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- Abstract
- 1. Introduction
- 2. Background
- 3. Relative Potency Functions
- 4. Application to Enzyme Activity Data
- 5. Discussion
- Supplementary Material
- References

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Regul Toxicol Pharmacol. Author manuscript; available in PMC 2012 August 1.

Published in final edited form as:

Published online 2011 May 13. doi: 10.1016/j.yrtph.2011.05.002

PMCID: PMC3134169

NIHMSID: NIHMS297378

Gregg E. Dinse, Biostatistics Branch, National Institute of Environmental Health Sciences, Mail Drop A3-03, P.O. Box 12233, Research Triangle Park, NC 27709 USA;

Gregg E. Dinse: vog.hin.shein@esnid; David M. Umbach: vog.hin.shein@hcabmu

Corresponding Author Information: Gregg E. Dinse, Biostatistics Branch, Mail Drop A3-03, P.O. Box 12233, National Institute of Environmental Health Sciences, Building 101, Room A-349, 111 T.W. Alexander Drive, Research Triangle Park, NC 27709-2233 USA, vog.hin.shein@esnid, Telephone: 919-541-4931, Fax: 919-541-4498

The publisher's final edited version of this article is available at Regul Toxicol Pharmacol

See other articles in PMC that cite the published article.

Relative potency plays an important role in toxicology. Estimates of relative potency are used to rank chemicals by their effects, to calculate equivalent doses of test chemicals compared to a standard, and to weight contributions of constituent chemicals when evaluating mixtures. Typically relative potency is characterized by a constant dilution factor, even when non-similar dose-response curves indicate that constancy is inappropriate. Improperly regarding relative potency as constant may distort conclusions and potentially mislead investigators or policymakers. We consider a more general approach that allows relative potency to vary as a function of dose, response, or response quantile. Distinct functions can be defined, each generalizing different but equivalent descriptions of constant relative potency. When two chemicals have identical response limits, these functions all carry fundamentally equivalent information; otherwise, relative potency as a function of response quantile is distinct and embodies a modified definition of relative potency. Which definition is preferable depends on whether one views any differences in response limits as intrinsic to the chemicals or as extrinsic, arising from idiosyncrasies of data sources. We illustrate these ideas with constructed examples and real data. Relative potency functions offer a unified and principled description of relative potency for non-similar dose-response curves.

Toxicologists use estimates of relative potency for ranking chemicals (*e.g*., Glass *et al*, 1991) and for dose-conversion analyses that calculate the equivalent dose of one chemical that produces the same response as a specific dose of another (*e.g*., Putzrath, 1997). They also combine separate estimates of relative potency, from different studies and endpoints, to determine a single toxic equivalency factor (TEF) (*e.g*., Van den Berg *et al*, 2006). The National Toxicology Program may soon use relative potency estimates in prioritizing chemicals via high throughput screening.

Relative potency of a test compound compared to a reference compound is typically thought of as a ratio of doses (reference divided by test) that produce the same mean response for a given endpoint. The notion that relative potency should be constant has roots in analytical dilution assays where all preparations are regarded as dilutions of a standard preparation with an inert diluent (Finney, 1965). In that context, the ratio of doses producing the same mean response should be identical at every response level. Consequently, when dose-response curves for such preparations are plotted on a log-dose axis, the curves should be identical up to a horizontal shift (Figure 1). Even when preparations are not simple dilutions of a single active compound, there may be biological reasons (e.g., a common mechanism of toxicity) to expect them to behave as though they were and, thus, to have constant relative potencies. Chemicals with constant relative potency are said to have ‘similar’ dose-response curves (*i.e.*, equal up to a horizontal shift on the log-dose axis). Their relative potency is characterized by a single dilution (or concentration) factor, a feature that simplifies ranking chemicals and calculating equivalent doses.

Similar dose-response curves. The reference and test chemicals are *C*_{0} and *C*_{1}, respectively. With dose plotted on a logarithmic scale, the horizontal distance between curves, denoted *λ* and equal to log(*ρ*), is constant. Each arrow extends **...**

Researchers have long recognized that constant relative potency is not always reasonable because comparative assays often yield dose-response curves that differ in ways other than a horizontal shift (Cornfield, 1964; Cox and Leaverton, 1966; Rodbard, 1974; De Lean *et al*, 1978; Guardabasso *et al*, 1987; Guardabasso *et al*, 1988). When dose-response curves are not similar, regarding them as similar and estimating relative potency accordingly may distort conclusions.

