Last updated: 2022-03-08

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Knit directory: Padgett-Dissertation/

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# Load packages & utility functions
source("code/load_packages.R")
source("code/load_utility_functions.R")
# environment options
options(scipen = 999, digits=3)
# get code to simulate data
source("code/study_1/study_1_generate_data.R")

Simulated Data

# data parameters
paravec <- c(
  N = 500
  , J = 5 # N_items
  , C = 3 # N_cat
  , etaCor = .23
  , etasd1 = 1
  , etasd2 = sqrt(0.1)
  , lambda=0.7
  , nu=1.5
  , sigma.ei=0.25
  , rho1=0.1
)
# simulated then saved below
sim_tau <- matrix(
  c(-0.822, -0.751, -0.616, -0.392, -0.865,
    0.780, 0.882, 0.827, 1.030, 0.877),
  ncol=2, nrow=5
)
# Use parameters to simulate data
sim.data <- simulate_data_misclass(paravec, tau=sim_tau)

Describing the Observed (simulated) Data

d1 <- sim.data$Ysampled %>%
  as.data.frame() %>%
  select(contains("y")) %>%
  mutate(id = 1:n()) %>%
  pivot_longer(
    cols = contains("y"),
    names_to = c("item"),
    values_to = "Response"
  ) %>%
  mutate(item = ifelse(nchar(item) > 2, substr(item, 2, 3), substr(item, 2, 2)))
d2 <- sim.data$logt %>%
  as.data.frame() %>%
  select(contains("logt")) %>%
  mutate(id = 1:n()) %>%
  pivot_longer(
    cols = contains("logt"),
    names_to = c("item"),
    values_to = "Time"
  ) %>%
  mutate(item = ifelse(nchar(item) > 5, substr(item, 5, 6), substr(item, 5, 5)))
dat <- left_join(d1, d2)
Joining, by = c("id", "item")
dat_sum <- dat %>%
  select(item, Response, Time) %>%
  group_by(item) %>%
  summarize(
    p1 = table(Response)[1] / n(),
    p2 = table(Response)[2] / n(),
    p3 = table(Response)[3] / n(),
    M1 = mean(Response, na.rm = T),
    Mt = mean(Time, na.rm = T),
    SDt = sd(Time, na.rm = T)
  )

colnames(dat_sum) <-
  c(
    "Item",
    "Prop. R == 1",
    "Prop. R == 2",
    "Prop. R == 3",
    "Mean Response",
    "Mean Response Time",
    "SD Response Time"
  )
dat_sum$Item <- paste0("item_", 1:N_items)

kable(dat_sum, format = "html", digits = 3) %>%
  kable_styling(full_width = T)
Item Prop. R == 1 Prop. R == 2 Prop. R == 3 Mean Response Mean Response Time SD Response Time
item_1 0.308 0.404 0.288 1.98 1.39 0.597
item_2 0.310 0.414 0.276 1.97 1.43 0.618
item_3 0.338 0.386 0.276 1.94 1.43 0.613
item_4 0.362 0.384 0.254 1.89 1.40 0.592
item_5 0.292 0.422 0.286 1.99 1.36 0.582 
# covariance among items
cov(sim.data$Ysampled)
       y1     y2     y3     y4     y5
y1 0.5968 0.0634 0.0428 0.0640 0.0319
y2 0.0634 0.5860 0.0440 0.0364 0.0258
y3 0.0428 0.0440 0.6114 0.0394 0.0457
y4 0.0640 0.0364 0.0394 0.6055 0.0655
y5 0.0319 0.0258 0.0457 0.0655 0.5791
# correlation matrix
psych::polychoric(sim.data$Ysampled)
Call: psych::polychoric(x = sim.data$Ysampled)
Polychoric correlations 
   y1   y2   y3   y4   y5  
y1 1.00                    
y2 0.14 1.00               
y3 0.09 0.09 1.00          
y4 0.13 0.08 0.08 1.00     
y5 0.07 0.05 0.09 0.14 1.00

 with tau of 
       1    2
y1 -0.50 0.56
y2 -0.50 0.59
y3 -0.42 0.59
y4 -0.35 0.66
y5 -0.55 0.57

