Eta-squared and partial eta-squared

About eta-squared

• A measure of relationship; like a correlation coefficient it tells you on a scale 0 to 1 how much of variance in DV can be account for by each IV.
• Analogous to r² and can be thought of as a % on a scale 0-100.
• It is a useful addition to just being told if a relationship or difference is significant.
• Eta-squared reflects the percentage of DV variance explained by the IVs in the sample data.  As an estimate of variance explained in the population it is upwardly biased (i.e., an overestimate).  Thus, omega-squared is a recommended alternative.

How to calculate eta-squared

• Factorial ANOVA:
  Example calculation of eta-squared from factorial ANOVA SPSS output (1 page handout; pdf)
• Mixed ANOVA:  
  Example calculation of eta-squared from SPSS mixed ANOVA output (2 page handout; pdf)

Eta-squared

• h² = SS_between / SS_total
• There will be one h² per effect (i.e. per IV plus for an interaction).
• The h²s will sum to 1.
• Not supplied by SPSS.
• Preferred to h_p²
• % of variance explained by each IV.

Partial eta-squared

• h_p² = SS_between / SS_{total +} SS_error
• Supplied by SPSS.
• Do not sum 1, and therefore can be difficult to interpret.
• Generally not recommended.

Designs

• In 1-way ANOVA, eta-squared and partial eta-squared are the same.
• For more complex designs, the partial eta-squared will generally be larger than eta-squared.
• For a mixed ANOVA, eta-squareds needs to be calculated separately within the context of the within-subject effects ANOVA table and the between-subject effects ANOVA table.  In this situation, the eta-squareds will sum to 1 for within-subjects, and the eta-squared will to 1 for the between-subject effects.  But they cannot all be combined to equal 1.