I have a problem now that we want to compare sample groups and wanted to come up with a function out of it, so we moved from ANOVA to ANCOVA. The problem now is, the world is not perfect and we do not have a normal distribution of residuals.

This journal [1] suggests few different approaches for ANCOVA. Kurskall Wallis as a non parametric alternative was first considered and then from the journal Quade's non-parametric ANCOVA, and Puri and Sen's non-parametric ANCOVA as well. The 'problem' is that all those three seems to fall under moving all the data to ranks, which seems to lead to loss of inference power.

The question thus are:

1. Is there to the day any other method that is suggested for not needing to moving the data to ranks while not needing the normal distribution?

2. Among the 3 (or other you would like to suggest), is there any of them that is more used than the other? And if yes, for which reasons?

2.1 Could you point me to a source that show their assumptions and constraints? Im having issues finding a book that talks about this on them. It can be a complicated book for a rookie, I don't mind.

3. I saw a technique called resampling and bootstrapping [2], would this be of any use in this specific situation?

4. Does Chi Square has anything to do with what is intended here?

I know I did not provide any data here, but please assume that what is needed here is what I stated on the title: An ANCOVA for non-normal distributions, if possible (not sure if this is absurd since all I have as background is a very basic statistics course) not moving the data to ranks (if not possible I would appreciate to know why as well).

Thank you!

[1]: http://www.jstor.org/stable/2987939

[2]: http://bcs.whfreeman.com/pbs/cat_160/PBS18.pdf