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multilevel_alpha.R
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multilevel_alpha.R
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# R function multilevel_alpha(), a wrapper for computing
# the reliability indices discussed in
# Lai, M. H. C. (2021). Composite reliability of multilevel data:
# It’s about observed scores and construct meanings.
# Psychological Methods, 26(1), 90–102.
# https://doi.org/10.1037/met0000287
# Copyright (C) 2021 Lai, Hok Chio (Mark)
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program. If not, see <https://www.gnu.org/licenses/>.
multilevel_alpha <- function(data, id, nsim = 5000, conf_level = .95,
se = "robust.huber.white",
pa_args = list(fa = "pc"), ...) {
if (!require(lavaan)) stop("The lavaan package needs to be installed.")
if (!require(psych)) stop("The psych package needs to be installed.")
nitem <- ncol(data)
ynames <- paste0("y", seq_len(nitem))
colnames(data) <- ynames
data <- cbind(data, id = id)
hmean_cs <- 1 / mean(1 / table(id))
# Alpha
# Generate syntax for saturated model
sat_syntax <- (function(y) {
if (length(y) <= 1) {
return(NULL)
}
paste(
c(paste(y[1], "~~", paste(y[-1], collapse = " + ")),
Recall(y[-1])),
collapse = "\n "
)
})(ynames)
msat <- paste0("level: 1\n ", sat_syntax, "\nlevel: 2\n ", sat_syntax)
msat_fit <- cfa(msat, data = data, cluster = "id", se = se,
test = "none", h1 = FALSE, baseline = FALSE, ...)
coef_msat <- coef(msat_fit, type = "user")
vcov_msat <- vcov(msat_fit)
vw <- names(coef_msat)[
with(msat_fit@ParTable, which(op == "~~" & lhs == rhs & level == 1))]
cvw <- names(coef_msat)[
with(msat_fit@ParTable, which(op == "~~" & lhs != rhs & level == 1))]
vb <- names(coef_msat)[
with(msat_fit@ParTable, which(op == "~~" & lhs == rhs & level == 2))]
cvb <- names(coef_msat)[
with(msat_fit@ParTable, which(op == "~~" & lhs != rhs & level == 2))]
Sw <- sum(coef_msat[vw], 2 * coef_msat[cvw])
Sb <- sum(coef_msat[vb], 2 * coef_msat[cvb])
alpha_const <- nitem / (nitem - 1)
alphaw <- alpha_const * sum(2 * coef_msat[cvw]) / Sw
alpha2l <- alpha_const * sum(2 * coef_msat[c(cvw, cvb)]) / (Sb + Sw)
alphab <- alpha_const * sum(2 * coef_msat[cvb]) / (Sb + Sw / hmean_cs)
sim_coef_msat <- MASS::mvrnorm(nsim,
mu = coef_msat[c(vw, vb, cvw, cvb)],
Sigma = vcov_msat[c(vw, vb, cvw, cvb),
c(vw, vb, cvw, cvb)])
sim_Sw <- rowSums(cbind(sim_coef_msat[ , vw], 2 * sim_coef_msat[ , cvw]))
sim_Sb <- rowSums(cbind(sim_coef_msat[ , vb], 2 * sim_coef_msat[ , cvb]))
sim_alphaw <- alpha_const * rowSums(2 * sim_coef_msat[ , cvw]) / sim_Sw
sim_alpha2l <- alpha_const * rowSums(2 * sim_coef_msat[ , c(cvw, cvb)]) /
(sim_Sb + sim_Sw)
sim_alphab <- alpha_const * rowSums(2 * sim_coef_msat[ , cvb]) /
(sim_Sb + sim_Sw / hmean_cs)
sim_alpha_cis <- lapply(list(alpha2l = sim_alpha2l,
alphab = sim_alphab,
alphaw = sim_alphaw),
quantile,
probs = .5 + c(- conf_level, conf_level) / 2)
# Omega
loading_labels <- paste0("l", seq_len(nitem))
g_syntax <- paste(loading_labels, "*", ynames,
collapse = " + ")
mcfa <- paste0("level: 1\n fw =~ ",
g_syntax,
"\nlevel: 2\n fb =~ ",
g_syntax)
mcfa_fit <- cfa(mcfa, data = data, cluster = "id", se = se,
test = "none", h1 = TRUE, baseline = FALSE, ...)
