Making Sense of Interactions in Factorial Designs, Explained

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Making Sense of Interactions in Factorial Designs

Research Methods in Psychology · Chapter 13: Factorial, Within-Subject, and Small-n DesignsSection 13.2 · lesson 78 of 949 min · free with a free account

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In this lesson

Reading the patterns that single-factor designs cannot reveal.

  • Define an interaction and say why one study is required
  • Distinguish ordinal from disordinal interactions on a graph
  • Use simple-effects analysis to clarify an interaction
  • Avoid the common errors in interpreting one

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Everything said in the video, section by section (1,224 words). Read along, or search it.

Factorial, Within-Subject, and Small-n Designs

Making Sense of Interactions in Factorial Designs

The interaction is the payoff of the factorial design. It is also the part most often misread in published papers. By the end of this section you will define an interaction effect precisely. And explain why separate single-factor experiments can never detect one. You will distinguish ordinal from disordinal interactions, and recognize each on a graph. You will use simple-effects analysis to clarify the structure of an interaction. You will avoid the common errors, including confusing a main effect with an interaction. And you will communicate factorial findings clearly, using graphs and supporting tables.

What It Says

An Interaction Is a Statement About Contingency

An interaction is a statistical statement about contingency. The effect of one independent variable depends on the level of another. That word, depends, is the whole concept. And the everyday language of contingency is already familiar to you. Whether a strategy works depends on the situation. Whether a treatment helps depends on the patient. Factorial designs translate that intuition into measurable form. Now notice what follows for prediction. When two variables interact, knowing the level of one is not enough. The level of the other matters too, and they must be considered together.

Why Factorials Exist

The Crossing Forces the Encounter

Interactions are the reason factorial designs were invented in the first place. Consider what two separate single-factor experiments can give you. They describe the average effect of each variable, in isolation. Each one is a clean answer to a narrow question. But neither can reveal how the variables behave when they meet. Nothing in either dataset contains that information. The factorial structure forces the encounter. Every level of every factor is crossed with every level of every other factor. And the interaction effect is what emerges from that crossing.

Reading the Graph

Parallelism Is the Visual Signature

The fastest way to read an interaction is visually. An interaction graph plots the cell means of a factorial design. One factor sits on the horizontal axis, and the other is drawn as separate lines. Parallel lines carry a specific message. They say the effect of the first factor is the same at every level of the second. In other words, there is no interaction. When the lines diverge, or converge, or cross, that changes. The effect of one factor varies with the level of the other, so an interaction is present. The visual cue is parallelism, and only parallelism. Deviations from it are the signature you are looking for.

Pattern One

Ordinal Interactions

Not all interactions look alike, and two patterns recur often enough to have names. The first is the ordinal interaction. The lines on the graph are not parallel, but they do not cross. The effect of the first factor runs in the same direction at every level of the second. What varies is the magnitude of that effect, not its sign. Here is a study that would produce one. Exercise improves mood for both younger and older participants. But it improves mood more strongly for the younger ones. Both groups improve. The difference is in how much. Ordinal interactions are common, and main effects survive them. You can usually interpret main effects in their presence, with reasonable care.

Pattern Two

Disordinal Interactions

The second pattern is the disordinal interaction, and it is a different animal. Here the lines on the interaction graph actually cross. The effect of the first factor reverses direction across levels of the second. A teaching method might improve outcomes in small classes. And worsen them in large classes. A drug might reduce symptoms at low doses, and make them worse at high doses. Disordinal interactions are theoretically dramatic. They say the variables do not simply combine. They actively transform each other's effects. And they make main effects close to meaningless. An average effect that hides opposite directions tells a story that is not in the data.

The Cancellation Trap

Zero Main Effect, Enormous Result

That last point deserves its own moment, because it is a genuine trap. In a strong disordinal interaction, the marginal averages can be near zero. Those are the averages used to compute main effects. The positive and negative effects simply cancel each other out. So imagine a researcher who reports only the main effect. They would conclude there was no effect at all. And they would be missing the most interesting result in the entire study. The habit that prevents this costs about ten seconds. Always plot the interaction before reporting main effects. The graph tells you whether the marginal averages are summarizing, or hiding, the structure.

