Atul Pandey , I'm not certain about nesting, but Aligned Rank Transform can be used for non-parametric style ANOVA. This page links to a paper and a program called ARTool that can be used (I've used it) in getting the data ready for analysis.

https://depts.washington.edu/madlab/proj/art/

Having said all of this, I'm increasingly less concerned about normality, but when I am I typically look into bootstrap solutions.

How did you check for normality? How large is your samle size? What is the exact model you are trying to calculate? What transformation did you use? I think before switching to other methods, you should be sure that you cant use an ANOVA model.

Friedman's test is a non-parametric alternative to ANOVA with two variables.

Kosta Sotiroski the Friedman test is only designed to handle one independent variable/factor with more than two levels, but not two independent variables/factors. Your advise is not correct.

You want me to give you literature where it is claimed and examples solved.? I am a statistician and not a psychologist, and I have been dealing with statistics for a long time and there is no need to wrongly advise anyone.

Rainer Duesing : You want me to give you literature where it is claimed and examples solved.? I am a statistician and not a psychologist, and I have been dealing with statistics for a long time and there is no need to wrongly advise anyone.

Kosta Sotiroski please provide literature that it IS possible to do a 2-factorial Friedman test. In his original article (Friedman, 1937, see file) he calls the data format a "two-way table", but which is only a row (subjects) by column (dependent measures) table. He did not describe the possibility to calculate a 2-factorial design with two separate factors and an interaction of these two factors. None of the textbooks I know of (e.g. Howell, 2013; Field, 2018) or textbooks on robust techniques (e.g. Wilcox, 2012, 2017) or nonparametric techniques (e.g. Bortz, Lienert, & Boehnke, 2008; Cliff, 1996; Corder & Foreman, 2014) mention a possibility to use Friedman in a two-factorial design. Even Sheskin (2004) and Linebach, Tesch, and Kovacsiss (2014), who use the phrase “two way analysis of variance”, only show the classical application with one repeated measures factor with more than two factor levels. So, please provide literature on a two-factorial design for the Friedman test. Maybe I missed a crucial publication on this topic.

-Bortz, J., Lienert, G. A., & Boehnke, K. (2008). Verteilungsfreie methoden in der biostatistik. Springer-Verlag.

-Cliff, N. (1996). Ordinal methods for behavioral data analysis.

-Corder, G. W., & Foreman, D. I. (2014). Nonparametric Statistics: A Step-by-Step Approach Wiley, 2nd ed: Hoboken.

-Field, A. (2018). Discovering statistics using IBM SPSS statistics 5th ed.. Sage.

-Howell, D. C. (2013). Statistical methods for psychology. Cengage Learning.

-Linebach, J. A., Tesch, B. P., & Kovacsiss, L. M. (2014). Nonparametric statistics for applied research. New York, NY: Springer.

-Sheskin, D. J. (2004). Handbook of parametric and nonparametric statistical procedures. crc Press.

-Wilcox, R. R. (2012). Modern statistics for the social and behavioral sciences: A practical introduction. CRC press.

-Wilcox, R. R. (2017). Introduction to robust estimation and hypothesis testing, 4th ed. Academic press.

Rainer Duesing , Thanks for nice discussions. Usually I am testing typical Sqrt, log, ln, ....transformations for my samples to check normality which has sample size of around 20-25 individuals per group. In general, we have 2-3 trials, 4 treatment groups and 4 ranks within each treatment groups, where ranks are an independent factor affecting the treatment. We tried two-factorial general linear model for analysis of variance following post hoc analysis of Bonferroni corrections . For this I did ln(X+1) transformation of my data to achieve the normal distribution. We are also using the two-way nested analyses of variance following Bonferroni corrections. While, seems like this way we can present a logical analysis of our data but the same time I was wondering if there are some similar Non-parametric tests available which can test our data?

Amir D.Aczel: Complete Business Statistics, IRWIN, 1989, page 778 : “Similarly, the Fridman test, the subject of this section, is a nonparametric version of the randomized block design ANOVA. Sometimes this design referred to as a two-way ANOVA with one item per cell because it is possible to view the blocks as one factor and the and the treatment levels as the other.”

Rainer Duesing Please check this book: Amir D.Aczel: Complete Business Statistics, IRWIN, 1989, page 778 : “Similarly, the Fridman test, the subject of this section, is a nonparametric version of the randomized block design ANOVA. Sometimes this design referred to as a two-way ANOVA with one item per cell because it is possible to view the blocks as one factor and the and the treatment levels as the other.” There is an example. My counseling was based on what I have read and is just advice.

Dear Kosta Sotiroski,

Any (one-way) repeated measures design can be considered to be a "special" type of two-way design (subjects x measurements) with a single observation per cell. That does not make it a factorial or multi-way design. Single observation per cell means that you cannot test the interaction between the "two" factors. Therefore, it would not be appropriate to call it a factorial design.

Dear Sercan Dolma,

I only transferred what is written in the book, I did not give my own comment, in order to help the question asked.