Does the ICC require that my data are normally distributed?
If more information about my data is needed, just ask.
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Depends on the method - if you're using the ANOVA (general linear) method, then yes. But you can also calculate the ICC using generalized linear methods (I'm not sure about non-parametric methods - anyone else?).
Nakagawa and Schielzeth wrote the user-friendly rptR package for R, which uses general or generalized linear mixed effects modelling (for normal, binomial and count data) to calculate repeatability within classes using ANOVA, REML or MCMC methods. I think this is what you want to do?