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Hello I am doing some stats on some research I did and was having trouble picking the correct statistical tests to use.

The question I am trying to answer is, I have two cell types (I, II) and two treatments (M, WT). I would like to know how they compare to one another, specifically:

I-WT - control for I
I-M - does this still look like I-M?
II-WT - control for II
II-M - does this look more like II-WT or Is?

To answer this it was recommended that I use a two-way anova, however according to Shapiro-Wilk my data is not normal (p < 10e-06 for all features). Is there a recommended non-parametric test for comparisons between 4 groups?

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You can use the Kruskall-Wallis test which generalizes the Mann-Witney test to multiple groups.

Unfortunately this only gives you a p-value, not a meaningful effect size. So in this case you might want to do a comparison between group I and its control, as well as a comparison between group II and its control, also.

For the two-group comparisons, if the data are not sufficiently close to being normal distributed that you are happy to use a t-test, you might consider some transformation (for example logarithm), or you can provide confidence intervals for the medians in each group.

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