For example, consider the activity of a liver enzyme in response to TCDD, PCB126, PeCDF and their TEF-based mixture (National Toxicology Program, 2006a–d). For illustration purposes, we regard the mixture as a fourth chemical. When a separate dose-response model is fit to each chemical and the ratio of median effective doses (*ED*_{50}s) is used to assess relative potency despite non-similarity (Figure 2a), the estimated potencies relative to TCDD suggest ranking the chemicals (from most to least potent) as: TCDD > Mixture > PCB126 > PeCDF. When the dose-response models are constrained to enforce similarity (Figure 2b), the constant relative potencies, determined as ratios of revised *ED*_{50}s, suggest the ranking: Mixture > TCDD > PeCDF > PCB126. These rankings differ, in part, because the chemicals vary substantially in their apparent upper limits of response. Both rankings indicate that TCDD and the mixture are more potent than PeCDF and PCB126, but they give opposite results for TCDD versus the mixture and for PeCDF versus PCB126. In both the constrained and unconstrained analyses, the mixture’s fitted curve lies above TCDD’s at all doses (Figure 2), indicating that the mixture is the more potent and, in turn, demonstrating that the use of *ED*_{50}s for estimating relative potency is problematic when dose-response curves are not similar. The unconstrained dose-response curves for TCDD and PeCDF cross twice, suggesting that their relative potency changes across response levels and doses; consequently, using any single value of relative potency for ranking or dose conversion would be an oversimplification.

Estimated dose-response curves for TCDD, PCB126, PeCDF and their TEF-based mixture. Panels: (a) separate Hill model fit for each chemical; and (b) Hill models constrained by assuming all four chemicals have similar dose-response functions. Curves: solid, **...**

Despite recognition of these issues, approaches to analysis that explicitly allow non-constant relative potencies are rare. Cornfield (1964) relaxed the similarity constraint, but he assumed that response was a linear function of log-dose (see, also, Cox and Leaverton, 1966); typically linearity is reasonable for only a portion of the dose-response curve. Under a more general dose-response model, DeVito *et al* (2000) assumed relative potency was constant for doses below some point and otherwise was linear in the reciprocal of dose. Several researchers suggested accommodating non-similar dose-response curves by simply reporting estimates of relative potency at a few selected doses (Putzrath, 1997; Villeneuve *et al*, 2000). Recently, Ritz *et al* (2006) formulated relative potency as a function of response. These proposals notwithstanding, toxicologists often estimate relative potency by the ratio of *ED*_{50}s despite non-similarity of the dose-response curves (Rodbard and Frazier, 1975; Villeneuve *et al*, 2000).

This article builds on previous proposals for quantifying relative potency for non-similar dose-response curves. We define functions that describe relative potency as depending on dose, response, or response quantile. In general, the ratio of doses used to define relative potency depends on where along the dose-response curves the ratio is taken; these functions reflect that dependence. We develop our proposals conceptually and graphically, describing their utility and limitations, and we defer technical details to an appendix. We postpone a full discussion of the statistical analysis of relative potency functions as beyond our present scope. We illustrate our proposals with the NTP enzyme activity data introduced earlier.

A dose-response model expresses the mean (or average) response as a mathematical function of dose. For a given endpoint, we use the notation *f* (*d;θ*) to represent the mean response elicited by dose *d* of the chemical of interest, where *f* is a monotone function of *d* that depends on a vector *θ* of unknown parameters. We consider dose-response curves where the mean response increases from a lower limit, *L*, at a dose value of zero (*d* = 0) to an upper limit, *U*, at an infinite dose (*d* = ∞). If the opposite is true, so that the curve decreases from an upper limit, the ideas that we describe still apply after reversing the definitions of *L* and *U*.

Various functions can be used for *f*. A sigmoid function commonly used for dose-response relationships is the Hill (1910) model:

$$f(d;\theta )=L+(U-L){d}^{S}/({M}^{S}+{d}^{S}).$$

(1)

The parameter vector is *θ* = (*L*, *U*, *S*, *M*), where *S* governs the shape (or steepness) of the dose-response curve and *M* represents the median effective dose (*ED*_{50}). All dose-response curves in our figures are based on Hill models, plotted with dose on a logarithmic scale, though our relative potency concepts are applicable to any pair of monotone dose-response models.

Suppose that a reference chemical *C _{0}* and a test chemical

For any monotone function *f*, similarity implies that (Finney, 1965):

$$f(d;{\theta}_{1})=f(\rho d;{\theta}_{0})\phantom{\rule{0.38889em}{0ex}}\text{for}\phantom{\rule{0.16667em}{0ex}}\text{every}\phantom{\rule{0.16667em}{0ex}}\text{dose}\phantom{\rule{0.16667em}{0ex}}d>0;$$

(2)

*ρ* is a constant representing the relative potency of *C _{1}* compared to

When similar dose-response curves are plotted with dose on a logarithmic scale, the magnitude of the logarithm of *ρ* equals the horizontal distance between the curves at any mean response level *μ*; and the sign of the logarithm depends on the direction from the test chemical’s curve to the reference chemical’s curve (left being negative and corresponding to relative potencies less than 1) (Figure 1). (Relative potency is not defined for values of *μ* below *L* or above *U*.) For non-similar dose-response curves, this signed horizontal distance varies with mean response *μ* (Figure 3a). Accordingly, one can express how the value of *ρ* changes with *μ* using the notation of mathematical functions (Ritz *et al,* 2006). We call the corresponding relative potency function ρ_{μ} (μ) to reflect the dependence of *ρ* on *μ*; in addition to *μ*, it depends on the parameters *θ _{0}* and

