### Introduction

In the one-way analysis of variance (Section 44.1) we consider the effect of one factor on the values taken by a variable. Very often, in engineering investigations, the effects of two or more factors are considered simultaneously.

The two-away ANOVA deals with the case where there are two factors. For example, we might compare the fuel consumptions of four car engines under three types of driving conditions (e.g. urban, rural, motorway). Sometimes we are interested in the effects of both factors. In other cases one of the factors is a ‘nuisance factor’ which is not of particular interest in itself but, if we allow for it in our analysis, we improve the power of our test for the other factor.

We can also allow for interaction effects between the two factors.

#### Prerequisites

- be familiar with the general techniques of hypothesis testing
- be familiar with the $F$ -distribution
- be familiar with the one-way ANOVA calculations

#### Learning Outcomes

- state the concepts and terminology of two-way ANOVA
- perform two-way ANOVA
- interpret the results of two-way ANOVA calculations

#### Contents

1 Two-way ANOVA without interaction1.1 Example of variance in data

2 Two-way ANOVA with interaction

2.1 Fixed and random effects

3 Two-way ANOVA versus one-way ANOVA