ANOVA (Analysis of Variance) › Effect Size: Eta-Squared

Michael BrenndoerferJanuary 5, 20261 min read
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Part 2 of 9Hypothesis Testing
Chapter 17ANOVA (Analysis of Variance) › Effect Size: Eta-Squared
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Statistical significance tells us whether an effect exists, but not how large it is. Effect size measures quantify practical significance.

Eta-Squared (η2\eta^2)

The most common effect size for ANOVA is eta-squared:

η2=SSBetweenSSTotal\eta^2 = \frac{SS_{Between}}{SS_{Total}}

This represents the proportion of total variance explained by group membership. It ranges from 0 to 1:

η2\eta^2Interpretation
0.01Small effect
0.06Medium effect
0.14+Large effect

In our fertilizer example, η2=0.72\eta^2 = 0.72 is a very large effect: fertilizer type explains 72% of the variation in plant growth.

Omega-Squared (ω2\omega^2)

Eta-squared tends to overestimate the population effect size. Omega-squared provides a less biased estimate:

ω2=SSBetween−(k−1)×MSWithinSSTotal+MSWithin\omega^2 = \frac{SS_{Between} - (k-1) \times MS_{Within}}{SS_{Total} + MS_{Within}}
In[9]:
Code
# Calculate both effect sizes
eta_sq = ss_between / ss_total
omega_sq = (ss_between - (k - 1) * ms_within) / (ss_total + ms_within)

print("Effect Size Measures:")
print("-" * 40)
print(f"Eta-squared (eta^2):     {eta_sq:.3f}")
print(f"Omega-squared (omega^2): {omega_sq:.3f}")
print()
print("Both indicate a very large effect.")
print("Omega-squared is generally preferred as it's less biased.")
Out[9]:
Console
Effect Size Measures:
----------------------------------------
Eta-squared (eta^2):     0.752
Omega-squared (omega^2): 0.726

Both indicate a very large effect.
Omega-squared is generally preferred as it's less biased.