Non-similar dose-response curves. The reference and test chemicals are *C*_{0} and *C*_{1}, respectively. With dose plotted on a logarithmic scale, the horizontal distance between curves is not constant. Panel (a) illustrates how the log relative potency’s **...**

Because each value of *μ* corresponds to a specific value of either *d _{1}* or

When the dose-response curves for *C _{0}* and

The relative potency function ρ_{π} (π) embodies the usual concept of relative potency only when the response limits for the two chemicals are the same; only then will a specified value of *π* represent the same value of *μ* for both chemicals so that the resulting concept of relative potency coincides with the usual concept. When the response limits differ, one value of *π* corresponds to different values of *μ* for each chemical (despite representing the same response quantile), thereby modifying the usual concept of relative potency. This modified concept is closely related to the idea of standardizing response variables by shifting and rescaling them to have a range from 0 to 1. A mathematical development of the relative potency functions ρ_{μ} (μ), ρ_{d0} (*d*_{0}), ρ_{d1} (*d*_{1}) and ρ_{π} (π) for both general dose-response models and Hill models is available in the Appendix.

We illustrate some characteristics of the different relative potency functions using three examples of dose-response curves for a reference chemical *C _{0}* and a test chemical

Examples of non-similar dose-response curves and corresponding relative potency functions. First column, labeled *f*(*d;θ*): dose-response curves for reference chemical *C*_{0} (solid) and test chemical *C*_{1} (dashed); vertical dotted lines indicate *ED*_{50} **...**

First, we consider two chemicals whose dose-response functions have the same response limits (*L*_{0} = *L*_{1} and *U*_{0} = *U*_{1}) but differ in shape (*S*_{0} > *S*_{1}) and *ED*_{50} (*M*_{0} < *M*_{1}) (Figure 4, row a). The dose-response curve for *C _{0}* is to the right of the curve for C

Next, consider two chemicals with different lower and upper limits of mean response (*L _{0}* >

Our third example considers two chemicals with the same *ED*_{50} values but different shape parameters (*S*_{0} > *S*_{1}) and different response limits such that those for *C*_{0} fall between those for *C*_{1} (*L*_{1} < *L*_{0} < *U*_{0} < *U*_{1}) (Figure 4, row c). The dose-response curves cross three times and the more potent chemical switches with each crossing. Whenever the dose-response curves cross, the relative potency functions ρ_{μ} (μ) and ρ* _{di}* (

Readers interested in visualizing dose-response curves and relative potency functions for different combinations of Hill model parameters can access a JAVA program called Hill Viewer at <http://www.niehs.nih.gov/research/atniehs/labs/bb/resources.cfm>.

With multiple relative potency functions available, the issue of how to choose among them arises. Generally speaking, if both chemicals have the same response limits, all these functions span the same range of relative potency values and carry equivalent information; the only difference among the graphs of the functions is how the horizontal axis is stretched or deformed as the function’s argument changes from mean response *μ* to dose *d _{0}* of

Because ρ_{μ} (μ), ρ_{d0} (*d*_{0}) and ρ_{d1} (*d*_{1}) represent an extension of the usual definition of relative potency based on the horizontal shift in the dose-response curves, whereas ρ_{π} (π) represents a slight modification of that definition, care must be taken in choosing between them. The fundamental issue involves the source or nature of any differences between the upper or lower limits of the dose-response curves for *C _{0}* and

On the other hand, if differences in response limits are extrinsic to the chemicals for the response under test, those differences should not influence relative potency and ρ_{π} (π) , which effectively rescales the responses to the same range, should be preferred. For example, if the dose-response experiment for each chemical was conducted separately with respect to some factor that should be irrelevant to relative potency itself (*e.g*., in two different labs or on separate cultures of cells grown days apart within the same lab), the difference in upper or lower response limits might not be related to the chemicals themselves but instead to different experimental conditions. In that situation, an investigator might justifiably believe that differences in response limits were extrinsic to the relative potency of the chemicals. The function ρ_{π} (π) honors the extrinsic nature of the response limits by, in effect, enforcing the idea that the mean response, all else being equal, would have the same range for both chemicals.

If differences in the upper and lower limits of mean response are regarded as intrinsic, the choice between ρ_{μ} (μ) and one of the ρ* _{di}* (

Relative potency functions were conceived as a way to address a common problem: the desire to characterize potency for pairs of chemicals whose dose-response curves are not strictly similar and thus have non-constant relative potency. In that setting, a single summary value such as the ratio of *ED*_{50}s is potentially misleading. Consequently, a principal use of relative potency functions will be graphical, displaying how the relative potency of two chemicals changes in the absence of similarity. As valuable as graphs are for this purpose, however, simple quantitative summaries and other numerical quantities derived from relative potency functions will likely be useful. We illustrate several ideas using ρ_{μ} (μ), though the concepts apply to the other relative potency functions as well.

