Applied Statistics

5Analysis of Variance (ANOVA) – Fixed Effects Models

5.1 Introduction

In the analysis of variance, we assume that parameters of random variables depend on non‐random variables, called factors. The values a factor can take we call factor levels or in short levels. We discuss cases where one, two or three factors have an influence on the observations.

An experimenter often has to find out in an experiment whether different values of several variables or of several factors have different results on the experimental material. If the effects of several factors have to be examined, the conventional method means to vary only one of these factors at once and to keep all other factors constant. To investigate the effect of p factors this way, p experiments have to be conducted. It can be that the results at the levels of a factor investigated depend on the constant levels of other factors, which means that interactions between the factors exist. The British statistician R. A. Fisher recommended experimental designs by varying the levels of all factors at the same time. For the statistical analysis of the experimental results of such designs (they are called factorial experiments), Fisher developed a statistical procedure: the analysis of variance. The first publication about this topic stemmed from Fisher and Mackenzie (1923), a paper about the analysis of field trials in Fisher's workplace at Rothamsted Experimental Station in Harpenden, UK. A good overview is given in Scheffé (1959) ...