ICC for count models

[Nanco Bonnevieri]

Stata does not estimate ICC for count models, and I saw in a different post here that it can't be done. However, Finch, Bolin & Kelley Multilevel modeling using R (2014, p.154-155) estimate ICC for random intercept Poisson regression (see my red boxes in attachment). Is this possible to estimate?
Throughout the book (e.g. p. 44-45), they seem to plug in standard deviations and not variances when estimating ICC. Can anyone explain why? Finch_multilevel-modeling-using-r_p44-45_154-155.pdf

[Leonardo Guizzetti]

With multilevel Poisson regression, there isn't such a thing as the ICC, like there are with mixed models of the linear and logistic sort. My understanding of this is that it is because the model doesn't assume a common variance scale for all observations, or said differently, there isn't a latent variable formation of the model. Instead, there is an ICC for each covariate pattern in the model, which limits the utility of computing the ICC. You can read more from the below article, as well as other suggestions for describing cluster-level heterogeneity from these models.

That said, the formulation for ICC you have linked is wrong and I wouldn't use it.

Austin PC, Stryhn H, Leckie G, Merlo J. Measures of clustering and heterogeneity in multilevel Poisson regression analyses of rates/count data. Stat Med. 2018 Feb 20;37(4):572-589. doi: 10.1002/sim.7532.