is there any non parametric Welch's ANOVA? I need to test for differences between non-normally distributed and heteroscedastic data.
I know that I can use the Kruskal-Wallis test to assess significant differences between groups but one assumption of this test is homoscedasticity. I tested for homoscedasticity (i.e similar variance around groups) with Levene and Bartlett test and my data was found not to be homoscedastic.
One way ANOVA is not the solution because my data aren't normally distributed even when log-transforming the data.
I thought on applying Welch's ANOVA but the data doesn't follow a normal distribution either.
So, is there any test to asses significant differences between groups given a heteroscedastic and non-normally distributed data ?

These are the assumptions for the Kruskal-Wallis test given by Conover, 1999, Practical Nonparametric Statistics, 3rd.

*

*All samples are random samples from their respective populations.

*In additional to independence within each sample, there is mutual independence among the various samples.

*The measurement scale is at least ordinal.

*Either the k population distribution functions are identical, or else some of the populations tend to yield larger values than other populations do.

I think some authors will sometimes substitute a homoscedasticity assumption for assumption 4 given here.
Assumption 4 seems like an odd assumption, but I think it's necessary because if you start with two populations that have the same location and the same distribution shape, but different variances, when you sample these, you are likely to get two samples that the K-W test may detect as different, even though they are stochastically similar at the population level.  The upshot being that without assumption 4, you can get an inflated type-I error rate in these cases. I'm not entirely sure about this; I'd have to double check with some simulations.
If I understand all this correctly, assumption 4 is superior to having a homoscedasticity assumption. The homoscedasticity assumption is too restrictive because it prevents you from addressing those populations that are heteroscedastic and that have differences in stochastic dominance that the K-W test could detect.
Hopefully that all makes some sense, and I'm not too far from accurate.
