No data is normally distributed. Never! Why? Because it would require an outcome measure that ranges from minus to plus infinity. All measures have natural boundaries. e.g. response times or counts cannot be negative. Ther closer measures get to this boundary, the stronger the skew and the smaller the variance.

The solution to your problem is Generalized Linear Models. In the GLM framework you have the choice of various response distributions other than the Normal. For example, Poisson distributions are adequate for count data.

In my book draft (ch. 7) I explain GLMs and give recommendations for various types of outcome measures.

https://www.researchgate.net/project/Book-New-statistics-for-the-design-researcher

You can use an aligned ranks test such as the ARTool:

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

or the art3.anova by Haiko Lupsen:

http://www.uni-koeln.de/~a0032/R/

or the F1.LD.F1 SAS program or R program from the nparLD R package by Brunner and colleagues: https://cran.r-project.org/web/packages/nparLD/nparLD.pdf

See attachments.

You can run R scripts online: https://rdrr.io/snippets/

SAS online:

https://www.sas.com/en_us/software/on-demand-for-academics.html

Martin Schmettow Martin Schmettow

I'm a bit confused and I want to talk to you about details. In the GLM for example in r with glm function we just can use one dependent variable. for example glm(y~group,family=poisson), however in the repeated measure we have more than two variables which are related together. In my data, I have three columns for three time-points of data called var1, var2, var3; all these variables are related and were measured in three time points. How can I restructure these three variables into one column? How the glm function knows these three variables are related?

Jos Feys Thank you

Which one do you recommend?

If you know how to implement a R function, then you could use the art3.anova by Haiko Luepsen. If not you can try the F1_LD_F1 either from the SAS macro or the nparLD package. In attachment the R script for 2 data sets. It includes the robust anova by Wilcox. This may cause some problems.

Jos Feys

Thank you dear Jos.

What is your idea about GEE (Generalized Estimation Equation)? I can use it and there is no assumption like normality in this method.

One can run an ANOVA using multiple regression. But, as far is I know, there is no good alternative to the nonparametric methods I mentioned, when the data are very skewed. Besides, even when the data are normaly distributed, it is quite complicated to specify the model in a GEE context.

PS: the npIntFactRep R package is of very limited use. It only tests whether the interaction (BetweenX Within) is sign. or not., and this only for 2-way mixed (between x within/repeated measure) designs.

In attachment an example of a SAS program with the F1_LD_F1 SAS macro. As I mentioned, you can use SAS online for free:

https://www.sas.com/en_us/software/on-demand-for-academics.html

On top of this program, the data in wide format are converted to long format for the SAS macro. The macro is invoqued at the very bottom. Just skip all the strange SAS stateùents, go to the bottom to see how the macro is invoked:

title1'LogCFU analysis nparLD'; * addapt the title;

title2'Analysis using the F1_LD_F1 macro';

%F1_LD_F1(data=long, factor=group, var=response, time=time, subject=subj);

run;

If you have repeated measures you should use either a multilevel model (preferred) or a GEE. Both can be used with various response distributions.

Martin Schmettow Thank you so much