Chapter 9: Power for a Two versus One Factor Model Example (p. 169)
Phil Wood
2025-07-26
Chapter 9 Statistical Power for all parameters via Monte Carlo simulation (p. .
Begin with a Plausible Model
As explained in the book, one simple way to proceed is to specify a model with plausible values. In this case, variables with unit variance and intra-class correlations of .49. The correlation between the factors is assumed to be 0.8
Plausible Initial Model
# ============================================================
# Monte Carlo in R (lavaan) with:
# - Data generation from hand-entered Sigma (MVN)
# - "True values" parsed from pop_model_raw (value*LABEL*var style),
# including labeled intercepts like: Alc3M~0*IAlc*1;
# 1.) Specify missingness
# 2.) Specify the Covariance Model
# 3.) Specify the true model values
# 4.) Specify the (possibly mis-specified) model
# - For ALL labeled target parameters, report:
# * Mean & Median estimates
# * Coverage (95% CI) + MCSE
# * # significant + proportion
# * Bias reporting (gold-standard):
# - Relative Bias (%) when true != 0
# - Scaled Bias (×100) when true == 0 [= 100*(est-true)]
# - Absolute Bias [= mean(est-true)]
# - RMSE [= sqrt(mean((est-true)^2))]
# - Std. Bias [= (mean(est-true))/sd(est)] (flag if sd=0)
# * MCSE for Absolute Bias and RMSE (via SD/sqrt(m) approximation)
# and MCSE for Coverage (binomial)
# - Grouped APA-ish table (flextable), with blocks
# ============================================================
library(MASS) # mvrnorm
library(lavaan)## Warning: package 'lavaan' was built under R version 4.5.2## This is lavaan 0.6-21
## lavaan is FREE software! Please report any bugs.library(flextable)## Warning: package 'flextable' was built under R version 4.5.2set.seed(12345)
# ---------------- SETTINGS ----------------
N <- 100
R <- 1000 # Use 1000+ for stable MCSEs
alpha <- 0.05
# 1.) Optional MCAR missingness. Use zeros if no missing values are anticipated
pmiss <- c(Alc3M = 0.1, FivePlus = 0.1, Thurs = 0.1, Fri = 0.1, Sat = 0.1)
make_mcar_missing <- function(dat, pmiss) {
for (v in names(pmiss)) dat[runif(nrow(dat)) < pmiss[[v]], v] <- NA
dat
}
# ---------------- HAND-ENTERED COVARIANCE MATRIX ----------------
vars <- c("Alc3M", "FivePlus", "Thurs", "Fri", "Sat")
# 2.) Use the predicted covariance matrix from Onys as a quick way to get this
Sigma <- matrix(
c(
1.0000, 0.4900, 0.3920, 0.3920, 0.3920,
0.4900, 1.0000, 0.3920, 0.3920, 0.3920,
0.3920, 0.3920, 1.0000, 0.4900, 0.4900,
0.3920, 0.3920, 0.4900, 1.0000, 0.4900,
0.3920, 0.3920, 0.4900, 0.4900, 1.0000
),
nrow = 5, byrow = TRUE,
dimnames = list(vars, vars)
)
mu <- setNames(rep(0, length(vars)), vars)
# 3,)- POPULATION PARAMETER MODEL (TRUTH) Label the parameters with start values as shown
pop_model_raw <- "
Use=~0.7*L1*Alc3M
Use=~0.7*L2*FivePlus
PastWl=~0.7*L3*Thurs
PastWl=~0.7*L4*Fri
PastWl=~0.7*L5*Sat
Alc3M ~~ 0.51*VAR_Alc3M*Alc3M
FivePlus ~~ 0.51*VAR_FivePlus*FivePlus
Thurs ~~ 0.51*VAR_Thurs*Thurs
Fri ~~ 0.51*VAR_Fri*Fri
Sat ~~ 0.51*VAR_Sat*Sat
Use ~~ 1.0*Use
