One-Way ANOVA and Post-Hoc Tests: A Guide for Your Seminar Paper
When to use a one-way ANOVA, why not several t-tests, how to choose a post-hoc test (Tukey, Bonferroni, Games-Howell), how to read the SPSS output and how to report it in APA.
Part of the series: Statistical Analysis for a Seminar Paper or Thesis: The Complete Guide
A one-way ANOVA compares the means of three or more groups on one continuous variable - study satisfaction among BA, MA and PhD students, say, or stress in three hospital departments.
Why not just several t-tests?
Three groups mean three pairwise comparisons. Each t-test carries a 5% Type I error risk; together it grows to about 14%, and with five groups (ten comparisons) to about 40%. ANOVA first asks one question - is there any difference between the groups? Only if there is do you look for where, with post-hoc tests that correct for the risk.
Assumptions
- A continuous dependent variable and a categorical independent variable with 3+ levels.
- Each participant in one group only (otherwise it's a repeated-measures ANOVA).
- Roughly normal distribution in each group, or groups that aren't tiny (about 20+ each).
- Equal variances - Levene's test. If significant, report Welch's F instead of the ordinary F and use Games-Howell for comparisons.
With badly violated assumptions in a small sample, the non-parametric alternative is Kruskal-Wallis - see How to choose a statistical test.
In SPSS
Analyze → Compare Means → One-Way ANOVA. Outcome in Dependent List, group in Factor. Under Options tick Descriptive, Homogeneity of variance test and Welch; under Post Hoc choose a test (see below).
Reading the output, step by step
- Descriptives: each group's mean and SD - you'll report them.
- Test of Homogeneity of Variances: Levene p > .05 → use the ordinary F; p < .05 → use the Robust Tests table (Welch).
- ANOVA table: F, the two df (between and within groups) and Sig. If p < .05, some difference exists.
- Post Hoc: only now look at which pair differs.
Which post-hoc test
| Situation | Test |
|---|---|
| Equal variances, similar group sizes | Tukey HSD - the default |
| A few comparisons planned in advance | Bonferroni - conservative |
| Unequal variances (Levene significant) | Games-Howell |
| Very different group sizes | Gabriel or Hochberg's GT2 |
Decide before you look at the results, and say in the method chapter which test and why.
Effect size
Report η² (eta squared): between-groups sum of squares divided by the total. 0.01 small, 0.06 medium, 0.14 large. Recent SPSS versions print it under ANOVA Effect Sizes.
Reporting in APA
"Satisfaction differed significantly between BA (M = 3.21, SD = 0.74), MA (M = 3.58, SD = 0.66) and PhD students (M = 3.89, SD = 0.59), F(2, 147) = 11.42, p < .001, η² = .13. Tukey comparisons showed PhD students were more satisfied than BA students (p < .001), while the MA-PhD difference was not significant (p = .061)."
With Welch: "Welch's F(2, 89.3) = 10.87, p < .001".
Common mistakes
- Going straight to post-hoc tests when F is not significant and hunting for a difference there.
- Using Tukey when Levene is significant instead of Games-Howell.
- Reporting only "a difference was found" without saying between whom and in which direction.
- Confusing it with a two-way ANOVA: two independent variables (e.g. gender and degree) and their interaction is a different analysis.
FAQ
Only two groups? Then it's a t-test. An ANOVA with two groups gives the identical result (F = t²).
Unequal group sizes? Not necessarily a problem. Check equal variances; with large size gaps prefer Welch and Games-Howell.
Where does it fit? Step 4 of 7 - see Statistical analysis for a seminar paper: the complete guide. Need a two-way, repeated-measures or ANCOVA? Tell us about it.
Tell us what you need, and we'll come back with an expert in Statistical analysis and a quote.
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