I read here that if group sizes are equal, ANOVA is robust against the violation of the assumptions of normality and homoscedasticity.
I am wondering if this is the case, and if so why?
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Welch one-way ANOVA. Maybe so, but I think it is better to use a Welch version of the one-way ANOVA---as shown briefly below in R. This test is similar to the Welch two-sample t test, specifically designed to accommodate unequal variances.
x1 = rnorm(10, 50, 2); x2 = rnorm(10, 50, 4) x3 = rnorm(10, 55, 4); x4 = rnorm(10, 60, 5) x = c(x1, x2, x3, x4); gp = rep(1:4, each=10)
These data are simulated to have among-group differences in both means and variances.
boxplot(x ~ gp, col="skyblue2", pch=20)
oneway.test(x ~ gp) One-way analysis of means (not assuming equal variances) data: x and gp F = 28.896, num df = 3.000, denom df = 18.466, p-value = 3.463e-07
Note: In a standard on-way ANOVA one would have a
of $4(10 - 1) = 36$ degrees of freedom.
References: This web page discussed ad-hoc tests and other relevant details. This excellent Answer by @gung on our own site explores this test and other related issues. Perhaps explore other "Related" links provided in the right-hand margin of this page.