Parameterizing dose-response models to estimate relative potency functions directly

Statistical inferences on nonconstant relative potency with quantal response data

Relative potency is widely used in toxicological and pharmacological studies to characterize potency of chemicals. The relative potency of a test chemical compared to a standard chemical is defined as the ratio of equally effective doses (standard divided by test). This classical concept relies on the assumption that the two chemicals are toxicologically similar-that is, they have parallel dose-response curves on log-dose scale-and thus have constant relative potency. Nevertheless, investigators are often faced with situations where the similarity assumption is deemed unreasonable, and hence the classical idea of constant relative potency fails to hold; in such cases, simply reporting a single constant value for relative potency can produce misleading conclusions. Relative potency functions, describing relative potency as a function of the mean response (or other quantities), is seen as a useful tool for handling nonconstant relative potency in the absence of similarity. Often, investigators are interested in assessing nonconstant relative potency at a finite set of some specific response levels for various regulatory concerns, rather than the entire relative potency function; this simultaneous assessment gives rise to multiplicity, which calls for efficient statistical inference procedures with multiplicity adjusted methods. In this paper, we discuss the estimation of relative potency at multiple response levels using the relative potency function, under the log-logistic dose-response model. We further propose and evaluate three approaches to calculating multiplicity-adjusted confidence limits as statistical inference procedures for assessing nonconstant relative potency. Monte Carlo simulations are conducted to evaluate the characteristics of the simultaneous limits.

Comparison of multi-stage dose-response mixture models, with applications

This article concerns the analysis of a stochastic model that we propose for the population that generates a response (response measure) to the dose with the multi-stage model. The parameter uncertainty is dealt with via random dose and random size of the population at risk. The response measure is modeled by a random sum of mixed Bernoulli random variables with arbitrary distribution for the mixing parameters. Some extensions of the model are defined by functionals of the infection probability, fulfilling some convexity properties. We analyze the response by stochastic comparisons under different stochastic relations on the random dosages and the random sizes of the population at risk; or on the random infection rates. We provide stochastic exact bounds of the mixture model for the response, using inequalities and the positive quadrant dependence. Numerical bounds of the response by a dose having a scalar value or having an exponential or uniform distributions are obtained. Some conclusions are derived: the lower estimation of the response measure in the increasing convex order sense by replacing the dosages by their means; effects of the variation of the dose on the magnitude of the probability distribution of the response; effects of parameter correlation on the degree of variability of the response to any random dose; the low-dose region assessment; and also, the classical multi-stage model is compared versus the mixture model featuring independence and versus that with positive quadrant dependence.