## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 36, Number 6 (1965), 1815-1828.

### 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.

#### Article information

**Source**

Ann. Math. Statist., Volume 36, Number 6 (1965), 1815-1828.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177699811

**Digital Object Identifier**

doi:10.1214/aoms/1177699811

**Mathematical Reviews number (MathSciNet)**

MR189191

**Zentralblatt MATH identifier**

0136.40705

**JSTOR**

links.jstor.org

#### Citation

Ogawa, Junjiro; Ishii, Goro. The Relationship Algebra and the Analysis of Variance of a Partially Balanced Incomplete Block Design. Ann. Math. Statist. 36 (1965), no. 6, 1815--1828. doi:10.1214/aoms/1177699811. https://projecteuclid.org/euclid.aoms/1177699811