we found that sepa-
rate Hill models for each chemical fit adequately, but that no
simpler models were consistent with the observations. In par-
ticular, PeCDF and TCDD appeared to have distinct response
limits. If we view the difference in response limits as intrinsic
to the chemicals, the relative potency function ρ(d1), embody-
ing the classical definition of relative potency, is more complex
than our proposed power-function model and must be estimated
in two stages from all eight parameters of the fitted Hill models.
This relative potency function was defined on an interval of
doses and exhibited strong curvature and vertical asymptotes as
it approached the limits of that range. Alternatively, if we view
the difference in response limits as extrinsic to the chemicals,
the relative potency function ρ*(d1) can be estimated explic-
itly via a reparameterization of the eight-parameter model. This
relative potency function was non-constant and was defined
for any dose; it suggested that the potency of PeCDF became
smaller relative to TCDD as dose increased.
For liver A4H activity, we found that the dose-response rela-
tionships were adequately modeled by two Hill functions con-
strained to have the same lower and same upper response limits
(six-parameter model). When response limits are the same for
both chemicals, ρ(d1) and ρ*(d1) are identical and can be esti-
mated explicitly by a reparameterization of the six-parameter
dose-response model. The resulting relative potency function
was nearly constant and, in fact, fit no better than a model
enforcing constant relative potency.
Relative potency functions offer a promising way to relax
the assumption of constant relative potency while still quan-
tifying the potency relationships among chemicals. Instead of
estimating relative potency functions by plugging parameter
estimates from fitted dose-response functions into appropriate
expressions, we proposed instead modeling relative potency as
a function of the test chemical’s dose with a power function and
estimating its parameters directly as part of the dose-response
fitting process. The proposed approach is widely applicable,
but not universally so. In our examples, the power-function
form proved convenient and amenable to use with ρ*(d1) and
ρ(d1), though its use with the latter requires both chemicals to
have the same response limits.
SUPPLEMENTARY DATA
Supplementary data are available online at http://toxsci.
oxfordjournals.org/.
FUNDING
This research was supported by the Intramural Research
Program of the NIH, National Institute of Environmental
Health Sciences (Z01-ES-102685).
ACkNOWLEDGMENTS
We are grateful to Grace Kissling for constructive comments.
The authors declare that there are no conflicts of interest.
REFERENCES
Cornfield, J. (1964). Comparative assays and the role of parallelism. J.
Pharmacol. Exp. Ther. 144, 143–149.
De Lean, A., Munson, P. J., and Rodbard, D. (1978). Simultaneous analysis of
families of sigmoidal curves: Application to bioassay, radioligand assay, and
physiological dose-response curves. Am. J. Physiol. 235, E97–E102.
DeVito, M. J., Menache, M. G., Diliberto, J. J., Ross, D. G., and Birnbaum,
L. S. (2000). Dose-response relationships for induction of CYP1A1 and
CYP1A2 enzyme activity in liver, lung, and skin in female mice follow-
ing subchronic exposure to polychlorinated biphenyls. Toxicol. Appl.
Pharmacol. 167, 157–172.
Dinse, G. E. and Umbach, D. M. (2011). Characterizing non-constant relative
potency. Reg. Toxicol. Pharmacol. 60, 342–353.
Finney, D. J. (1965). The meaning of bioassay. Biometrics 21, 785–798.
Guardabasso, V., Munson, P. J., and Rodbard, D. (1988). A versatile method for
simultaneous analysis of families of curves. FASEB J. 2, 209–215.
Hill, A. V. (1910). The possible effects of the aggregation of the molecules of
haemoglobin on its dissociation curves. J. Physiol. 40(Suppl.), iv–vii.
National Toxicology Program (2006a). NTP Technical Report on the Toxicology
and Carcinogenesis Studies of 2,3,7,8-Tetrachlorodibenzo-p-dioxin (TCDD)
(CAS No. 1746-01-6) in Female Harlan Sprague Dawley Rats (Gavage
Studies). Technical Report Series No. 521, NIH Publication No. 06-4468.
U.S. Department of Health and Human Services, Public Health Service,
National Institutes of Health, RTP, NC.
National Toxicology Program (2006b). NTP Technical Report on the Toxicology
and Carcinogenesis Studies of 2,3,4,7,8-Pentachlorodibenzofuran (PeCDF)
(CAS No. 57117-31-4) in Female Harlan Sprague Dawley Rats (Gavage
Studies). Technical Report Series No. 525, NIH Publication No. 06-4461.
U.S. Department of Health and Human Services, Public Health Service,
National Institutes of Health, RTP, NC.
Ritz, C., Cedergreen, N., Jensen, J. E. and Streibig, J.C. (2006). Relative
potency in nonsimilar dose-response curves. Weed Sci. 54, 407–412.
Seber, G. and Wild, C. (1989). Nonlinear Regression. John Wiley & Sons, New
York.
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