For example, if a relative potency function is not constant but does not intersect the line of equipotency (*ρ* = 1), then one chemical is everywhere more potent than the other and having a single numerical summary of that relationship might be useful. One possible summary value that can be derived from a relative potency function is the closest value of ρ_{μ} (μ) to the null value of 1, denoted here by _{μ}. Two chemicals are equipotent for at least one response level if _{mu;} = 1; otherwise, _{μ} serves as an index of relative potency. Whenever the upper and lower limits of response are the same for both chemicals, the value of _{μ} equals the values of _{d0}, _{d1} and _{π} calculated using the other relative potency functions; for the Hill model, each is 1 if *S _{0}* ≠

An alternative summary that addresses this difficulty is an average, although problems arise when ρ_{μ} (μ) is infinite or undefined for some values of μ. We suggest using the geometric mean of ρ_{μ} (μ) over a specified interval (*a*,*b*) of finite response levels. This geometric mean, denoted
${\stackrel{\u2323}{\rho}}_{\mu}^{(a,b)}$, is calculated as the anti-log of a definite integral of the log relative potency function (Appendix). A drawback is that the geometric mean relative potency can differ depending on which relative potency function is used for the calculation, reflecting rules for change of variables in integration. Interestingly, for Hill dose-response curves, the geometric mean of ρ_{π} (π) for any interval (*a*,*b*) centered on *π* = 0.5 always equals the ratio of the *ED*_{50}s (Appendix), perhaps providing a rationale for favoring the geometric mean over the arithmetic mean when averaging relative potency functions. Of course, no single number can fully describe a non-constant relative potency function, so care must be taken when interpreting such summaries.

Another potentially useful summary is the set of mean response levels where relative potency lies within a specified interval (*l,m*), that is, the set
${D}_{\mu}^{(l,m)}=\{\mu :l<{\rho}_{\mu}(\mu )<m\}$. For example, if we are interested in the range of responses where one chemical is more potent than another, the set
${D}_{\mu}^{(1,\infty )}$ or its complement
${D}_{\mu}^{(0,1)}$ would be useful. Alternatively, if we want to determine a range of responses over which the potencies of two chemicals differ by at most some small amount *δ*, which corresponds to an ‘indifference zone’ or an ‘equivalence region’ near equipotency, we could use
${D}_{\mu}^{(1-\delta ,1+\delta )}$ for an appropriately chosen *δ*.

Often, investigators want to compare several test chemicals to the same reference chemical. Consider two test chemicals, *C _{1}* and

The U.S. National Toxicology Program (NTP) recently evaluated the relative potency of dioxin-like compounds with respect to toxicity and carcinogenicity endpoints (NTP, 2006a-d). They studied 2,3,7,8-tetrachlorodibenzo-*p*-dioxin (TCDD), 3,3′,4,4′,5-pentachlorobiphenyl (PCB126), 2,3,4,7,8-pentachlorodibenzofuran (PeCDF), and a TEF-based mixture of the three. We used a subset of these data (Online Supplement) to illustrate several concepts introduced earlier. Specifically, we focused on the activity of cytochrome *P*450 1A1-associated 7-ethoxyresorufin-*O*-deethylase (EROD) as measured in liver tissue of female Harlan Sprague-Dawley rats treated by oral gavage for 53 weeks (Toyoshiba *et al*, 2004). There were 8 rats per dose for each chemical, though dose levels and numbers of doses varied across chemicals. To illustrate our ideas, we treated the mixture as a fourth chemical, ignoring its composition.

We used SAS Proc NLIN (version 9.00, SAS Institute Inc., Cary, NC, USA) to fit Hill models to log-transformed EROD activity using unweighted least squares. Initially, we fit an unrestricted model, with four parameters for each of the four chemicals (Table 2). Individual Hill models appeared to fit the data well, as evidenced by the proximity of the estimated dose-response curves to the dose-specific mean EROD levels (Figure 5).

EROD-activity dose-response data and estimated Hill-model dose-response curves for four chemicals. Panels: (a) TCDD, (b) PCB126, (c) PeCDF, and (d) their TEF-based mixture. Symbols: individual response (open circle), and dose-specific average response **...**

Parameter estimates (standard errors) by chemical under unrestricted, partially similar, and fully similar Hill-model dose-response curves for the NTP liver enzyme (EROD) activity data.

We assessed similarity by allowing each dose-response curve to have a distinct *ED _{50}* but restricting the lower response limits, upper response limits and shape parameters to be the same across chemicals (Table 2). Chemical-specific dose-response curves estimated after enforcing similarity did not fit the data as well as the unrestricted curves (Figure 5); the hypothesis that all four chemicals had similar dose-response curves with respect to EROD activity was rejected (

Following convention, we treated TCDD as the reference and estimated the relative potency functions of the other chemicals compared to TCDD. In accord with the preceeding results, the Hill models for PeCDF and the mixture were unrestricted, but those for TCDD and PCB126 were constrained to produce similar dose-response curves (Table 2, Figure 6). We estimated relative potency functions by substituting the parameter estimates from the fitted Hill models into the appropriate formulae (Appendix).

