How to calculate two-way mixed ICC? I want to calculate ICC for two-way mixed model, single rater, and either for consistency and absolute agreement. I found two packages for calculating ICC (intra-class correlation coefficient), that is, "irr" and "psych". However, from their argument description, I cannot sure that whether they are suitable for the use. I'll describe my concerns as following.

If they are indeed not suitable, hope you can recommend package or
provide some work example or steps for this use case.

"irr" package: icc
The "irr" package, it provides detailed argument for user to customize their situation, includes model, type and units, which is all parameters we need for selecting correct ICC form [ref]. However, I'm not sure whether this is a suitable package for the mixed model, that is, when judges are fixed and NOT randomly selected, since it only describe the use of "random effect" in the document

model：a character string specifying if a '"oneway"' model (default) with row effects random, or a '"twoway"' model with column and row effects random should be applied.

"psych" package: ICC
This is rather a simpler function, what we only need to do is input our data and it'll shows kinds of report under different ICC parameters. However, though this time it clearly tell that "ICC3" is for "fixed set of k judges" (i.e. two-way mixed model) and report both results for single and mean of k raters, I'm not sure whether the result comes from "consistency" or "absolute agreement".

From the "psych" package documentation, under the ICC section, I found this:

As M&W say "The importance of the random-fixed effects distinction is in its effect on the interpretation, but not calculation, of an ICC. Namely, when levels of the column factor are randomly sampled, one can generalize beyond one’s data, but not when they are fixed. In either case, however,the value of the ICC is the same," (p 37)

So, you can use the results from the "irr" package to report absolute agreement in a two way mixed model(as the "psych" package returns consistency for the two way mixed model). You would only need to mention that the raters were not randomly sampled but were fixed.
I used both packages to check ICC on some of my data and present the results below. You will find that the icc value and 95% CI are similar for a two way model with absolute agreement based on a single rater in both packages (randomised effects for "irr" and ICC3 for "psych")
"irr" results

Subjects = 119; Raters = 2; ICC(A,1) = 0.942


F-Test, H0: r0 = 0 ; H1: r0 > 0
F(118,87.8) = 35.2 , p = 5.55e-46


95%-Confidence Interval for ICC Population Values:
0.914 < ICC < 0.96

"psych" results

Single fixed raters (ICC3): ICC value (0.94); 95% CI (0.92- 0.96)