Model 1: Traditional IFA

Model details

cat(read_file(paste0(w.d, "/code/study_1/model_1.txt")))
model {
### Model
  for(p in 1:N){
    for(i in 1:nit){
      # data model
      y[p,i] ~ dcat(pi[p,i, ])

      # LRV
      ystar[p,i] ~ dnorm(lambda[i]*eta[p], 1)

     # Pr(nu = 3)
      pi[p,i,3] = phi(ystar[p,i] - tau[i,2])
      # Pr(nu = 2)
      pi[p,i,2] = phi(ystar[p,i] - tau[i,1]) - phi(ystar[p,i] - tau[i,2])
      # Pr(nu = 1)
      pi[p,i,1] = 1 - phi(ystar[p,i] - tau[i,1])

    }
  }
  ### Priors
  # person parameters
  for(p in 1:N){
    eta[p] ~ dnorm(0, 1) # latent ability
  }

  for(i in 1:nit){
    # Thresholds
    tau[i, 1] ~ dnorm(0.0,0.1)
    tau[i, 2] ~ dnorm(0, 0.1)T(tau[i, 1],)
    # loadings
    lambda[i] ~ dnorm(0, .44)T(0,)
    # LRV total variance
    # total variance = residual variance + fact. Var.
    theta[i] = 1 + pow(lambda[i],2)
    # standardized loading
    lambda.std[i] = lambda[i]/pow(theta[i],0.5)
  }

  # compute omega
  lambda_sum[1] = lambda[1]
  for(i in 2:nit){
    #lambda_sum (sum factor loadings)
    lambda_sum[i] = lambda_sum[i-1]+lambda[i]
  }
  reli.omega = (pow(lambda_sum[nit],2))/(pow(lambda_sum[nit],2)+nit)
}

Model results

# Save parameters
jags.params <- c("tau", "lambda", "theta", "reli.omega", "lambda.std")
# initial-values
jags.inits <- function(){
    list(
      "tau"=matrix(c(-0.822, -0.751, -0.616, -0.392, -0.865,
                     0.780, 0.882, 0.827, 1.030, 0.877),
                   ncol=2, nrow=5),
      "lambda"=rep(0.7,5),
      "eta"=sim.data$eta[,1,drop=T],
      "ystar"=t(sim.data$ystar)
    )
  }
# data
mydata <- list(y = sim.data$Ysampled,
               N = nrow(sim.data$Ysampled),
               nit = ncol(sim.data$Ysampled))
model.fit <-  R2jags::jags(
  model = paste0(w.d, "/code/study_1/model_1.txt"),
  parameters.to.save = jags.params,
  inits = jags.inits,
  data = mydata,
  n.chains = 4,
  n.burnin = 5000,
  n.iter = 10000
)
module glm loaded
Compiling model graph
   Resolving undeclared variables
   Allocating nodes
Graph information:
   Observed stochastic nodes: 2500
   Unobserved stochastic nodes: 3015
   Total graph size: 25550