Estimated Hill-model dose-response curves for TCDD, PCB126, PeCDF, and their TEF-based mixture and corresponding relative potency functions (relative to TCDD). Hill models were fit separately for PeCDF and the mixture but were constrained by assuming **...**

Because TCDD and PCB126 were forced to have similar dose-response curves, relative potency functions for those chemicals were constant (Figures 6b, 6c and 6d) and equal to the ratio of their *ED*_{50}s ( = 0.19, 95% CI: 0.16, 0.22), indicating that PCB126 is less potent than TCDD. The chemicals allowed to have non-similar dose-response curves had estimated relative potency functions that deviated substantially from constant functions.

If one considers the differences between the lower and upper response limits for these chemicals as intrinsic to the comparison, attention should focus on ρ_{μ}(μ) or one of the ρ* _{di}* (

On the other hand, if one considers the differences between the lower and upper response limits as extrinsic to the comparison, attention should focus on ρ_{π}(π) . Only for PCB126 does _{π} (π) fail to cross the equipotency line, though the crossing point for PeCDF at *π* = 0.001 is near the edge of the range. For either PCB126 or PeCDF, _{π} (π) suggests that the respective compounds are less potent than TCDD, whereas _{π} (π) for the mixture crosses the equipotency line at *π* = 0.2, with a negative slope, suggesting that the mixture is more potent than TCDD at low quantiles of response but less potent at higher quantiles. The average values of _{π} (π) over the interval 0 < *π* < 1 were ρ̆_{π}= 0.19, 0.05, and 0.76 for PCB126, PeCDF and the mixture, respectively, relative to TCDD, in qualitative agreement with the ordering suggested by the curves themselves. The qualitative ordering of the chemicals based on _{π} (π) differs from that based on _{μ} (μ) or _{d0} (*d*_{0} ) , so the issue of whether evident differences in the response limits are intrinsic or extrinsic to the potency comparison has a definite impact.

The relative potency of chemicals is a fundamental concern of toxicology. The usual approach is to report a single constant value for relative potency. This approach is sound only when the chemicals have similar dose-response curves. Investigators have, however, long recognized that the ideal of similarity is not always realized in comparative assays. Some analysts simply enforce similarity in fitting curves so that relative potency is constant and estimated as the *ED*_{50} ratio for the similar curves. Others estimate relative potency as the *ED*_{50} ratio without enforcing similarity even for non-similar curves (Rodbard and Frazier, 1975; Villeneuve *et al*, 2000). Still other analysts ‘standardize’ the data for each chemical by subtracting a data-based estimate of *L* from every observed response and then dividing by a data-based estimate of the response range *U – L* (*e.g*., Streibig *et al*, 1995). None of these approaches are completely satisfactory for chemicals with non-similar dose-response curves. Enforcing similarity when it does not hold can misrepresent dose-response relationships. Using a single relative potency value can be misleading when relative potency is not constant, especially if dose-response curves cross. Standardizing the data affects statistical fitting procedures by inducing correlations that are unaccounted for in the usual analyses.

To address some of these issues, Ritz *et al* (2006) expressed relative potency as a function of mean response (ρ_{μ}(μ) in our notation). We elaborated on this extension of the concept of relative potency from a constant to a function. Regarded as a constant, relative potency represents a global comparison of potency between two chemicals; regarded as a function, it captures a changing local characterization of the potency relation between the chemicals. When chemicals have similar dose-response curves, a global summary of relative potency is appropriate and its graph would be a horizontal line, that is, a constant function of mean response, of the dose of either the reference or test chemical, or of response quantile. For non-constant relative potency, one needs a convenient descriptor for how relative potency changes. To describe changing relative potency, then, one naturally considers indexing it to those same quantities: mean response, dose, and response quantile. Using hypothetical examples and real data, we illustrated pairs of dose-response curves whose relative potency functions depart from a flat line and indicated some consequences of ignoring such departures. For simplicity, we used the same functional form, the Hill model, for both chemicals; however, the concepts we introduced would apply even if the dose-response models had distinct functional forms. Our hypothetical examples support our contention that, when chemicals do not have similar dose-response curves, use of a constant value for relative potency (typically based on a ratio of *ED*_{50}s) may be seriously misleading. We reanalyzed NTP EROD-activity data to show the potential value of estimating relative potency by a function rather than by *ED*_{50} ratios alone or by forcing dose-response curves to be similar.