mcfa_pt <- partable(mcfa_fit)
coef_mcfa <- coef(mcfa_fit)
vcov_mcfa <- vcov(mcfa_fit)
coef_mcfa[loading_labels]
ld <- names(coef_mcfa)[with(mcfa_pt, free[which(op == "=~" &
level == 1)])]
evw <- names(coef_mcfa)[with(mcfa_pt,
free[which(op == "~~" &
lhs == rhs &
lhs != "fw" &
level == 1)])]
fvw <- names(coef_mcfa)[with(mcfa_pt, free[which(op == "~~" &
lhs == "fw")])]
evb <- names(coef_mcfa)[with(mcfa_pt,
free[which(op == "~~" &
lhs == rhs & lhs != "fb" &
level == 2)])]
fvb <- names(coef_mcfa)[with(mcfa_pt, free[which(op == "~~" &
lhs == "fb")])]
sumldsq <- sum(1, coef_mcfa[ld])^2
sumevw <- sum(coef_mcfa[evw])
sumevb <- sum(coef_mcfa[evb])
omegaw <- sumldsq * coef_mcfa[[fvw]] /
(sumldsq * coef_mcfa[[fvw]] + sumevw)
omega2l <- sum(sumldsq * coef_mcfa[c(fvw, fvb)]) /
sum(sumldsq * coef_mcfa[c(fvw, fvb)], sumevw, sumevb)
omegab <- sumldsq * coef_mcfa[[fvb]] /
(sumldsq * (coef_mcfa[[fvb]] + coef_mcfa[[fvw]] / hmean_cs) +
sumevb + sumevw / hmean_cs)
sim_coef_mcfa <- MASS::mvrnorm(nsim,
mu = coef_mcfa[c(ld, fvw, fvb, evw, evb)],
Sigma = vcov_mcfa[c(ld, fvw, fvb, evw, evb),
c(ld, fvw, fvb, evw, evb)])
sim_sumldsq <- (1 + rowSums(sim_coef_mcfa[ , ld]))^2
sim_sumevw <- rowSums(sim_coef_mcfa[ , evw])
sim_sumevb <- rowSums(sim_coef_mcfa[ , evb])
sim_omegaw <- sim_sumldsq * sim_coef_mcfa[ , fvw] /
(sim_sumldsq * sim_coef_mcfa[ , fvw] + sim_sumevw)
sim_omega2l <- rowSums(sim_sumldsq * sim_coef_mcfa[ , c(fvw, fvb)]) /
(rowSums(sim_sumldsq * sim_coef_mcfa[ , c(fvw, fvb)]) +
sim_sumevw + sim_sumevb)
sim_omegab <- sim_sumldsq * sim_coef_mcfa[ , fvb] /
(sim_sumldsq * (sim_coef_mcfa[ , fvb] +
sim_coef_mcfa[ , fvw] / hmean_cs) +
sim_sumevb + sim_sumevw / hmean_cs)
sim_omega_cis <- lapply(list(omega2l = sim_omega2l,
omegab = sim_omegab,
omegaw = sim_omegaw),
quantile,
probs = .5 + c(- conf_level, conf_level) / 2)
# resid_corb <- resid(mcfa_fit, type = "cor")$id$cov
# diag(resid_corb) <- 1
# psych::KMO(resid_corb)$MSA
# psych::fa.parallel(resid_corb, fm = "pa", fa = "fa",
# n.obs = lavTech(msat_fit, "nclusters")[[1]],
# n.iter = 30 * nitem,
# plot = FALSE)$nfact
# Dimensionality
corw <- lavTech(msat_fit, what = "cor.ov")$within
corb <- lavTech(msat_fit, what = "cor.ov")$id
paw <- do.call(fa.parallel,
args = c(list(x = corw,
n.obs = nobs(msat_fit) -
lavTech(msat_fit, "nclusters")[[1]],
n.iter = 30 * nitem,
plot = FALSE), pa_args))
pab <- do.call(fa.parallel,
args = c(list(x = corw,
n.obs = lavTech(msat_fit, "nclusters")[[1]],
n.iter = 30 * nitem,
plot = FALSE), pa_args))
if (pa_args$fa == "pc") {
ncompw <- paw$ncomp
ncompb <- pab$ncomp
} else if (pa_args$fa == "fa") {
ncompw <- paw$nfact
ncompb <- pab$nfact
}
list(alpha = c(alpha2l = alpha2l, alphab = alphab, alphaw = alphaw),
alpha_ci = do.call(rbind, sim_alpha_cis),
omega = c(omega2l = omega2l, omegab = omegab, omegaw = omegaw),
omega_ci = do.call(rbind, sim_omega_cis),
ncomp = c(within = ncompw, between = ncompb))
}