Initializing model
print(model.fit, width=1000)
Inference for Bugs model at "C:/Users/noahp/Documents/GitHub/Padgett-Dissertation/code/study_1/model_1.txt", fit using jags,
 4 chains, each with 10000 iterations (first 5000 discarded), n.thin = 5
 n.sims = 4000 iterations saved
               mu.vect sd.vect     2.5%      25%      50%      75%    97.5% Rhat n.eff
lambda[1]        0.573   0.234    0.213    0.420    0.542    0.690    1.108 1.02   500
lambda[2]        0.422   0.172    0.121    0.303    0.408    0.522    0.803 1.00   760
lambda[3]        0.389   0.166    0.097    0.277    0.377    0.482    0.767 1.01   400
lambda[4]        0.553   0.217    0.209    0.408    0.528    0.667    1.021 1.01   360
lambda[5]        0.389   0.160    0.106    0.277    0.379    0.487    0.733 1.00   750
lambda.std[1]    0.477   0.135    0.208    0.387    0.476    0.568    0.742 1.01   580
lambda.std[2]    0.377   0.129    0.120    0.290    0.377    0.463    0.626 1.00   860
lambda.std[3]    0.352   0.127    0.097    0.267    0.353    0.434    0.609 1.02   410
lambda.std[4]    0.466   0.131    0.205    0.378    0.467    0.555    0.714 1.01   480
lambda.std[5]    0.352   0.124    0.105    0.267    0.354    0.438    0.591 1.00   760
reli.omega       0.516   0.063    0.377    0.478    0.523    0.561    0.620 1.01   500
tau[1,1]        -0.771   0.102   -0.978   -0.834   -0.767   -0.702   -0.584 1.01  1100
tau[2,1]        -0.735   0.091   -0.918   -0.797   -0.732   -0.673   -0.562 1.00  2500
tau[3,1]        -0.618   0.088   -0.798   -0.677   -0.614   -0.559   -0.448 1.00  3900
tau[4,1]        -0.540   0.095   -0.732   -0.597   -0.538   -0.479   -0.362 1.01  1000
tau[5,1]        -0.808   0.091   -0.983   -0.871   -0.806   -0.746   -0.631 1.00  1300
tau[1,2]         0.861   0.109    0.669    0.789    0.855    0.926    1.094 1.01   750
tau[2,2]         0.884   0.095    0.700    0.821    0.883    0.946    1.078 1.00   850
tau[3,2]         0.879   0.092    0.706    0.816    0.877    0.938    1.063 1.00  1100
tau[4,2]         1.011   0.108    0.815    0.940    1.006    1.075    1.238 1.00  1400
tau[5,2]         0.832   0.090    0.661    0.772    0.831    0.890    1.013 1.00  4000
theta[1]         1.383   0.365    1.045    1.177    1.293    1.476    2.228 1.06   200
theta[2]         1.208   0.168    1.015    1.092    1.166    1.273    1.645 1.01   470
theta[3]         1.179   0.157    1.009    1.077    1.142    1.232    1.589 1.03   250
theta[4]         1.353   0.317    1.044    1.167    1.279    1.445    2.042 1.04   180
theta[5]         1.177   0.143    1.011    1.077    1.144    1.237    1.537 1.00   830
deviance      3947.897  68.595 3811.240 3902.434 3948.085 3993.949 4080.362 1.00   760

For each parameter, n.eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor (at convergence, Rhat=1).

DIC info (using the rule, pD = var(deviance)/2)
pD = 2345.3 and DIC = 6293.2
DIC is an estimate of expected predictive error (lower deviance is better).

Posterior Distribution Summary

# extract for plotting
jags.mcmc <- as.mcmc(model.fit)
a <- colnames(as.data.frame(jags.mcmc[[1]]))
fit.mcmc <- data.frame(as.matrix(jags.mcmc, chains = T, iters = T))
colnames(fit.mcmc) <- c("chain", "iter", a)
fit.mcmc.ggs <- ggmcmc::ggs(jags.mcmc) # for GRB plot

Categroy Thresholds (\(\tau\))

# tau
bayesplot::mcmc_areas(fit.mcmc, regex_pars = "tau", prob = 0.8); ggsave("fig/study1_model1_tau_dens.pdf")

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bayesplot::mcmc_acf(fit.mcmc, regex_pars = "tau"); ggsave("fig/study1_model1_tau_acf.pdf")

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bayesplot::mcmc_trace(fit.mcmc, regex_pars = "tau"); ggsave("fig/study1_model1_tau_trace.pdf")

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ggmcmc::ggs_grb(fit.mcmc.ggs, family = "tau") + theme_bw()+theme(panel.grid = element_blank()); ggsave("fig/study1_model1_tau_grb.pdf")

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Factor Loadings (\(\lambda\))

bayesplot::mcmc_areas(fit.mcmc, regex_pars = "lambda", prob = 0.8)

bayesplot::mcmc_acf(fit.mcmc, regex_pars = "lambda")

bayesplot::mcmc_trace(fit.mcmc, regex_pars = "lambda")

ggmcmc::ggs_grb(fit.mcmc.ggs, family = "lambda")

bayesplot::mcmc_areas(fit.mcmc, regex_pars = "lambda.std", prob = 0.8); ggsave("fig/study1_model1_lambda_dens.pdf")

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bayesplot::mcmc_acf(fit.mcmc, regex_pars = "lambda.std"); ggsave("fig/study1_model1_lambda_acf.pdf")

