Planning incomplete block experiments when treatments are genetically related.

Selection trials in plant and animal breeding, in incomplete blocks, are described by linear models with random effect parameters associated with treatments with known genetic covariance structure. It is now well known that the information on relatives can improve the analysis and many extensions of this model have been proposed, but no studies have been done on the consequences of this genetical relatedness among treatments for the optimality of block designs. Using a suitable optimality criterion, we show that the knowledge on relatedness may imply that the optimal design is not in the class of designs which are optimal for unrelated treatments. Implications for practical applications are discussed.