In the previous two chapters we have looked at situations in which we've tried to test for differences between groups when there has been a single independent variable (i.e., one variable has been manipulated). However, at the beginning of Chapter 11 I said that one of the advantages of ANOVA was that we could look at the effects of more than one independent variable (and how these variables interact). This chapter extends what we already know about ANOVA to look at situations where there are two independent variables. We've already seen in the previous chapter that it's very easy to incorporate a second variable into the ANOVA framework when that variable is a continuous variable (i.e., not split into groups), but now we'll move onto to situations where there is a second independent variable that has been systematically manipulated by assigning people to different conditions.

Factorial designs

In the previous two chapters we have explored situations in which we have looked at the effects of a single independent variable on some outcome. However, independent variables often get lonely and want to have friends. Scientists are obliging individuals and often put a second (or third) independent variable into their designs to keep the others company. When an experiment has two or more independent variables it is known as a factorial design (this is because, as we have seen, variables are sometimes referred to as factors). There are several types of factorial design:

Independent factorial design: In this type of experiment there are several independent variables or predictors and each has been measured using different entities (between groups). We discuss this design in this chapter.

Repeated-measures (related) factorial design: This is an experiment in which several independent variables or predictors have been measured, but the same entities have been used in all conditions. This design is discussed in Chapter 14.

Mixed design: This is a design in which several independent variables or predictors have been measured; some have been measured with different entities whereas others used the same entities. This design is discussed in Chapter 15.

As you might imagine, analysing these types of experiments can get quite complicated. Fortunately, we can extend the ANOVA model that we encountered in the previous two chapters to deal with these more complicated situations. When we use ANOVA to analyse a situation in which there are two or more independent variables it is sometimes called factorial ANOVA; however, the specific names attached to different ANOVAs reflect the experimental design that they are being used to analyse (see Jane Superbrain Box 13.1). This section extends the one-way ANOVA model to the factorial case (specifically when there are two independent variables). In subsequent chapters we will look at repeated-measures designs, factorial repeated-measures designs and finally mixed designs.