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bayesplot::mcmc_trace(fit.mcmc, regex_pars = "lambda.std", nrow=1); ggsave("fig/study1_model1_lambda_trace.pdf")
Warning: The following arguments were unrecognized and ignored: nrow

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ggmcmc::ggs_grb(fit.mcmc.ggs, family = "lambda.std") + theme_bw()+theme(panel.grid = element_blank()); ggsave("fig/study1_model1_lambda_grb.pdf")

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Latent Response Total Variance (\(\theta\))

bayesplot::mcmc_areas(fit.mcmc, regex_pars = "theta", prob = 0.8); ggsave("fig/study1_model1_theta_dens.pdf")

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bayesplot::mcmc_acf(fit.mcmc, regex_pars = "theta"); ggsave("fig/study1_model1_theta_acf.pdf")

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bayesplot::mcmc_trace(fit.mcmc, regex_pars = "theta"); ggsave("fig/study1_model1_theta_trace.pdf")

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ggmcmc::ggs_grb(fit.mcmc.ggs, family = "theta") + theme_bw()+theme(panel.grid = element_blank()); ggsave("fig/study1_model1_theta_grb.pdf")

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Factor Reliability Omega (\(\omega\))

bayesplot::mcmc_areas(fit.mcmc, regex_pars = "reli.omega", prob = 0.8); ggsave("fig/study1_model1_omega_dens.pdf")

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bayesplot::mcmc_acf(fit.mcmc, regex_pars = "reli.omega"); ggsave("fig/study1_model1_omega_acf.pdf")

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bayesplot::mcmc_trace(fit.mcmc, regex_pars = "reli.omega"); ggsave("fig/study1_model1_omega_trace.pdf")

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ggmcmc::ggs_grb(fit.mcmc.ggs, family = "reli.omega") + theme_bw()+theme(panel.grid = element_blank()); ggsave("fig/study1_model1_omega_grb.pdf")

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# extract omega posterior for results comparison
extracted_omega <- data.frame(model_1 = fit.mcmc$reli.omega)
write.csv(x=extracted_omega, file=paste0(getwd(),"/data/study_1/extracted_omega_m1.csv"))

Posterior Predictive Distributions

# Posterior Predictive Check
Niter <- 200
model.fit$model$recompile()
Compiling model graph
   Resolving undeclared variables
   Allocating nodes
Graph information:
   Observed stochastic nodes: 2500
   Unobserved stochastic nodes: 3015
   Total graph size: 25550

Initializing model
fit.extra <- rjags::jags.samples(model.fit$model, variable.names = "pi", n.iter = Niter)
NOTE: Stopping adaptation
N <- model.fit$model$data()[[1]]
nit <- 5
nchain=4
C <- 3
n <- i <- iter <- ppc.row.i <- 1
y.prob.ppc <- array(dim=c(Niter*nchain, nit, C))
for(chain in 1:nchain){
  for(iter in 1:Niter){
    # initialize simulated y for this iteration
    y <- matrix(nrow=N, ncol=nit)
    # loop over item
    for(i in 1:nit){
      # simulated data for item i for each person
      for(n in 1:N){
        y[n,i] <- sample(1:C, 1, prob = fit.extra$pi[n, i, 1:C, iter, chain])
      }
      # computer proportion of each response category
      for(c in 1:C){
        y.prob.ppc[ppc.row.i,i,c] <- sum(y[,i]==c)/N
      }
    }
    
    # update row of output
    ppc.row.i = ppc.row.i + 1
  }
}

yppcmat <- matrix(c(y.prob.ppc), ncol=1)
z <- expand.grid(1:(Niter*nchain), 1:nit, 1:C)
yppcmat <- data.frame(  iter = z[,1], nit=z[,2], C=z[,3], yppc = yppcmat)

