There is nothing like "nonparametric data".

What you mean is that you would data that can be resonably described by a normal-distributed random variable. If this is the case, than one would expect that the frequency distribution of the data can be approximately described by the normal distribution model.

The other thing is: The rmANOVA is not assuming that the response variable is normal distributed. It assumes that the response variable is conditionally normal distributed, conditional on the explanatory factors in the model. In the case of the rmANOVA (and other models assuming conditionally normal distributed response variables), this can be checked by the distribution of the residuals of the fitted model. Make a normal-quantile-quantile plot of the residuals to see if the assumption of a conditional normal distribution is appropriate for your data.

Instead of transforming the the observed data to fit it into a particular distribution model it might be more sensible to use a better distribution model and an appropriate link function. Look for "generalized linear models", in your case even "generalized linear mixed models". Consult a statistician to help you.

I suppose you could use a non-parametric equivalent of the factorial repeated measures anova, with 1 between factor and 1 within factor (i.e., a 2-way between-within design). There are such non-parametric tests:

You can use an aligned ranks test such as the ARTool. This is a repeated measures ANOVA on the aligned ranks. See:

http://depts.washington.edu/acelab/proj/art/index.html

There is also the f1_LD_F1 test which is part of the R package nparLD or the f1_LD_f1 SAS macro. See attachments. Shah & Madden explain and illustrate the method underlying the nparLD method.

Luepsen has also developed an R script art3.anova, which is relevant here.See: http://www.uni-koeln.de/~luepsen/R/

You could also use the ART R package:

https://cran.r-project.org/web/packages/ART/ART.pdf

There is nothing like "nonparametric data".

What you mean is that you would data that can be resonably described by a normal-distributed random variable. If this is the case, than one would expect that the frequency distribution of the data can be approximately described by the normal distribution model.

The other thing is: The rmANOVA is not assuming that the response variable is normal distributed. It assumes that the response variable is conditionally normal distributed, conditional on the explanatory factors in the model. In the case of the rmANOVA (and other models assuming conditionally normal distributed response variables), this can be checked by the distribution of the residuals of the fitted model. Make a normal-quantile-quantile plot of the residuals to see if the assumption of a conditional normal distribution is appropriate for your data.

Instead of transforming the the observed data to fit it into a particular distribution model it might be more sensible to use a better distribution model and an appropriate link function. Look for "generalized linear models", in your case even "generalized linear mixed models". Consult a statistician to help you.

I should opt for "generalized linear models" or "generalized linear mixed models", they are more useful in the interpretation of the results, instead to “force” data with transformation.