One-Way ANOVA
Foundations of Statistics
Comparing Multiple Group Means Simultaneously
One-way ANOVA tests whether three or more group means differ significantly, controlling the family-wise error rate. The F-statistic compares between-group variance to within-group variance, with post-hoc tests identifying which specific groups differ.
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Pharmaceutical Research — Compare efficacy across multiple drug dosages
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Agriculture — Test crop yield differences across fertilizer types
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Education — Evaluate teaching effectiveness across multiple methods
The F-test asks whether any group stands out — post-hoc tests reveal which ones.
Motivation
When comparing group means, conducting all pairwise -tests inflates the Type I error rate. With groups, there are pairwise comparisons. If each is tested at , the family-wise error rate can exceed — far above the nominal 5%.
ANOVA (Analysis of Variance) tests all group means simultaneously with a single test, controlling the overall Type I error at .
The Statistical Model
Hypotheses
states that all group means are equal (no treatment effect). is the composite alternative that at least one group mean differs from the others.
The ANOVA Decomposition
The F-Statistic
The intuition: if is true, MSB and MSW both estimate , so . If is false, MSB overestimates (it includes the treatment effect), so is large.
Effect Size: Eta-Squared
| | Interpretation |
|----------|---------------|
| | Negligible effect |
| | Small effect |
| | Medium effect |
| | Large effect |
Assumptions
| Assumption | What It Means | How to Check |
|-----------|---------------|-------------|
| Independence | Observations are independent within and between groups | Study design (randomization) |
| Normality | Within each group, the errors are normally distributed | Shapiro–Wilk test; Q–Q plots |
| Homogeneity of variance | is constant across groups | Levene's test; Bartlett's test |
Post-Hoc Tests
When is rejected, post-hoc tests determine which pairs of groups differ:
| Method | Controls | Best For |
|--------|----------|----------|
| Tukey HSD | Family-wise error rate | All pairwise comparisons |
| Bonferroni | Family-wise error rate | Small number of comparisons |
| Scheffé | Family-wise error rate (conservative) | All possible contrasts |
| Dunnett | Family-wise error rate | Comparisons with a control group |
| Games–Howell | Family-wise error rate | Unequal variances and group sizes |