Interaction Plots and Their Meaning
Interaction effects in Two-Way ANOVA tell you whether the effect of one factor on your outcome depends on the level of another factor. The primary tool for understanding these effects is the interaction plot, and learning to read one correctly is essential for interpreting any two-way model.
Visualizing Interaction Effects
An interaction plot displays the mean of the response variable for every combination of levels of your two factors. Typically, one factor goes on the x-axis, the response variable goes on the y-axis, and separate lines represent the levels of the second factor.
The key to reading these plots is the parallelism of the lines:
- Parallel lines indicate no interaction. The effect of Factor A on the response is the same regardless of the level of Factor B.
- Non-parallel lines suggest an interaction is present. The more the lines diverge or converge, the stronger the interaction effect.
Consider a study on plant growth with two factors: fertilizer type (organic vs. synthetic) and soil pH (acidic vs. neutral). If the lines for organic and synthetic fertilizer run roughly parallel across pH levels, fertilizer affects growth the same way regardless of soil pH. If the lines converge or cross, the benefit of one fertilizer over the other changes depending on pH.
Interpreting Interaction Plots
When the lines are non-parallel, look at two things:
- The slopes of the lines. Different slopes tell you that the effect of the x-axis factor differs across levels of the other factor. The greater the difference in slopes, the stronger the interaction.
- Whether the lines cross. This distinction (crossing vs. not crossing) determines whether the interaction is ordinal or disordinal, which has major consequences for interpretation (covered below).
An interaction plot won't tell you whether the interaction is statistically significant. You still need the F-test for the interaction term in your ANOVA table. But the plot tells you the shape and direction of the effect, which the F-test alone cannot.
Interaction Effects on Main Effects
Why Main Effects Become Misleading
When a significant interaction is present, the main effects of each factor are averages across the levels of the other factor. Those averages can hide or distort what's actually happening.
Take a study on exercise type (cardio vs. strength training) and diet (low-carb vs. high-carb) on weight loss. Suppose cardio paired with low-carb produces large weight loss, while strength training paired with high-carb produces large weight loss, but the other two combinations show little effect. The main effect of exercise, averaged across both diets, might look small or nonsignificant, even though exercise clearly matters once you account for diet. Reporting only the main effect would be misleading.
Conditional Nature of Main Effects
In the presence of a significant interaction, each main effect is conditional on the level of the other factor. This means you can't make a blanket statement like "Factor A increases the response" without specifying at which level of Factor B that holds.
To properly unpack the interaction, you use:
- Simple effects tests: These examine the effect of one factor at each individual level of the other factor. For example, testing the effect of exercise separately for the low-carb group and the high-carb group.
- Post-hoc pairwise comparisons: These compare specific cell means to identify exactly which combinations differ from each other.
The practical rule is straightforward: if the interaction term is significant, interpret the interaction first. Report simple effects rather than (or in addition to) main effects, because the main effects alone don't capture the full picture.
Ordinal vs. Disordinal Interactions
The distinction between ordinal and disordinal interactions determines how strongly the interaction qualifies your conclusions about main effects.
Ordinal Interactions
An ordinal interaction occurs when the lines in the interaction plot do not cross. The rank order of the factor levels stays the same across all levels of the other factor; only the size of the difference changes.
For example, in a study of two fertilizers across three levels of soil moisture, an ordinal interaction might show that Fertilizer A always produces higher yield than Fertilizer B, but the advantage is large in moist soil and small in dry soil. The lines diverge but never cross.
With an ordinal interaction, you can still make a directional statement about the main effect (e.g., "Fertilizer A generally outperforms B"), but you need to qualify it by noting that the magnitude of the advantage depends on the other factor.
Disordinal (Crossover) Interactions
A disordinal interaction occurs when the lines in the interaction plot cross. The rank order of one factor's levels actually reverses depending on the level of the other factor.
For example, in a study of two learning strategies across simple and complex tasks, Strategy 1 might produce better performance on simple tasks while Strategy 2 produces better performance on complex tasks. The lines cross, and neither strategy is universally better.
Disordinal interactions make main effects essentially uninterpretable on their own. Saying "Strategy 1 is better on average" would be meaningless if it's only better for one type of task and worse for the other. Simple effects analysis becomes critical here.
Implications for Interpretation
Ordinal interaction: You can cautiously discuss main effects, but qualify them. The direction of the effect is consistent; only the magnitude changes.
Disordinal interaction: Main effects are misleading. The direction of the effect reverses across levels, so you must report simple effects to give an accurate picture.
Identifying which type of interaction you have is not just a visual exercise. It directly shapes what conclusions your analysis can support and how you should communicate your findings.