Eta squared (η²) is a measure of effect size commonly used in analysis of variance (ANOVA).
It quantifies how much of the total variability in the response variable is attributable to a particular factor or interaction in the model.
While a p-value answers “Is there evidence of an effect?”, eta squared answers “How large is that effect?”
Definition
Eta squared measures the proportion of total variance explained by a specific effect:
$\eta^2 = \dfrac{SS_{\text{effect}}}{SS_{\text{total}}}$
where:
- $SS_{\text{effect}}$ is the sum of squares associated with a specific factor or interaction
- $SS_{\text{total}}$ is the total sum of squares in the ANOVA model
Interpretation
The value of $\eta^2$ lies in the interval $0 \le \eta^2 \le 1$:
- $\eta^2 = 0$ means the factor explains none of the variance
- $\eta^2 = 1$ means the factor explains all of the variance
A commonly used rule of thumb is:
- $\eta^2 \approx 0.01$ → Small effect
- $\eta^2 \approx 0.06$ → Medium effect
- $\eta^2 \ge 0.14$ → Large effect
These thresholds are context-dependent, but they provide a practical scale for interpretation.
Example: Two-Way ANOVA (Exercise × Gender)
Suppose we study whether exercise intensity and gender affect weight loss.
Participants are assigned to one of three exercise levels (none, light, intense), and both factors are included in a two-way ANOVA.
The ANOVA summary is:
| Source | Df | Sum Sq | Mean Sq | F value | p value |
|---|---|---|---|---|---|
| gender | 1 | 15.8 | 15.80 | 9.916 | 0.00263 |
| exercise | 2 | 505.6 | 252.78 | 158.61 | < 2e−16 |
| residuals | 56 | 89.2 | 1.59 |
Step 1: Compute Total Sum of Squares
$SS_{\text{total}} = 15.8 + 505.6 + 89.2 = 610.6$
Step 2: Compute Eta Squared for Each Effect
Gender:
$\eta^2_{\text{gender}} = \dfrac{15.8}{610.6} \approx 0.026$
→ Small effect size
Exercise:
$\eta^2_{\text{exercise}} = \dfrac{505.6}{610.6} \approx 0.828$
→ Very large effect size
Interpretation of Results
- Exercise intensity explains about 83% of the total variance in weight loss
→ This is an extremely strong effect. - Gender explains about 2.6% of the variance
→ This is a statistically detectable but practically small effect.
Although both factors are statistically significant, their practical importance differs greatly.
Why Eta Squared Matters
- A p-value only tells you whether an effect exists under a null hypothesis.
- Eta squared tells you how important that effect actually is.
This example illustrates a critical statistical insight:
A variable can be statistically significant and yet explain very little variance.
Effect size measures like $\eta^2$ prevent overinterpretation of small but statistically significant effects, especially in large samples.
Key Takeaways
- $\eta^2$ measures proportion of explained variance
- It complements p-values by quantifying strength of association
- Large samples can produce small p-values for trivial effects
- Eta squared helps distinguish statistical significance from practical significance