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  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 , 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.
"The Relationship Algebra and the Analysis of Variance of a Partially Balanced Incomplete Block Design." Ann. Math. Statist. 36 (6) 1815 - 1828, December, 1965. https://doi.org/10.1214/aoms/1177699811