Understanding R² for Type 3 Sums of Squares ANOVA
When conducting an analysis of variance (ANOVA), you often encounter the term "R²," also known as the coefficient of determination. This value represents the proportion of variance in the dependent variable that is explained by the independent variables in your model. However, in situations where you have a complex experimental design with multiple factors, you may need to consider Type 3 sums of squares for a more accurate interpretation of R².
What is Type 3 Sums of Squares?
Type 3 sums of squares is a method for calculating the effect of each factor in an ANOVA model, taking into account all other factors in the model. It essentially asks: "What is the effect of this factor, controlling for all other factors in the model?" This is particularly important when dealing with interactions between factors or when factors are not orthogonal (independent).
Why R² Matters for Type 3 Sums of Squares
While R² generally provides a measure of the overall fit of your model, its interpretation can be nuanced with Type 3 sums of squares. Here's why:
- Hierarchical Models: In models with hierarchical structures, where factors are nested within others, Type 3 sums of squares often result in a lower R² compared to Type 1 or Type 2 sums of squares. This is because Type 3 considers the effects of all factors simultaneously, potentially reducing the explained variance attributable to any individual factor.
- Interactions: When interactions are present, Type 3 sums of squares might lead to a smaller R² for main effects. This is because the interactions capture a portion of the variance that would otherwise be attributed to the main effects alone.
Interpreting R² with Type 3 Sums of Squares
Despite the potential for a lower R² when using Type 3 sums of squares, it is crucial for understanding the unique contribution of each factor to the overall model. Here's a key point to remember:
- Focus on Individual Effects: Rather than solely relying on the overall R², focus on the partial R² values associated with each factor in your model. These values represent the proportion of variance explained by that specific factor, after accounting for all other factors in the model.
Practical Example
Imagine a study examining the impact of two factors, fertilizer type and irrigation method, on crop yield.
- Scenario 1: No interaction: If there's no interaction between fertilizer type and irrigation method, the R² for the model might be relatively high, indicating a good overall fit.
- Scenario 2: Interaction: If there is an interaction, Type 3 sums of squares might lead to a lower overall R², but the partial R² values for fertilizer and irrigation might be lower compared to a model without interactions. This suggests that the individual effects of each factor are diminished when the interaction is taken into account.
Conclusion
When you use Type 3 sums of squares in ANOVA, remember that the overall R² may not be the best indicator of your model's fit. Instead, focus on the partial R² values for each factor to understand their unique contribution to the explained variance. This provides a more accurate assessment of the effects of each factor, especially in complex models with interactions and hierarchical structures.