Applied Statistics

6Analysis of Variance – Models with Random Effects

6.1 Introduction

Whereas in Chapter 5 the general structure of analysis of variance models are introduced and investigated for the case that all effects are fixed real numbers (ANOVA model I), we now consider the same models but assume that all factor levels have randomly been drawn from a universe of factor levels. We call this the ANOVA model II. Therefore, effects (except the overall mean μ) are random variables and not parameters, which have to be estimated. Instead of estimating the random effects, we estimate and test the variances of these effects – called variance components. The terms main effect and the interaction effect are defined as in Chapter 5 (but these effects are now random variables).

6.2 One‐Way Classification

We characterise methods of variance component estimation for the simplest case, the one‐way ANOVA, and demonstrate most of them by some data set. For this, we assume that a sample of a levels of a random factor A has been drawn from the universe of factor levels, which is assumed to be large. In order to be not too abstract let us assume that the levels are sires. From the ith sire a random sample of n_i daughters is drawn and their milk yield y_ij recorded. This case is called balanced if for each of the sires the same number n of daughters has been selected. If the n_i are not all equal to n we called it an unbalanced design.

We consider the model

6.1

The a_i are the main effects of the levels A_i