[R] Data With Ordinal Responses: Calculate ICC & Assessing Model Fit

[Sidoti, Salvatore A.]

To begin with, I'm not a fan of cross-posting. However, I posted my question on Stack Exchange more than two weeks ago, but I have yet to receive a sufficient answer:

https://stats.stackexchange.com/questions/479600/data-with-ordinal-responses-calculate-icc-assessing-model-fit
 
Here's what I've learned since then (hopefully):
 
1) ICC of a CLMM:
Computed like this:
(variance of the random effect) / (variance of the random effect + 1)
If this is correct, I would love to see a reference/citation for it.
 
2) 95% Confidence Interval for the ICC from a CLMM Model
To my current understanding, a confidence interval for an ICC is only obtainable via simulation. I've conducted simulations with GLMM model objects ('lme4' package) and the bootMer() function. Unfortunately, bootMer() will not accept a CLMM model ('ordinal' package).
 
3) Model Fit of a CLMM
Assuming that the model converges without incident, the model summary includes a condition number of the Hessian ('cond.H'). This value should be below 10^4 for a "good fit". This is straightforward enough. However, I am not as sure about the value for 'max.grad', which needs to be "well below 1". The question is, to what magnitude should max.grad < 1 for a decent model fit? My reference is linked below (Christensen, 2019), but it does not elaborate further on this point:
 
https://documentcloud.adobe.com/link/track?uri=urn:aaid:scds:US:b6a61fe2-b851-49ce-b8b1-cd760d290636
 
3) Effect Size of a CLMM
The random variable's effect is determined by a comparison between the full model to a model with only the fixed effects via the anova() function. I found this information on the 'rcompanion' package website:
 
https://rcompanion.org/handbook/G_12.html
 
The output of this particular anova() will include a value named 'LR.stat', the likelihood ratio statistic. The LR.stat is twice the difference of each log-likelihood (absolute value) of the respective models. Is LR.stat the mixed-model version of an "effect size"? If so, how does one determine if the effect is small, large, in-between, etc?

Cheers,
Sal

Salvatore A. Sidoti
PhD Candidate
Behavioral Ecology
The Ohio State University