ANOVA

ANOVA, short for analysis of variance, is a statistical method used to compare means across two or more groups by partitioning total variability into components attributed to different sources. It is applied to determine whether observed differences among group means are likely to have occurred by random chance or reflect real effects.

One-way ANOVA compares means across k groups defined by a single factor. Two-way ANOVA extends this to two factors and can test the main effects of each factor as well as their interaction. Repeated-measures ANOVA applies when the same subjects are measured under multiple conditions or times, introducing correlations among measurements within subjects. MANOVA, a related family, analyzes multiple dependent variables simultaneously.

The core idea is that if group means differ, the variation between groups will be large relative to the variation within groups. The F statistic, defined as F = MS_between / MS_within, summarizes this ratio; a large F leads to a small p-value and potential rejection of the null hypothesis that all group means are equal.

ANOVA relies on several assumptions: independence of observations, normal distribution of residuals within groups, and homogeneity of variances across groups. Violations may affect results; diagnostics include Levene’s test for equal variances and normality checks for residuals, with corrections for sphericity in repeated-measures designs (e.g., Greenhouse-Geisser, Huynh-Feldt).

Results are presented in an ANOVA table with sources (between, within, total), sums of squares, degrees of freedom, mean squares, F-statistic, and p-values. When the null is rejected, post hoc tests (e.g., Tukey’s HSD, Bonferroni) identify which specific group means differ. Effect sizes such as eta-squared or partial eta-squared quantify the magnitude of effects.

Extensions include nonparametric alternatives like Kruskal-Wallis for one-way designs and Friedman for repeated measures. ANOVA is widely used across psychology, agriculture, medicine, and social sciences.