For instance, the academic achievement of a student depends on study habits of the student as well as home environment. We may have two simple experiments, one to study the effect of study habits and another for home environment.
Independence of Factors
But these experiments will not give us any information about the dependence or independence of the two factors, namely study habit and home environment.
In such cases, we resort to Factorial ANOVA which not only helps us to study the effect of two or more factors but also gives information about their dependence or independence in the same experiment. There are many types of factorial designs like 22, 23, 32 etc. The simplest of them all is the 22 or 2 x 2 experiment.
In these experiments, the factors are applied at different levels. In a 2 x 2 factorial design, there are 2 factors each being applied in two levels.
Let us illustrate this with the help of an example. Suppose that a new drug has been developed to control hypertension.
We want to test the effect of quantity of the drug taken and the effect of gender. Here, the quantity of the drug is the first factor and gender is the second factor (or vice versa).
Suppose that we consider two quantities, say 100 mg and 250 mg of the drug (1 / 2). These two quantities are the two levels of the first factor.
Similarly, the two levels of the second factor are male and female (A / B).
Thus we have two factors each being applied at two levels. In other words, we have a 2 x 2 factorial design.
Here we have 4 different treatment groups, one for each combination of levels of factors - by convention, the groups are denoted by A1, A2, B1, B2. These groups mean the following.
A1 : 100mg of the drug applied on male patients
A2 : 250mg of the drug applied on male patients
B1 : 100mg of the drug applied on female patients
B2 : 250mg of the drug applied on female patients.
A main effect is an outcome that can show consistent difference between levels of a factor.
In our example, there are two main effects - quantity and gender.
Factorial ANOVA also enables us to examine the interaction effect between the factors. An interaction effect is said to exist when differences on one factor depend on the level of other factor.
However, it is important to remember that interaction is between factors and not levels. We know that there is no interaction between the factors when we can talk about the effect of one factor without mentioning the other factor.