PastWl ~~ 1.0*PastWl
Use ~~ 0.8*Corr*PastWl
Alc3M~0*IAlc*1;
FivePlus~0*I5Plus*1;
Thurs~0*ITh*1;
Fri~0*IFr*1;
Sat~0*ISa*1;
"
# ============================================================
# Parse TRUE VALUES from pop_model_raw, including labeled intercepts
# ============================================================
parse_true_values <- function(raw) {
lines <- trimws(unlist(strsplit(raw, "\n", fixed = TRUE)))
lines <- lines[nzchar(lines)]
lines <- lines[!grepl("^!", lines)]
lines <- gsub(";", "", lines, fixed = TRUE)
lines <- gsub("\\s+", " ", lines)
true <- list()
load_re <- "^([A-Za-z][A-Za-z0-9_]*)\\s*=~\\s*([+-]?[0-9]*\\.?[0-9]+)\\s*\\*\\s*(L[0-9]+)\\s*\\*\\s*([A-Za-z][A-Za-z0-9_]*)$"
resv_re <- "^([A-Za-z][A-Za-z0-9_]*)\\s*~~\\s*([+-]?[0-9]*\\.?[0-9]+)\\s*\\*\\s*(VAR_[A-Za-z0-9_]+)\\s*\\*\\s*\\1$"
cov_re <- "^([A-Za-z][A-Za-z0-9_]*)\\s*~~\\s*([+-]?[0-9]*\\.?[0-9]+)\\s*\\*\\s*([A-Za-z][A-Za-z0-9_]*)\\s*\\*\\s*([A-Za-z][A-Za-z0-9_]*)$"
int_lab_re <- "^([A-Za-z][A-Za-z0-9_]*)\\s*~\\s*([+-]?[0-9]*\\.?[0-9]+)\\s*\\*\\s*([A-Za-z][A-Za-z0-9_]*)\\s*\\*\\s*1$"
for (ln in lines) {
if (grepl("=~", ln, fixed = TRUE)) {
rr <- regmatches(ln, regexec(load_re, ln))[[1]]
if (length(rr) == 5) true[[ rr[4] ]] <- as.numeric(rr[3]) # Lk
next
}
if (grepl("~~", ln, fixed = TRUE) && grepl("VAR_", ln, fixed = TRUE)) {
rr <- regmatches(ln, regexec(resv_re, ln))[[1]]
if (length(rr) == 4) true[[ rr[4] ]] <- as.numeric(rr[3]) # VAR_*
next
}
if (grepl("~~", ln, fixed = TRUE) && grepl("\\*", ln)) {
rr <- regmatches(ln, regexec(cov_re, ln))[[1]]
if (length(rr) == 5) true[[ rr[4] ]] <- as.numeric(rr[3]) # Corr (or other)
next
}
if (grepl("~", ln, fixed = TRUE) && !grepl("~~", ln, fixed = TRUE) && !grepl("=~", ln, fixed = TRUE)) {
rr <- regmatches(ln, regexec(int_lab_re, ln))[[1]]
if (length(rr) == 4) true[[ rr[4] ]] <- as.numeric(rr[3]) # IAlc etc.
next
}
}
unlist(true)
}
true_by_label <- parse_true_values(pop_model_raw)
# 4.) Specify the ANALYSIS MODEL (to be estimated each replication) ----------------
analysis_model <- "
Use =~ L1*Alc3M + L2*FivePlus
PastWl =~ L3*Thurs + L4*Fri + L5*Sat
Alc3M ~~ VAR_Alc3M*Alc3M
FivePlus ~~ VAR_FivePlus*FivePlus
Thurs ~~ VAR_Thurs*Thurs
Fri ~~ VAR_Fri*Fri
Sat ~~ VAR_Sat*Sat
Use ~~ 1*Use
PastWl ~~ 1*PastWl
Use ~~ Corr*PastWl
Alc3M ~ IAlc*1
FivePlus ~ I5Plus*1
Thurs ~ ITh*1
Fri ~ IFr*1
Sat ~ ISa*1
"
extract_labeled <- function(fit) {
pe <- parameterEstimates(fit, ci = TRUE)
pe <- pe[pe$label != "", ]
data.frame(
label = pe$label,
est = pe$est,
p = pe$pvalue,
ci.lower = pe$ci.lower,
ci.upper = pe$ci.upper,
stringsAsFactors = FALSE
)
}
# Determine targets (labels) from a probe fit, then intersect with true values
fit_probe <- lavaan(
analysis_model,
sample.cov = Sigma,
sample.mean = as.numeric(mu),
sample.nobs = 1e5,
meanstructure = TRUE,
fixed.x = FALSE
)
targets <- extract_labeled(fit_probe)$label
targets <- targets[targets %in% names(true_by_label)]
K <- length(targets)
if (K == 0) stop("No overlapping parameter labels between analysis model and population model.")
