I have Likert-scale data from an experiment with 5 conditions, each of which has a different treatment and one is a control group. The goal is to compare the effect of the treatment between the conditions.

I tested the normality of distributions with the Shapiro-Wilk test. The result shows that the data is not normally distributed. Therefore, I used a non-parametric equivalent to ANOVA, in this case, Kruskal-Wallis test.

But then I tested the homogeneity of variance with Levene's test. The result shows that the variances are homogeneous.

Considering the results from Shapiro-Wilk test (non-normal distribution) and the results from Levene's test (variances are homogeneous), should I use the non-parametric Kruskal-Wallis or ANOVA?

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    $\begingroup$ With a Likert scale, or any distribution that is a few discrete spikes, whether it's normal is arguably not relevant. Even if data pass a Shapiro-Wilk test, the comparison is of very unlike things. Feeding that to an ANOVA essentially asserts a measured (interval) scale. Minimally, you need to flag that assumption and be prepared for dissent. Nearly equal variances aren't assured by a bounded scale say 1 to 5, but they don't seem surprising. Again, even calculating variance is treating the scale as interval. Watch out: people disagree on whether this is acceptable or objectionable. $\endgroup$ – Nick Cox Oct 26 '17 at 16:57
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    $\begingroup$ Wilk = Martin B. Wilk. Use that spelling to avoid confusion or conflation with Samuel S. Wilks. Edited accordingly. $\endgroup$ – Nick Cox Oct 26 '17 at 16:58
  • $\begingroup$ Thank you, Nick Cox. What would you recommend me to do? Use ANOVA or non-parametric test? Interestingly, the results are almost the same. Thank you. $\endgroup$ – Shimano Oct 26 '17 at 18:04
  • $\begingroup$ I don't have a recommendation. It's your choice on whether the scale is deserving of being treated as measured. Similar results are reassuring, however. $\endgroup$ – Nick Cox Oct 26 '17 at 18:13

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