Eta Squared

Interpreting and Calculating Eta Squared

Eta squared measures the strength of a relationship between two variables and is in the same family of effects sizes as the Pearson product moment coefficient (r). This family is used in statistical tests that measure association between variables. Eta squared is analogous to the coefficient of determination (r2): both represent the proportion of variance in a single continuous (interval or ratio) dependent variable that can be explained by a single independent variable, except η2 is used when the independent variable is categorical (nominal or ordinal) and r2 is used when the independent variable is also continuous.

Eta squared can be interpreted using the same guidelines Jacob Cohen provides for interpreting r as an index of effect size, except given that η2 is analogous to r2, Cohen’s guidelines must be squared. Based on his guidelines for the behavioral sciences, η2 = .01 (1% of the variance in the dependent variable can be explained by the variance in the attributes of the dependent variable) would correspond to a small effect size; η2 = .09 (9% of the variance in the dependent variable can be explained by the variance in the attributes of the dependent variable) would correspond to a medium effect size; and η2 = .25 (25% of the variance in the dependent variable can be explained by the variance in the attributes of the dependent variable) would correspond to a large effect size.

[Page 447]In an ANOVA, η2 can be calculated by dividing the between-group sum of squares by the total sum of squares. The total sum of squares can be calculated with the formula $\mathrm{\Sigma }{({\mathrm{X}}_i-m_{\mathrm{grand}})}^2$, where Ʃ represents the sum over all n observations of the dependent variable (x1 through xn) and mgrand is the grand mean of all n observations. In an ANOVA with k groups of the independent variable, the between-group sum of squares can be calculated with the formula $\mathrm{\Sigma }{n}_j{(m_j-m_{\mathrm{grand}})}^2$, where Ʃ represents the sum over all k groups, where nj represents the sample size for the jth group (n1 through nk) and mj is the mean of the observations of the dependent variable in the jth group (m1 through mk).

...