This function calculates the intraclass-correlation
(icc) - sometimes also called variance partition coefficient
(vpc) - for random intercepts of mixed effects models.
objects are supported.
Fitted mixed effects model (of class
More fitted model objects, to compute multiple intraclass-correlation coefficients at once.
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 Poisson multilevel models, please refere to Stryhn et al. (2006). Aly et al. (2014) describe computation of ICC for negative binomial models.
There is a
comp-argument set to
print(x, comp = "var")
(see 'Examples'). The
re_var-function is a convenient wrapper.
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)
A numeric vector with all random intercept intraclass-correlation-coefficients, or a list of numeric vectors, when more than one model were used as arguments. Furthermore, between- and within-group variances as well as random-slope variance are returned as attributes.
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).
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).
Caution: For three-level-models, depending on the nested structure of the model, the ICC only reports the proportion of variance explained for each grouping level. However, 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
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) / sum(get_re_var(fit))
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)
Aly SS, Zhao J, Li B, Jiang J. 2014. Reliability of environmental sampling culture results using the negative binomial intraclass correlation coefficient. Springerplus [Internet] 3. Available from: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3916583/
Grace-Martion K. The Intraclass Correlation Coefficient in Mixed Models, web
Hox J. 2002. Multilevel analysis: techniques and applications. Mahwah, NJ: Erlbaum
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
Stryhn H, Sanchez J, Morley P, Booker C, Dohoo IR. 2006. Interpretation of variance parameters in multilevel Poisson regression models. Proceedings of the 11th International Symposium on Veterinary Epidemiology and Economics, 2006 Available at http://www.sciquest.org.nz/node/64294
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:
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library(lme4) fit0 <- lmer(Reaction ~ 1 + (1 | Subject), sleepstudy) icc(fit0) fit1 <- lmer(Reaction ~ Days + (Days | Subject), sleepstudy) icc(fit1) sleepstudy$mygrp <- sample(1:45, size = 180, replace = TRUE) fit2 <- lmer(Reaction ~ Days + (1 | mygrp) + (Days | Subject), sleepstudy) icc(fit2) # return icc for all models at once icc(fit0, fit1, fit2) icc1 <- icc(fit1) icc2 <- icc(fit2) print(icc1, comp = "var") print(icc2, comp = "var")
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