Optimality and efficiency of two-treatment repeated measurements designs

Abstract

For an arbitrary covariance matrix and any number of treatments and periods, necessary and sufficient conditions for the optimality of a repeated measurements, i.e. crossover or changeover, design were given in the, as yet, unpublished report by H. B. Kushner ‘Optimal repeated measurements designs: the linear optimality equations’ (Kushner, report). Here we specialise the results to the two-treatment case. We give linear equations whose solutions are the proportions of subjects allocated to each treatment sequence in the most general optimal design. The treatment effects information matrix of an optimal design is also presented. The results are applied to several situations. For arbitrary covariance matrices, we show that designs consisting of at most two treatment sequences and their duals must necessarily be dual balanced to be optimal. For autoregressive covariance matrices, we confirm that designs of Matthews (1987) for three and four periods are optimal and we derive new optimal designs for five and six periods. Some optimality results of Laska & Meisner (1985) are extended. A simple formula for the efficiency of a dual balanced nonoptimal design is provided and used to compute the exact efficiencies of some of Kunert's (1991) designs.