Gibbs_2PNO function

Gibbs Implementation of 2PNO

Implement Gibbs 2PNO Sampler

Gibbs_2PNO(Y, mu_xi, Sigma_xi_inv, mu_theta, Sigma_theta_inv, burnin,
  chain_length = 10000L)

Arguments

• Y: A N by J matrix of item responses.
• mu_xi: A two dimensional vector of prior item parameter means.
• Sigma_xi_inv: A two dimensional identity matrix of prior item parameter VC matrix.
• mu_theta: The prior mean for theta.
• Sigma_theta_inv: The prior inverse variance for theta.
• burnin: The number of MCMC samples to discard.
• chain_length: The number of MCMC samples.

Returns

Samples from posterior.

Examples

# simulate small 2PNO dataset to demonstrate function
J = 5
N = 100

# Population item parameters
as_t = rnorm(J,mean=2,sd=.5)
bs_t = rnorm(J,mean=0,sd=.5)

# Sampling gs and ss with truncation
gs_t = rbeta(J,1,8)
ps_g = pbeta(1-gs_t,1,8)
ss_t = qbeta(runif(J)*ps_g,1,8)
theta_t = rnorm(N)
Y_t = Y_4pno_simulate(N,J,as=as_t,bs=bs_t,gs=gs_t,ss=ss_t,theta=theta_t)

# Setting prior parameters
mu_theta = 0
Sigma_theta_inv = 1
mu_xi = c(0,0)
alpha_c = alpha_s = beta_c = beta_s = 1
Sigma_xi_inv = solve(2*matrix(c(1,0,0,1), 2, 2))
burnin = 1000

# Execute Gibbs sampler. This should take about 15.5 minutes
out_t = Gibbs_4PNO(Y_t,mu_xi,Sigma_xi_inv,mu_theta,Sigma_theta_inv,
                    alpha_c,beta_c,alpha_s, beta_s,burnin,
                    rep(1,J),rep(1,J),gwg_reps=5,chain_length=burnin*2)

# Summarizing posterior distribution
OUT = cbind(
    apply(out_t$AS[, -c(1:burnin)], 1, mean),
    apply(out_t$BS[, -c(1:burnin)], 1, mean),
    apply(out_t$GS[, -c(1:burnin)], 1, mean),
    apply(out_t$SS[, -c(1:burnin)], 1, mean),
    apply(out_t$AS[, -c(1:burnin)], 1, sd),
    apply(out_t$BS[, -c(1:burnin)], 1, sd),
    apply(out_t$GS[, -c(1:burnin)], 1, sd),
    apply(out_t$SS[, -c(1:burnin)], 1, sd)
)
OUT = cbind(1:J, OUT)
colnames(OUT) = c('Item','as','bs','gs','ss','as_sd','bs_sd',
                  'gs_sd','ss_sd')
print(OUT, digits = 3)

Author(s)

Steven Andrew Culpepper