When two dose-response functions have different upper or lower limits of response, the relative potency functions ρ _{μ} (μ),ρ _{d0} (*d*_{0} ) and ρ _{d1} (*d*_{1} ) express a similar concept of relative potency but one distinct from that expressed by ρ_{π} (π) . Which of these two concepts makes more sense in any given situation depends crucially on whether the discrepancies in response limits between the chemicals are regarded as intrinsic to the chemicals themselves for that response or as extrinsic, deriving instead from sources like the dose-response experiments being conducted on different days or in different labs or on different strains of organism. If the source of discrepancies in response limits is extrinsic, then arguably ρ _{π} (π) is the more suitable relative potency function because it ignores those discrepancies. It is conceptually related to data standardization (Appendix) without the inherent drawbacks. On the other hand, if the source of discrepancies in response limits is intrinsic, then one of the other relative potency functions should be preferred because they honor unequal response limits. They also maintain the classical concept of relative potency but apply it locally rather than globally.

Our presentation has stressed the concept of relative potency functions, describing what they are and suggesting ways that they might be used. We have treated them as mathematical objects, not as estimates from data. Consequently, we have not presented methods for the statistical analysis of these functions. For example, users will want to estimate not only the functions themselves, as we have done in our example, but also to derive confidence bands for the curves or to test hypotheses about them. In addition, statistical inference procedures will be needed for our proposed summary measures. Availability of valid and efficient inference procedures will enhance the utility of relative potency functions. Because the relative potency functions are defined by the parameters of dose-response models, confidence limits and statistical tests about relative potency functions and related quantities follow, in principle, from information about the dose-response parameters themselves. After the parameters of the dose-response curves and their variance-covariance matrix are estimated using standard statistical methods, those estimates can be translated into inferences about a relative potency function. We are currently investigating such inference procedures.

A toxic equivalency factor (TEF) summarizes the potency of one chemical relative to another by a single number that is independent not only of dose and response but also of endpoint, laboratory, and other sources of variation. TEFs are commonly derived for classes of chemicals that share a common mechanism of toxicity, such as the dioxin-like compounds in our example. Under an assumption of dose additivity, TEFs are used to predict the toxic effects of a mixture of chemicals based on knowledge about the potencies of its constituents. Relative to a reference, a chemical’s TEF is a summary of available estimates of constant relative potency, even when similarity may not hold for all endpoints. One potential value of relative potency functions is to extend the TEF approach to better cope with chemicals having non-similar dose-response curves. Putzrath (1997) advocated expanding the TEF framework in a related way. This task will not be easy, however, as study- and endpoint-specific estimates of relative potency can vary by orders of magnitude (Walker *et al*, 2005), complicating broad summaries of potency.

In closing, researchers may draw incorrect conclusions about relative potency if their inferences are based on *ED*_{50} ratios alone or on dose-response curves forced to be similar when data suggest that they are not. Consequently, we recommend expressing relative potency as a function of dose, response, or response quantile and making inferences within this more general framework. Which relative potency function to use depends primarily on whether the response limits differ between chemicals and, if so, whether those differences are viewed as intrinsic or extrinsic to the comparison. Graphical displays of relative potency functions provide a useful visual summary; but, when appropriate, single-number or set-based summaries of the relative potency function may succinctly describe important aspects of relative potency.

We are grateful to M. DeVito, J. Haseman, S. Peddada, and N. Walker for their constructive comments and discussions. We thank S. Harris for programming the Hill Viewer application.

**Funding and Conflict of Interest Statement**

This research was supported by the Intramural Research Program of the NIH, National Institute of Environmental Health Sciences (Z01-ES-102685). The NIH had no involvement in the study design; collection, analysis and interpretation of data; writing of the manuscript; or decision to submit the manuscript for publication. The authors declare that there are no conflicts of interest.

- CI
- confidence interval
- ED
_{50} - median effective dose
- EROD
- 7-ethoxyresorufin-
*O*-deethylase - NTP
- National Toxicology Program
- PCB126
- 3,3′,4,4′,5-pentachlorobiphenyl
- PeCDF
- 2,3,4,7,8-pentachlorodibenzofuran
- TCDD
- 2,3,7,8-tetrachlorodibenzo-
*p*-dioxin - TEF
- toxic equivalency factor

Using the notation in the manuscript, we denote the mean response to dose *d _{i}* of chemical

For similar dose-response curves, as noted in the manuscript, whenever doses *d _{0}* and

$${\rho}_{\mu}(\mu ;{\theta}_{0},{\theta}_{1})={f}^{-1}(\mu ;{\theta}_{0})/{f}^{-1}(\mu ;{\theta}_{1}).$$

(A.1)

Here, *ρ _{μ}*(

Let *L _{i}* and

Consider a Hill dose-response model. Setting the right-hand side of equation (1) in the main text equal to *μ* and solving for *d* in terms of *μ* gives a Hill-model-specific inverse function:

$$d=M{[(\mu -L)/(U-\mu )]}^{1/S}={f}^{-1}(\mu ;\theta ).$$

(A.2)

