A 3 way interaction refers to a type of statistical interaction between 3 variables in a regression model or experimental design. Specifically, it is an interaction between 2 factors, where the effect of one factor depends on the level of a third factor. 3 way interactions are complex, but can provide insight into how different variables work together to influence an outcome.
What is an interaction in statistics?
In statistics, an interaction occurs when the effect of one independent variable on a dependent variable differs depending on the level of another independent variable. There are different orders of interactions:
- A main effect is when an independent variable affects the dependent variable by itself, regardless of other variables.
- A 2-way interaction is between 2 independent variables – the effect of one depends on the level of the other.
- A 3-way interaction is between 3 independent variables – the effect of two variables depends on the level of a third variable.
Higher order interactions are possible with more variables, but become complex and difficult to interpret. 3-way interactions are usually the highest order interaction that would be examined and interpreted.
Example of a 3 way interaction
Here is a hypothetical example of a 3 way interaction from an experiment:
- Independent Variable 1: Drug (2 levels – Drug A vs Placebo)
- Independent Variable 2: Exercise (2 levels – Exercise vs No Exercise)
- Independent Variable 3: Gender (2 levels – Male vs Female)
- Dependent Variable: Anxiety symptoms
In this example, there is a 3 way interaction between drug, exercise, and gender on anxiety symptoms. Specifically:
- For males, Drug A reduces anxiety more than placebo when they exercise, but less than placebo when no exercise.
- For females, Drug A reduces anxiety more than placebo regardless of exercise.
So the effect of Drug A vs placebo depends on the level of Exercise, but this interaction itself depends on the level of Gender. This demonstrates a 3 way interaction between the variables.
Visualizing 3 way interactions
3 way interactions are best visualized using a graph. Here is an example graph of the hypothetical 3 way interaction described above:
| Male | Female | |
|---|---|---|
| Exercise | ||
| No Exercise |
This shows how the effect of Drug A vs Placebo on anxiety differs depending on the combination of exercise and gender. The graph makes the 3 way interaction clear.
Interpreting 3 way interactions
To interpret a 3 way interaction, examine the effect of the primary independent variables at each level of the third variable. In this example:
- For males, the effect of exercise differs depending on drug – exercise reduces anxiety for Drug A but increases it for placebo.
- For females, exercise has the same effect regardless of drug – it reduces anxiety for both Drug A and placebo.
So the effect of exercise depends on drug, but differently for males and females. This demonstrates the complex interplay of the 3 variables.
Testing for 3 way interactions
In an analysis of variance (ANOVA), the main effects, 2-way interactions, and 3-way interactions can be tested. For example, in a 2x2x2 ANOVA:
- Main effects of A, B, and C are tested
- 2-way interactions of A*B, A*C, and B*C are tested
- The 3-way interaction of A*B*C is tested
The 3-way interaction term directly tests if there is a significant 3 way interaction. This is done by comparing a model with just main effects and 2-way interactions, to a model that also includes the 3-way interaction term. If the 3-way model is a significantly better fit, there is evidence of a 3 way interaction.
3 way interactions in linear regression
In multiple linear regression, you can also test for 3 way interactions by adding an interaction term between 3 predictors. For example, with predictors X1, X2, and X3, you would add the term:
X1*X2*X3
This directly tests if the interaction between X1 and X2 depends on the level of X3. The interaction term would be assessed for statistical significance, controlling for all lower order effects.
Implications of 3 way interactions
3 way interactions demonstrate complex relationships between variables. Some implications are:
- The effect of one independent variable can reverse depending on the levels of the other two variables.
- Interpreting any one variable’s effect is contingent on the values of the other two.
- Simple interpretations of main effects are inadequate in the presence of 3-way interactions.
- Predicting outcomes requires knowing levels of all 3 interacting variables.
Therefore, 3 way interactions imply that each variable’s influence is interdependent on the others. This can make results more nuanced and predictions more complex.
Follow-up analyses
When a significant 3-way interaction is found, follow-up tests are required to fully probe the interaction. Useful follow-ups include:
- Testing simple effects of each variable at specific levels of the other two.
- Graphing data split out by the different variable combinations.
- Conducting separate analyses for each level of the third variable.
- Exploring interaction effects in different subgroups.
These help unpack the interaction and clarify the relationship dynamics. The end goal is a more precise understanding of how the variables interrelate in the 3-way interaction.
Examples of 3 way interactions
Here are some real-world examples of 3 way interactions found in various fields of research:
Psychology
A study on rewards found a 3-way interaction between reward timing, reward type, and impulsivity on motivation. Immediate rewards increased motivation more than delayed for impulsive people, but only when rewards were consumables rather than money.
Medicine
A clinical trial found a 3-way interaction between two chemotherapy drugs and cancer stage on tumor response. The synergistic effect of the drug combination depended on cancer stage.
Education
Research on tutoring showed an interaction between instruction method, tutor expertise, and student knowledge. Novice tutors best helped lower knowledge students with problem-based learning, while expert tutors used direct instruction more effectively.
Marketing
An ad effectiveness study revealed an interaction between ad appeal type, product type, and consumer lifestyle. Emotional ads boosted sales more for indulgence products with impulsive lifestyles, while functional ads worked better on necessities and prudent lifestyles.
Limitations of 3 way interactions
While 3 way interactions can provide insight, they also have limitations:
- They require larger sample sizes to have sufficient power to detect real effects.
- They can be difficult to interpret and communicate clearly.
- Spurious interactions can occur through chance with multiple testing.
- Interactions may only apply under certain conditions or contexts.
- Understanding interactions requires further probing and follow-up.
Thus, caution must be taken when claiming an interaction is meaningful, generalizable, or predictive. Replication in confirmatory studies is important.
Conclusion
In summary, a 3 way interaction refers to when the interaction between two variables depends on the level of a third variable. While complex, it demonstrates the interdependent nature of multiple factors in influencing outcomes. Proper statistical tests can detect 3 way interactions, and follow-up probes are needed to fully understand their implications. When interpreted thoughtfully, 3 way interactions can provide unique insights into the dynamics between variables.