ANOVA (Analysis of Variance)

ANOVA (Analysis of Variance)

1. Definition & overview

• Analysis of variance (ANOVA) is the family of techniques used to test the null hypothesis that the means of two or more populations (groups/treatments) are equal, by partitioning the total variability in the data into identifiable components and comparing them. ([NIST] e-Handbook §7.4.3)
• The core insight that gives ANOVA its name: a hypothesis about means is tested by analysing variances — specifically by comparing the variance between group means against the variance within groups. If the groups truly differ, the between-group variance is inflated relative to the within-group variance. ([NIST] §7.4.3)
• The general technique "can be used to test the hypothesis that the means among two or more groups are equal, under the assumption that the sampled populations are normally distributed." ([NIST] §7.4.2 / §7.4.3)
• ANOVA is the natural extension of the two-sample t-test to ≥3 groups. With exactly 2 groups, one-way ANOVA is algebraically equivalent to the two-sample t-test, and F = t² (the ANOVA F-statistic on 1 numerator degree of freedom equals the square of the corresponding t-statistic). For the two-group continuous-data comparison the usual test is the Student's t-test (Machin Ch.5, p.47–48). ([NIST] §7.4.3; Machin 3e Ch.5 pp.47–8)
• Why not just run many t-tests? Performing all pairwise t-tests across several groups inflates the overall (familywise) Type I error far above the nominal α — the multiple-comparisons problem (§7 below). ANOVA provides a single omnibus test of "are any of the means different?" at a controlled α before any pairwise probing. ([PMID 25984481]; [NIST] §7.4.3)
• ANOVA answers only the omnibus question. A significant F tells you at least one group mean differs from the others, but not which — that requires post-hoc / multiple-comparison procedures (§7). ([NIST] §7.4.3; [PMID 25984481])
• Pharmacology relevance: ANOVA is the default analysis whenever a continuous outcome (blood pressure fall, enzyme concentration, AUC, pain score, receptor density, % inhibition) is compared across ≥3 treatment arms — multi-dose comparisons, dose–response designs, several actives vs placebo, multi-group preclinical experiments, and repeated-measures pharmacodynamic time-course studies (Machin Ch.6 pp.58–64). ([Machin 3e Ch.6 pp.58–64])