Substituting equation (A.2), suitably subscripted to distinguish test and reference chemicals, into equation (A.1) and simplifying yields the Hill-model-specific relative potency function:

$${\rho}_{\mu}(\mu )=({M}_{0}/{M}_{1}){[(\mu -{L}_{0})/({U}_{0}-\mu )]}^{1/{S}_{0}}/{[(\mu -{L}_{1})/({U}_{1}-\mu )]}^{1/{S}_{1}}.$$

(A.3)

Whenever *C*_{0} and *C*_{1} have the same lower and upper response limits, say *L*_{0} = *L*_{1} = *L* and *U*_{0} = *U*_{1} = *U*, equation (A.3) reduces to:

$${\rho}_{\mu}(\mu )=({M}_{0}/{M}_{1}){[(\mu -L)/(U-\mu )]}^{(1/{S}_{0})-(1/{S}_{1})}.$$

If *S*_{0} = *S*_{1} as well, thenρ_{μ}(μ) is constant and equal to the ratio of *ED*_{50}s, namely, *M*_{0}/*M*_{1}.

Because, with a monotone dose-response function, each value of *μ* corresponds to a unique value of *d _{1}*, we can re-express ρ

$${\rho}_{{d}_{1}}({d}_{1};{\theta}_{0},{\theta}_{1})={f}^{-1}(f({d}_{1};{\theta}_{1});{\theta}_{0})/{d}_{1}.$$

(A.4)

Analogously for *d _{0}*, we can substitute

$${\rho}_{{d}_{0}}({d}_{0};{\theta}_{0},{\theta}_{1})={d}_{0}/{f}^{-1}(f({d}_{0};{\theta}_{0});{\theta}_{1}).$$

(A.5)

Each *ρ _{di}* (

The values where each ρ* _{di}* (

Substituting explicit expressions for the Hill dose-response model from equation (1) and its inverse from equation (A.2), suitably subscripted to distinguish test and reference chemicals, into equations (A.4) and (A.5) and simplifying yields the following Hill-model-specific relative potency functions:

$${\rho}_{{d}_{1}}({d}_{1})=\frac{{M}_{0}}{{d}_{1}}{\left[\frac{({L}_{1}-{L}_{0})({M}_{1}^{{S}_{1}}+{d}_{1}^{{S}_{1}})+({U}_{1}-{L}_{1}){d}_{1}^{{S}_{1}}}{({U}_{0}-{L}_{1})({M}_{1}^{{S}_{1}}+{d}_{1}^{{S}_{1}})+({U}_{1}-{L}_{1}){d}_{1}^{{S}_{1}}}\right]}^{1/{S}_{0}},$$

(A.6)

$${\rho}_{{d}_{0}}({d}_{0})=\frac{{d}_{0}}{{M}_{1}}{\left[\frac{({U}_{1}-{L}_{0})({M}_{0}^{{S}_{0}}+{d}_{0}^{{S}_{0}})-({U}_{0}-{L}_{0}){d}_{0}^{{S}_{0}}}{({L}_{0}-{L}_{1})({M}_{0}^{{S}_{0}}+{d}_{0}^{{S}_{0}})+({U}_{0}-{L}_{0}){d}_{0}^{{S}_{0}}}\right]}^{1/{S}_{1}}.$$

(A.7)

Whenever *C*_{0} and *C*_{1} have the same lower and upper response limits, equations (A.6) and (A.7), respectively, reduce to:

$${\rho}_{{d}_{1}}({d}_{1})=({M}_{0}/{M}_{1}){[{d}_{1}/{M}_{1}]}^{({S}_{1}/{S}_{0})-1},$$

(A.8)

$${\rho}_{{d}_{0}}({d}_{0})=({M}_{0}/{M}_{1}){[{M}_{0}/{d}_{0}]}^{({S}_{0}/{S}_{1})-1}.$$

(A.9)

From equations (A.8) and (A.9), we see that ρ* _{di}* (

$$log\left({\rho}_{{d}_{1}}({d}_{1})\right)=\left[log({M}_{0})-({S}_{1}/{S}_{0})log({M}_{1})\right]+[({S}_{1}/{S}_{0})-1]log({d}_{1}).$$

The same argument applied to equation (A.9) shows that log(ρ_{d0} (*d _{0}* )) is linear in log(

Finally, we can define relative potency as the ratio of *ED*_{100π}*s* for *C _{0}* and

$${\rho}_{\pi}(\pi ;{\theta}_{0},{\theta}_{1})={f}^{-1}({L}_{0}+({U}_{0}-{L}_{0})\pi ;{\theta}_{0})/{f}^{-1}({L}_{1}+({U}_{1}-{L}_{1})\pi ;{\theta}_{1}).$$

(A.10)

Here, *ρ _{π}*(

The function ρ_{π} (π) , defined as the ratio of *ED*_{100π}*s* , embodies the usual concept of relative potency only when *C _{0}* and

