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There are lots of options starting with the realization that if you have enough cases normality may not even matter because of the central limit theorem. How did you test your non-normality (and what form did it take). Data can be non-normal in many ways and solutions depends on how and why it is non-normal.

Which transformations did you try, Box Cox?

This is the distribution of the dependent variable:

You should ask a second opinion, I rarely worry about normality since I have thousands of points and so normality is not a great concern to me. Parametric methods are so much better supported and common that I would try non-parametric approaches only as a last concern.

You might read this about the need for normality (which remember only impacts the p values)

http://rctdesign.org/techreports/arphnonnormality.pdf

Statisticians disagree, but I think it is common not to worry about non-normality with large sample sizes.

What statistic did you run to get residuals. I assumed the non-normality was in the raw data, but probably would have given the same advice.

If you ran regression I would look at your df beta and some measure of leverage. I suspect you have some extreme outliers.

http://rctdesign.org/techreports/arphnonnormality.pdf

Statisticians disagree, but I think it is common not to worry about non-normality with large sample sizes.

What statistic did you run to get residuals. I assumed the non-normality was in the raw data, but probably would have given the same advice.

If you ran regression I would look at your df beta and some measure of leverage. I suspect you have some extreme outliers.

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Boxplot of the dependent variable: