Last updated: 2022-01-16

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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)
# generate data for study 1
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 3: IFA with RT only

Model details

cat(read_file(paste0(w.d, "/code/study_1/model_3.txt")))
model {
### Model
  for(p in 1:N){
    for(i in 1:nit){
      # data model
      y[p,i] ~ dcat(omega[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])

      # MISCLASSIFICATION MODEL
      for(c in 1:ncat){
        # generate informative prior for misclassificaiton
        #   parameters based on RT
        for(ct in 1:ncat){
          alpha[p,i,ct,c] <- ifelse(c == ct,
                                    ilogit(lrt[p,i]),
                                    (1/(ncat-1))*(1-ilogit(lrt[p,i]))
          )
        }
        # sample misclassification parameters using the informative priors
        gamma[p,i,c,1:ncat] ~ ddirch(alpha[p,i,c,1:ncat])
        # observed category prob (Pr(y=c))
        omega[p,i, c] = gamma[p,i,c,1]*pi[p,i,1] +
          gamma[p,i,c,2]*pi[p,i,2] +
          gamma[p,i,c,3]*pi[p,i,3]
      }

    }
  }
  ### 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),
      "gamma"=sim.data$gamma
    )
  }
mydata <- list(
  y = sim.data$Ysampled,
  lrt = sim.data$logt,
  N = nrow(sim.data$Ysampled),
  nit = ncol(sim.data$Ysampled),
  ncat = 3
)
model.fit <-  R2jags::jags(
  model = paste0(w.d, "/code/study_1/model_3.txt"),
  parameters.to.save = jags.params,
  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: 10515
   Total graph size: 108063

Initializing model
print(model.fit, width=1000)
Inference for Bugs model at "C:/Users/noahp/Documents/GitHub/Padgett-Dissertation/code/study_1/model_3.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]        0.894   0.427    0.281    0.611    0.823    1.078    2.005 1.09    48
lambda[2]        0.858   0.385    0.297    0.584    0.794    1.067    1.737 1.01   190
lambda[3]        0.466   0.220    0.082    0.310    0.451    0.600    0.952 1.04    74
lambda[4]        0.902   0.383    0.327    0.627    0.843    1.105    1.804 1.05    60
lambda[5]        0.432   0.230    0.056    0.262    0.407    0.577    0.947 1.01   390
lambda.std[1]    0.621   0.158    0.271    0.521    0.635    0.733    0.895 1.07    60
lambda.std[2]    0.610   0.157    0.285    0.504    0.622    0.730    0.867 1.01   270
lambda.std[3]    0.403   0.157    0.082    0.296    0.411    0.514    0.689 1.04    83
lambda.std[4]    0.630   0.150    0.311    0.531    0.644    0.742    0.875 1.04    69
lambda.std[5]    0.376   0.167    0.056    0.253    0.377    0.500    0.688 1.01   390
reli.omega       0.709   0.060    0.570    0.674    0.717    0.752    0.802 1.01  2200
tau[1,1]        -0.889   0.208   -1.411   -0.991   -0.860   -0.752   -0.575 1.13    41
tau[2,1]        -0.886   0.178   -1.303   -0.987   -0.867   -0.763   -0.594 1.01   430
tau[3,1]        -0.629   0.124   -0.874   -0.713   -0.626   -0.543   -0.396 1.01   210
tau[4,1]        -0.543   0.152   -0.879   -0.630   -0.530   -0.444   -0.271 1.01   220
tau[5,1]        -0.875   0.137   -1.162   -0.961   -0.871   -0.779   -0.619 1.00  4000
tau[1,2]         1.085   0.221    0.756    0.944    1.056    1.181    1.653 1.08    53
tau[2,2]         1.159   0.210    0.828    1.015    1.132    1.267    1.647 1.01   210
tau[3,2]         1.044   0.139    0.783    0.949    1.041    1.134    1.324 1.00   660
tau[4,2]         1.312   0.239    0.938    1.152    1.284    1.429    1.867 1.06    55
tau[5,2]         0.961   0.135    0.700    0.869    0.960    1.052    1.224 1.00  4000
theta[1]         1.982   1.065    1.079    1.373    1.677    2.163    5.018 1.16    34
theta[2]         1.885   0.840    1.088    1.341    1.630    2.138    4.018 1.03   130
theta[3]         1.266   0.237    1.007    1.096    1.204    1.360    1.906 1.05    58
theta[4]         1.960   0.865    1.107    1.394    1.710    2.222    4.255 1.06    56
theta[5]         1.240   0.239    1.003    1.069    1.166    1.333    1.897 1.01   480
deviance      4248.466  72.869 4105.088 4199.490 4247.851 4297.012 4395.539 1.00   870

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 = 2648.1 and DIC = 6896.6
DIC is an estimate of expected predictive error (lower deviance is better).