# ---------------- STORAGE ----------------
conv <- logical(R)
est_store <- matrix(NA, R, K, dimnames = list(NULL, targets))
cover <- matrix(NA, R, K, dimnames = list(NULL, targets))
sig <- matrix(NA, R, K, dimnames = list(NULL, targets))
# For bias/RMSE we also store raw error (est-true)
err_store <- matrix(NA, R, K, dimnames = list(NULL, targets))
# ---------------- MONTE CARLO ----------------
tv_all <- as.numeric(true_by_label[targets])
for (r in 1:R) {
X <- MASS::mvrnorm(n = N, mu = mu, Sigma = Sigma)
dat <- as.data.frame(X); names(dat) <- vars
dat <- make_mcar_missing(dat, pmiss)
fit <- tryCatch(
lavaan(analysis_model, data = dat, missing = "fiml", fixed.x = FALSE),
error = function(e) NULL
)
if (is.null(fit) || !inspect(fit, "converged")) next
conv[r] <- TRUE
est <- extract_labeled(fit)
est <- est[match(targets, est$label), ]
est_store[r, ] <- est$est
err_store[r, ] <- est$est - tv_all
cover[r, ] <- (tv_all >= est$ci.lower) & (tv_all <= est$ci.upper)
sig[r, ] <- (est$p < alpha)
}## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.
## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.
## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.## Warning: lavaan->lav_data_full():
## some cases are empty and will be ignored: 65.## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.
## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.
## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.
## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.
## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.
## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.## Warning: lavaan->lav_object_post_check():
## some estimated ov variances are negative
## Warning: lavaan->lav_object_post_check():
## some estimated ov variances are negative## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.
## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.## Warning: lavaan->lav_data_full():
## some cases are empty and will be ignored: 25.## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.
## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.
## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.
## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.
## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.
## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.
## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.
## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.
## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.
## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.
## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.
## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.
## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.
## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.
## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.
## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.
## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.
## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.
## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.
## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.
## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.
## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.
