Does aligned rank transform ANOVA require normally distributed data? Initially, I wanted to do a two-factorial (4x3) repeated measures ANOVA in order to analyse my data.
To be more precise, I do have two factors with factor1 having 4 levels (A, B, C, D), factor2 having 3 levels (t1, t2, t3). In addition, it is a repeated measures design as all 28 subjects (s) are confronted with all the 4 levels (A, B, C, D) within the 3 timepoints (t1, t2, t3). However, it turned out, that according to Shapiro Wilk test my data are not normally distributed. That is why now I wanted to use the aligned rank transform ANOVA as a non-parametric alternative for a two-factorial rmANOVA.
Within this method, data are aligned before doing the ANOVA.
The question is: After alignment, do the aligned data need to be normally distributed before performing the ANOVA for a valid outcome?
I am not sure about this, as the ART seems to be a non-parametric method, but still includes an ANOVA (with normally distributed data as a premise).
I am using the R-library "ARTool" and have come up with the following code:
library(ARTool)
aligned_data <- art(dependentVariable ~ factor1 * factor2 + Error(s), data=data)

Now does the following Shapiro Test need to tell me, that my data is normally distributed in order to continue and trust the results?
shapiro.test(residuals(aligned_data))

anova(aligned_data)

Thanks in advance and any help is appreciated.

This does not answer your question but what about using a two-way repeated measures ANOVA on trimmed means instead? Does not require data normality and/or homogeneity of variances. In R you can do this with the function wwtrim from the WRS package.
Besides that the Shapiro-Wilk test will be often significant even if your data is normally distributed provided that you have a large enough sample size.
