## 3.1 Introduction

The problems presented in this chapter can be considered a generalization of the one-sample and two-sample location problems, in the sense that more than two samples are compared. The primary interest here consists in comparing the location of three or more samples. Under the null hypothesis the sample data are supposed to come from the same population. Under the alternative hypothesis data are supposed to come from different populations (or groups). In the analysis of variance problem (ANOVA) the variability of data is broken down into two components: (1) between groups variability, due to the different locations of the populations; (2) within groups variability, due to the variability in the populations. The focus of the analysis is on the significance of the first type of variability.

When the populations are defined according to different levels of one factor (or different treatments) we are in the so called one-way ANOVA layout. In this case the dataset is represented by {*X*_{ji}; *i* = 1, …, *n*_{j}; *j* = 1, …, *C*} , where *n*_{j} is the sample size of the *j*th group and *C* > 2 is the number of groups or samples. The factor levels are represented by the integers {1, …, *C*}. In the presence of two factors, the problem takes the name of two-way ANOVA. The dataset is {*X*_{jri}; *i* = 1, …, *n*_{jr}; *j* = 1, …, *C*_{1}; *r* = 1, …, *C*_{2}}, where *n*_{jr} is the number of sample observations in the group where the first factor is at level *j* and the second factor at level ...