Optimal Completely Randomized Designs For A Cyclic Covariate Model With Even Replication Number

We consider a linear model for the comparison \(V \geq 2\) treatments (or one treatment at \(V\) levels) in a completely randomized statistical setup, making \(r\) (the replication number) observations per treatment level in the presence of \(K\) continuous covariates with values on the \(K\)-cube. The main interest is restricted to cyclic designs characterized by the property that the allocation matrix of each treatment level is obtained through cyclic permutation of the columns of the allocation matrix of the first treatment level. The \(D\)-optimality criterion is used for estimating all the parameters of this model.

By studying the nonperiodic autocorrelation function of circulant matrices, we develop an exhaustive algorithm for constructing \(D\)-optimal cyclic designs with even replication number. We apply this algorithm for \(r = 4, 16 \leq V \leq 24, N=rV \equiv 0 \mod 4\), for \(r=6, 12\leq V \leq 24, N =rV \equiv 0 \mod 4\), for \(r =6, V =m.n, m\) is a prime, \(N =rV \equiv 2 \mod 4\) and the corresponding cyclic designs are given.