Two-way ANOVA tests the main effects of two factors and, optionally, the interaction between them.
Two-way ANOVA partitions variance into three (or four with interaction) sources: factor A, factor B, A × B interaction, and residual. The main effects test whether the marginal means differ across each factor's levels; the interaction tests whether A's effect depends on B (equivalently, whether the cell pattern departs from a strictly additive model).
Interactions are usually the most interesting term. A significant A × B means you can't talk about A's effect in isolation — it differs across B's levels — and you should focus on simple effects (A within each level of B) rather than the marginal A effect.
For unbalanced designs (cells with different n) the result depends on the SS type. Our implementation uses Type I (sequential) SS by default; Type II / III are common in social-science software. With balanced designs all three types give the same answer.
Numeric response + two categorical factors + interaction toggle.
F, df, p, η², partial η², ω² per term (A, B, A × B, residual).
Always plot the cell means before interpreting. Parallel lines ⇒ no interaction; crossing or fanning lines ⇒ interaction.
Partial η² ignores the variance from other terms in the model, so it's better than plain η² when you want each term's local effect size.