Optimal Designs for Quantile Regression Models

Abstract

Despite their importance, optimal designs for quantile regression models have not been developed so far. In this article, we investigate the D-optimal design problem for nonlinear quantile regression analysis. We provide a necessary condition to check the optimality of a given design and use it to determine bounds for the number of support points of locally D-optimal designs. The results are illustrated, determining locally, Bayesian and standardized maximin D-optimal designs for quantile regression analysis in the Michaelis–Menten and EMAX model, which are widely used in such important fields as toxicology, pharmacokinetics, and dose–response modeling.

KEY WORDS:

• EMAX model
• Heteroscedasticity
• Locally optimal design
• Michaelis–Menten model
• Robust designs

Acknowledgments

This work has been supported in part by the Collaborative Research Center “Statistical modeling of nonlinear dynamic processes” (SFB 823, Teilprojekt C2) of the German Research Foundation (DFG). The authors would like to thank three anonymous referees and the Associate Editor for their constructive comments on an earlier version of this article, and Martina Stein for typing parts of this manuscript with considerable technical expertise.

Notes

NOTE: The corresponding minimum D-efficiency (2.9) is listed in the columns labeled “eff.”

NOTE: All saturated designs are Bayesian D-optimal in the class of all designs.

NOTE: Exemplarily, we also present results for the estimation of the minimum effective dose (MED) at Δ = 10, 20, and 30.