## Warning: lavaan->lav_object_post_check():
## covariance matrix of latent variables is not positive definite ; use
## lavInspect(fit, "cov.lv") to investigate.idx <- which(conv)
m <- length(idx)
cat("Replications:", R, "\n")## Replications: 1000cat("Converged:", m, "\n")## Converged: 1000cat("Convergence rate:", round(mean(conv), 3), "\n\n")## Convergence rate: 1if (m == 0) stop("No converged replications.")
# ---------------- SUMMARIES ----------------
mean_est <- colMeans(est_store[idx, , drop = FALSE], na.rm = TRUE)
median_est <- apply(est_store[idx, , drop = FALSE], 2, median, na.rm = TRUE)
# Coverage + MCSE
coverage <- colMeans(cover[idx, , drop = FALSE], na.rm = TRUE)
mcse_cov <- sqrt(coverage * (1 - coverage) / m)
# Significance
n_sig <- colSums(sig[idx, , drop = FALSE], na.rm = TRUE)
prop_sig <- n_sig / m
# Errors and bias metrics
err_mat <- err_store[idx, , drop = FALSE]
abs_bias <- colMeans(err_mat, na.rm = TRUE) # mean(est-true)
rmse <- sqrt(colMeans(err_mat^2, na.rm = TRUE)) # sqrt(mean((est-true)^2))
# MCSE for absolute bias: sd(errors)/sqrt(m)
mcse_abs_bias <- apply(err_mat, 2, function(x) {
x <- x[is.finite(x)]
if (length(x) <= 1) return(NA_real_)
sd(x) / sqrt(length(x))
})
# MCSE for RMSE: delta-method-ish approximation using SD of squared error
mcse_rmse <- apply(err_mat, 2, function(x) {
x <- x[is.finite(x)]
if (length(x) <= 2) return(NA_real_)
se_mse <- sd(x^2) / sqrt(length(x))
mse <- mean(x^2)
if (!is.finite(mse) || mse <= 0) return(NA_real_)
se_mse / (2 * sqrt(mse))
})
# Standardized bias: mean(error)/sd(est)
sd_est <- apply(est_store[idx, , drop = FALSE], 2, sd, na.rm = TRUE)
std_bias <- ifelse(sd_est > 0, abs_bias / sd_est, NA_real_)
# Relative bias (%) for true != 0, scaled bias (×100) for true==0
true_vals <- tv_all
rel_bias_pct <- ifelse(abs(true_vals) > .Machine$double.eps,
(mean_est - true_vals) / true_vals * 100,
NA_real_)
scaled_bias_100 <- ifelse(abs(true_vals) <= .Machine$double.eps,
100 * (mean_est - true_vals),
NA_real_)
# MCSE for relative/scaled bias computed from replication-level quantities
# (use stored errors + scaling rules)
mcse_rel_or_scaled <- function(j) {
tv <- true_vals[j]
e <- err_mat[, j]
e <- e[is.finite(e)]
if (length(e) <= 1) return(NA_real_)
if (abs(tv) > .Machine$double.eps) {
z <- (e / tv) * 100
} else {
z <- 100 * e
}
sd(z) / sqrt(length(z))
}
mcse_rel_scaled <- vapply(seq_len(K), mcse_rel_or_scaled, numeric(1))
# ---------------- GROUP PARAMETER BLOCKS ----------------
param_block <- function(lbl) {
if (grepl("^L\\d+$", lbl)) return("Loadings (λ)")
if (lbl == "Corr") return("Factor covariance (ψ)")
if (grepl("^VAR_", lbl)) return("Residual variances (θ)")
if (grepl("^I", lbl)) return("Intercepts (τ)") # IAlc, I5Plus, ...
"Other"
}
results <- data.frame(
Block = vapply(targets, param_block, character(1)),
Parameter = targets,
TrueValue = true_vals,
MeanEstimate = mean_est,
MedianEstimate = median_est,
Coverage95 = coverage,
MCSE_Coverage = mcse_cov,
N_Significant = n_sig,
Prop_Significant = prop_sig,
RelBiasPct = rel_bias_pct, # only for true != 0
ScaledBias_x100 = scaled_bias_100, # only for true == 0
MCSE_RelOrScaled = mcse_rel_scaled,
AbsBias = abs_bias,
MCSE_AbsBias = mcse_abs_bias,
RMSE = rmse,
