## Note on the ANOVA of a completely confounded factorial experiment

### Volume 32 / 2005

#### Abstract

The purpose of this paper is to present a modern approach to the analysis of variance (ANOVA) of disconnected resolvable group divisible partially balanced incomplete block (GDPBIB) designs with factorial structure and with some interaction effects completely confounded. A characterization of a factorial experiment with completely confounded interaction is given. The treatment effect estimators and some relations between the matrix $\textbf{F}$ of the reduced normal equations and the information matrix $\textbf{A}$ are given. Moreover the ANOVA of the sum of squares for adjusted treatment effects and the matrix $\textbf{F}$ with its eigenvalues and orthonormal eigenvectors for the case of a completely confounded factorial experiment are presented. A special form of a generalized inverse ($g$-inverse) of $\textbf{F}$ is introduced (Theorems 3.2.1–3.2.4). The corresponding numerical example has been worked out by Oktaba (1956) and Oktaba, Rejmak and Warteresiewicz (1956) by applying Galois fields and congruences.