This paper is concerned with a two-factor analysis of variance situation, the factors being conveniently referred to as blocks and treatments. One of the treatments has the role of a control. Attention is focused on inference about the treatment parameters, the block parameters being regarded as nuisance parameters. With a general multivariate normal form for the distribution of errors and for the prior distribution on the block and treatment parameters, the posterior distribution of the treatment parameters is derived. With a quadratic loss function an algorithm is derived for the optimum allocation of treatments over a given sample with known blocking. In special cases the optimum allocation can be written down immediately and the algorithm need not be resorted to.
R. J. Owen. "The Optimum Design of a Two-Factor Experiment Using Prior Information." Ann. Math. Statist. 41 (6) 1917 - 1934, December, 1970. https://doi.org/10.1214/aoms/1177696693