Optimal and Efficient Repeated-Measurements Designs for Uncorrelated Observations

Abstract

In repeated measurements (crossover or changeover) designs, experimental subjects (units) are repeatedly exposed to treatments. Effects of treatments from one period that extend into the next are called carryover, or residual, effects. In exact design theory, designs that are universally optimal in the presence of carryover effects have been found when a subject's responses are uncorrelated. However, the results are valid only for restricted values of t, the number of treatments, p, the number of periods, and N, the number of subjects. In other optimality results, the competing class of designs is a subset of the full class of repeated-measurements designs. These restrictions hinder applicability. In the setting of approximate design theory, Kushner (1997b) developed a new approach, valid even for correlated responses. Here the results are specialized to the case of uncorrelated responses. Linear equations are given that determine all universally (or A- and D-) optimal repeated-measurements designs in approximate design theory. There are no limitations on p or t, but there are limitations on N for applications to exact design theory. A number of examples give designs that illustrate the theory. A formula for the treatment effects information matrix of a universally optimal design is presented. The single equation that determines all “symmetric” optimal designs is given. Results from the literature are reconsidered and extended. Issues of special interest to a practitioner—efficient nonoptimal designs and the variance of best linear unbiased estimators (BLUE) from differing optimal ones—are also studied. Joint (treatment and carryover effects) optimality is briefly treated.