Follow-Up

Simple-Effects Analysis

When a factorial design produces a significant interaction, you are not finished. Follow-up analyses are needed to clarify what the interaction actually consists of. The standard approach is simple-effects analysis. You examine the effect of one factor at each level of the other, separately. Take the teaching design again, crossing method with class size. First, compute the lecture versus discussion difference within small classes alone. Then compute the same difference within large classes alone. Now compare what those two comparisons say. Maybe the difference is significant in one condition and not the other. Or, more dramatically, maybe it runs in opposite directions. Either way, the structure of the interaction has become interpretable.

Which Direction

Both Decompositions Are Valid

Simple effects can be computed in either direction, and this choice is often overlooked. You can ask about the effect of factor A within levels of B. Or the effect of B within levels of A. The choice depends on your theoretical question, not on convention. A study primarily interested in teaching method asks how method varies across class sizes. A study primarily interested in class size asks how size varies across methods. Same data. Same interaction. Two different narratives. Both decompositions are statistically valid. And reporting both can sometimes be genuinely useful to a reader.

Errors One to Three

How Interactions Get Misread

Five errors recur when people interpret interactions. Start with three. The first is reporting main effects without checking for interactions. A non-significant main effect gets read as no effect, while an interaction sits underneath. The fix is the habit from a moment ago. Always plot the interaction graph before interpreting main effects. The second error is treating an ordinal interaction as disordinal. That means describing a magnitude difference as a directional reversal. So check whether the lines actually cross within the range you studied. The third error is decomposing an interaction in only one direction. You miss alternative theoretical interpretations that the other decomposition would surface. Compute simple effects both ways when both are conceptually meaningful.

Errors Four and Five

Significance Is Not Importance

The remaining two errors are about weight rather than direction. The fourth is ignoring effect size in favor of significance. A significant but tiny interaction gets reported as theoretically central. The remedy is to report standardized effect sizes alongside significance tests. The fifth is over-interpreting three-way interactions from underpowered designs. Theoretical claims end up resting on a few cells with high variability. So treat higher-order interactions as exploratory. Unless the design was specifically powered to detect them. Both of these errors survive peer review regularly. A significance test looks like a verdict, even when very little data stands behind it.

Practice

The Graph Is an Analytic Tool

One habit ties this whole section together. Statistical tables alone are usually insufficient for understanding interactions. The interaction graph is not a presentation tool you add at the end. It is a core analytic tool. It makes patterns visible that no table reveals at a glance. So generate the graph early in your analysis. Before running the formal tests, not after. Doing that shapes which simple-effects comparisons are theoretically meaningful. As opposed to the ones that are merely available to compute.

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FAQ

Questions students ask us

About Making Sense of Interactions in Factorial Designs

What is an interaction effect in a factorial design?
An interaction effect occurs when the effect of one independent variable depends on the level of another. For example, discussion might produce better outcomes than lecture in small classes but worse outcomes in large classes. When variables interact, knowing the level of one is not enough to predict the outcome. Interactions can be detected only when the variables are crossed in the same experiment, not studied in separate single-factor experiments.
How can you tell if there is an interaction on a graph?
Look at whether the lines are parallel. An interaction graph plots cell means with one factor on the horizontal axis and the other factor drawn as separate lines. Parallel lines mean the first factor has the same effect at every level of the second, so there is no interaction. Lines that converge, diverge, or cross suggest an interaction, which the statistical test then evaluates.
What is the difference between an ordinal and a disordinal interaction?
In an ordinal interaction the lines are not parallel but do not cross, so an effect runs in the same direction at every level of the other factor and only its size changes. Exercise improving mood for both younger and older adults, but more for younger ones, is an example. In a disordinal interaction the lines cross and the effect reverses direction, which can make main effects close to meaningless.
What is a simple effects analysis?
Simple effects analysis follows up a significant interaction by testing the effect of one factor separately at each level of the other. In a study crossing teaching method with class size, the lecture versus discussion difference is tested within small classes and again within large classes. The results show whether the effect appears in only one condition or reverses direction. The breakdown can run either way, chosen by the theoretical question.

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