dina (version 1.0.2)

DINA_Gibbs: Generate Posterior Distribution with Gibbs sampler

Description

Function for sampling parameters from full conditional distributions. The function returns a list of arrays or matrices with parameter posterior samples. Note that the output includes the posterior samples in objects named: CLASSES = individual attribute profiles, PIs = latent class proportions, SigS = item slipping parameters, and GamS = item guessing parameters.

Usage

DINA_Gibbs(Y, Amat, Q, chain_length = 10000L)

Arguments

Y

A N by J matrix of observed responses.

Amat

A C by K matrix of latent classes.

Q

A N by K matrix indicating which skills are required for which items.

chain_length

Number of MCMC iterations.

Value

A list with samples from the posterior distribution.

See Also

DINAsim

Examples

# NOT RUN {
###################################
#Tatsuoka Fraction Subtraction Data
###################################
require(CDM)
data(fraction.subtraction.data)
Y_1984 = as.matrix(fraction.subtraction.data)
Q_1984 = as.matrix(fraction.subtraction.qmatrix)
K_1984 = ncol(fraction.subtraction.qmatrix)
J_1984 = ncol(Y_1984)
    
# Creating matrix of possible attribute profiles
As_1984 = rep(0,K_1984)
for(j in 1:K_1984){
    temp = combn(1:K_1984,m=j)
    tempmat = matrix(0,ncol(temp),K_1984)
    for(j in 1:ncol(temp)) tempmat[j,temp[,j]] = 1
    As_1984 = rbind(As_1984,tempmat)
}
As_1984 = as.matrix(As_1984)
            
# Generate samples from posterior distribution
# May take 8 minutes
chainLength <- 5000
burnin <- 1000
outchain_1984 <- DINA_Gibbs(Y = Y_1984, Amat = As_1984,
                            Q_1984, chain_length = chainLength)
                
# Summarize posterior samples for g and 1-s
mgs_1984 = apply(outchain_1984$GamS[,burnin:chainLength],1,mean)
sgs_1984 = apply(outchain_1984$GamS[,burnin:chainLength],1,sd)
mss_1984 = 1-apply(outchain_1984$SigS[,burnin:chainLength],1,mean)
sss_1984 = apply(outchain_1984$SigS[,burnin:chainLength],1,sd)
output_1984=cbind(mgs_1984,sgs_1984,mss_1984,sss_1984)
colnames(output_1984) = c('g Est','g SE','1-s Est','1-s SE')
rownames(output_1984) = colnames(Y_1984)
print(output_1984,digits=3)
                    
# Summarize marginal skill distribution using posterior samples for latent class proportions
marg_PIs = t(As_1984)\%*\%outchain_1984$PIs
PI_Est = apply(marg_PIs[,burnin:chainLength],1,mean)
PI_Sd = apply(marg_PIs[,burnin:chainLength],1,sd)
PIoutput = cbind(PI_Est,PI_Sd)
colnames(PIoutput) = c('EST','SE')
rownames(PIoutput) = paste0('Skill ',1:K_1984)
print(PIoutput,digits=3)

#####################################################
#de la Torre (2009) Simulation Replication w/ N = 200
#####################################################
N = 200
K = 5
J = 30
delta0 = rep(1,2^K)

#Creating Q matrix
Q = matrix(rep(diag(K),2),2*K,K,byrow=TRUE)
for(mm in 2:K){
    temp = combn(1:K,m=mm)
    tempmat = matrix(0,ncol(temp),K)
    for(j in 1:ncol(temp)) tempmat[j,temp[,j]] = 1
    Q = rbind(Q,tempmat)
}
Q = Q[1:J,]
    
# Setting item parameters and generating attribute profiles
ss = gs = rep(.2,J)
PIs = rep(1/(2^K),2^K)
CLs = c((1:(2^K))\%*\%rmultinom(n=N,size=1,prob=PIs) )
        
# Defining matrix of possible attribute profiles
As = rep(0,K)
for(j in 1:K){
    temp = combn(1:K,m=j)
    tempmat = matrix(0,ncol(temp),K)
    for(j in 1:ncol(temp)) tempmat[j,temp[,j]] = 1
    As = rbind(As,tempmat)
}
As = as.matrix(As)
                
# Sample true attribute profiles
Alphas = As[CLs,]
                
# Simulate data under DINA model 
gen = DINAsim(Alphas,Q,ss,gs)
Y_sim = gen$Y
                    
# Execute MCMC
# NOTE small chain length used to reduce computation time for pedagogical example.
chainLength = 200
burnin = 100
                    
outchain <- DINA_Gibbs(Y_sim,Amat=As,Q,chain_length=chainLength)
                        
# Summarize posterior samples for g and 1-s  
mGs = apply(outchain$GamS[,burnin:chainLength],1,mean)
sGs = apply(outchain$GamS[,burnin:chainLength],1,sd)
m1mSS = 1 - apply(outchain$SigS[,burnin:chainLength],1,mean)
s1mSS = apply(outchain$SigS[,burnin:chainLength],1,sd)
output = cbind(mGs,sGs,m1mSS,s1mSS)
colnames(output) = c('g Est','g SE','1-s Est','1-s SE')
rownames(output) = paste0('Item ',1:J)
print(output, digits=3)
                            
# Summarize marginal skill distribution using posterior samples for latent class proportions
PIoutput = cbind(apply(outchain$PIs,1,mean),apply(outchain$PIs,1,sd))
colnames(PIoutput) = c('EST','SE')
rownames(PIoutput) = apply(As,1,paste0,collapse='')
print(PIoutput,digits=3)
# }