The Levene's test for my set of data is >0.05 but the Shapiro Wilk test is
Think and try to understand your data.
What distributional assumptions are reasonable, which are not?
What statistical hypotheses do you really want to test? Are they meaningful?
The current discussion here may be helpful: https://www.researchgate.net/post/What_alternative_for_ANOVA_test_when_levene_test_shows_a_violation_of_the_homogeneity_assumption
Generally, in case both methods apply the results of a no parametric test are more conservative in favor of null hypothesis.
It is advised to go ahead with appropriate transformations of your data like a log transformation, square root transformation, or power transformations by Box Cox etc. Check for the normality assumption and you may then take a call. Also, check for your p-value in this case. If it's very close to 0.05, the violation of the normality assumption can be regarded as not a serious one. Identify the potential outliers(if any), remove them, and check for the normality assumption.
I have few reservations for Kruskal-Wallis test. This test is very less sensitive. As far as ANOVA is concerned, it is generally robust to normality. So, in my opinion, you may try log transformation or reciprocal transformation, or square root transformation of your data and see if you are getting nonsignificant Shapiro. If it fails, you may proceed with ANOVA even if it does not satisfy normality criteria.
Anuraj Nayarisseri , I'm sure you're aware of what's problematic in your answer. You say to try some transformations, and if they don't help, don't worry about it because anova is robust to violations of its assumptions. ... Anova may be _somewhat_ robust against the conditional normality assumption. But it certainly isn't entirely robust against the assumption. Otherwise, why would we bother talking about it ? It's certainly not good advice to say to just ignore the assumptions of the analysis.
Thank you all for your responses- very helpful. Salvatore S. Mangiafico thank you for the link. Anuraj Nayarisseri thank you for the explanation
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