Warning: The above code chunk cached its results, but it won’t be re-run if previous chunks it depends on are updated. If you need to use caching, it is highly recommended to also set knitr::opts_chunk$set(autodep = TRUE) at the top of the file (in a chunk that is not cached). Alternatively, you can customize the option dependson for each individual chunk that is cached. Using either autodep or dependson will remove this warning. See the knitr cache options for more details.

Posterior Distribution Summary

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

# save posterior draws for later
write.csv(x=fit.mcmc, file=paste0(getwd(),"/data/study_1/posterior_draws_m3.csv"))

Categroy Thresholds (\(\tau\))

# tau
bayesplot::mcmc_areas(fit.mcmc, regex_pars = "tau", prob = 0.8); ggsave("fig/study1_model3_tau_dens.pdf")

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bayesplot::mcmc_acf(fit.mcmc, regex_pars = "tau"); ggsave("fig/study1_model3_tau_acf.pdf")

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bayesplot::mcmc_trace(fit.mcmc, regex_pars = "tau"); ggsave("fig/study1_model3_tau_trace.pdf")

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ggmcmc::ggs_grb(fit.mcmc.ggs, family = "tau"); ggsave("fig/study1_model3_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_model3_lambda_dens.pdf")

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bayesplot::mcmc_acf(fit.mcmc, regex_pars = "lambda.std"); ggsave("fig/study1_model3_lambda_acf.pdf")

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bayesplot::mcmc_trace(fit.mcmc, regex_pars = "lambda.std"); ggsave("fig/study1_model3_lambda_trace.pdf")

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ggmcmc::ggs_grb(fit.mcmc.ggs, family = "lambda.std"); ggsave("fig/study1_model3_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_model3_theta_dens.pdf")

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bayesplot::mcmc_acf(fit.mcmc, regex_pars = "theta"); ggsave("fig/study1_model3_theta_acf.pdf")

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bayesplot::mcmc_trace(fit.mcmc, regex_pars = "theta"); ggsave("fig/study1_model3_theta_trace.pdf")

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ggmcmc::ggs_grb(fit.mcmc.ggs, family = "theta"); ggsave("fig/study1_model3_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_model3_omega_dens.pdf")

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bayesplot::mcmc_acf(fit.mcmc, regex_pars = "reli.omega"); ggsave("fig/study1_model3_omega_acf.pdf")

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bayesplot::mcmc_trace(fit.mcmc, regex_pars = "reli.omega"); ggsave("fig/study1_model3_omega_trace.pdf")

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ggmcmc::ggs_grb(fit.mcmc.ggs, family = "reli.omega"); ggsave("fig/study1_model3_omega_grb.pdf")

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# extract omega posterior for results comparison
extracted_omega <- data.frame(model_3 = fit.mcmc$reli.omega)
write.csv(x=extracted_omega, file=paste0(getwd(),"/data/study_1/extracted_omega_m3.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: 10515
   Total graph size: 108063

