Three Way ANOVA

Three-way ANOVA is a factorial analysis of variance that tests the main effects of three categorical factors and all their two-way and three-way interactions on a continuous outcome.

삼원분산분석은 세 범주형 요인의 주효과와 모든 이원·삼원 상호작용이 연속형 결과변수에 미치는 영향을 검정하는 요인설계 분산분석이다.

I have three categorical factors; how do I test their effects and interactions on one outcome?

범주형 요인이 세 개인데, 각 효과와 상호작용을 하나의 결과변수에 대해 어떻게 검정하나요?

Three-way ANOVA partitions the outcome variance into three main effects, three two-way interactions and one three-way interaction, each tested against the residual mean square. With unbalanced data, Type II sums of squares test each effect after the other effects that do not contain it.

When to use Three Way ANOVA

• An experiment crosses three categorical factors.
• Interactions among the factors are of substantive interest.
• Every factor combination has at least two observations.

When to use something else

• One or more predictors are continuous; use a linear regression or ANCOVA.
• Factors are measured repeatedly on the same participants; use a mixed ANOVA or linear mixed model.

Assumptions and requirements

• Observations are independent.
• Residuals are approximately normal.
• Variances are equal across cells.
• All factor combinations are observed for the full model; with empty cells the three-way (and if needed two-way) interactions are dropped.

What the analysis reports

• ANOVA table with sums of squares, df, F and p for each effect
• Partial eta squared for each effect
• Residual mean square
• Model R squared

How to report it

A three-way ANOVA showed a significant [factor A] × [factor B] interaction, F([df], [df_error]) = [F], p = [p], partial η² = [eta].

Related methods

• Two way ANOVA
• ANCOVA
• Mixed ANOVA

Run Three Way ANOVA on Quorant