# Simulation Study with cIRT

#### 2025-09-26

# 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 differing \(\theta\) 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")

    # Load the Library
    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
    load("a_b_true.rda")

    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)

      load(paste0(mname, ".rda"))

      d = get(mname)

      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")