This function calculates the intraclass-correlation
(icc) - sometimes also called variance partition coefficient
(vpc) - for random intercepts of mixed effects models. Currently,
merMod
, glmmTMB
,
stanreg
and brmsfit
objects are supported.
icc(x, ...) # S3 method for merMod icc(x, adjusted = FALSE, ...) # S3 method for glmmTMB icc(x, adjusted = FALSE, ...) # S3 method for stanreg icc(x, re.form = NULL, typical = "mean", prob = 0.89, ppd = FALSE, ...) # S3 method for brmsfit icc(x, re.form = NULL, typical = "mean", prob = 0.89, ppd = FALSE, ...)
x | Fitted mixed effects model (of class |
---|---|
... | Currently not used. |
adjusted | Logical, if |
re.form | Formula containing group-level effects to be considered in
the prediction. If |
typical | Character vector, naming the function that will be used as
measure of central tendency for the ICC. The default is "mean". See
|
prob | Vector of scalars between 0 and 1, indicating the mass within
the credible interval that is to be estimated. See |
ppd | Logical, if |
A numeric vector with all random intercept intraclass-correlation-coefficients.
Furthermore, if adjusted = FALSE
, between- and within-group variances
as well as random-slope variance are returned as attributes.
For stanreg
or brmsfit
objects, the HDI for each statistic
is also included as attribute.
The ICC is calculated by dividing the between-group-variance (random
intercept variance) by the total variance (i.e. sum of between-group-variance
and within-group (residual) variance).
The calculation of the ICC for generalized linear mixed models with binary outcome is based on
Wu et al. (2012). For other distributions (negative binomial, poisson, ...),
calculation is based on Nakagawa et al. 2017.
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 a level-2 variable explains the portion of variation in the grouping
structure (random intercept).
ICC for random-slope models
Caution: For models with random slopes and random intercepts,
the ICC would differ at each unit of the predictors. Hence, the ICC for these
kind of models cannot be understood simply as proportion of variance
(see Goldstein et al. 2010). For convenience reasons, as the
icc()
function also extracts the different random effects
variances, the ICC for random-slope-intercept-models is reported
nonetheless, but it is usually no meaningful summary of the
proportion of variances.
ICC for models with multiple or nested random effects
Caution: By default, for three-level-models, depending on the
nested structure of the model, or for models with multiple random effects,
icc()
only reports the proportion of variance explained for each
grouping level. Use adjusted = TRUE
to calculate the adjusted and
conditional ICC.
Adjusted and conditional ICC
If adjusted = TRUE
, an adjusted and conditional ICC are calculated,
which take all sources of uncertainty (of all random effects)
into account to report an "adjusted" ICC, as well as the conditional ICC.
The latter also takes the fixed effects variances into account (see
Nakagawa et al. 2017). If random effects are not nested and not
cross-classified, the adjusted (adjusted = TRUE
) and unadjusted
(adjusted = FALSE
) ICC are identical.
ICC for specific group-levels
To calculate 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 computed
manually. Use get_re_var
to get the 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(get_re_var(fit)) / (sum(get_re_var(fit)) + get_re_var(fit, "sigma_2"))
or for the ICC between level 2 and 3:
get_re_var(fit)[2] / sum(get_re_var(fit))
ICC for Bayesian models
If ppd = TRUE
, 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 re.form
) 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 accross models, also by specifying different group-level
terms via the re.form
-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 a more robust measure
to calculate the central tendency of the variances. For example, use
typical = "median"
.
Some notes on why the ICC is useful, based on Grace-Martin:
It can help you determine whether or not a linear mixed model is even necessary. If you find that the correlation is zero, that means the observations within clusters are no more similar than observations from different clusters. Go ahead and use a simpler analysis technique.
It can be theoretically meaningful to understand how much of the overall variation in the response is explained simply by clustering. For example, in a repeated measures psychological study you can tell to what extent mood is a trait (varies among people, but not within a person on different occasions) or state (varies little on average among people, but varies a lot across occasions).
It can also be meaningful to see how the ICC (as well as the between and within cluster variances) changes as variable are added to the model.
In short, the ICC can be interpreted as “the proportion of the variance
explained by the grouping structure in the population” (Hox 2002: 15).