ymat <- model.fit$model$data()[[3]]
y.prob <- matrix(ncol=C, nrow=nit)
for(i in 1:nit){
  for(c in 1:C){
    y.prob[i,c] <- sum(ymat[,i]==c)/N
  }
}
yobsmat <- matrix(c(y.prob), ncol=1)
z <- expand.grid(1:nit, 1:C)
yobsmat <- data.frame(nit=z[,1], C=z[,2], yobs = yobsmat)
plot.ppc <- full_join(yppcmat, yobsmat)
Joining, by = c("nit", "C")
p <- plot.ppc %>%
  mutate(C    = as.factor(C),
         item = nit) %>%
  ggplot()+
  geom_boxplot(aes(x=C,y=y.prob.ppc), outlier.colour = NA)+
  geom_point(aes(x=C,y=yobs), color="red")+
  lims(y=c(0, 0.67))+
  labs(y="Posterior Predictive Category Proportion", x="Item Category")+
  facet_wrap(.~nit, nrow=1)+
  theme_bw()+
  theme(
    panel.grid = element_blank(),
    strip.background = element_rect(fill="white")
  )
p

ggsave(filename = "fig/study1_model1_ppc_y.pdf",plot=p,width = 6, height=4,units="in")
ggsave(filename = "fig/study1_model1_ppc_y.png",plot=p,width = 6, height=4,units="in")
ggsave(filename = "fig/study1_model1_ppc_y.eps",plot=p,width = 6, height=4,units="in")

Manuscript Table and Figures

Table

# print to xtable
print(
  xtable(
    model.fit$BUGSoutput$summary,
    caption = c("study1 Model 1 posterior distribution summary")
    ,align = "lrrrrrrrrr"
  ),
  include.rownames=T,
  booktabs=T
)
% latex table generated in R 4.0.5 by xtable 1.8-4 package
% Tue Mar 08 18:18:40 2022
\begin{table}[ht]
\centering
\begin{tabular}{lrrrrrrrrr}
  \toprule
 & mean & sd & 2.5\% & 25\% & 50\% & 75\% & 97.5\% & Rhat & n.eff \\ 
  \midrule
deviance & 3947.90 & 68.60 & 3811.24 & 3902.43 & 3948.08 & 3993.95 & 4080.36 & 1.00 & 760.00 \\ 
  lambda[1] & 0.57 & 0.23 & 0.21 & 0.42 & 0.54 & 0.69 & 1.11 & 1.02 & 500.00 \\ 
  lambda[2] & 0.42 & 0.17 & 0.12 & 0.30 & 0.41 & 0.52 & 0.80 & 1.00 & 760.00 \\ 
  lambda[3] & 0.39 & 0.17 & 0.10 & 0.28 & 0.38 & 0.48 & 0.77 & 1.01 & 400.00 \\ 
  lambda[4] & 0.55 & 0.22 & 0.21 & 0.41 & 0.53 & 0.67 & 1.02 & 1.01 & 360.00 \\ 
  lambda[5] & 0.39 & 0.16 & 0.11 & 0.28 & 0.38 & 0.49 & 0.73 & 1.00 & 750.00 \\ 
  lambda.std[1] & 0.48 & 0.14 & 0.21 & 0.39 & 0.48 & 0.57 & 0.74 & 1.01 & 580.00 \\ 
  lambda.std[2] & 0.38 & 0.13 & 0.12 & 0.29 & 0.38 & 0.46 & 0.63 & 1.00 & 860.00 \\ 
  lambda.std[3] & 0.35 & 0.13 & 0.10 & 0.27 & 0.35 & 0.43 & 0.61 & 1.02 & 410.00 \\ 
  lambda.std[4] & 0.47 & 0.13 & 0.20 & 0.38 & 0.47 & 0.55 & 0.71 & 1.01 & 480.00 \\ 
  lambda.std[5] & 0.35 & 0.12 & 0.11 & 0.27 & 0.35 & 0.44 & 0.59 & 1.01 & 760.00 \\ 
  reli.omega & 0.52 & 0.06 & 0.38 & 0.48 & 0.52 & 0.56 & 0.62 & 1.01 & 500.00 \\ 
  tau[1,1] & -0.77 & 0.10 & -0.98 & -0.83 & -0.77 & -0.70 & -0.58 & 1.01 & 1100.00 \\ 
  tau[2,1] & -0.73 & 0.09 & -0.92 & -0.80 & -0.73 & -0.67 & -0.56 & 1.00 & 2500.00 \\ 
  tau[3,1] & -0.62 & 0.09 & -0.80 & -0.68 & -0.61 & -0.56 & -0.45 & 1.00 & 3900.00 \\ 
  tau[4,1] & -0.54 & 0.09 & -0.73 & -0.60 & -0.54 & -0.48 & -0.36 & 1.01 & 1000.00 \\ 
  tau[5,1] & -0.81 & 0.09 & -0.98 & -0.87 & -0.81 & -0.75 & -0.63 & 1.00 & 1300.00 \\ 
  tau[1,2] & 0.86 & 0.11 & 0.67 & 0.79 & 0.85 & 0.93 & 1.09 & 1.01 & 750.00 \\ 
  tau[2,2] & 0.88 & 0.10 & 0.70 & 0.82 & 0.88 & 0.95 & 1.08 & 1.00 & 850.00 \\ 
  tau[3,2] & 0.88 & 0.09 & 0.71 & 0.82 & 0.88 & 0.94 & 1.06 & 1.00 & 1100.00 \\ 
  tau[4,2] & 1.01 & 0.11 & 0.81 & 0.94 & 1.01 & 1.07 & 1.24 & 1.00 & 1400.00 \\ 
  tau[5,2] & 0.83 & 0.09 & 0.66 & 0.77 & 0.83 & 0.89 & 1.01 & 1.00 & 4000.00 \\ 
  theta[1] & 1.38 & 0.36 & 1.05 & 1.18 & 1.29 & 1.48 & 2.23 & 1.06 & 200.00 \\ 
  theta[2] & 1.21 & 0.17 & 1.01 & 1.09 & 1.17 & 1.27 & 1.65 & 1.01 & 470.00 \\ 
  theta[3] & 1.18 & 0.16 & 1.01 & 1.08 & 1.14 & 1.23 & 1.59 & 1.03 & 250.00 \\ 
  theta[4] & 1.35 & 0.32 & 1.04 & 1.17 & 1.28 & 1.45 & 2.04 & 1.04 & 180.00 \\ 
  theta[5] & 1.18 & 0.14 & 1.01 & 1.08 & 1.14 & 1.24 & 1.54 & 1.00 & 830.00 \\ 
   \bottomrule
\end{tabular}
\caption{study1 Model 1 posterior distribution summary} 
\end{table}

