Two-Way Analysis of Variance (Two-Way ANOVA)

A significant interaction means the effect of Factor A differs depending on the level of Factor B (and vice versa), so the main effects alone do not tell the full story. The standard approach is to produce an interaction plot to visualise the crossing or diverging lines, and then conduct simple effects analysis — that is, test the effect of one factor separately at each level of the other factor, applying an appropriate correction for multiple comparisons.

In a balanced design (equal cell sizes) the order does not matter because the factors are orthogonal. In unbalanced designs, however, Type I (sequential) sums of squares depend on the order of entry, which can lead to different conclusions. Always use Type III sums of squares for unbalanced data, as implemented by default in most software when a Type is specified explicitly.

Running two separate one-way ANOVAs inflates the experiment-wise error rate, ignores the interaction between the factors, and uses only the marginal data rather than the full cross-classified structure. Two-Way ANOVA tests both main effects and their interaction simultaneously within a single model, preserving the nominal significance level and providing the additional, often critical, interaction test.

Partial eta-squared (η²p) is the conventional choice for factorial ANOVA. It expresses each effect's sum of squares as a proportion of that effect's SS plus the error SS, making it interpretable on its own: values around 0.01 are considered small, 0.06 medium, and 0.14 large. Omega-squared (ω²) is a less biased estimator and is preferred for small samples or when generalising beyond the specific study.