## The Annals of Statistics

- Ann. Statist.
- Volume 12, Number 4 (1984), 1488-1499.

### On Analysis of Variance in the Mixed Model

#### Abstract

An analysis of variance (ANOVA) is defined to be a partition of the total sum of squares into independent terms which, when suitably scaled, are chi-squared variables. A partition of less than the total sum of squares, but with these properties, will often suffice and is referred to as a partial ANOVA. Conditions for an ANOVA, and for partial ANOVAs selected to contain only specific parameters, are given. Implications for estimation of variance components from an ANOVA are also discussed. These results are largely an extension of work by Graybill and Hultquist (1961). With unbalanced data, conditions for an ANOVA and the number of terms in it both can depend on which effects in the model are fixed and which are random. This is not taken into account by those procedures for partitioning a sum of squares which distinguish between random and fixed effects only in the calculation of expected mean squares. Several examples are given.

#### Article information

**Source**

Ann. Statist., Volume 12, Number 4 (1984), 1488-1499.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176346805

**Digital Object Identifier**

doi:10.1214/aos/1176346805

**Mathematical Reviews number (MathSciNet)**

MR760701

**Zentralblatt MATH identifier**

0558.62062

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62J10: Analysis of variance and covariance

Secondary: 62E10: Characterization and structure theory

**Keywords**

Analysis of variance mixed model chi-squared distribution variance components

#### Citation

Brown, K. G. On Analysis of Variance in the Mixed Model. Ann. Statist. 12 (1984), no. 4, 1488--1499. doi:10.1214/aos/1176346805. https://projecteuclid.org/euclid.aos/1176346805