Figure

plot.dat <- fit.mcmc %>%
  select(!c("iter", "deviance", "reli.omega", paste0("lambda[",1:5,"]")))%>%
  pivot_longer(
    cols= !c("chain"),
    names_to="variable",
    values_to="value"
  ) %>%
  mutate(
    variable = factor(
      variable,
      levels = c(
        "lambda.std[1]", "theta[1]", "tau[1,1]", "tau[1,2]",
        "lambda.std[2]", "theta[2]", "tau[2,1]", "tau[2,2]",
        "lambda.std[3]", "theta[3]", "tau[3,1]", "tau[3,2]",
        "lambda.std[4]", "theta[4]", "tau[4,1]", "tau[4,2]",
        "lambda.std[5]", "theta[5]", "tau[5,1]", "tau[5,2]"
      ), ordered = T
    )
  )

p <- ggplot(plot.dat, aes(x=value, group=variable))+
  geom_density(adjust=2)+
  facet_wrap(variable~., scales="free_y", ncol=4) +
  lims(x=c(-2, 4))+
  labs(x="Magnitude of Parameter",
       y="Posterior Density")+
  theme_bw()+
  theme(
    panel.grid = element_blank(),
    strip.background = element_rect(fill="white"),
    axis.text.y = element_blank(),
    axis.ticks.y = element_blank() 
  )
p
Warning: Removed 20 rows containing non-finite values (stat_density).

ggsave(filename = "fig/study1_model1_posterior_dist.pdf",plot=p,width = 7, height=5,units="in")
Warning: Removed 20 rows containing non-finite values (stat_density).
ggsave(filename = "fig/study1_model1_posterior_dist.png",plot=p,width = 7, height=5,units="in")
Warning: Removed 20 rows containing non-finite values (stat_density).
ggsave(filename = "fig/study1_model1_posterior_dist.eps",plot=p,width = 7, height=5,units="in")
Warning: Removed 20 rows containing non-finite values (stat_density).

