Variance Explained: Eta-squared and R-squared
How much variability the model explains
Variance-explained effect sizes express, as a proportion between 0 and 1, how much of the total variability is accounted for by the model. In ANOVA designs, eta-squared and partial eta-squared serve this purpose; omega-squared is a less biased alternative. In regression, R-squared and adjusted R-squared play the same role. A significant p-value only confirms that an effect exists; these measures reveal how large that effect is in practical terms.
The Concept and Its Formula Logic
The variance-explained ratio is fundamentally computed as SS_effect / SS_total, where SS stands for sum of squares. In ANOVA, eta-squared (η²) divides the sum of squares for a factor by the total sum of squares. Partial eta-squared narrows the denominator to only the factor of interest plus its error term, excluding other factors. In regression, R-squared is the ratio of the regression sum of squares to the total sum of squares, indicating how well the model predicts variability in the dependent variable.
Computing and Reporting
Most statistical software produces these values automatically, but the specific measure reported must be clearly identified. Partial eta-squared is the default in SPSS and JASP outputs and should not be confused with eta-squared. Omega-squared (ω²) corrects for sample size bias and is preferred, especially with small samples. In regression, adjusted R-squared should be used in models with multiple predictors; it penalizes the inclusion of unnecessary predictors and provides a more realistic estimate of explained variance.
Common Misuses and Misconceptions
The most common error is reporting partial eta-squared as eta-squared; in factorial designs, partial eta-squared values can sum above 1. Another misconception is treating a high R-squared as proof of causation; it reflects association only. R-squared never decreases when predictors are added, so the unadjusted value can be misleading in model comparisons. Also, small effect sizes can be statistically significant while large effect sizes may not be; p-values and effect sizes convey independent pieces of information.
Why It Matters and How to Interpret It
Variance-explained measures are the primary tool for assessing practical importance beyond statistical significance. According to Cohen's general guidelines, values of 0.01, 0.06, and 0.14 for η² or R-squared represent small, medium, and large effects respectively; however, these thresholds should be interpreted alongside field-specific expectations. A researcher should always report the effect size, identify which measure was used, and include a confidence interval. This information contributes to meta-analyses and enhances the reproducibility of findings.