ANOVA (Analysis of Variance) › Summary

Analysis of Variance (ANOVA) is a powerful technique for comparing means across three or more groups while maintaining control over the Type I error rate.

Core concepts:

• ANOVA compares variances to test means: between-group variation vs. within-group variation
• If group means truly differ, between-group variance will be inflated relative to within-group variance
• The F-statistic is the ratio of these variance estimates

The ANOVA decomposition:

$$SS_{Total} = SS_{Between} + SS_{Within}$$

The F-statistic:

$$F = \frac{MS_{Between}}{MS_{Within}} = \frac{SS_{Between}/(k-1)}{SS_{Within}/(N-k)}$$

Key results:

• ANOVA tests the omnibus hypothesis: are all means equal?
• A significant F-test indicates at least one mean differs, but not which ones
• Post-hoc tests (like Tukey's HSD) identify specific pairwise differences
• Effect sizes ($\eta^2$, $\omega^2$) quantify the magnitude of group differences

Assumptions:

• Independence of observations
• Normality within groups (moderate departures are tolerated)
• Equal variances across groups (Levene's test to check)

When assumptions fail:

• Unequal variances: Welch's ANOVA
• Non-normality: Kruskal-Wallis test