Parameterizing Dose-Response Models to Estimate Relative Potency Functions Directly

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 identical 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 horizontal 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 general 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 function 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 parameters 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 relative potency function contains no fundamentally new information 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 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 reference 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 experimental procedure (extrinsic differences). We find that, when differences 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 extrinsic, compatible relative potency functions are easy to specify for commonly used dose-response models. Thus, the explicit specification of relative potency functions is widely, though not universally, applicable. We use data from U.S. National Toxicology Program (NTP) bioassays to illustrate the direct estimation of relative potency functions.