Chapter 9 Statistical Power for Nested Model (Reference Figure 8.6 on page 130).

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
Plausible Initial Model
require(lavaan)
## Loading required package: 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(MASS)
library(lavaan)

set.seed(12345)

# --- Population Sigma (your matrix taken from the covariance matrix in Onyx) ---
#Variable Names
vars <- c("Alc3M", "FivePlus", "Thurs", "Fri", "Sat")
#Assumed covariance matrix (taken from plausible model)
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)
)
#Sample size
N <- 100
#Number of Replications
R <- 1000

# MCAR missingness rates
pmiss <- c(Alc3M = 0.50, FivePlus = 0.40, Thurs = 0.20, Fri = 0.20, Sat = 0.20)

make_mcar_missing <- function(dat, pmiss) {
  for (v in names(pmiss)) {
    dat[runif(nrow(dat)) < pmiss[[v]], v] <- NA
  }
  dat
}

mu <- rep(0, length(vars)); names(mu) <- vars

# --- H1: free factor correlation ---
model_free <- "
  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    ~~ PastWl   # FREE correlation/covariance

  Alc3M    ~ 1
  FivePlus ~ 1
  Thurs    ~ 1
  Fri      ~ 1
  Sat      ~ 1
"

# --- H0: correlation fixed to 1 ---
model_r1 <- gsub("Use    ~~ PastWl", "Use    ~~ 1*PastWl", model_free, fixed = TRUE)

alpha <- 0.05

reject <- logical(R)
converged <- logical(R)
rho_hat <- rep(NA_real_, R)

#Do the replications (Notice that sometimes all variables are missing for some observations in the data)
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_free <- tryCatch(lavaan(model_free, data = dat, missing = "fiml", fixed.x = FALSE),
                       error = function(e) NULL)
  fit_r1   <- tryCatch(lavaan(model_r1,   data = dat, missing = "fiml", fixed.x = FALSE),
                       error = function(e) NULL)
  
  if (!is.null(fit_free) && !is.null(fit_r1) &&
      inspect(fit_free, "converged") && inspect(fit_r1, "converged")) {
    
    converged[r] <- TRUE
    
    # grab estimated factor correlation (standardized latent covariance)
    # (this is handy for reporting, not needed for the LRT)
    rho_hat[r] <- lavInspect(fit_free, "cor.lv")[ "Use", "PastWl" ]
    
