When should I use a Two-Way ANOVA?

Use a two-way ANOVA when you have two independent categorical factors and one continuous dependent variable. It tests main effects of each factor and their interaction. Example: testing how teaching method (lecture vs discussion) and gender (male vs female) affect exam scores.

What is an interaction effect?

An interaction effect occurs when the effect of one factor depends on the level of the other factor. For example, a drug might work well for males but not females - this means drug and gender interact. Interaction is tested by the F-statistic for the A x B term.

Should I interpret main effects when the interaction is significant?

When the interaction is significant, main effects may be misleading because the effect of one factor changes across levels of the other. In this case, examine simple main effects instead. If the interaction is not significant, main effects can be interpreted independently.

What is partial eta-squared?

Partial eta-squared (eta-squared-p) is an effect size measure calculated as SS_effect / (SS_effect + SS_error). Unlike eta-squared, it excludes variance from other effects, making it appropriate for multi-factor designs. Benchmarks: 0.01 = small, 0.06 = medium, 0.14+ = large.

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連続的な結果変数に対する2つの独立変数の主効果と交互作用を検討します。結果にはF統計量、偏η²、交互作用プロットが含まれます。

What is Two-Way ANOVA (Factorial ANOVA)?

Two-way ANOVA, also known as factorial ANOVA, is a statistical method that examines the simultaneous effects of two independent categorical variables (factors) on a continuous dependent variable. Unlike one-way ANOVA, which tests a single factor, two-way ANOVA tests three distinct hypotheses: the main effect of Factor A, the main effect of Factor B, and the interaction between Factor A and Factor B. This makes it one of the most widely used analytical tools in experimental research across psychology, medicine, education, and the social sciences.

When to Use Two-Way ANOVA

Use a two-way ANOVA when your study design includes two independent categorical factors, each with two or more levels, and a single continuous dependent variable measured on an interval or ratio scale. Common scenarios include experimental designs examining the combined effects of treatment type and demographic group, dose and delivery method, or any two grouping variables measured simultaneously.

Two-Way ANOVA vs. One-Way ANOVA

Feature	One-Way ANOVA	Two-Way ANOVA
Factors	1	2
Tests	1 main effect	2 main effects + 1 interaction
Interaction	Not applicable	Tested
Effect size	η²	Partial η²
Design complexity	Simple	Factorial (A × B)

Worked Example: 2 × 2 Factorial Design

A researcher tests the effects of study method (Method A vs. Method B) and test difficulty (Easy vs. Hard) on exam scores. Five students are randomly assigned to each of the four cells.

Results

Study method: F(1, 16) = 52.27, p < .001, η²_p = .77

Difficulty: F(1, 16) = 36.82, p < .001, η²_p = .70

Interaction: F(1, 16) = 0.33, p = .576, η²_p = .02

Both main effects are significant, but the interaction is not, meaning the advantage of Method A over Method B is consistent across difficulty levels.

Assumptions of Two-Way ANOVA

Before interpreting your results, verify these four assumptions:

1. Normality

The dependent variable should be approximately normally distributed within each cell of the design. Assess with Shapiro-Wilk tests or Q-Q plots. ANOVA is robust to moderate violations when cell sizes are equal and reasonably large.

2. Homogeneity of Variance

Variances should be approximately equal across all cells. Use Levene's test to check. When group sizes are unequal and variances differ, results may be unreliable.

3. Independence of Observations

Each observation must be independent. Random assignment to cells ensures independence. If observations are nested or repeated, use mixed-effects models instead.

4. Interval or Ratio Data

The dependent variable must be continuous (interval or ratio scale). For ordinal or categorical outcomes, consider non-parametric alternatives such as the Aligned Rank Transform.

Interpreting the Interaction Effect

The interaction is arguably the most informative part of a two-way ANOVA. A significant interaction means the effect of one factor depends on the level of the other factor. When the interaction is significant, main effects should be interpreted cautiously because average differences across one factor may mask opposite patterns at different levels of the other factor. In such cases, report simple main effects (the effect of Factor A at each level of Factor B, and vice versa) rather than overall main effects.

How to Report Two-Way ANOVA in APA Format

Report each effect (Factor A, Factor B, and the interaction) separately, including F-statistic, degrees of freedom, p-value, and partial eta-squared:

Example Report

A 2 × 2 between-subjects ANOVA was conducted. There was a significant main effect of study method, F(1, 16) = 52.27, p < .001, η²_p = .77. The main effect of difficulty was also significant, F(1, 16) = 36.82, p < .001, η²_p = .70. The interaction between study method and difficulty was not significant, F(1, 16) = 0.33, p = .576, η²_p = .02.

Note: Always report partial η² (not regular η²) for factorial designs. Italicize F, p, and η². Report degrees of freedom for both the effect and the residual.

Common Mistakes to Avoid

Ignoring a significant interaction: If the interaction is significant, interpreting main effects in isolation can be misleading. Always check the interaction first.
Running separate one-way ANOVAs: Analyzing each factor separately misses the interaction effect and wastes statistical power. Use two-way ANOVA instead.
Unbalanced designs without adjustment: When cell sizes are very unequal, standard Type I sums of squares can give misleading results. Consider Type III sums of squares or ensure balanced designs when possible.
Confusing eta-squared with partial eta-squared: In factorial designs, always report partial η², which removes variance due to other effects from the denominator.
Too many levels without sufficient sample size: A 4 × 4 design has 16 cells. With only 3 observations per cell, the error degrees of freedom are very low, reducing statistical power.

Calculation Accuracy

StatMate's two-way ANOVA calculations have been validated against R's aov() and SPSS GLM output. The implementation uses balanced-formula sums of squares with the jstat library for the F-distribution. All F-statistics, p-values, and partial eta-squared values match R and SPSS output. Degrees of freedom use standard formulas: df_A = a − 1, df_B = b − 1, df_AB = (a − 1)(b − 1), df_error = N − ab.

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