The random effect variances indicate the between- and within-group
variances as well as random-slope variance and random-slope-intercept
correlation. The components are denoted as following:
Within-group (residual) variance: sigma_2
Between-group-variance: tau.00 (variation between individual intercepts and average intercept)
Random-slope-variance: tau.11 (variation between individual slopes and average slope)
Random-Intercept-Slope-covariance: tau.01
Random-Intercept-Slope-correlation: rho.01
Aguinis H, Gottfredson RK, Culpepper SA. 2013. Best-Practice Recommendations for Estimating Cross-Level Interaction Effects Using Multilevel Modeling. Journal of Management 39(6): 1490–1528 (doi: 10.1177/0149206313478188 )
Goldstein H, Browne W, Rasbash J. 2010. Partitioning Variation in Multilevel Models. Understanding Statistics, 1:4, 223-231 (doi: 10.1207/S15328031US0104_02 )
Grace-Martion K. The Intraclass Correlation Coefficient in Mixed Models, web
Hox J. 2002. Multilevel analysis: techniques and applications. Mahwah, NJ: Erlbaum
Nakagawa S, Johnson P, Schielzeth H (2017) The coefficient of determination R2 and intra-class correlation coefficient from generalized linear mixed-effects models revisted and expanded. J. R. Soc. Interface 14. 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 SW, Bryk AS. 2002. Hierarchical linear models: applications and data analysis methods. 2nd ed. Thousand Oaks: Sage Publications
Wu S, Crespi CM, Wong WK. 2012. Comparison of methods for estimating the intraclass correlation coefficient for binary responses in cancer prevention cluster randomized trials. Contempory Clinical Trials 33: 869-880 (doi: 10.1016/j.cct.2012.05.004 )
Further helpful online-ressources:
CrossValidated (2012) Intraclass correlation (ICC) for an interaction?
CrossValidated (2014) Interpreting the random effect in a mixed-effect model
CrossValidated (2014) how to partition the variance explained at group level and individual level
#> #> Linear mixed model #> #> Family : gaussian (identity) #> Formula: Reaction ~ 1 + (1 | Subject) #> #> ICC (Subject): 0.3949# note: ICC for random-slope-intercept model usually not # meaningful - see 'Note'. fit1 <- lmer(Reaction ~ Days + (Days | Subject), sleepstudy) icc(fit1)#>#> #> Linear mixed model #> #> Family : gaussian (identity) #> Formula: Reaction ~ Days + (Days | Subject) #> #> ICC (Subject): 0.4831sleepstudy$mygrp <- sample(1:45, size = 180, replace = TRUE) fit2 <- lmer(Reaction ~ Days + (1 | mygrp) + (1 | Subject), sleepstudy) icc(fit2)#> #> Linear mixed model #> #> Family : gaussian (identity) #> Formula: Reaction ~ Days + (1 | mygrp) + (1 | Subject) #> #> ICC (mygrp): 0.0060 #> ICC (Subject): 0.5899icc(fit2, adjusted = TRUE)#> #> Intra-Class Correlation Coefficient for Generalized Linear Mixed Model #> #> Family : gaussian (identity) #> Formula: Reaction ~ Days + (1 | mygrp) + (1 | Subject) #> #> Adjusted ICC: 0.5959 #> Conditional ICC: 0.4292 #>icc1 <- icc(fit1)#>icc2 <- icc(fit2) print(icc1, comp = "var")#> #> Linear mixed model #> #> Family : gaussian (identity) #> Formula: Reaction ~ Days + (Days | Subject) #> #> Within-group-variance: 654.941 #> Between-group-variance: 612.090 (Subject) #> Random-slope-variance: 35.072 (Subject.Days) #> Slope-Intercept-covariance: 9.604 (Subject.(Intercept)) #> Slope-Intercept-correlation: 0.066 (Subject)print(icc2, comp = "var")#> #> Linear mixed model #> #> Family : gaussian (identity) #> Formula: Reaction ~ Days + (1 | mygrp) + (1 | Subject) #> #> Within-group-variance: 946.474 #> Between-group-variance: 14.082 (mygrp) #> Between-group-variance: 1381.596 (Subject)# NOT RUN { # compute ICC for Bayesian mixed model, with an ICC for each # sample of the posterior. The print()-method then shows # the median ICC as well as 89% HDI for the ICC. # Change interval with print-method: # print(icc(m, posterior = TRUE), prob = .5) if (requireNamespace("brms", quietly = TRUE)) { library(dplyr) sleepstudy$mygrp <- sample(1:5, size = 180, replace = TRUE) sleepstudy <- sleepstudy %>% group_by(mygrp) %>% mutate(mysubgrp = sample(1:30, size = n(), replace = TRUE)) m <- brms::brm( Reaction ~ Days + (1 | mygrp / mysubgrp) + (1 | Subject), data = sleepstudy ) # by default, 89% interval icc(m) # show 50% interval icc(m, prob = .5) # variances based on posterior predictive distribution icc(m, ppd = TRUE) } # }