Approximate and exact designs for total effects

Abstract

This paper considers both approximate and exact designs for estimating the total effects under one crossover and two interference models. They are different from the traditional block designs in the sense that the assigned treatments also affect their neighboring plots, hence a design is understood as a collection of sequences of treatments. A notable result in literature is that the circular neighbor balanced design (CNBD) is optimal among designs, which do not allow treatments to be neighbors of themselves. However, we find it necessary to allow self-neighboring, and further show that it is the best to allocate each treatment in a subblock of adjacent plots with equal or almost equal numbers of replications. This explains why the efficiency of CNBD drops down to $50\mathrm{\%}$ as the sequence length, say k, increases. Unlike CNBD or the designs for direct effects, our proposed designs do not try to put as many treatments in a sequence as possible. The optimal number of distinct treatments in a sequence is around $\sqrt{2\mathit{k}}$ for crossover designs and $\sqrt{\mathit{k}/0.96}$ for interference models, whenever they are smaller than the total number of treatments under consideration. We systematically study necessary and sufficient conditions for any design to be universally optimal under the approximate design framework, based on which algorithms for deriving optimal or efficient exact designs are proposed. This hybrid nature of cohesively combining theories with algorithms makes our method more flexible than existing ones in the following aspects. (i) Not only symmetric designs are studied, general procedures for producing asymmetric designs are also provided. (ii) Our method applies to any form of within-block covariance matrix instead of specific forms. (iii) We cover all configurations of the numbers of treatments and sequence lengths, especially for large values of them when purely computational methods are not applicable. (iv) On top of the latter, we cover a continuous spectrum of the number of sequences instead of special numbers decided by combinatorial constraints.