Graph decomposition methods for variance balanced block designs with correlated errors

Block designs have an important historical footing in the design of experiments. Let theta be the treatment mean vector in the statistical model with fixed block effects. It is desirable to have the covariance of the least squares estimator of the treatment effects, Cov(theta circumflex accent ), be a completely symmetric matrix. When this occurs, we have a variance balanced design (VBD). It is known that a pairwise balanced design can be converted into a VBD by choosing blocks with a multiplicity inversely proportional to the block size. This assumes that experimental errors are uncorrelated. However, this need not be the case, especially in ecological experiments where blocks have spatial correlation or laboratory settings with temporal correlation. This paper proposes the use of graph decompositions as a generalization of balanced incomplete block designs for the construction of VBDs in the presence of correlated errors. A general result is obtained to construct VBDs for various error correlation structures. Several applications are given to illustrate the relationship between graph decompositions and VBDs. (C) 2022 Elsevier B.V. All rights reserved.