MCSE_RMSE = mcse_rmse,
StdBias = std_bias,
row.names = NULL,
check.names = FALSE
)
block_order <- c("Loadings (λ)", "Factor covariance (ψ)", "Residual variances (θ)", "Intercepts (τ)", "Other")
results$Block <- factor(results$Block, levels = block_order)
results <- results[order(results$Block, results$Parameter), ]
# If the true value is zero (as for the intercepts here) specify a bias just as the difference (i.e., not divided by true value)
print(results, digits = 3, row.names = FALSE)## Block Parameter TrueValue MeanEstimate MedianEstimate
## Loadings (λ) L1 0.70 0.686509 0.68856
## Loadings (λ) L2 0.70 0.698018 0.69998
## Loadings (λ) L3 0.70 0.693350 0.69675
## Loadings (λ) L4 0.70 0.693870 0.69269
## Loadings (λ) L5 0.70 0.700019 0.69959
## Factor covariance (ψ) Corr 0.80 0.794285 0.79475
## Residual variances (θ) VAR_Alc3M 0.51 0.499778 0.50429
## Residual variances (θ) VAR_FivePlus 0.51 0.487079 0.48662
## Residual variances (θ) VAR_Fri 0.51 0.493988 0.49555
## Residual variances (θ) VAR_Sat 0.51 0.496877 0.49743
## Residual variances (θ) VAR_Thurs 0.51 0.500974 0.50065
## Intercepts (τ) I5Plus 0.00 -0.004116 -0.00479
## Intercepts (τ) IAlc 0.00 -0.000308 0.00594
## Intercepts (τ) IFr 0.00 0.001470 0.00248
## Intercepts (τ) ISa 0.00 0.001687 0.00212
## Intercepts (τ) ITh 0.00 -0.000609 0.00208
## Coverage95 MCSE_Coverage N_Significant Prop_Significant RelBiasPct
## 0.944 0.00727 1000 1.000 -1.92731
## 0.942 0.00739 1000 1.000 -0.28307
## 0.945 0.00721 1000 1.000 -0.94998
## 0.940 0.00751 1000 1.000 -0.87570
## 0.950 0.00689 1000 1.000 0.00268
## 0.955 0.00656 1000 1.000 -0.71439
## 0.947 0.00708 929 0.929 -2.00423
## 0.969 0.00548 922 0.922 -4.49440
## 0.919 0.00863 992 0.992 -3.13952
## 0.939 0.00757 990 0.990 -2.57308
## 0.930 0.00807 992 0.992 -1.76983
## 0.936 0.00774 64 0.064 NA
## 0.941 0.00745 59 0.059 NA
## 0.949 0.00696 51 0.051 NA
## 0.953 0.00669 47 0.047 NA
## 0.943 0.00733 57 0.057 NA
## ScaledBias_x100 MCSE_RelOrScaled AbsBias MCSE_AbsBias RMSE MCSE_RMSE
## NA 0.545 -1.35e-02 0.00381 0.121 0.00278
## NA 0.558 -1.98e-03 0.00391 0.123 0.00268
## NA 0.498 -6.65e-03 0.00349 0.110 0.00239
## NA 0.503 -6.13e-03 0.00352 0.112 0.00263
## NA 0.489 1.88e-05 0.00343 0.108 0.00245
## NA 0.453 -5.72e-03 0.00362 0.115 0.00296
## NA 0.874 -1.02e-02 0.00446 0.141 0.00373
## NA 0.816 -2.29e-02 0.00416 0.134 0.00310
## NA 0.705 -1.60e-02 0.00360 0.115 0.00290
## NA 0.688 -1.31e-02 0.00351 0.112 0.00271
## NA 0.713 -9.03e-03 0.00364 0.115 0.00273
## -0.4116 0.339 -4.12e-03 0.00339 0.107 0.00241
## -0.0308 0.329 -3.08e-04 0.00329 0.104 0.00233
## 0.1470 0.321 1.47e-03 0.00321 0.101 0.00228
## 0.1687 0.324 1.69e-03 0.00324 0.102 0.00229
## -0.0609 0.327 -6.09e-04 0.00327 0.103 0.00237
## StdBias
## -0.111893
## -0.016042
## -0.060268
## -0.055005
## 0.000173
## -0.049883
## -0.072537
## -0.174131
## -0.140731
## -0.118230
## -0.078522
## -0.038362
## -0.002956
## 0.014497
## 0.016455
## -0.005883# ---------------- APA-ish table (flextable) ----------------
apa_sim_table <- function(df, digits = 3) {
tab <- df
# round numeric columns
num_cols <- setdiff(names(tab), c("Block", "Parameter"))
for (cc in num_cols) tab[[cc]] <- round(tab[[cc]], digits)
