Two-Way ANOVA: Understanding and interpreting the coefficients
One of the most powerful statistical methods for examining the interaction effects between two factors on a dependent variable is Two-Way ANOVA (two-factorial analysis of variance). This method is often used in experiments to test how different groups (e.g. treatment and control groups) respond to a dependent variable under different conditions (e.g. different time points or environments).
What is Two-Way ANOVA?
Two-way ANOVA allows us to analyze the effects of two independent variables (factors) on a dependent variable simultaneously. A major advantage of this method is that it analyzes both the main effects of each factor and the interaction effects between the factors.
- Main effect: The influence of one factor on the dependent variable, independent of the other factor.
- Interaction effect: How the combination of factors together influences the dependent variable. It indicates whether the influence of one UV on the AV is dependent on the other UV. In other words, it indicates whether the influence of the diet on weight is dependent on the training plan or vice versa, or whether the influence of the training plan on weight is dependent on the diet.
Example of a two-way ANOVA
Suppose we want to investigate how two different diets (diet A and diet B) and two different training plans (plan 1 and plan 2) affect the weight of participants. The dependent variable here is weight.
A possible example output of the Two-Way ANOVA could look as follows:
| Source of Variation | Sum of Squares | Degrees of Freedom (DF) | Mean Square | F-Value | P-Value |
|---|---|---|---|---|---|
| Diet | 40.32 | 1 | 40.32 | 5.67 | 0.031 |
| Training plan | 55.21 | 1 | 55.21 | 7.76 | 0.015 |
| Diet * Training plan | 25.68 | 1 | 25.68 | 3.61 | 0.065 |
| Error | 287.00 | 40 | 7.18 | ||
| Total | 408.21 | 43 |
Interpretation of the coefficients
- Main effect of diet: The F-value of 5.67 and the p-value of 0.031 indicate that diet has a significant effect on participants' weight, with a p-value of less than 0.05 indicating statistical significance.
- Main effect of the training plan: The training plan also shows a significant effect (F = 7.76, p = 0.015), indicating that the training plan influences weight independently of the diet.
- Interaction effect (diet * training plan): The interaction effect has an F-value of 3.61 and a p-value of 0.065. Although the p-value here is just over 0.05, which is close to the significance threshold, this suggests that there may be an interaction between diet and exercise plan that affects weight. In a different context or at a lower significance threshold, this could be considered significant.
The plot illustrates that there is a difference in weight between the two training plans (different colors). You can also see that the weight is lower with diet B than with diet A. However, we can see that the difference between training plan 1 and training plan 2 is not very dependent on the diet, as the two lines in the plot run almost parallel to each other. The two significant main effects and the non-significant interaction effect in the ANOVA output confirm this.
Conclusion
The interpretation of the coefficients in a two-way ANOVA requires an understanding of the main effects and possible interactions between the factors. The p-values help to determine whether these effects are statistically significant. When applied to real data, it is crucial not only to assess statistical significance, but also to consider the practical significance of the effects.
Two-Way ANOVA is a valuable tool when you want to examine multiple independent variables in an experiment and understand how they together influence the dependent variable. This allows for deeper analysis and the identification of interactions that might be overlooked in simple ANOVA models.
