HLM_ICC_rWG {bruceR}  R Documentation 
Compute ICC(1) (nonindependence of data), ICC(2) (reliability of group means), and rWG/rWG(J) (withingroup agreement for singleitem/multiitem measures) in multilevel analysis (HLM).
HLM_ICC_rWG( data, group, icc.var, rwg.vars = icc.var, rwg.levels = 0, nsmall = 3 )
data 
Data frame. 
group 
Grouping variable. 
icc.var 
Key variable for analysis (usually the dependent variable). 
rwg.vars 
Default is

rwg.levels 
As rWG/rWG(J) compares the actual group variance to the expected random variance (i.e., the variance of uniform distribution, σ_EU^2), it is required to specify which type of uniform distribution is.

nsmall 
Number of decimal places of output. Default is 3. 
σ_{u0}^2: betweengroup variance (i.e., tau00)
σ_{e}^2: withingroup variance (i.e., residual variance)
n_k: group size of the kth group
K: number of groups
σ^2: actual group variance of the kth group
σ_{MJ}^2: mean value of actual group variance of the kth group across all J items
σ_{EU}^2: expected random variance (i.e., the variance of uniform distribution)
J: number of items
ICC(1) = var.u0 / (var.u0 + var.e) = σ_{u0}^2 / (σ_{u0}^2 + σ_{e}^2))
ICC(1) is the ICC we often compute and report in multilevel analysis (usually in the Null Model, where only the random intercept of group is included). It can be interpreted as either "the proportion of variance explained by groups" (i.e., heterogeneity between groups) or "the expectation of correlation coefficient between any two observations within any group" (i.e., homogeneity within groups).
ICC(2) = mean(var.u0 / (var.u0 + var.e / n.k)) = Σ[σ_{u0}^2 / (σ_{u0}^2 + σ_{e}^2 / n_k)] / K
ICC(2) is a measure of "the representativeness of grouplevel aggregated means for withingroup individual values" or "the degree to which an individual score can be considered a reliable assessment of a grouplevel construct".
rWG = 1  σ^2 / σ_{EU}^2
rWG(J) = 1  (σ_{MJ}^2 / σ_{EU}^2) / [J * (1  σ_{MJ}^2 / σ_{EU}^2) + σ_{MJ}^2 / σ_{EU}^2]
rWG/rWG(J) is a measure of withingroup agreement or consensus. Each group has an rWG/rWG(J).
Invisibly return a list of results.
Bliese, P. D. (2000). Withingroup agreement, nonindependence, and reliability: Implications for data aggregation and Analysis. In K. J. Klein & S. W. Kozlowski (Eds.), Multilevel theory, research, and methods in organizations (pp. 349381). San Francisco, CA: JosseyBass, Inc.
James, L.R., Demaree, R.G., & Wolf, G. (1984). Estimating withingroup interrater reliability with and without response bias. Journal of Applied Psychology, 69, 8598.
data=lme4::sleepstudy # continuous variable HLM_ICC_rWG(data, group="Subject", icc.var="Reaction") data=lmerTest::carrots # 7point scale HLM_ICC_rWG(data, group="Consumer", icc.var="Preference", rwg.vars="Preference", rwg.levels=7) HLM_ICC_rWG(data, group="Consumer", icc.var="Preference", rwg.vars=c("Sweetness", "Bitter", "Crisp"), rwg.levels=7)