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ANOVA (three-way)

Three-way ANOVA tests the main effects of three factors together with any subset of their two-way and three-way interactions.

What is ANOVA (three-way)?

Three-way ANOVA partitions variance into up to seven terms: three main effects (A, B, C), three two-way interactions (A:B, A:C, B:C), and one three-way (A:B:C). Each can be toggled on or off — start with all interactions on, then drop the highest-order term if it's non-significant and you have substantive reason to.

The three-way interaction is rarely significant in real data; when it is, the two-way interactions become hard to interpret in isolation. Most papers report up to two-way only unless theory demands the three-way.

When should I use ANOVA (three-way)?

Three crossed factors with a continuous outcome.
Factorial experimental designs where every cell has observations.

What data does it need?

Numeric response + three categorical factors + per-term interaction toggles.

What does it report?

ANOVA table with F / df / p / η² / partial η² / ω² for every requested term.

What does it assume?

Independent observations.
Approximately normal residuals.
Equal variance across cells.
Balanced design (or accept Type-I-SS depending on order).

How do I interpret the result?

Always plot cell means and look for non-parallel patterns before interpreting interaction p-values — the visual story matters more than the F.