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 supported by
`insight::get_variance()`

. For models fitted with the
**brms**-package, `icc()`

might fail due to the large variety of
models and families supported by the **brms**-package. In such cases, an
alternative to the ICC is the `variance_decomposition()`

, which is based
on the posterior predictive distribution (see 'Details').

## Usage

```
icc(model, by_group = FALSE, tolerance = 1e-05)
variance_decomposition(model, re_formula = NULL, robust = TRUE, ci = 0.95, ...)
```

## Arguments

- model
A (Bayesian) mixed effects model.

- by_group
Logical, if

`TRUE`

,`icc()`

returns the variance components for each random-effects level (if there are multiple levels). See 'Details'.- tolerance
Tolerance for singularity check of random effects, to decide whether to compute random effect variances or not. Indicates up to which value the convergence result is accepted. The larger tolerance is, the stricter the test will be. See

`performance::check_singularity()`

.- re_formula
Formula containing group-level effects to be considered in the prediction. If

`NULL`

(default), include all group-level effects. Else, for instance for nested models, name a specific group-level effect to calculate the variance decomposition for this group-level. See 'Details' and`?brms::posterior_predict`

.- robust
Logical, if

`TRUE`

, the median instead of mean is used to calculate the central tendency of the variances.- ci
Credible interval level.

- ...
Arguments passed down to

`brms::posterior_predict()`

.

## Value

A list with two values, the adjusted and conditional ICC. For
`variance_decomposition()`

, a list with two values, the decomposed
ICC as well as the credible intervals for this ICC.

## Details

### Interpretation

The ICC can be interpreted as “the proportion of the variance explained by the grouping structure in the population”. The grouping structure entails that measurements are organized into groups (e.g., test scores in a school can be grouped by classroom if there are multiple classrooms and each classroom was administered the same test) and ICC indexes how strongly measurements in the same group resemble each other. 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.

### Calculation

The ICC is calculated by dividing the random effect variance,
σ^{2}_{i}, by
the total variance, i.e. the sum of the random effect variance and the
residual variance, σ^{2}_{ε}.

### Adjusted and conditional ICC

`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
`?insight::get_variance`

.

### ICC for unconditional and conditional models

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).

### ICC for specific group-levels

The proportion of variance for specific levels related to the overall model
can be computed by setting `by_group = TRUE`

. The reported ICC is
the variance for each (random effect) group compared to the total
variance of the model. For mixed models with a simple random intercept,
this is identical to the classical (adjusted) ICC.

### Variance decomposition for brms-models

If `model`

is of class `brmsfit`

, `icc()`

might fail due to
the large variety of models and families supported by the **brms**
package. In such cases, `variance_decomposition()`

is an alternative
ICC measure. The function 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_formula = NA)`

) are calculated as
well as draws from this distribution *conditioned* on *all random
effects* (by default, unless specified else in `re_formula`

) 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 `re_formula`

-argument.

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`

.

## References

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.

## Examples

```
if (require("lme4")) {
model <- lmer(Sepal.Length ~ Petal.Length + (1 | Species), data = iris)
icc(model)
}
#> # Intraclass Correlation Coefficient
#>
#> Adjusted ICC: 0.910
#> Conditional ICC: 0.311
# ICC for specific group-levels
if (require("lme4")) {
data(sleepstudy)
set.seed(12345)
sleepstudy$grp <- sample(1:5, size = 180, replace = TRUE)
sleepstudy$subgrp <- NA
for (i in 1:5) {
filter_group <- sleepstudy$grp == i
sleepstudy$subgrp[filter_group] <-
sample(1:30, size = sum(filter_group), replace = TRUE)
}
model <- lmer(
Reaction ~ Days + (1 | grp / subgrp) + (1 | Subject),
data = sleepstudy
)
icc(model, by_group = TRUE)
}
#> # ICC by Group
#>
#> Group | ICC
#> ------------------
#> subgrp:grp | 0.017
#> Subject | 0.589
#> grp | 0.001
```