Levene post hoc test in R - r

Question

I have a problem regarding my data analysis in R. One of my hypothesis is basically that my groups will differ in terms of spread of the scores, indicating that there would be a difference in extremity between the groups.
I decided to check my hypothesis with Levenes test, which turned out significant and should thus highlight that the standard deviations is significantly different between the groups.
But I do not know of any post hoc tests for Levenes test, and after reading up on possible post hoc analyses I decided to conduct an ANOVA on the residuals, and then do a post hoc test on the ANOVA.
This is the code I've tried so far:
leveneTest(SS_mean~RA01, DF)
DF$residuals <- abs(DF$SS_mean - DF$SS_mean_big) #SS_mean = Participants score,
#SS_mean_big = mean for each group.
My test and post hoc test looks like this:
levene.anova<-aov(residuals~RA01, DF) #RA01 is the groups. Four in total
summary(levene.anova)
TukeyHSD(levene.anova)
The ANOVA on the residuals turned out significant as well, but the p-value changed from 0.04 (Levenes test) to 0.01 (ANOVA on residuals).
When reading about it, it seemed like Levene test is just an ANOVA on the resiudals, and thus it should give me the same results. And I am also unsure what post hoc test i should use. I thought about Dunnett as well as it includes a baseline, which corresponds to one of my groups.
Lastly, I did a leveneTest on the residuals as well "leveneTest(residuals~RA01)", which turned out significant. Is it better for me to use a non-parametric test, e.g. Kruskal-Wallis h-test and conduct a post hoc test on my kruskal wallis test instead? And if this is the case, what would be the appropriate test? Should I use a pairwise Mann Whitney u-test or Dunn test?
As this is the first time im doing something like this, I'm unsure about if this is a legitimate analysis, I would really appreciate your help or input!

Answer 1

Levene's test should indeed give the same p-value as an ANOVA on the residuals.
See for example this code:
data("mtcars")
mtcars$cyl <- as.factor(mtcars$cyl)
# Calculate means and add them to data
cyl_means <- aggregate(disp ~ cyl, data = mtcars, FUN = mean)
colnames(cyl_means)[2] <- "disp_mean"
mtcars2 <- merge(mtcars, cyl_means, by = "cyl")
# Residuals and anova
mtcars2$residuals <- abs(mtcars2$disp - mtcars2$disp_mean)
res.aov <- aov(residuals ~ cyl, data = mtcars2)
summary(res.aov)
# Levene's test
lawstat::levene.test(mtcars$disp, mtcars$cyl, location = "mean")
Maybe you accidentially ran the Brown–Forsythe test instead, which is the default in lawstat::levene.test, and which uses the median instead of the mean to calculate residuals.
Use Dunnett's if you are only interested in comparing the groups to one baseline group.
Use TukeyHSD if you want all pairwise comparisons among groups.