CLMM: Calculate ICC & Assessing Model Fit

It turns out that the statistical contribution of each animal in my study is quite 
profound, so a mixed model is essential in my case.

Do you really need the cauchit? What does the distribution across 9 levels of the trait
look like? We regularly fit ordinal mixed models in genetic contexts, and one rule of thumb is that if there are more than 5 levels, then treating the variable as continuous usually gives much the same answers as the ordinal model, but interpretation of effects, variance explained etc is a lot more straightforward. Aside from residuals, one can get model goodness-of-fit chi-squares from the pairwise (and higher order) contingency tables (predicted v. observed) eg Visit1 v. Visit2 summed across all subjects (where the probit model parameters are the 8 thresholds and 1 polychoric correlation, so you have 64-9 degrees of freedom for whether a latent bivariate normal fits).

Cheers, David Duffy.