Optimal designs for mixed models in experiments based on ordered units

TL;DR

This paper develops optimal experimental designs for mixed models with ordered units, incorporating trend effects and random components, using a maximin approach to ensure robustness across unknown variance parameters.

Contribution

It introduces universally optimal designs for mixed models with ordered units, extending existing results and employing a maximin strategy for robustness.

Findings

Derived maximin universally optimal designs based on semibalanced arrays.

Studied robustness of these designs under variance parameter uncertainty.

Connected special cases to existing literature results.

Abstract

We consider experiments for comparing treatments using units that are ordered linearly over time or space within blocks. In addition to the block effect, we assume that a trend effect influences the response. The latter is modeled as a smooth component plus a random term that captures departures from the smooth trend. The model is flexible enough to cover a variety of situations; for instance, most of the effects may be either random or fixed. The information matrix for a design will be a function of several variance parameters. While data will shed light on the values of these parameters, at the design stage, they are unlikely to be known, so we suggest a maximin approach, in which a minimal information matrix is maximized. We derive maximin universally optimal designs and study their robustness. These designs are based on semibalanced arrays. Special cases correspond to results available in the literature.