Eta Squared (η²): Effect Size in ANOVA

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?”

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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

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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.

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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

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Step 1: Compute Total Sum of Squares

$SS_{\text{total}} = 15.8 + 505.6 + 89.2 = 610.6$

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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

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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.

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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.

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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