Short Answer
ANOVA (Analysis of Variance) is used to compare the means of three or more groups in a single test, and it prevents the Type I error inflation that results from running a separate t-test for each pairwise comparison. If there is one independent variable and independent groups, one-way ANOVA is chosen; if there are two factors and their interactions, two-way ANOVA; and if the same individuals are measured multiple times, repeated measures ANOVA. The assumptions of normality and homogeneity of variance (Levene) must be met, and if they are not, you move to alternatives such as Kruskal-Wallis or Games-Howell. The most common mistake is stopping when ANOVA turns out significant; the F test only says "at least one group is different," and it is post-hoc tests such as Tukey and Bonferroni that show which groups differ. The second common mistake is looking only at the p value without reporting the effect size (eta-squared).
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When you want to test the differences in means among more than two groups, running a separate t-test for each pairwise comparison may seem tempting. However, this approach substantially increases the risk of Type I error. ANOVA is designed to solve this problem.
The Basic Logic of ANOVA
ANOVA (Analysis of Variance) tests whether groups come from the same population by comparing the between-group variance with the within-group variance.
Null hypothesis: The means of all groups are equal (μ₁ = μ₂ = μ₃ = ...)
F statistic = Between-group variance / Within-group variance
If the F ratio is greater than 1 and significant, the conclusion is that at least one group differs from the others; but which one differs is determined by post-hoc tests.
Which ANOVA?
One-Way ANOVA
One independent variable, three or more independent groups.
Example: Comparison of blood pressure across three different drug groups.
Two-Way ANOVA
Two independent variables and the interaction between them.
Example: The effect of drug type and sex factors on blood pressure; and the joint effect of these two factors (interaction).
Repeated Measures ANOVA
The same individuals measured multiple times.
Example: Comparison of pain scores before treatment, at month 1, and at month 3.
Mixed ANOVA
Both independent groups and repeated measures together.
ANOVA Assumptions
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Normality: The data in each group should be normally distributed
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Homogeneity of variance: The variances of the groups should be close to each other (Levene test)
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Independence: Observations should be independent of each other
If normality is not met, the non-parametric alternative is the Kruskal-Wallis test.
For repeated measures, the Friedman test.
Post-Hoc Tests
After ANOVA turns out significant, post-hoc tests are performed to find out which groups differ from each other.
Tukey HSD: The most commonly used. It performs all pairwise comparisons and is powerful with balanced samples.
Bonferroni: More conservative. It may be preferred when the number of comparisons is small.
Scheffé: The most conservative, detecting only large differences.
Games-Howell: Used when homogeneity of variance is not met.
Eta-Squared: Effect Size
It is not enough for ANOVA to be significant; the magnitude of the effect should also be reported.
- η² (eta-squared): 0.01 = small, 0.06 = medium, 0.14 = large effect
- Partial η² is more appropriate in multivariate models
High-impact journals do not accept articles without an effect size.
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Where Do People Get Stuck Most in This Analysis?
- ANOVA turned out significant but you do not know which post-hoc test to choose, and each one gives a different result.
- In repeated measures ANOVA, the sphericity assumption is violated, and it is unclear when the Greenhouse-Geisser correction is required.
- You need to report the effect size (eta-squared) but cannot find it in the SPSS output.