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I have three variables X (IV), Y (DV) and Z, where there is a significant association between X and Z and, as well as Z and Y, but not between X and Y.

X is a categorical variables with 2 levels Y is a ordinal variabel Z is a ordinal variable

I have used both ordinal regression and non-parametric tests (Mann-Whitney and Kruskal-Wallis), and found the same results. Can I use any of these to say something about mediation? If not, which test could I use?

Is it possible that Z mediates the relationship between X and Y, when X-Y is not significant? If not, how should one interpret such results?

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There is not universal agreement on what exactly "mediation" means and your question highlights one of the areas of dispute. My own view is that mediation exists when the mediator changes the relationship between the IV and the DV in a way that is important.

So, my practice is to look at the X Y relationship by itself and after controlling for Z and see if the X Y relationship changes a lot.

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