A two-way ANOVA is similar to a one-way ANOVA in that it analyzes the variance between groups and within groups. The logic is the same: if either of the variables has an effect, the variance within the group should be greater than the variance between the groups.As with the one way ANOVA, an F ratio is formed by dividing between-group variance by between group variance. The difference is that in two ways ANOVA , between groups variance may be attributable to factor A ( one of the independent variables in a study), to factor B(the second independent variable in the study), and to the interaction of factors A and B.With two independent variables, there is a possibility of main effect for each variable, and an F ratio is calculated to represent each of these effects. In addition there is a possibility of an interaction effect, and an F ratio is also needed to represent this effect.Thus, with a two way ANOVA, there are three F ratios to calculate and ultimately to interpret.

In a 2x2 factorial design, such as the one we have been looking at in this chapter, there are three null and alternative hypotheses. The null hypothesis for factor A states that there is no main effect of factor A( the differences observed between the groups are greater than what would be expected based chance). In other words, the null hypothesis states that the population means represented by sample means are not from the same population. A second null hypothesis states that there is no main effect in factor B, and the alternative hypothesis states that there is no interaction of factors A and B, and the alternative hypothesis states that there is an interaction effect.