Can I assess the relationship between a normally and non-normally distributed variable? My study is related to the visual attractiveness of route-plans in a logistics context. In practice, route-plans are rejected based on the fact that they "do not look nice". I have conducted an experiment where I showed participants of one group a route-plan with certain characteristics, while other groups were shown route-plans with different characteristics, however they both had the same quality ( in terms of total distance). Three main concepts I studied were: beauty 1=ugly - 10=beautiful), perceived quality (0 - 100% of shortest route-plan possible), and choice to accept a route-plan or not (yes/no).
I wanted to analyze the relationship between beauty (independent) and perceived quality (dependent), however the data for perceived quality is strongly skewed to the left. I tried several transformations, but I wasn't able to normalize the data for perceived quality. To be clear, multiple linear regression does not require the variables to be normally distributed, however the residuals should be. The residuals look normally distributed, but just not in the middle of the distribution. 
Is there anything I can do or is it just not possible to test the relationship between those variables reliably? Perhaps non-parametric tests? If so, which ones?
 A: I do not have enough reputation to add a comment, so I'll give my answer despite some question or remarks I have. And I'll add the questions and remarks at the end of my answer.
It is not really clear what you want to do. Do you want to check if there is a "relationship between beauty and perceived quality" or do you want to find what this relationship is?


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*If you want to check if there is a relationship
there are multiple options depending on what relation you expect. Since you were talking about linear regression, I'll assume that you at least expect some kind of strictly monotonic function (if not you'll have to give us more information about the data). Then I'd ineed suggest a non-parametric test, like Spearman's correlation coefficient (and test it's significance with a hypothesis test).

*If, however, you want to know what the relationship is
you will have to come up with some possible models and test how they fit. If you want to us to help with suggestions for models, you'll have to show the data.
I would like to point out, though, that it is rather unlikely that you will find a good fit with linear regression (as your residuals probably show), since your dependent variable is strongly skewed, but your independent variable not (I infer from what you said). This would mean that both variables don't have the same type of distribution, and since any linear transformation will always preserve the type of distribution, both variables cannot be linearly related.
Questions/remarks:


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*Could you show us your data (at least their distributions)? It is difficult to help without

*Do you want to know if there is a relationship or what it is?

*Those residuals also don't seem normally distributed, they seem skewed.

*This may be sound pedantic, but just so I understand your situation correctly, why do you want to do multiple linear regression when you have only one independent variable?

