Hi,

The Friedman test.

https://www.researchgate.net/post/Non-parametric-alternative-for-Mixed-ANOVA

Freidman Test. "The Friedman test is a non-parametric statistical test developed by Milton Friedman that used to detect differences in treatments across multiple test attempts. The procedure involves ranking each row (or block) together, then considering the values of ranks by columns".

(https://stats.libretexts.org/Courses/Taft_College/PSYC_2200%3A_Elementary_Statistics_for_Behavioral_and_Social_Sciences_(Oja)/Unit_2%3A_Mean_Differences/12%3A_RM_ANOVA/12.04%3A_Non-Parametric_RM_ANOVA)

Samuel John Estenor Parreño , Bhogaraju Anand , Friedman's test works only for unreplicated complete block design. In general, it isn't analogous to a non-parametric mixed repeated measure ANOVA.

Imen Zribi , aligned ranks transformation anova may be what you are looking for. ... If your data are in fact arranged in unreplicated complete block design, you could use Quade test or Friedman test.

Bhogaraju Anand Samuel John Estenor Parreño Sal Mangiafico . Thank you for answering my inquiries. I am working with different treatments and for each treatment, I have 30 replicates. I have treatment as a fixed factor between subject-effect and time as the within-subjects effect. I guess probably the aligned rank transformation will be more adequate in my study case.

Friedmann's test compares the medians of three or more dependent groups in the nonparametric equivalent of the two-way ANOVA. So in case the residuals are not normally distributed and there is no constant variance then you should opt for Friedmann's test.

Johnson Jeremia Mshiu , Friedman's test is not analogous to a two-way anova. Simply, Friedman's test can be used only with unreplicated complete block design. A two-way anova usually implies that there are multiple observations in each treatment x block combination.

Friedman's test is also not really an analysis of variance but an extension of the sign test that can often have low power. There was a 2x2 mixed ANOVA in Ray Meddis' book Statistics Using Ranks and there are other published approach. Alternatively one could just rank the data and run parametric ANOVA on the ranks. The main issue is that interpreting an interaction among ranks is tricky as additivity or non-additivity of ranks doesn't imply the same for raw scores. So I'd actually advise a different approach such as bootstrapping a robust regression or a generalized linear model if you are interested in the underlying metrics on a particular scale (e.g., for prediction).

I've rarely seen a data set where a rank based approach is the best option (with maybe the exception maybe of very small data sets where the robustness of the rank transformation can be very useful in noisy/messy data).

Johnson Jeremia Mshiu Sal Mangiafico Thom Baguley Thank you for your answers.

I guess Brunner and Langer or GLMM are appropriate.

Also try Ranked Alignment ANCOVA. You can implement this in R.

https://digitalcommons.wayne.edu/cgi/viewcontent.cgi?article=1149&=&context=jmasm&=&sei-redir=1&referer=https%253A%252F%252Fwww.google.com%252Furl%253Fq%253Dhttps%253A%252F%252Fdigitalcommons.wayne.edu%252Fcgi%252Fviewcontent.cgi%25253Farticle%25253D1149%252526context%25253Djmasm%2526sa%253DU%2526ved%253D2ahUKEwjK-8XlgoX9AhVzjokEHaqbBVwQFnoECAcQAQ%2526usg%253DAOvVaw3w_UV_ys3dqypzjtdiSpEH#search=%22https%3A%2F%2Fdigitalcommons.wayne.edu%2Fcgi%2Fviewcontent.cgi%3Farticle%3D1149%26context%3Djmasm%22