sessionInfo()
R version 4.0.5 (2021-03-31)
Platform: x86_64-w64-mingw32/x64 (64-bit)
Running under: Windows 10 x64 (build 22000)

Matrix products: default

locale:
[1] LC_COLLATE=English_United States.1252 
[2] LC_CTYPE=English_United States.1252   
[3] LC_MONETARY=English_United States.1252
[4] LC_NUMERIC=C                          
[5] LC_TIME=English_United States.1252    

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
 [1] car_3.0-10           carData_3.0-4        mvtnorm_1.1-1       
 [4] LaplacesDemon_16.1.4 runjags_2.2.0-2      lme4_1.1-26         
 [7] Matrix_1.3-2         sirt_3.9-4           R2jags_0.6-1        
[10] rjags_4-12           eRm_1.0-2            diffIRT_1.5         
[13] statmod_1.4.35       xtable_1.8-4         kableExtra_1.3.4    
[16] lavaan_0.6-10        polycor_0.7-10       bayesplot_1.8.0     
[19] ggmcmc_1.5.1.1       coda_0.19-4          data.table_1.14.0   
[22] patchwork_1.1.1      forcats_0.5.1        stringr_1.4.0       
[25] dplyr_1.0.5          purrr_0.3.4          readr_1.4.0         
[28] tidyr_1.1.3          tibble_3.1.0         ggplot2_3.3.5       
[31] tidyverse_1.3.0      workflowr_1.6.2     

loaded via a namespace (and not attached):
 [1] minqa_1.2.4        TAM_3.5-19         colorspace_2.0-0   rio_0.5.26        
 [5] ellipsis_0.3.1     ggridges_0.5.3     rprojroot_2.0.2    fs_1.5.0          
 [9] rstudioapi_0.13    farver_2.1.0       fansi_0.4.2        lubridate_1.7.10  
[13] xml2_1.3.2         splines_4.0.5      mnormt_2.0.2       knitr_1.31        
[17] jsonlite_1.7.2     nloptr_1.2.2.2     broom_0.7.5        dbplyr_2.1.0      
[21] compiler_4.0.5     httr_1.4.2         backports_1.2.1    assertthat_0.2.1  
[25] cli_2.3.1          later_1.1.0.1      htmltools_0.5.1.1  tools_4.0.5       
[29] gtable_0.3.0       glue_1.4.2         reshape2_1.4.4     Rcpp_1.0.7        
[33] cellranger_1.1.0   jquerylib_0.1.3    vctrs_0.3.6        svglite_2.0.0     
[37] nlme_3.1-152       psych_2.0.12       xfun_0.21          ps_1.6.0          
[41] openxlsx_4.2.3     rvest_1.0.0        lifecycle_1.0.0    MASS_7.3-53.1     
[45] scales_1.1.1       ragg_1.1.1         hms_1.0.0          promises_1.2.0.1  
[49] parallel_4.0.5     RColorBrewer_1.1-2 curl_4.3           yaml_2.2.1        
[53] sass_0.3.1         reshape_0.8.8      stringi_1.5.3      highr_0.8         
[57] zip_2.1.1          boot_1.3-27        rlang_0.4.10       pkgconfig_2.0.3   
[61] systemfonts_1.0.1  evaluate_0.14      lattice_0.20-41    labeling_0.4.2    
[65] tidyselect_1.1.0   GGally_2.1.1       plyr_1.8.6         magrittr_2.0.1    
[69] R6_2.5.0           generics_0.1.0     DBI_1.1.1          foreign_0.8-81    
[73] pillar_1.5.1       haven_2.3.1        withr_2.4.1        abind_1.4-5       
[77] modelr_0.1.8       crayon_1.4.1       utf8_1.1.4         tmvnsim_1.0-2     
[81] rmarkdown_2.7      grid_4.0.5         readxl_1.3.1       CDM_7.5-15        
[85] pbivnorm_0.6.0     git2r_0.28.0       reprex_1.0.0       digest_0.6.27     
[89] webshot_0.5.2      httpuv_1.5.5       textshaping_0.3.1  stats4_4.0.5      
[93] munsell_0.5.0      viridisLite_0.3.0  bslib_0.2.4        R2WinBUGS_2.1-21