    # LRT statistic (H0 nested in H1)
    LR <- as.numeric(lavTestLRT(fit_r1, fit_free)$Chisq[2])
    
    # boundary-corrected (50:50 mixture) p-value
    p_mixture <- 0.5 * (1 - pchisq(LR, df = 1))
    
    reject[r] <- (p_mixture < alpha)
  }
}
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 70.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 70.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 37.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 37.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 55.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 55.
## 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: 91.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 91.
## 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_data_full():  
##    some cases are empty and will be ignored: 41.
## 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: 41.
## 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: 44 78.
## Warning: lavaan->lav_object_post_check():  
##    some estimated ov variances are negative
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 44 78.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 94.
## 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: 94.
## 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_object_post_check():  
##    some estimated ov variances are negative
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 76.
## 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: 76.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 94.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 94.
## 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_data_full():  
##    some cases are empty and will be ignored: 38.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 38.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 54.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 54.
## 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: 47.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 47.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 71.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 71.
## 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: 11.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 11.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 83.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 83.
## 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: 38.
## 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: 38.
## 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_data_full():  
##    some cases are empty and will be ignored: 65.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 83.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 83.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 43.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 43.
## 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():  
##    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: 14 51.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 14 51.
## 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_data_full():  
##    some cases are empty and will be ignored: 67.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 67.
## 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_data_full():  
##    some cases are empty and will be ignored: 69.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 69.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 66.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 66.
## 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: 29.
## 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: 29.
## Warning: lavaan->lav_object_post_check():  
##    some estimated ov variances are negative
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 38.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 38.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 26.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 26.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 49.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 49.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 54.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 54.
## Warning: lavaan->lav_object_post_check():  
##    some estimated ov variances are negative
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 5.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 5.
## Warning: lavaan->lav_object_post_check():  
##    some estimated ov variances are negative
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 91.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 91.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 32.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 32.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 42.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 42.
## 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_data_full():  
##    some cases are empty and will be ignored: 95.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 95.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 61.
## 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: 61.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 85.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 85.
## 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_data_full():  
##    some cases are empty and will be ignored: 69 72.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 69 72.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 79.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 79.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 18.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 18.
## 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():  
##    some estimated ov variances are negative
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 53.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 53.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 65.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 65.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 46.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 46.
## 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: 6.
## 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: 6.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 46.
## 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: 46.
## 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: 45.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 45.
## 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_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: 55.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 55.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 47.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 47.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 86.
## 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: 86.
## 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: 12.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 12.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 23.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 23.
## Warning: lavaan->lav_object_post_check():  
##    some estimated ov variances are negative
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 40.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 40.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 31.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 31.
## 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: 96.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 96.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 8.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 8.
## 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: 25.
## 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_data_full():  
##    some cases are empty and will be ignored: 90.
## Warning: lavaan->lav_object_post_check():  
##    some estimated ov variances are negative
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 90.
## 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: 28.
## 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: 28.
## 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: 30 85.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 30 85.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 10.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 10.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 16.
## 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: 16.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 41.
## 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: 41.
## 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_data_full():  
##    some cases are empty and will be ignored: 97.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 97.
## 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_data_full():  
##    some cases are empty and will be ignored: 23.
## 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: 23.
## Warning: lavaan->lav_object_post_check():  
##    some estimated ov variances are negative
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 69.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 69.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 47.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 47.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 25.
## 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_data_full():  
##    some cases are empty and will be ignored: 92.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 92.
## 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_data_full():  
##    some cases are empty and will be ignored: 92.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 92.
## 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: 57.
## Warning: lavaan->lav_object_post_check():  
##    some estimated ov variances are negative
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 57.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 59.
## Warning: lavaan->lav_object_post_check():  
##    some estimated ov variances are negative
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 59.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 69.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 69.
## 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_data_full():  
##    some cases are empty and will be ignored: 95.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 95.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 47.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 47.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 75.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 75.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 91.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 91.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 50.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 50.
## 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_data_full():  
##    some cases are empty and will be ignored: 67.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 67.
## 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_data_full():  
##    some cases are empty and will be ignored: 86.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 86.
## Warning: lavaan->lav_object_post_check():  
##    some estimated ov variances are negative
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 93.
## 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: 93.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 88.
## 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: 88.
## 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_data_full():  
##    some cases are empty and will be ignored: 11.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 11.
## Warning: lavaan->lav_object_post_check():  
##    some estimated ov variances are negative
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 24 78.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 24 78.
## 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():  
##    some estimated ov variances are negative
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 11 48.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 11 48.
## 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():  
##    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: 77.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 77.
## 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_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():  
##    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_data_full():  
##    some cases are empty and will be ignored: 35.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 35.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 92.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 92.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 58.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 58.
## Warning: lavaan->lav_object_post_check():  
##    some estimated ov variances are negative
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 2.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 2.
## 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_data_full():  
##    some cases are empty and will be ignored: 36.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 36.
## 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: 53.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 53.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 10.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 10.
## Warning: lavaan->lav_object_post_check():  
##    some estimated ov variances are negative
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 68.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 68.
## 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():  
##    some estimated ov variances are negative
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 70.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 70.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 52.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 52.
## Warning: lavaan->lav_object_post_check():  
##    some estimated ov variances are negative
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 78.
## 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: 78.
## 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: 30.
## Warning: lavaan->lav_object_post_check():  
##    some estimated ov variances are negative
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 30.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 50.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 50.
## Warning: lavaan->lav_object_post_check():  
##    some estimated ov variances are negative
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 81.
## Warning: lavaan->lav_object_post_check():  
##    some estimated ov variances are negative
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 81.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 79.
## 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: 79.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 50.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 50.
## 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_data_full():  
##    some cases are empty and will be ignored: 72.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 72.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 23.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 23.
## 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: 53.
## Warning: lavaan->lav_object_post_check():  
##    some estimated ov variances are negative
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 53.
## 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_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: 73.
## 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: 73.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 36 55.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 36 55.
## 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: 5.
## 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: 5.
## 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():  
##    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: 74.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 74.
## Warning: lavaan->lav_object_post_check():  
##    some estimated ov variances are negative
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 76.
## Warning: lavaan->lav_object_post_check():  
##    some estimated ov variances are negative
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 76.
## 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_data_full():  
##    some cases are empty and will be ignored: 3 61 77.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 3 61 77.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 28.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 28.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 21.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 21.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 79.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 79.
## 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():  
##    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: 58.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 58.
## 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_data_full():  
##    some cases are empty and will be ignored: 41.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 41.
## Warning: lavaan->lav_object_post_check():  
##    some estimated ov variances are negative
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 26.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 26.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 57.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 57.
## 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: 68.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 68.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 69.
## 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: 69.
## 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: 92.
## Warning: lavaan->lav_object_post_check():  
##    some estimated ov variances are negative
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 92.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 5.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 5.
## 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: 48.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 48.
## 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: 83.
## 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: 83.
## Warning: lavaan->lav_object_post_check():  
##    some estimated ov variances are negative
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 15.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 15.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 77.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 77.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 18.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 18.
## 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: 61.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 61.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 67.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 67.
## Warning: lavaan->lav_object_post_check():  
##    some estimated ov variances are negative
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 41.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 41.
## 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: 90.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 90.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 3.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 3.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 36.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 36.
## 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_data_full():  
##    some cases are empty and will be ignored: 97.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 97.
## Warning: lavaan->lav_object_post_check():  
##    some estimated ov variances are negative
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 59.
## Warning: lavaan->lav_object_post_check():  
##    some estimated ov variances are negative
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 59.
## Warning: lavaan->lav_object_post_check():  
##    some estimated ov variances are negative
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 66.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 66.
## 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_data_full():  
##    some cases are empty and will be ignored: 55.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 55.
## 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_data_full():  
##    some cases are empty and will be ignored: 94.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 94.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 26.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 26.
## 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():  
##    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: 69.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 69.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 71.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 71.
## Warning: lavaan->lav_lavaan_step11_estoptim():  
##    the optimizer warns that a solution has NOT been found!
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 57.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 57.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 20.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 20.
## 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():  
##    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: 33.
## Warning: lavaan->lav_data_full():  
##    some cases are empty and will be ignored: 33.
## 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_object_post_check():  
##    covariance matrix of latent variables is not positive definite ; use 
##    lavInspect(fit, "cov.lv") to investigate.
cat("Replications:", R, "\n")
## Replications: 1000
cat("Converged:", sum(converged), "\n")
## Converged: 999
cat("Convergence rate:", mean(converged), "\n")
## Convergence rate: 0.999
cat("Estimated power (reject H0: rho=1):", mean(reject[converged]), "\n\n")
## Estimated power (reject H0: rho=1): 0.8648649
cat("Summary of estimated rho (free model), converged reps:\n")
## Summary of estimated rho (free model), converged reps:
print(summary(rho_hat[converged]))
##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
##   0.2081   0.6932   0.7980   0.9545   0.9013 138.3335