Applied Statistics

7Analysis of Variance – Mixed Models

7.1 Introduction

In mixed models, as well fixed effects as in Chapter 5, random effects as in Chapter 6 also occur, i.e. mixed models are models where in the model equation at least one, but not all, effects are random variables.

In mixed models, we discuss problems of variance component estimation and of estimating and testing fixed effects.

Therefore, we need expected mean squares from the tables of Chapter 5 if the effect defining a row is fixed and the tables of Chapter 6 if the effect defining a row is random.

An interaction is random if at least one of the factors involved is random. In discussing the different classifications, we use the same procedure as in Chapters 5 and 6. Of course, in a one‐way classification a mixed model is impossible. We therefore start with the two‐way classification.

7.2 Two‐Way Classification

We discuss here the cross‐classification where we consider the factor A to be fixed without loss of generality. If the factor B is fixed, we rename both factors. In the nested classification, two‐mixed models occur when the super‐ordinate factor or the nested factor is random. We write random factors with their elements in bold.

7.2.1 Balanced Two‐Way Cross‐Classification

We consider two cross‐classified factors A (fixed) and B (random) and their interactions AB.

The model equation of the balanced case is

7.1

with ...