Optimal designs for quadratic regression with random block effects: The case of block size two

• We obtain an essentially complete class of designs with a simple structure.
• The class contains locally and Bayesian ϕpϕp-optimal designs.
• We present some highly efficient designs for arbitrary intra-block correlations.

Optimal approximate designs for quadratic regression with random block effects in the case of block size two are considered. We obtain, with respect to the Schur ordering, an essentially complete class consisting of designs with a simple structure. The locally D- and A-optimal designs given in Cheng (1995a) and Atkins and Cheng (1999) belong to this class. We explicitly identify locally E  -optimal designs and show that for each pp, −∞≤p≤1−∞≤p≤1, there is a unique ϕpϕp-design in this class. Bayesian ϕpϕp-optimal designs are also considered.