Initializing model
fit.extra <- rjags::jags.samples(model.fit$model, variable.names = "omega", n.iter = Niter)
NOTE: Stopping adaptation
N <- model.fit$model$data()[[1]]
nit <- 5
nchain=4
C <- 3
n <- i <- iter <- ppc.row.i <- 1
y.prob.ppc <- array(dim=c(Niter*nchain, nit, C))
for(chain in 1:nchain){
  for(iter in 1:Niter){
    # initialize simulated y for this iteration
    y <- matrix(nrow=N, ncol=nit)
    # loop over item
    for(i in 1:nit){
      # simulated data for item i for each person
      for(n in 1:N){
        y[n,i] <- sample(1:C, 1, prob = fit.extra$omega[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()[["y"]]
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_model3_ppc_y.pdf",plot=p,width = 6, height=4,units="in")
ggsave(filename = "fig/study1_model3_ppc_y.png",plot=p,width = 6, height=4,units="in")
ggsave(filename = "fig/study1_model3_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 3 posterior distribution summary")
    ,align = "lrrrrrrrrr"
  ),
  include.rownames=T,
  booktabs=T
)
% latex table generated in R 4.0.5 by xtable 1.8-4 package
% Sun Jan 16 13:44:12 2022
\begin{table}[ht]
\centering
\begin{tabular}{lrrrrrrrrr}
  \toprule
 & mean & sd & 2.5\% & 25\% & 50\% & 75\% & 97.5\% & Rhat & n.eff \\ 
  \midrule
deviance & 4248.47 & 72.87 & 4105.09 & 4199.49 & 4247.85 & 4297.01 & 4395.54 & 1.01 & 870.00 \\ 
  lambda[1] & 0.89 & 0.43 & 0.28 & 0.61 & 0.82 & 1.08 & 2.00 & 1.09 & 48.00 \\ 
  lambda[2] & 0.86 & 0.38 & 0.30 & 0.58 & 0.79 & 1.07 & 1.74 & 1.02 & 190.00 \\ 
  lambda[3] & 0.47 & 0.22 & 0.08 & 0.31 & 0.45 & 0.60 & 0.95 & 1.04 & 74.00 \\ 
  lambda[4] & 0.90 & 0.38 & 0.33 & 0.63 & 0.84 & 1.11 & 1.80 & 1.05 & 60.00 \\ 
  lambda[5] & 0.43 & 0.23 & 0.06 & 0.26 & 0.41 & 0.58 & 0.95 & 1.01 & 390.00 \\ 
  lambda.std[1] & 0.62 & 0.16 & 0.27 & 0.52 & 0.64 & 0.73 & 0.89 & 1.07 & 60.00 \\ 
  lambda.std[2] & 0.61 & 0.16 & 0.28 & 0.50 & 0.62 & 0.73 & 0.87 & 1.01 & 270.00 \\ 
  lambda.std[3] & 0.40 & 0.16 & 0.08 & 0.30 & 0.41 & 0.51 & 0.69 & 1.04 & 83.00 \\ 
  lambda.std[4] & 0.63 & 0.15 & 0.31 & 0.53 & 0.64 & 0.74 & 0.87 & 1.04 & 69.00 \\ 
  lambda.std[5] & 0.38 & 0.17 & 0.06 & 0.25 & 0.38 & 0.50 & 0.69 & 1.01 & 390.00 \\ 
  reli.omega & 0.71 & 0.06 & 0.57 & 0.67 & 0.72 & 0.75 & 0.80 & 1.01 & 2200.00 \\ 
  tau[1,1] & -0.89 & 0.21 & -1.41 & -0.99 & -0.86 & -0.75 & -0.58 & 1.13 & 41.00 \\ 
  tau[2,1] & -0.89 & 0.18 & -1.30 & -0.99 & -0.87 & -0.76 & -0.59 & 1.01 & 430.00 \\ 
  tau[3,1] & -0.63 & 0.12 & -0.87 & -0.71 & -0.63 & -0.54 & -0.40 & 1.01 & 210.00 \\ 
  tau[4,1] & -0.54 & 0.15 & -0.88 & -0.63 & -0.53 & -0.44 & -0.27 & 1.01 & 220.00 \\ 
  tau[5,1] & -0.87 & 0.14 & -1.16 & -0.96 & -0.87 & -0.78 & -0.62 & 1.00 & 4000.00 \\ 
  tau[1,2] & 1.08 & 0.22 & 0.76 & 0.94 & 1.06 & 1.18 & 1.65 & 1.09 & 53.00 \\ 
  tau[2,2] & 1.16 & 0.21 & 0.83 & 1.01 & 1.13 & 1.27 & 1.65 & 1.01 & 210.00 \\ 
  tau[3,2] & 1.04 & 0.14 & 0.78 & 0.95 & 1.04 & 1.13 & 1.32 & 1.00 & 660.00 \\ 
  tau[4,2] & 1.31 & 0.24 & 0.94 & 1.15 & 1.28 & 1.43 & 1.87 & 1.06 & 55.00 \\ 
  tau[5,2] & 0.96 & 0.13 & 0.70 & 0.87 & 0.96 & 1.05 & 1.22 & 1.00 & 4000.00 \\ 
  theta[1] & 1.98 & 1.07 & 1.08 & 1.37 & 1.68 & 2.16 & 5.02 & 1.16 & 34.00 \\ 
  theta[2] & 1.88 & 0.84 & 1.09 & 1.34 & 1.63 & 2.14 & 4.02 & 1.03 & 130.00 \\ 
  theta[3] & 1.27 & 0.24 & 1.01 & 1.10 & 1.20 & 1.36 & 1.91 & 1.05 & 58.00 \\ 
  theta[4] & 1.96 & 0.86 & 1.11 & 1.39 & 1.71 & 2.22 & 4.25 & 1.06 & 56.00 \\ 
  theta[5] & 1.24 & 0.24 & 1.00 & 1.07 & 1.17 & 1.33 & 1.90 & 1.01 & 480.00 \\ 
   \bottomrule
\end{tabular}
\caption{study1 Model 3 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 448 rows containing non-finite values (stat_density).

ggsave(filename = "fig/study1_model3_posterior_dist.pdf",plot=p,width = 7, height=5,units="in")
Warning: Removed 448 rows containing non-finite values (stat_density).
ggsave(filename = "fig/study1_model3_posterior_dist.png",plot=p,width = 7, height=5,units="in")
Warning: Removed 448 rows containing non-finite values (stat_density).
ggsave(filename = "fig/study1_model3_posterior_dist.eps",plot=p,width = 7, height=5,units="in")
Warning: Removed 448 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_0.6-7         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