ICC for mixed-effects location scale models

[simoncolumbus]

I am wondering how performance::icc() computes ICCs for mixed-effects location scale models (MELSMs) in brms. Using the function on a MELSM yields adjusted and conditional ICCs, but I could not find any documentation for this.

I am not too familiar with ICCs for MELSMs, but know that Hedeker et al. (2012) propose an ICC for MELSM,

$\frac{exp(\alpha_0)}{exp(\alpha_0) + exp(t_0 + 0.5\sigma_\omega^2)}$,

where $\alpha_0$ is the BS variance of the intercept, $t_0$ is the WS variance, and $\sigma_\omega^2$ is the BS variance of the scale. The results from icc() are quite far from that.

I am also not sure why the adjusted and conditional effects differ, given that the model (see MWE below) does not include fixed effects.

# Libraries
library(data.table)
library(brms)
library(performance)

# Generate some data
dmixed <- withr::with_seed(
  seed = 12345, code = {
    nGroups <- 100
    nObs <- 20
    theta.location <- matrix(rnorm(nGroups * 2), nrow = nGroups, ncol = 2)
    theta.location[, 1] <- theta.location[, 1] - mean(theta.location[, 1])
    theta.location[, 2] <- theta.location[, 2] - mean(theta.location[, 2])
    theta.location[, 1] <- theta.location[, 1] / sd(theta.location[, 1])
    theta.location[, 2] <- theta.location[, 2] / sd(theta.location[, 2])
    theta.location <- theta.location %*% chol(matrix(c(1.5, -.25, -.25, .5^2), 2))
    theta.location[, 1] <- theta.location[, 1] - 2.5
    theta.location[, 2] <- theta.location[, 2] + 1
    dmixed <- data.table(
      x = rep(rep(0:1, each = nObs / 2), times = nGroups))
    dmixed[, ID := rep(seq_len(nGroups), each = nObs)]
    
    for (i in seq_len(nGroups)) {
      dmixed[ID == i, y := rnorm(
        n = nObs,
        mean = theta.location[i, 1] + theta.location[i, 2] * x,
        sd = exp(1 + theta.location[i, 1] + theta.location[i, 2] * x))
      ]
    }
    copy(dmixed)
  })

# Mixed-effects model
lm.me <- brm(bf(
  y ~ (1|ID)),
  family = "gaussian",
  data = dmixed, seed = 1234,
  chains = 2, cores = 2)

# Location scale model
ls.me <- brm(bf(
  y ~ (1|a|ID),
  sigma ~ (1|a|ID)),
  family = "gaussian",
  data = dmixed, seed = 1234,
  chains = 2, cores = 2)

# Get ICCs
icc(lm.me)
icc(ls.me)

Reference
Hedeker, D., Mermelstein, R. J., & Demirtas, H. (2012). Modeling between‐subject and within‐subject variances in ecological momentary assessment data using mixed‐effects location scale models. Statistics in medicine, 31(27), 3328-3336.