Eta Squared

η2 is a commonly used effect size estimate. It describes the proportion of the total variability in a data set that is associated with an effect. Its value is zero when there is no effect and 1.0 when the effect accounts for 100% of the total variability. η2 is most often used in association with analysis of variance and can be calculated from the analysis of variance summary table.

η2 = Sum of squares effect Sum of squares total.

[Page 607]η2 can also be calculated from published F ratios, as long as all F ratios in the design are reported. Jacob Cohen provided general guidelines for what constitutes small (η2 = .01), medium (η2 = .06), and large (η2 = .14) effect sizes in many areas of psychological research.

The symbol R2 is sometimes used rather than η2, to conform to the modern convention that Greek letters are reserved for population parameters, but its use can lead to confusion. R2 is more commonly used with multiple regression. Like r2 (or R2), η2 describes the proportion of variability in one variable (the dependent variable) that is related to another variable (the independent variable). Unlike r2, η2 accounts for both nonlinear and linear relationships. η2 is obtainable from many statistical software packages.

With more than one independent variable (factor), partial ${\eta }^2\left({\eta }_p^2\right)$ is often reported rather than η2. ${\eta }_p^2$ does not address the total variability in the data set; it excludes variability associated with factors and interactions other than the one under consideration. It describes the proportion of variability associated with an effect when variability associated with all other effects is excluded from consideration. It can be calculated from the analysis of variance summary table or from the published F ratio:

$$\begin{array}{l}{\eta }_p^2=\text{Sum of squares effect sum of squares effect}+\\ \text{Sum of squares error.}\\ {\eta }_p^2=df\text{effect}\times F\text{effect}df\text{effect}\times F\text{effect}+df\text{error.}\end{array}$$

The value of any effect size statistic is influenced by the design of the research, which can increase or decrease error variability. With complex factorial designs, η2 and ${\eta }_p^2$ must be interpreted with care because each factor and interaction account for some of the variability present, increasing the value of ${\eta }_p^2$ and decreasing the value of η2. Thus, η2 and ${\eta }_p^2$ may not be comparable across studies. Generalized ${\eta }^2\left({\eta }_G^2\right)$ is a similar statistic, designed to facilitate comparisons across designs.

Based on the sample, η2 and ${\eta }_p^2$ provide a point estimate of the population parameter. To identify likely values for the population effect size, it is essential to know the confidence interval. Unfortunately, these confidence intervals are not centered on the statistic, making calculation difficult. There are, however, downloadable utilities for determining these confidence intervals. Daniel Lakens’s blog, The 20% Statistician, is a good resource.

The η2 family of effect size statistics are optimistically biased; that is, they overestimate the population effect size. This overestimate is arguably no more than that found with conventional correlations, but more realistic estimates can be obtained using the corresponding ω2 statistics. Unfortunately, ω2 cannot be accurately calculated for studies involving repeated measures.

See also Analysis of Variance; Effect Size; Interaction; Multiple Linear Regression