Compute ICC(1) (non-independence of data), ICC(2) (reliability of group means), and rWG/rWG(J) (within-group agreement for single-item/multi-item measures) in multilevel analysis (HLM).
HLM_ICC_rWG(
data,
group,
icc.var,
rwg.vars = icc.var,
rwg.levels = 0,
digits = 3,
nsmall = digits
)Data frame.
Grouping variable.
Key variable for analysis (usually the dependent variable).
Default is icc.var. It can be:
A single variable (single-item measure), then computing rWG.
Multiple variables (multi-item measure), then computing rWG(J), where J = the number of items.
As rWG/rWG(J) compares the actual group variance to the expected random variance (i.e., the variance of uniform distribution, \(\sigma_EU^2\)), it is required to specify which type of uniform distribution is.
For continuous uniform distribution, \(\sigma_EU^2 = (max - min)^2 / 12\).
Then rwg.levels is not useful and will be set to 0 (the default).
For discrete uniform distribution, \(\sigma_EU^2 = (A^2 - 1) / 12\),
where A is the number of response options (levels).
Then rwg.levels should be provided (= A in the above formula).
For example, if the measure is a 5-point Likert scale, you should set rwg.levels=5.
Number of decimal places of output. Default is 3.
Invisibly return a list of results.
\(\sigma_{u0}^2\): between-group variance (i.e., tau00)
\(\sigma_{e}^2\): within-group variance (i.e., residual variance)
\(n_k\): group size of the k-th group
\(K\): number of groups
\(\sigma^2\): actual group variance of the k-th group
\(\sigma_{MJ}^2\): mean value of actual group variance of the k-th group across all J items
\(\sigma_{EU}^2\): expected random variance (i.e., the variance of uniform distribution)
\(J\): number of items
ICC(1) = var.u0 / (var.u0 + var.e) = \(\sigma_{u0}^2 / (\sigma_{u0}^2 + \sigma_{e}^2)\))
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)) = \(\Sigma[\sigma_{u0}^2 / (\sigma_{u0}^2 + \sigma_{e}^2 / n_k)] / K\)
ICC(2) is a measure of "the representativeness of group-level aggregated means for within-group individual values" or "the degree to which an individual score can be considered a reliable assessment of a group-level construct".
rWG = \(1 - \sigma^2 / \sigma_{EU}^2\)
rWG(J) = \(1 - (\sigma_{MJ}^2 / \sigma_{EU}^2) / [J * (1 - \sigma_{MJ}^2 / \sigma_{EU}^2) + \sigma_{MJ}^2 / \sigma_{EU}^2]\)
rWG/rWG(J) is a measure of within-group agreement or consensus. Each group has an rWG/rWG(J).
Bliese, P. D. (2000). Within-group agreement, non-independence, and reliability: Implications for data aggregation and Analysis. In K. J. Klein & S. W. Kozlowski (Eds.), Multilevel theory, research, and methods in organizations (pp. 349-381). San Francisco, CA: Jossey-Bass, Inc.
James, L.R., Demaree, R.G., & Wolf, G. (1984). Estimating within-group interrater reliability with and without response bias. Journal of Applied Psychology, 69, 85-98.
# NOT RUN {
data=lme4::sleepstudy # continuous variable
HLM_ICC_rWG(data, group="Subject", icc.var="Reaction")
data=lmerTest::carrots # 7-point 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)
# }
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