Two-Way ANOVA calculator

Compare means across two independent factors. Calculate main effects and interaction effects online with a detailed ANOVA table.

What it does

Tests two categorical factors at once and, crucially, whether they interact — whether the effect of one factor depends on the level of the other.

When to use it

A factorial design: two grouping variables crossed with each other, and a continuous outcome.

Worked example

You test a drug against placebo in both men and women. Two-way ANOVA tells you whether the drug works, whether outcomes differ by sex, and — the reason to use it — whether the drug works differently in men than in women, which neither one-way test could show.

Cautions

• Read the interaction first. If it is significant, the main effects describe averages over conditions where the effect genuinely differs, and quoting them alone is misleading.
• Badly unbalanced cells make the sums of squares depend on the order terms enter the model — interpret with care, and be explicit about which type of sums of squares you report.
• Empty cells (a combination nobody is in) mean that part of the interaction is not estimable at all.

Alternatives

• One-way ANOVA — you only have one factor, or the second is a nuisance variable you would rather adjust for
• ANCOVA — the second variable is continuous rather than categorical

How to read the output

The interaction row — read this first

Whether the effect of one factor changes across levels of the other. This row governs how the rest of the table may be described. If the interaction is significant, the two main-effect rows are averages over a difference that is not constant, and quoting them as 'the effect of treatment' misdescribes the data. Report the simple effects — the effect of one factor within each level of the other — instead.

Main effect rows (SS, df, F, p)

Each factor's effect after adjusting for the other factor but not for the interaction — Type II sums of squares. Which factor you entered first does not change them, unlike the sequential (Type I) decomposition some packages default to. Type II is the most powerful choice when the interaction is negligible and the wrong summary when it is not — which is why the interaction row is read first. It also means these numbers can differ from a Type III table in another package on the same unbalanced data; state which you used.

Partial η²

The share of variance attributable to that effect once the other effects are removed — the effect size to report alongside each F. Partial η² values do not sum to 1 across rows, by construction. Comparing them across studies with different designs is not meaningful.

p-value

How surprising a difference at least this large would be IF the groups really were identical on this measure. A small p means the data would be surprising under that assumption. It is not the probability that the result is due to chance, and not the probability that the groups are the same — those read the conditional backwards. It also answers 'would this be surprising?', never 'is this big enough to matter?': with a large sample a trivial difference returns a small p, and the result looks identical to an important one.

How this calculator is validated · Which statistical test should I use?