The majority of my research on this project was performed in five areas: (1) generalizing the concept of relative potency, (2) developing statistical methods for making inferences about generalized relative potency functions, (3) specifying a power model to express relative potency as a function of dose, (4) studying methods that rely on dose additivity, and (5) identifying situations in which Hill model parameters are not uniquely estimable. These five areas of research are described in more detail below. Area 1: 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. Within a class of chemicals having "similar" dose-response curves, the relative potency of one chemical compared to another is the ratio of doses producing the same toxicity response, and this ratio is constant across all levels of response. If the dose-response curves are non-similar, however, relative potency need not be constant and typically varies according to where along the dose-response curves the dose ratio is calculated. In practice, relative potency is usually equated to 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 developed a more general approach that allows relative potency to vary as a function of the dose of a chemical, the level of a specific response, or the percentage of the range of possible response levels. Distinct functions can be defined, each generalizing different but equivalent descriptions of constant relative potency. These relative potency functions are constructed from dose-response curves for test and reference chemicals, and they all provide fundamentally equivalent information if the chemicals have the same lower and upper limits of response. In fact, if two chemicals differ only with respect to their ED50s (i.e., their dose-response curves are similar), then all of the relative potency functions are constant and equal to the ratio of the ED50s. Otherwise, if the response limits differ, relative potency as a function of the response-range percentage is distinct from the other functions and embodies a modified definition of relative potency. Non-constant relative potency functions may cross the baseline value of 1.0, indicating that one chemical is more potent than another for some doses, responses, or response-range percentages. If chemicals have non-similar dose-response curves, then inferences based on ratios of ED50s or based on models that force the other parameters to be identical can be misleading. Thus, we recommend using relative potency functions, where the preferred function depends on the application (e.g., chemical ranking or dose conversion) and whether one views differences in response limits as intrinsic to the chemicals or as extrinsic, arising from idiosyncrasies of data sources. Relative potency functions offer a unified and principled description of relative potency for non-similar dose-response curves. We recently wrote a book chapter on this topic, which is currently under review. Also, I gave an invited talk about this research at the Conference on Risk Assessment and Evaluation of Predictions in Silver Spring, MD on October 13, 2011. Area 2: In ongoing research, we are working on formal statistical methods for analyzing relative potency functions. First, we will describe techniques for estimating parameters in the underlying dose-response models (e.g., Hill models), assessing model adequacy, quantifying variability of parameter estimates, constructing confidence intervals for model parameters, and testing hypotheses about model parameters. Then, based on specific models for the dose-response curves, we will develop procedures for making inferences about the resulting relative potency functions and any summaries obtained from these functions. These procedures will deal with function estimation, variance estimation, construction of pointwise confidence intervals and simultaneous confidence bands, and testing of hypotheses. Area 3: Recently, we developed an approach that parameterizes dose-response models in a way that enables relative potency functions to be estimated directly. For example, when analyzing data on a reference chemical and a test chemical, rather than specifying models for both dose-response functions and then using these to derive a formula for the relative potency function, we specified a dose-response model for the reference chemical and a relative potency model. This approach provides direct estimates of the relative potency function (and indirect estimates of the dose-response function for the test chemical). In particular, we used a power function in dose as a relative potency model, which is equivalent to a log-linear model (in dose) for the log relative potency function. This model keeps the dose-response functions for the two chemicals within the same family of models for those models typically used in toxicology. When differences in the response limits for the test and reference chemicals are attributable to the chemicals themselves, the indirect approach is the more convenient. When differences in response limits are attributable to other features of the experimental protocol or when response limits do not differ, our new direct approach is straightforward to apply with nonlinear regression methods and simplifies calculation of simultaneous confidence bands. We published an article describing this work (see reference 1). Area 4: Analyses of chemical mixtures often rely on an assumption of dose additivity. Under this assumption, each test chemical can be expressed in terms of equivalent units of a reference chemical, which allows a dose of some mixture of these chemicals to be expressed as a weighted sum of the doses of the constituent chemicals. For much of the past year, a small working group has been reading papers on dose additivity and studying various methods that assume dose additivity. This group is composed of four researchers: Cynthia Rider (NTP), Jane Ellen Simmons (EPA), David Umbach (BB), and myself. We meet for a few hours every two or three weeks; we discuss the papers assigned at the last meeting; and we work towards a comprehensive review paper on dose additivity. This work is ongoing and we hope to have a draft manuscript completed by late 2012. Area 5: Finally, I have also been studying ways to determine the number of uniquely estimable parameters when a Hill model is fitted to binary data. It is well known that there are identifiability problems if all or none of the subjects respond. Similar problems arise if all subjects receiving a dose below a certain level do not respond and all subjects receiving a higher dose do respond. On the other hand, if there are several observed response rates and they tend to be ordered with respect to dose, then typically the Hill model parameters are uniquely estimable. For intermediate cases, however, the number of uniquely estimable parameters appears to be related to the number of distinct nonparametric estimates obtained under a monotonicity constraint on the dose-response curve (i.e., the number of level sets in a nonparametric isotonic regression analysis).