Yes, an interaction effect can be statistically significant even when the main effects of the individual factors are not.
This is a crucial concept in statistical analysis, particularly in factorial designs. It occurs when the effect of one independent variable on a dependent variable is not consistent across all levels of another independent variable. The most compelling and illustrative example of this is a "crossover" or "disordinal" interaction, where the direction of the effect for one factor reverses depending on the level of the other factor.
Deconstructing main effects and interactions
To understand how this is possible, it is important to clearly distinguish between a main effect and an interaction effect.
- Main effect: A main effect is the overall effect of a single independent variable on the dependent variable, averaging across the levels of all other independent variables in the study. It answers the question, "What is the average effect of Factor A, regardless of Factor B?".
- Interaction effect: An interaction effect occurs when the effect of one independent variable changes depending on the level of another independent variable. It reveals that the simple, main effects do not tell the whole story.
The results of a statistical test, such as an Analysis of Variance (ANOVA), are based on these distinctions. A non-significant main effect merely means that the average effect of that variable across all conditions is not statistically different from zero. However, this average can mask a very real and significant interaction.
The crossover interaction: A definitive example
The clearest demonstration of a significant interaction with non-significant main effects is a crossover or disordinal interaction.
Hypothetical Study: Treatment and Age on RecoveryImagine a clinical trial studying the effectiveness of a new drug compared to a placebo on a patient's recovery score. The researchers also want to see if the drug's effect differs for young adults versus older adults.
- Independent Variable 1: Treatment (Drug vs. Placebo)
- Independent Variable 2: Age Group (Young vs. Old)
- Dependent Variable: Recovery Score (higher scores = better recovery)
The Interaction EffectThe results of the study show a significant interaction effect between treatment and age group. A post-hoc analysis reveals the following:
- For young adults, the drug leads to a significantly higher recovery score than the placebo.
- For older adults, the drug leads to a significantly lower recovery score than the placebo.
This reversal in the direction of the treatment effect across age groups is the hallmark of a crossover interaction.
The Non-Significant Main EffectsNow, let's consider the main effects in this same scenario:
- Main Effect of Treatment: To calculate this, the average recovery score for the "Drug" group is compared to the average score for the "Placebo" group, averaging across the age groups. Since the drug is very beneficial for young adults and very harmful for older adults, the positive and negative effects may cancel each other out. The overall average effect of the drug across all participants could be close to zero, leading to a non-significant main effect for treatment.
- Main Effect of Age Group: Similarly, to determine the main effect of age, the average recovery score of all young adults (drug + placebo) is compared to the average score of all older adults (drug + placebo). If the placebo effects are similar across age groups and the drug effects cancel out, the overall average scores for young and old adults may be statistically indistinguishable, leading to a non-significant main effect for age.
This example illustrates that while neither variable shows a consistent effect on its own, their combined effect is highly significant and meaningful.
Implications for interpretation and research
When an interaction is significant, and especially when main effects are not, it has profound implications for how the results are interpreted and reported.
The conditional nature of effects
A significant interaction effect fundamentally changes the interpretation of any main effects involved. If an interaction is significant, you should not interpret the main effects in isolation, as they can be misleading. Instead, the interaction reveals that the effect of one variable is conditional upon the level of the other. The focus of the interpretation must shift to the combined effect of the variables.
The importance of visualization
Plotting the data becomes essential for understanding a significant interaction. A line plot, with one independent variable on the x-axis and separate lines for each level of the other independent variable, can visually illustrate the interaction. For a crossover interaction, the lines will physically cross, clearly showing the reversal of effect.
The concept of simple effects
To fully understand a significant interaction, researchers often conduct follow-up analyses of "simple effects." This involves looking at the effect of one variable at each specific level of the other variable. In the clinical trial example, a simple effects analysis would:
- Test the effect of the treatment only within the young adult group.
- Test the effect of the treatment only within the older adult group.
These tests would reveal the specific nature of the interaction and pinpoint where the effects are statistically different from zero.
Other scenarios and nuances
While crossover interactions are the clearest example, there are other reasons for a significant interaction without significant main effects:
- Marginal vs. Conditional Effects: A main effect is a marginal average. It is possible for the conditional effects (the effects within each level of the other variable) to be strong and significant, yet average out to a non-significant marginal effect.
- Lack of Power for Main Effects: In some cases, the study might have enough power to detect the interaction effect but lack sufficient power to detect the average main effects, especially if they are small or masked by the stronger interaction.
- Qualitative vs. Quantitative Interactions: A crossover interaction is a "qualitative" interaction because the nature of the effect (positive vs. negative) changes. In contrast, a "quantitative" interaction occurs when the direction of the effect is consistent, but its magnitude changes. A significant quantitative interaction might also occur with non-significant main effects, especially if the average effect size is small.
Conclusion
The presence of a significant interaction effect alongside non-significant main effects is not contradictory but rather a rich and meaningful statistical finding. It highlights the complex interplay between variables and provides a more nuanced understanding of the phenomenon under study. It serves as a powerful reminder that averaged effects can be misleading, and the true story lies in the conditional relationship between factors. When confronted with this result, the proper analytical path is to focus on understanding and interpreting the interaction, rather than dismissing the non-significant main effects as evidence of no effect.