What Is ANOVA? One-Way, Two-Way, and Post Hoc Tests
Daniel Ha · Seoul National University PhD StudentWhat Is ANOVA?
ANOVA uses the ratio of between-group variance to within-group variance (the F statistic) to test whether the means of three or more groups are all equal. It avoids the error inflation that comes from repeating t-tests many times.
When Do You Use ANOVA Instead of a T-Test?
Use it whenever you compare three or more groups. With two groups, a t-test is sufficient.
What Types of ANOVA Are There?
Types differ by the number of independent variables (factors) and the study design.
| Type | Number of factors | Example |
|---|---|---|
| One-way ANOVA | 1 | Score differences across three grade levels |
| Two-way ANOVA | 2 | Grade level x gender, including the interaction |
| Repeated measures ANOVA | Same cases measured repeatedly | Pre, mid, and post score change |
How Do You Run an ANOVA?
Check assumptions, run the F-test, then run post hoc tests.
Step 1: Check Assumptions
Confirm normality and equality of variances (Levene's test). If the equal-variance assumption fails, use Welch's ANOVA.
Step 2: Run the F-Test
Judge whether group means differ using the F statistic and its p-value. A significant result means at least one pair of means differs.
Step 3: Run Post Hoc Tests
Identify which groups differ by running post hoc comparisons, and report an effect size such as eta squared (η²) as well.
Why Are Post Hoc Tests Necessary?
Because the F-test only tells you that a difference exists somewhere, not which groups differ. Tukey HSD, Scheffé, or Bonferroni comparisons pinpoint the specific pairs.
Summary
ANOVA tests mean differences across three or more groups with an F-test, and when significant, post hoc tests reveal the specific differences. For two groups use a t-test, and to estimate effects on a continuous outcome use regression analysis.