I thought about analyzing results from Likert scale using Kruskal-Wallis, but the more I read about it, the more confused I get. People apparently do not agree about how to use Kruskal-Wallis...

So, I have a questions like this:

- What is influence of factor A, factor B and factor C on factor X?

I am using a 5 points Likert scale (5 - very high influence ... 1 - very low influence). I want to evaluate if there is difference between the influence of factor A, factor B and factor C on factor X.

First of all, if I use Kruskal-Wallis, am I comparing medians or not? I have read people saying the Kruskal-Wallis compares medians and other saying that this is not true If I am not comparing medians, am I still analyzing if there is difference between the influence of each of the three factors (A, B and C) on factor X?

Second, can I use Kruskal-Wallis? This is a survey result. So, the same group of people that responded about the influence of factor A on factor X, responded about the other relationships (factor B on factor X, and factor C on factor X). Is this breaking the assumption of mutual independence among samples or independence among samples?

Finally, if the sample sizes are 50 and I only have a 5 point Likert scale, how "rigorous" I need to be regarding the assumption of "populations distribution have the same shape"?

Hope getting some insights soon...

Cheers