One of the golden standards of experimental design in both the physical and social sciences is a random controlled experiment with only one dependent variable. The limitation to that design is that it overlooks the effects multiple variables may have with one another. In statistics, there is a special design that not only deals with multiple independent variables, but reveals effects that they may have with one another. This design is called **factorial design analysis of variance** or factorial ANOVA for short. Let’s explore some of the finer points of this powerful tool.

For our example, let’s say that you have just finished a college class in statistics (congratulations!) and that beast of a final exam has gotten everyone talking about which majors do best, science or arts. Now, the class is mixed between art and science majors *and* underclassmen and upperclassmen. You decide to analyze the means of the final exam scores to figure out who is the best.

## The Idea Behind Factorial Design

In your statistics class example, there are two variables that have an effect on the outcome: major and college experience, and each has two levels in it. This means that there are two independent variables and one dependent variable (final exam scores). Factorial design was born to handle this kind of design.

Factorial ANOVA compares groups that may interact with one another. Instead of comparing two groups (majors and experience), you are actually comparing *four* groups. You can see the groups in this diagram

What you have here is an example of 2 x 2 factorial design ANOVA. This means that there are two *factors* (what we consider independent variables) with two levels of treatment each. So there will be four groups based on the combination of these factors.

Just like ANOVA, you will compare the means using the variances of each group and group level. However, you cannot simply do a series of ANOVAs because that would introduce too much error to confidently say there is a significant difference. You’re simply looking for a significant difference between the groups. This means looking for main effects of each independent variable and how they potentially interact.

## Main Effects and Interactions

The real hallmark of factorial design is the ability to distinguish between main effects and potential interactions in the groups.

As in one-way ANOVA, a **main effect** is present when the groups *within* a factor demonstrate a significant difference from the grand mean. In your example, a main effect would be a significant difference between upper and underclassmen or a difference between the arts and the sciences. Note, you are considering the two independent variables separately as if it were a one-way ANOVA.

An **interaction** is present when the dependent variable of the group is affected by a combination of both factors (i.e., the level of one is affected by the level of the other). In your example, a possible interaction would be between underclassmen status and being a science major. This would mean that there is a significant difference between this group and the others. Factorial design reveals a difference that may exist in the subgroups of the design.

## Visual Differences

When you read about factorial designs in studies and textbooks, the results are usually accompanied by some sort of graph that promotes quick interpretation. In the case of our 2 x 2 design, there are two lines; the red one represents the arts majors and the blue one represents the science majors.

In the example above, you can see that the line are close and almost parallel. This means that there is most likely no significant difference between majors, between college experience, and no interaction.

This example indicates that there is a main effect. We know this because, even though the lines are still approximately parallel, the mean final exam scores represents a difference. There is no interaction in this diagram.

Now we have an example of an interaction. The lines are no longer parallel, so there is something going on between the two factors. This is an example of both a main effect and an interaction. You know there is a main effect because the mean final exam scores are different between under and upperclassmen. Since the lines cross (or would if we extend them) there is an interaction between major and college experience.

## The Takeaways

A factorial design is a method for testing multiple levels of independent variables similar to ANOVA. However, the advantage of factorial design is that it copes with multiple independent variables too. Not only does it test for the differences within factors (called main effects) but it also tests for whether those factors interact with one another to make a difference. I hope this helps clarify some things about factorial design. I look forward to seeing your questions below. Happy statistics!

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