Intraclass correlation used to assess variability of lower-order relationships across higher-order processes/units.

A function and vignettes for computing the intraclass correlation described in Aguinis & Culpepper (2015). iccbeta quantifies the share of variance in an outcome variable that is attributed to heterogeneity in slopes due to higher-order processes/units.

icc_beta(x, ...)

# S3 method for lmerMod
icc_beta(x, ...)

# S3 method for default
icc_beta(x, l2id, T, vy, ...)

Arguments

x	A lmer model object or a design matrix with no missing values.
...	Additional parameters...
l2id	A vector that identifies group membership. The vector must be coded as a sequence of integers from 1 to J, the number of groups.
T	A matrix of the estimated variance-covariance matrix of a lmer model fit.
vy	The variance of the outcome variable.

Value

A list with:

• J

• means

• XcpXc

• Nj

• rho_beta

References

Aguinis, H., & Culpepper, S.A. (2015). An expanded decision making procedure for examining cross-level interaction effects with multilevel modeling. Organizational Research Methods. Available at: http://hermanaguinis.com/pubs.html

See also

lme4::lmer(), model.matrix(), lme4::VarCorr(), RLRsim::LRTSim(), iccbeta::Hofmann, and iccbeta::simICCdata

Examples

if (FALSE) {

if(requireNamespace("lme4") && requireNamespace("RLRsim")){

## Example 1: Simulated Data Example from Aguinis & Culpepper (2015) ----
data(simICCdata)
library("lme4")

# Computing icca
vy <- var(simICCdata$Y)
lmm0 <- lmer(Y ~ (1 | l2id), data = simICCdata, REML = FALSE)
VarCorr(lmm0)$l2id[1, 1]/vy

# Create simICCdata2
grp_means = aggregate(simICCdata[c('X1', 'X2')], simICCdata['l2id'], mean)
colnames(grp_means)[2:3] = c('m_X1', 'm_X2')
simICCdata2 = merge(simICCdata, grp_means, by='l2id')

# Estimating random slopes model
lmm1  <- lmer(Y ~ I(X1 - m_X1) + I(X2 - m_X2) +
                 (I(X1 - m_X1) + I(X2 - m_X2) | l2id),
              data = simICCdata2, REML = FALSE)

## iccbeta calculation on `lmer` object
icc_beta(lmm1)

## Manual specification of iccbeta

# Extract components from model.
X <- model.matrix(lmm1)
p <- ncol(X)
T1  <- VarCorr(lmm1)$l2id[1:p,1:p]

# Note: vy was computed under "icca"

# Computing iccb
# Notice '+1' because icc_beta assumes l2ids are from 1 to 30.
icc_beta(X, simICCdata2$l2id + 1, T1, vy)$rho_beta

## Example 2: Hofmann et al. (2000)   ----

data(Hofmann)
library("lme4")

# Random-Intercepts Model
lmmHofmann0 = lmer(helping ~ (1|id), data = Hofmann)
vy_Hofmann = var(Hofmann[,'helping'])

# Computing icca
VarCorr(lmmHofmann0)$id[1,1]/vy_Hofmann

# Estimating Group-Mean Centered Random Slopes Model, no level 2 variables
lmmHofmann1 <- lmer(helping ~ mood_grp_cent + (mood_grp_cent |id),
                    data = Hofmann, REML = FALSE)

## Automatic calculation of iccbeta using the lmer model
amod = icc_beta(lmmHofmann1)

## Manual calculation of iccbeta

X_Hofmann <- model.matrix(lmmHofmann1)
P <- ncol(X_Hofmann)
T1_Hofmann <- VarCorr(lmmHofmann1)$id[1:P,1:P]

# Computing iccb
bmod = icc_beta(X_Hofmann, Hofmann[,'id'], T1_Hofmann, vy_Hofmann)$rho_beta

# Performing LR test
library("RLRsim")
lmmHofmann1a <- lmer(helping ~ mood_grp_cent + (1 |id),
                     data = Hofmann, REML = FALSE)
obs.LRT <- 2*(logLik(lmmHofmann1) - logLik(lmmHofmann1a))[1]
X <- getME(lmmHofmann1,"X")
Z <- t(as.matrix(getME(lmmHofmann1,"Zt")))
sim.LRT <- LRTSim(X, Z, 0, diag(ncol(Z)))
(pval <- mean(sim.LRT > obs.LRT))
} else {
 stop("Please install packages `RLRsim` and `lme4` to run the above example.")
}

}