Optimal design for the precise estimation of an interaction threshold: the impact of exposure to a mixture of 18 polyhalogenated aromatic hydrocarbons

Abstract

Traditional additivity models provide little flexibility in modeling the dose-response relationships of the single agents in a mixture. While the flexible single chemical required (FSCR) methods allow greater flexibility, its implicit nature is an obstacle in the formation of the parameter covariance matrix, which forms the basis for many statistical optimality design criteria. The goal of this effort is to develop a method for constructing the parameter covariance matrix for the FSCR models, so that (local) alphabetic optimality criteria can be applied. Data from Crofton et al. are provided as motivation; in an experiment designed to determine the effect of 18 polyhalogenated aromatic hydrocarbons on serum total thyroxine (T(4)), the interaction among the chemicals was statistically significant. Gennings et al. fit the FSCR interaction threshold model to the data. The resulting estimate of the interaction threshold was positive and within the observed dose region, providing evidence of a dose-dependent interaction. However, the corresponding likelihood-ratio-based confidence interval was wide and included zero. In order to more precisely estimate the location of the interaction threshold, supplemental data are required. Using the available data as the first stage, the Ds-optimal second-stage design criterion was applied to minimize the variance of the hypothesized interaction threshold. Practical concerns associated with the resulting design are discussed and addressed using the penalized optimality criterion. Results demonstrate that the penalized Ds-optimal second-stage design can be used to more precisely define the interaction threshold while maintaining the characteristics deemed important in practice.

Fig. 1

Predicted T4 versus total dose of the mixture, adapted from Gennings et al.⁽¹²⁾

Fig. 2

Desirability functions for the penalized optimal second-stage design.

Fig. 3

The Ds-optimal second-stage design for the estimation of the interaction threshold. The vertical dotted line in (B) represents the location of the hypothesized interaction threshold. The triangles represent the location of the second-stage design points.

Fig. 4

The penalized optimal second-stage design for the estimation of the interaction threshold. The vertical dotted line in (B) represents the location of the hypothesized interaction threshold. The triangles represent the location of the second-stage design points.