Gibbs_4PNO: Gibbs Implementation of 4PNO

In fourPNO: Bayesian 4 Parameter Item Response Model

Description

Internal function to -2LL

Usage

Gibbs_4PNO(Y, mu_xi, Sigma_xi_inv, mu_theta, Sigma_theta_inv, alpha_c,
  beta_c, alpha_s, beta_s, burnin, cTF, sTF, gwg_reps,
  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.
alpha_c	The lower asymptote prior 'a' parameter.
beta_c	The lower asymptote prior 'b' parameter.
alpha_s	The upper asymptote prior 'a' parameter.
beta_s	The upper asymptote prior 'b' parameter.
burnin	The number of MCMC samples to discard.
cTF	A J dimensional vector indicating which lower asymptotes to estimate. 0 = exclude lower asymptote and 1 = include lower asymptote.
sTF	A J dimensional vector indicating which upper asymptotes to estimate. 0 = exclude upper asymptote and 1 = include upper asymptote.
gwg_reps	The number of Gibbs within Gibbs MCMC samples for marginal distribution of gamma. Values between 5 to 10 are adequate.
chain_length	The number of MCMC samples.

Value

Samples from posterior.

Author(s)

Steven Andrew Culpepper

Examples

# Simulate small 4PNO 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
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)

Example output

     Item   as     bs     gs     ss as_sd bs_sd  gs_sd  ss_sd
[1,]    1 2.82  1.055 0.0563 0.0808 0.654 0.431 0.0345 0.0697
[2,]    2 2.81  0.647 0.1141 0.0566 0.603 0.489 0.0582 0.0472
[3,]    3 2.80 -0.340 0.2325 0.0598 0.990 0.738 0.1443 0.0400
[4,]    4 2.19  0.682 0.2171 0.1477 0.715 0.522 0.0814 0.0873
[5,]    5 2.95  0.124 0.0489 0.0372 0.731 0.334 0.0427 0.0318

fourPNO documentation built on Sept. 24, 2019, 9:05 a.m.