Intraclass correlation for ordered logit models in brms

Modeling
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1

I am trying to retrieve the intraclass correlation for a random intercept in a cumulative-link ordinal model with a logistic link function. But I’m getting a little confused by the disc parameter being set to 1, as outlined in Paul and Matti’s paper. Can I use this as the residual variance in my calculations?

Here’s my attempt:

library(brms)
set.seed(2113)
d <- data.frame(y = sample(1:4, 100, replace = TRUE),
                group = rep(1:10, each = 10))
m <- brm(y ~ 1 + (1 | group), data = d, 
         family = cumulative("logit"), seed = 2113)
# since disc = 1, variance = (1/disc)^2 = 1 ?
hyp <- "sd_group__Intercept^2 / (sd_group__Intercept^2 + 1) = 0"
hypothesis(m, hyp, class = NULL)

This code returns an ICC of 0.13, 95% CI [0.00 0.50].

Alternatively, I have read elsewhere that the variance for the standard logistic distribution is (pi^2)/3, which is 3.289868.

hyp <- "sd_group__Intercept^2 / (sd_group__Intercept^2 + 3.289868) = 0"
hypothesis(m, hyp, class = NULL)

This code returns an ICC of 0.05, 95% CI [0.00 0.23], different from the previous value.

Which ICC estimate should I calculate? Thanks in advance.

2

From my understanding the disc term is the inverse standard deviation of the error term of the linear model. Therefore, the residual variance Ve = 1/(disc)^2 + (pi^2)/3.

and ICC = sd_group__Intercept^2 / (sd_group__Intercept^2 + Ve)

In cumulative ordinal models, by default the disc value is fixed to 1, thus, 1/(disc)^2 = 1

1 Like
3

Thank you!