12.3: Interpreting Main and Interaction Effects

In this part of the chapter, we will dig into interaction effects and how to detect and interpret them alongside main effects in factorial analyses. We will see that main effects can be detected using group means tables, and interactions can be detected using the tools of bar graphs and interaction plots.

We will also look at how to interpret three major scenarios: when we have significant main effects but no significant interaction; when we have a significant interaction, but no main effects; and when we have both interactions and main effects that turn out significant.

A main effect means that one of the factors explains a significant amount of variability in the data when taken on its own, independent of the other factor. You can tell (roughly) whether a main effect is likely to exist by looking at the data tables. Specifically, you want to look at the marginal means, or what we called the row and column means in the context of a two-way ANOVA.

Video: Factorial Designs: Main Effects & Interactions

Factorial Designs: Main Effects & Interactions on YouTube.

Some Possible Scenarios

Main Effects with No Interaction

The first possible scenario is that main effects exist with no interaction. This can be interpreted as the following: each factor independently influenced the dependent variable (or at least accounted for a sizeable share of variance). This can be seen in the row and column means of Table \(\PageIndex{1}\)

Table \(\PageIndex{1}\): Data Example - main effects with no interaction.

	Male	Female	Row means
Low dose of drug	40	20	30
High dose of drug	30	10	20
Column means	35	15

Going across, we can see a difference in the row means. People who receive the low dose have less pain than those who receive the high dose: this could be a significant main effect. Going down, we can see a difference in the column means as well: males report more pain than females, another likely main effect. So in this example, there is an apparent main effect of each factor, independent of the other factor. People with a low dose have lower pain scores if they are female. A similar pattern exists for the high dose as well. This similarity in pattern suggests there is no interaction. You can do the same test with the columns and reach the same conclusion. In other words, if you were to look at one factor at a time (like in Figure \(\PageIndex{1}\)), ignoring the other factor entirely, you would see that there was a difference in the dependent variable you were measuring between the levels of that factor.

Figure \(\PageIndex{1}\): Separate bar graphs of main effects.

Interaction with No Main Effects

Detecting interaction effects in a data table is tricky, but if you can see a clear X-pattern in the group means table (the four cell means), such that similar numbers connect in an “X”, then that is a sign that there is probably an interaction. If not, there may not be. In the first example (Table \(\PageIndex{1}\)), there is no clear X pattern. But in the next example (Table \(\PageIndex{2}\)), there is an X pattern if you connect similar numbers (20 with 20 and 10 with 10)—probably an interaction. Ask yourself: if you take one row at a time, is there a different pattern for each or a similar one?

The second possible scenario is that an interaction exists without main effects. We can interpret this as follows: each factor did not, in and of itself, influence the dependent variable.

In the row and column means of Table \(\PageIndex{2}\), you can see that neither dose nor sex marginal means differ – no main effects. But the non-parallel lines in the graph of cell means (the first plot in Figure \(\PageIndex{2}\)) indicate an interaction.

The best way to interpret an interaction is to start describing the patterns for each level of one of the factors. First, we will examine the low-dose group. They have lower average pain scores only if they are female. Now look at the high-dose group: they have lower average pain scores only if they are male – the opposite pattern. This means that the effect of the drug on pain depends on (or interacts with) sex.

Table \(\PageIndex{2}\): Data Example 1 - no main effects with an interaction.

	Male	Female	Row means
Low dose of drug	20	10	15
High dose of drug	10	20	15
Column means	15	15

Going across the data table, you can see the mean pain score measured in people who received a low dose of a drug and those who received a high dose. The marginal means are 15 vs. 15. This indicates there is clearly no difference between the two, so there is no main effect of drug dose. Now look top to bottom to find the comparison between male and female participants on average. 15 vs. 15 again, so no main effect of education level.

Main Effects and Interactions Exist

The third possible basic scenario in a dataset is that main effects and interactions exist. This means each factor independently accounted for variability in the dependent variable in its own right. But also, they interacted synergistically to explain variance in the dependent variable. Together, the two factors do something else beyond their separate, independent main effects.

In this example, at both low and high doses of the drug, pain levels are higher for males.

Table \(\PageIndex{2}\): Data Example 3 - main and interaction effects.

	Male	Female	Row means
Low dose of drug	40	20	30
High dose of drug	20	15	17.5
Column means	30	17.5

In this example, the higher dose is more effective at reducing pain than the lower dose for both sexes. But there is also an interaction, in that the difference between drug doses is much more accentuated in males.

Plotting Interaction Effects

It is far easier to tell at a glance whether an interaction exists if you graph the data. In a bar graph, look for a U- or inverted-U-shaped pattern across side-by-side bar graphs as an indication of an interaction. In the first bar graph in Figure \(PageIndex{2}\), there is clearly an interaction: look at the U shape the bars form. In the second graph, there is no such U shape. When you look at each set of bars in turn, the pattern displayed is similar, just a little higher overall for males. Clearly, there is no hint of an interaction.

Figure \(\PageIndex{2}\): Bar charts showing possible frequencies for two-factor ANOVA with factors dose and sex.

Interaction plots like those shown in Figure \(\PageIndex{3}\) make it even easier to see if an interaction exists in a dataset. If you were to connect the tops of like-colored bars of the graphs on the previous bar graphs, you would get line plots. If the two resulting lines are non-parallel, then there is likely an interaction, but if the lines are parallel or close to parallel, there is no interaction.

Figure \(\PageIndex{3}\): Example interaction plots for a two-factor ANOVA with factors dose and sex.

Finally, look at the difference in the slope of the lines in the interaction plot in \(\PageIndex{4}\). The lines are certainly non-parallel. So drug dose and sex matter, each in their own right, but also in their particular combination.

Figure \(\PageIndex{4}\): Interaction plot demonstrating an interaction between dose and sex.

You can probably imagine how such a pattern could arise. Perhaps males are more sensitive to pain, and thus require a higher dose to achieve relief. Or perhaps the higher body mass in males means a higher dose of the drug is required to be effective. For females, both doses are similar in their efficacy.

Now you have seen the same three example datasets displayed in three different ways (tables, bar graphs, interaction plots), each making it easy to see particular aspects of the patterns made by the data.

Test Your Knowledge

Question \(\PageIndex{1}\)