Identification of universally optimal circular designs for the interference model

Abstract

Many applications of block designs exhibit neighbor and edge effects. A popular remedy is to use the circular design coupled with the interference model. The search for optimal or efficient designs has been intensively studied in recent years. The circular neighbor balanced designs at distances 1 and 2 (CNBD2), including orthogonal array of type I ($\mathrm{OA}_{I}$) of strength $2$, are the two major designs proposed in literature for the purpose of estimating the direct treatment effects. They are shown to be optimal within some reasonable subclasses of designs. By using benchmark designs in approximate design theory, we show that CNBD2 is highly efficient among all possible designs when the error terms are homoscedastic and uncorrelated. However, when the error terms are correlated, these designs will be outperformed significantly by other designs. Note that CNBD2 fall into the special catalog of pseudo symmetric designs, and they only exist when the number of treatments is larger than the block size and the number of blocks is multiple of some constants. In this paper, we elaborate equivalent conditions for any design, pseudo symmetric or not, to be universally optimal for any size of experiment and any covariance structure of the error terms. This result is novel for circular designs and sheds light on other similar models in the search for optimal or efficient asymmetric designs.