Welch's ANOVA

Welch's ANOVA is a one-way analysis of variance that compares three or more group means without assuming equal population variances, weighting each group by its sample size divided by its variance.

Welch 분산분석은 집단 분산이 같다고 가정하지 않고, 표본 크기를 분산으로 나눈 가중치로 셋 이상 집단의 평균 차이를 검정하는 일원분산분석이다.

My groups have very different variances; how can I still test whether their means differ?

집단마다 분산이 많이 다른데, 그래도 평균 차이를 검정하려면 어떻게 해야 하나요?

Welch's ANOVA replaces the pooled error term of the classic F test with variance-weighted group means and an adjusted denominator degrees of freedom. It keeps the Type I error rate close to nominal when variances and group sizes are unequal.

When to use Welch's ANOVA

• Three or more independent groups are compared on a continuous outcome.
• Levene's test or inspection shows unequal group variances.
• Group sizes are unbalanced, especially when smaller groups have larger variances.

When to use something else

• The outcome is ordinal or strongly skewed with outliers; use the Kruskal-Wallis test or a robust ANOVA on trimmed means.
• Observations are repeated on the same participants; use a repeated measures ANOVA or a mixed model.

Assumptions and requirements

• Observations are independent within and between groups.
• The outcome is approximately normally distributed within each group.
• Each group has at least two observations with nonzero variance.

What the analysis reports

• Welch F statistic
• Numerator and adjusted denominator degrees of freedom
• p value
• Estimated omega squared
• Group means and standard deviations

How to report it

A Welch's ANOVA indicated that mean scores differed across groups, F([df1], [df2]) = [F], p = [p], est. ω² = [omega2].

Related methods

• One way ANOVA
• Games-Howell Post Hoc Test
• Brown-Forsythe Test for Equality of Means

Run Welch's ANOVA on Quorant