Mathematical Statistics

5Analysis of Variance (ANOVA) – Fixed Effects Models (Model I of Analysis of Variance)

5.1 Introduction

An experimenter often has to find out in an experiment whether different values of one variable or of several variables have different results on the experimental material. The variables investigated in an experiment are called factors; their values are called factor levels. 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. This approach is not only very labour intensive, but it can also be that the results at the levels of factor investigated depend on the constant levels of the remaining 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; see Chapter 12), Fisher developed a statistical procedure, the analysis of variance (ANOVA). 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).

The ANOVA is based on the decomposition of the sum of squared ...