# Choose a manuscript-friendly subset of columns
tab <- tab[, c(
"Block", "Parameter", "TrueValue", "MeanEstimate", "MedianEstimate",
"Coverage95", "MCSE_Coverage",
"Prop_Significant",
"RelBiasPct", "ScaledBias_x100", "MCSE_RelOrScaled",
"AbsBias", "MCSE_AbsBias",
"RMSE", "MCSE_RMSE",
"StdBias"
)]
ft <- flextable(tab)
ft <- theme_booktabs(ft)
ft <- align(ft, j = 1:2, align = "left", part = "all")
ft <- align(ft, j = 3:ncol(tab), align = "center", part = "all")
ft <- merge_v(ft, j = "Block")
ft <- valign(ft, j = "Block", valign = "top")
ft <- set_header_labels(
ft,
Block = "",
Parameter = "Parameter",
TrueValue = "True",
MeanEstimate = "Mean est.",
MedianEstimate = "Median est.",
Coverage95 = "Coverage",
MCSE_Coverage = "MCSE(cov)",
Prop_Significant = "Prop. sig",
RelBiasPct = "Rel. bias (%)",
ScaledBias_x100 = "Scaled bias (×100)",
MCSE_RelOrScaled = "MCSE(bias)",
AbsBias = "Abs. bias",
MCSE_AbsBias = "MCSE(abs)",
RMSE = "RMSE",
MCSE_RMSE = "MCSE(RMSE)",
StdBias = "Std. bias"
)
ft <- autofit(ft)
ft
}
ft <- apa_sim_table(results, digits = 3)
ftParameter | True | Mean est. | Median est. | Coverage | MCSE(cov) | Prop. sig | Rel. bias (%) | Scaled bias (×100) | MCSE(bias) | Abs. bias | MCSE(abs) | RMSE | MCSE(RMSE) | Std. bias | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Loadings (λ) | L1 | 0.70 | 0.687 | 0.689 | 0.944 | 0.007 | 1.000 | -1.927 | 0.545 | -0.013 | 0.004 | 0.121 | 0.003 | -0.112 | |
L2 | 0.70 | 0.698 | 0.700 | 0.942 | 0.007 | 1.000 | -0.283 | 0.558 | -0.002 | 0.004 | 0.123 | 0.003 | -0.016 | ||
L3 | 0.70 | 0.693 | 0.697 | 0.945 | 0.007 | 1.000 | -0.950 | 0.498 | -0.007 | 0.003 | 0.110 | 0.002 | -0.060 | ||
L4 | 0.70 | 0.694 | 0.693 | 0.940 | 0.008 | 1.000 | -0.876 | 0.503 | -0.006 | 0.004 | 0.112 | 0.003 | -0.055 | ||
L5 | 0.70 | 0.700 | 0.700 | 0.950 | 0.007 | 1.000 | 0.003 | 0.489 | 0.000 | 0.003 | 0.108 | 0.002 | 0.000 | ||
Factor covariance (ψ) | Corr | 0.80 | 0.794 | 0.795 | 0.955 | 0.007 | 1.000 | -0.714 | 0.453 | -0.006 | 0.004 | 0.115 | 0.003 | -0.050 | |
Residual variances (θ) | VAR_Alc3M | 0.51 | 0.500 | 0.504 | 0.947 | 0.007 | 0.929 | -2.004 | 0.874 | -0.010 | 0.004 | 0.141 | 0.004 | -0.073 | |
VAR_FivePlus | 0.51 | 0.487 | 0.487 | 0.969 | 0.005 | 0.922 | -4.494 | 0.816 | -0.023 | 0.004 | 0.134 | 0.003 | -0.174 | ||
VAR_Fri | 0.51 | 0.494 | 0.496 | 0.919 | 0.009 | 0.992 | -3.140 | 0.705 | -0.016 | 0.004 | 0.115 | 0.003 | -0.141 | ||
VAR_Sat | 0.51 | 0.497 | 0.497 | 0.939 | 0.008 | 0.990 | -2.573 | 0.688 | -0.013 | 0.004 | 0.112 | 0.003 | -0.118 | ||
VAR_Thurs | 0.51 | 0.501 | 0.501 | 0.930 | 0.008 | 0.992 | -1.770 | 0.713 | -0.009 | 0.004 | 0.115 | 0.003 | -0.079 | ||
Intercepts (τ) | I5Plus | 0.00 | -0.004 | -0.005 | 0.936 | 0.008 | 0.064 | -0.412 | 0.339 | -0.004 | 0.003 | 0.107 | 0.002 | -0.038 | |
IAlc | 0.00 | 0.000 | 0.006 | 0.941 | 0.007 | 0.059 | -0.031 | 0.329 | 0.000 | 0.003 | 0.104 | 0.002 | -0.003 | ||
IFr | 0.00 | 0.001 | 0.002 | 0.949 | 0.007 | 0.051 | 0.147 | 0.321 | 0.001 | 0.003 | 0.101 | 0.002 | 0.014 | ||
ISa | 0.00 | 0.002 | 0.002 | 0.953 | 0.007 | 0.047 | 0.169 | 0.324 | 0.002 | 0.003 | 0.102 | 0.002 | 0.016 | ||
ITh | 0.00 | -0.001 | 0.002 | 0.943 | 0.007 | 0.057 | -0.061 | 0.327 | -0.001 | 0.003 | 0.103 | 0.002 | -0.006 |