Anova of Balanced Variance Component Models

TL;DR

This paper extends the distributional theory of sums of squares in fixed effects linear models to models with random effects, using a simple result on noncentral chi-square distributions.

Contribution

It introduces an easy extension of Cochran's Theorem for models with random effects, enhancing understanding of variance components in balanced designs.

Findings

Extension of distributional theory to random effects models

Utilization of noncentral chi-square distribution results

Simplified approach for teaching variance component models

Abstract

Balanced linear models with fixed effects are taught in undergraduate programs of all universities. These occur in experimental designs such as one-way and two-way Anova, randomized complete block designs (RCBD) and split plot designs. The distributional theory for sums of squares in fixed effects models can be taught using the simplest form of Cochran's Theorem. The contribution provided here allows for an easy extension of the distributional theory to corresponding models with random effects. The main tool used is a simple result on noncentral chi-square distribution overlooked in textbooks at undergraduate and graduate levels.