Aligned Rank Transform ANOVA

Aligned rank transform ANOVA is a nonparametric factorial analysis that aligns the response for each effect, ranks the aligned values and runs an ANOVA on the ranks to test main effects and interactions.

정렬순위변환 ANOVA는 효과마다 반응값을 정렬한 뒤 순위로 바꾸어 분산분석을 하는 비모수 요인설계 분석으로, 주효과와 상호작용을 검정합니다.

My data are skewed ordinal ratings from a 2 × 3 design; how can I test the interaction without assuming normality?

2×3 설계에서 결과가 치우친 서열 척도인데, 정규성 가정 없이 상호작용을 검정하려면 어떻게 하나요?

For each main effect or interaction, the aligned rank transform first removes every other estimated effect from the response, then ranks the result and fits the full factorial ANOVA on those ranks, reading only the effect it was aligned for. This gives valid nonparametric tests of interactions, which ordinary rank transforms do not.

When to use Aligned Rank Transform ANOVA

• A factorial design has non-normal or ordinal outcomes and you need tests of interactions.
• Transformations do not fix skewness or outliers in a two- or three-factor design.
• Subjects are measured repeatedly within a factorial design (unit column).

When to use something else

• Residuals are close to normal with equal variances; use factorial ANOVA.
• There is a single factor; use the Kruskal-Wallis test.

Assumptions and requirements

• Observations are independent apart from the subject grouping.
• Every combination of factor levels is observed.
• Ranks of the aligned response satisfy approximately the usual ANOVA conditions.

What the analysis reports

• F test, degrees of freedom and p value for each main effect and interaction (Type III, effect coding)
• Partial eta squared on the ranks (between-subjects designs)
• Alignment check: F of other effects on the aligned data
• Wald chi-square p from a random-intercept model on ranks when a subject column is given

How to report it

An aligned rank transform ANOVA showed a significant [A] × [B] interaction, F([df1], [df2]) = [value], p = [value], and a main effect of [A], F([df1], [df2]) = [value], p = [value].

Related methods

• Two-way ANOVA
• Kruskal-Wallis Test
• Friedman Test

Run Aligned Rank Transform ANOVA on Quorant