Despite the appearance of response limits *L _{i}* and

$$f(d;\theta )=L+(U-L)g(d;\phi ),$$

(A.11)

where *g*(*d;*) is a monotone function of dose *d* with 0 < *g*(*d;*) <1, is a parameter vector, and *θ* = (*L*,*U*,). For a function that ranges from 0 to 1, multiplying by (*U* − *L*) stretches the range so the upper limit is (*U*− *L*); then adding *L* shifts both limits so the resulting range is from *L* to *U*. Thus, *f*(*d;θ*) is a stretched and shifted version of *g*(*d;*). We represent the inverse of function *μ ^{0}* =

$${f}^{-1}(\mu ;\theta )={g}^{-1}({\mu}^{o};\phi ).$$

(A.12)

Substituting *μ* for *f*(*d;θ*) and μ* ^{o}* for

$${f}^{-1}(L+(U-L){\mu}^{o};\theta )={g}^{-1}({\mu}^{o};\phi ).$$

(A.13)

Now, μ* ^{o}* and

$${\rho}_{\pi}(\pi ;{\theta}_{0},{\theta}_{1})={g}^{-1}(\pi ;{\phi}_{0})/{g}^{-1}(\pi ;{\phi}_{1}).$$

(A.14)

Hence, equation (A.14) shows that ρ_{π} (π) is free of response limits *L*_{i} and *U*_{i} (*i*=0,1) as claimed.

The preceding development also shows the relationship between ρ_{π} (π) and the approach of ‘standardizing’ the response variables by shifting and rescaling them to have range 0 to 1. A relative potency function properly calculated from such standardized variables would be ρ_{π} (π). Without standardization and for any dose-response model, the construction of ρ_{π} (π) , in effect, implicitly rescales the mean response of each chemical separately so that the corresponding dose-response curves have the same upper and lower limits.

Substituting the Hill-model-specific inverse formula of equation (A.2), with *μ* replaced by *L _{i}* + (

$${\rho}_{\pi}(\pi )=({M}_{0}/{M}_{1}){[\pi /(1-\pi )]}^{(1/{S}_{0})-(1/{S}_{1})}.$$

(A.15)

Equation (A.15) illustrates that, for a pair of Hill models, ρ_{π} (π) does not depend on *L _{i}* and

$$log({\rho}_{\pi}(\pi ))=log({M}_{0}/{M}_{1})+[(1/{S}_{0})-(1/{S}_{1})]log(\pi /(1-\pi )).$$

Define *h*(π) = log(π /(1−π)) and note that *h*(1/ 2) = 0 and *h*(π) = −*h*(1−π) for all 0 < *π* < 1. For any 0 <*ε* < ½, substituting ½ +*ε* for *π* in *h*(π) = −*h*(1−π) gives *h*(1/ 2+ε) = −*h*(1/ 2−ε). Thus, from the baseline value log(ρ_{π} (1/ 2)) = log(*M*_{0} / *M*_{1}), the function log(ρ_{π} (π)) increases (or decreases) by [(1/ *S*_{0} ) −(1/ *S*_{1} )] *h*(1/ 2+ε) units for every increment (or decrement) of size *ε*.

The geometric mean of ρ_{μ} (μ) over interval (*a*,*b*), denoted
${\stackrel{\u2323}{\rho}}_{\mu}^{(a,b)}$, is the anti-log of the mean of the log relative potency function. Let λ (μ) = log(ρ_{μ} (μ)) and define its mean over (*a*,*b*) as:

$${\overline{\lambda}}_{\mu}^{(a,b)}={(b-a)}^{-1}{\int}_{a}^{b}{\lambda}_{\mu}(x)dx.$$

(A.16)

The geometric mean relative potency over the interval (*a*,*b*) is
${\stackrel{\u2323}{\rho}}_{\mu}^{(a,b)}=exp\left({\overline{\lambda}}_{\mu}^{(a,b)}\right)$. A geometric mean relative potency based on ρ* _{di}* (

Also, because log(ρ_{π} (π)) is symmetric about log (*M*_{0} / *M*_{1}), we see from equation (A.16) that
${\overline{\lambda}}_{\pi}^{(a,b)}=log({M}_{0}/{M}_{1})$ and hence
${\stackrel{\u2323}{\rho}}_{\pi}^{(a,b)}={M}_{0}/{M}_{1}$ for any interval (*a,b*) centered at *π* = ½; that is, for any interval (*a,b*) = (½ − *ε*, ½ + *ε*) with 0 <*ε* < ½.

The liver EROD activity data used in our example are available online as an Excel spreadsheet.

Gregg E. Dinse, Biostatistics Branch, National Institute of Environmental Health Sciences, Mail Drop A3-03, P.O. Box 12233, Research Triangle Park, NC 27709 USA.

David M. Umbach, Biostatistics Branch, National Institute of Environmental Health Sciences, Mail Drop A3-03, P.O. Box 12233, Research Triangle Park, NC 27709 USA.

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