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I understand that ICC in binomial GLMMs with a logit link can be calculated via R, where the residual deviance is (pi ^ 2) / 3. However, this is assuming that the scale factor of the distribution is equals to 1 (according to Hox et al., 2017)

Given this, I suspect that this method of calculating ICC would not be appropriate if used in the context of beta-binomial GLMM. If so, is there a way to calculate the ICC of a beta-binomial GLMM?

I am pretty new to the world of GLMMs(and stackexchange) so please do bear with me if I misunderstood anything!

References Hox, J. J., Moerbeek, M., & van de Schoot, R. (2017). Multilevel Analysis: Techniques and Applications (3rd ed.). Routledge.

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