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One-way ANOVA

Compare three or more groups, with variance explained and every pair tested.

Paste at least two values into each of the three groups.

FormulaF = MS between ÷ MS within; ω² = (SS_b − df₁·MS_w) ÷ (SS_total + MS_w)
ModelOne-way ANOVA, Welch pairwise with Holm correction

If each treatment is being compared only against the control rather than against every other group, Dunnett’s test is more powerful than the all-pairs correction used here.

When to use this

Use this to compare three or more groups at once, and to follow up with pairwise comparisons that are corrected for the number of tests. Reach for it instead of running several t-tests, which inflates the error rate.

Worked example

Four treatment groups, five replicates each.

Groups
4
Replicates
5 per group

Result

F statistic with its p-value, and Holm-corrected pairwise comparisons

A significant F says the groups are not all alike; the pairwise table says which differ.

What people get wrong

  • Running every pairwise t-test instead. Six comparisons at α = 0.05 give roughly a one in four chance of a false positive somewhere.
  • Stopping at the F statistic. It tells you something differs, not what, which is why the pairwise comparisons are there.
  • Treating a non-significant F as evidence the groups are the same. It is evidence of nothing in particular, especially at small n.

Questions

+Why Holm rather than Bonferroni?

Holm controls the same family-wise error rate and is uniformly more powerful, so there is no reason to prefer plain Bonferroni.

+Do I need equal group sizes?

No, though unequal sizes make the test more sensitive to unequal variances.

+What if the assumptions do not hold?

Kruskal–Wallis is the rank-based alternative. As with Mann–Whitney, you trade the ability to state an effect size and interval for fewer assumptions.

Related tools

Science last reviewed .