This function calculates the intraclass-correlation coefficient
(ICC) - sometimes also called variance partition coefficient
(VPC) - for mixed effects models. The ICC can be calculated for all models
get_variance. For models fitted with
the brms-package, a variance decomposition based on the posterior
predictive distribution is calculated (see 'Details').
icc(model, ...) # S3 method for brmsfit icc(model, re.form = NULL, robust = TRUE, ci = 0.95, ...)
A (Bayesian) mixed effects model.
Currently not used.
Formula containing group-level effects to be considered in
the prediction. If
The Credible Interval level.
A list with two values, the adjusted and conditional ICC. For models
brmsfit, a list with two values, the decomposed ICC as well
as the credible intervals for this ICC.
The ICC can be interpreted as “the proportion of the variance explained by the grouping structure in the population”. This index goes from 0, if the grouping conveys no information, to 1, if all observations in a group are identical (Gelman \& Hill, 2007, p. 258). In other word, the ICC “can also be interpreted as the expected correlation between two randomly drawn units that are in the same group” (Hox 2010: 15), although this definition might not apply to mixed models with more complex random effects structures.
The ICC is calculated by dividing the random effect variance, σ2i, by the total variance, i.e. the sum of the random effect variance and the residual variance, σ2ε.
icc() calculates an adjusted and conditional ICC, which both take
all sources of uncertainty (i.e. of all random effects) into account. While
the adjusted ICC only relates to the random effects, the conditional ICC
also takes the fixed effects variances into account (see Nakagawa et al. 2017).
Typically, the adjusted ICC is of interest when the analysis of random
effects is of interest.
icc() returns a meaningful ICC also for more
complex random effects structures, like models with random slopes or nested
design (more than two levels) and is applicable for models with other distributions
than Gaussian. For more details on the computation of the variances, see
Usually, the ICC is calculated for the null model ("unconditional model"). However, according to Raudenbush and Bryk (2002) or Rabe-Hesketh and Skrondal (2012) it is also feasible to compute the ICC for full models with covariates ("conditional models") and compare how much, e.g., a level-2 variable explains the portion of variation in the grouping structure (random intercept).
The proportion of variance for specific levels related to each other
(e.g., similarity of level-1-units within level-2-units or level-2-units
within level-3-units) must be calculated manually. Use
to get the random intercept variances (between-group-variances) and residual
variance of the model, and calculate the ICC for the various level correlations.
For example, for the ICC between level 1 and 2:
(sum(insight::get_variance_intercept(model)) + insight::get_variance_residual(model))
For for the ICC between level 2 and 3:
model is of class
icc() calculates a
variance decomposition based on the posterior predictive distribution. In
this case, first, the draws from the posterior predictive distribution
not conditioned on group-level terms (
posterior_predict(..., re.form = NA))
are calculated as well as draws from this distribution conditioned
on all random effects (by default, unless specified else in
are taken. Then, second, the variances for each of these draws are calculated.
The "ICC" is then the ratio between these two variances. This is the recommended
way to analyse random-effect-variances for non-Gaussian models. It is then possible
to compare variances across models, also by specifying different group-level
terms via the
Sometimes, when the variance of the posterior predictive distribution is very large, the variance ratio in the output makes no sense, e.g. because it is negative. In such cases, it might help to use
robust = TRUE.
Hox, J. J. (2010). Multilevel analysis: techniques and applications (2nd ed). New York: Routledge.
Nakagawa, S., Johnson, P. C. D., & Schielzeth, H. (2017). The coefficient of determination R2 and intra-class correlation coefficient from generalized linear mixed-effects models revisited and expanded. Journal of The Royal Society Interface, 14(134), 20170213. doi: 10.1098/rsif.2017.0213
Rabe-Hesketh, S., & Skrondal, A. (2012). Multilevel and longitudinal modeling using Stata (3rd ed). College Station, Tex: Stata Press Publication.
Raudenbush, S. W., & Bryk, A. S. (2002). Hierarchical linear models: applications and data analysis methods (2nd ed). Thousand Oaks: Sage Publications.