Chapter 13

Two-Way Factorial ANOVA:
Using More than One Independent Variable

13.1 Introduction to Two-Way ANOVA

In the previous two chapters we introduced One-Way ANOVA designs that involve multiple levels of one independent variable (or factor). The present chapter will introduce a more sophisticated ANOVA model that allows the researcher to test the effectives of two independent variables; hence, this procedure is called Two-Way ANOVA (sometimes also called Factorial ANOVA). That is, factorial ANOVA improves on one-way ANOVA in that the researcher can simultaneously assess the effects of two (or more) independent variables on a single dependent variable within the same analysis.  Thus, factorial ANOVA yields the same information that two one-way ANOVA's would, but it does so in one analysis.

But that's not all.  Factorial ANOVA also allows the investigator to determine the possible combined effects of the independent variables.  That is, it also assesses the ways in which these variables interact with one another to influence scores on the dependent variable.  Although understanding such interaction effects can be a complex and difficult task, it is essential to the progress of science, since in the real world many variables interact with one another to determine behavior.  In this chapter we provide a basic introduction to factorial ANOVA and the SPSS program that performs this powerful statistical analysis. 

The conceptual basis of factorial ANOVA is essentially the same as that of one-way ANOVA, and the interpretation of the resulting F-values is also based on the same logic as in the one-way ANOVA. The difference is that where one-way ANOVA only generates one F-value, two-way ANOVA generates three F-values: one to test the main effects of each factor, and a third to test the interaction effect (i.e., the combined effect of the two factors).

• 13.1a The need for a factorial combination of Independent Variables

A basic requirement for factorial experimental design is that the levels of the two independent variables have been completely crossed in a factorial combination.  This means that each level of the first independent variable must be combined with each level of the other independent variable.  In the simplest two-way ANOVA (a 2 x 2 design), four different groups of participants would be needed.  If the first factor (Factor A) is GENDER (where level A1 is male employees and level A2 is female employees), and the second factor (Factor B) is MASC (where level B1 is low-masculine employees and level B2 is high-masculine employees), four combinations would be required to permit a factorial ANOVA. Each unique combination is referred to as a cell (see Table 13.1). In this table, the four cells (in boldface) represent these four possible combinations.

Thus, the four cells of the factorial combination shown in Table 13.1 for a 2 (Gender) x 2 (Masculinity) factorial design would require us to divide our participants into the following four groups:

(1) Low-masculine males (A1B1)
(2) High-masculine males (A1B2)
(3) Low-masculine females (A2B1)
(4) High-masculine females (A2B2)

This factorial combination will allow us to compare the scores of men vs. women (GENDER) and low-masculine vs. high-masculine employees (MASC) on a given dependent variable. This would be the same as if we did two separate studies and conducted two t-tests (one comparing the male vs. female scores, and one comparing low- vs. high-masculine employees' scores). But it would be more economical and efficient, because we would get the same information from one study and one analysis (the 2 x 2 ANOVA). What is crucial to the factorial combination of these two independent variables is that we are also able to assess the possible interaction effect of the two independent variables combined.