# Intro

The document addresses the simulation study that was performed to understand whether cIRT was able to effectively recover parameters under the proposed model. There are three sections to this document. The first section details the simulation setup and implementation. The second section deals with how we obtained overall measurements for each model realization. The third and final section depicts the output structure we used to create the tables within the publication.

# Simulation

The simulation study below is configured to generate results one might obtain from a pool of 1,000 subjects taking a 20 item test. We obtain summarize the results obtained from running the model 100 times over slightly differring $$\theta$$ and $$\eta$$.

### Variables
# Y = trial matix
# C = KN vector of binary choices
# N = #of subjects
# J = # of items
# K = # of choices
# atrue = true item discriminations
# btrue = true item locations
# thetatrue = true thetas/latent performance
# gamma = fixed effects coefficients
# Sig = random-effects variance-covariance
# subid = id variable for subjects

# install mvtnorm is necessary
# install.packages("mvtnorm")

library(cIRT)

# Simulate 2PNO Data
N = 1000  # Students
J = 20    # Total numbers of possible items per SA

# Set a seed for the random generation of a's and b's.
set.seed(1337)

# Randomly pick a's and b's

# Generate as, bs
atrue=runif(J)+1
btrue=2*runif(J)-1

save(atrue, btrue, file="a_b_true.rda")

# 2 Level Probit Data
K = 30

gam_notheta = c(.5,1)
gam_theta   = c(3,.25)
gamma = c(gam_notheta,gam_theta)

Sig = matrix(c(.25,0,0,.125),2,2)

# Number of replications
B = 100

# Begin the Simulation Study
for(b in 1:B){

# Provide user with state information
cat(paste0("On Iteration:",b,"\n"))
set.seed(b + 1234)

# True theta and etay
thetatrue = rnorm(N)
etay = outer(rep(1,N),atrue) * thetatrue - outer(rep(1,N),btrue)

# Generate Y for 2PNO model
p.correct = pnorm(etay)
Y = matrix(rbinom(N*J, 1, p.correct),N,J)

#################################################
# Simulating 2 level probit data
#################################################

subid = expand.grid(cid = 1:K,sid = 1:N)[,2]

pred = rnorm(K*N,0,1) # Pred

center_pred = center_matrix(as.matrix(pred))

Xnotheta = cbind(1,center_pred)

Xtheta = rep(thetatrue,each=K)*Xnotheta
X = cbind(Xnotheta,Xtheta)

zetas = mvtnorm::rmvnorm(N,mean=c(0,0),sigma=Sig) # mvtnorm environment accessed
W_veczeta = apply(Xnotheta*zetas[rep(1:N,each=K),],1,sum)

etac = X%*%gamma + W_veczeta
Zc = rnorm(N*K,mean=etac,sd=1)
C = 1*(Zc>0)

# Run the Choice Item Response Model
out1 = cIRT(subid,
Xnotheta,
c(1,2),
Xnotheta,
Y,
C,
5000)

mname = paste0("model_",b)

# Assign the data set name
assign(mname, out1)

# Save the out object
save(list=mname, file=paste0(mname,".rda"))

# Clean up Export
rm(list = c(mname,"out1"))
}

# Simulation Results

After we obtain the 100 different models, we will need to take averages over each models chain and then obtain an overall average. To do so, we used the following script:

# E[theta] - theta
bias = function(theta.star, theta.true){
matrix(mean(theta.star) - theta.true,ncol=1)
}

bias2 = function(theta.star, theta.true){
matrix(theta.star - theta.true,ncol=1)
}

# sqrt ( 1/n * sum( (y_i - y.hat_i)^2 )
RMSE = function(y,y.hat){
sqrt(  (y-y.hat)^2 )
}

# Change true values if needed
# True Values
gam_notheta = c(.5,1)
gam_theta   = c(3,.25)
gamma = c(gam_notheta,gam_theta)
# Loads a and b values

Sig = as.numeric(matrix(c(.25,0,0,.125),2,2))[c(1,2,4)]
B = 100

# Storage to hold bootstrap replications
a_result = matrix(0, B, 20)

b_result = matrix(0, B, 20)

gs0_result = matrix(0, B, 2)

beta_result = matrix(0, B, 2)

sig_result = array(NA, dim=c(2,2,B))

for(b in 1:B){

mname = paste0("model_",b)

a_result[i,] = apply(d$as, 2, FUN = mean) b_result[i,] = apply(d$bs, 2, FUN = mean)
gs0_result[i,] = apply(d$gs0, 2, FUN = mean) beta_result[i,] = apply(d$betas, 2, FUN = mean)
sig_result[,,i] = solve(apply(d$Sigma_zeta_inv, c(1,2), FUN = mean)) } # Obtain an overall mean for each of the following: m_a_result = apply(a_result, 2, FUN = mean) m_b_result = apply(b_result, 2, FUN = mean) m_gs0_result = apply(gs0_result, 2, FUN = mean) m_beta_result = apply(beta_result, 2, FUN = mean) m_sig_result = as.numeric(apply(sig_result, c(1,2), FUN = mean))[c(1,2,4)] # Perform a bias evaluation given the true values: a_bias = bias2(m_a_result,atrue) b_bias = bias2(m_b_result,btrue) gs0_bias = bias2(m_gs0_result,gamma[1:2]) beta_bias = bias2(m_beta_result,gamma[3:4]) sig_bias = bias2(m_sig_result, Sig) # Perform the RMSE under the supplied results: a_RMSE = RMSE(m_a_result,atrue) b_RMSE = RMSE(m_b_result,btrue) gs0_RMSE = RMSE(m_gs0_result,gamma[1:2]) beta_RMSE = RMSE(m_beta_result,gamma[3:4]) sig_RMSE = RMSE(m_sig_result, Sig) # Exporting Results for Publication After we obtain the different results, we opted to write export code to format the data in a way that the xtable could process and export it. # Make a results export results_a = cbind(m_a_result, atrue, a_bias, a_RMSE) rownames(results_a) = paste0("Item ",1:length(atrue)) colnames(results_a) = c("$a$Estimate", "$a$True", "$a$Bias", "$a$RMSE") results_b = cbind(m_b_result, btrue, b_bias, b_RMSE) rownames(results_b) = paste0("Item ",1:length(btrue)) colnames(results_b) = c("$b$Estimate", "$b$True", "$b$Bias", "$b$RMSE") results_gs0 = cbind(m_gs0_result,gamma[1:2],gs0_bias,gs0_RMSE) rownames(results_gs0) = paste0("$\\gamma_",1:2,"$") colnames(results_gs0) = c("$\\beta$Estimate", "$\\beta$True", "$\\beta$Bias", "$\\beta$RMSE") results_beta = cbind(m_beta_result,gamma[3:4],beta_bias,beta_RMSE) rownames(results_beta) = paste0("$\\beta_",1:2,"$") colnames(results_beta) = c("$\\beta$Estimate", "$\\beta$True", "$\\beta$Bias", "$\\beta$RMSE") results_sig_zeta = cbind(m_sig_result,Sig,sig_bias,sig_RMSE) rownames(results_sig_zeta) = c("$\\Zeta_{1,1}$","$\\Zeta_{2,1} = $\\Zeta_{1,2}$", "$\\Zeta_{2,2}$")
colnames(results_sig_zeta) = c("$\\Sigma_{\\Zeta}$ Estimate", "$\\beta$ True", "$\\beta$ Bias", "$\\beta$ RMSE")
save(results_a, results_b, results_gs0, results_beta, results_sig_zeta, file="sim_results.rda")