The Relationship Algebra and the Analysis of Variance of a Partially Balanced Incomplete Block Design

Abstract

The analysis of variance of a partially balanced incomplete block design is investigated in connection with its relationship algebra. This gives a somewhat clearer insight into the structure of the partition of the total sum of squares than before. A. T. James [5] dealt with the same problem with the balanced incomplete block design and this article should be regarded as a generalization of his work to the partially balanced incomplete block design. The definitions and necessary notations concerning a partially balanced incomplete block design (PBIBD) and its association algebra are briefly given in Section 1. Although the properties of the association algebra have been known in another expression [4], they are presented in Section 2 in the form fitting to our discussions on the relationship algebra. In Section 3, the definition and properties of the relationship algebra of a PBIBD are given and these are believed to be new. In Section 4, the analysis of variance of a PBIBD is considered and the partition of the sum of squares due to treatments adjusted pertinent to the association under consideration is given. Finally, Section 5 is devoted to the analysis of variance of a PBIBD of triangular type and a numerical illustration.