dina {dina}

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.

Usage

dina(Y, Q, chain_length = 10000)

Arguments

Y

A N \times J matrix of observed responses.

Q

A N \times 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 with each entry named:

• CLASSES = individual attribute profiles,

• PIs = latent class proportions,

• SigS = item slipping parameters, and

• GamS = item guessing parameters.

Author(s)

Steven Andrew Culpepper and James Joseph Balamuta

See Also

simcdm::sim_dina_items() and simcdm::attribute_classes()

Examples

## Not run: 

####################################
# Tatsuoka Fraction Subtraction Data
####################################

# This example requires data from the CDM package.
if(requireNamespace("CDM")) {

data(fraction.subtraction.data, package = "CDM")
data(fraction.subtraction.qmatrix, package = "CDM")
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
chain_samples = burnin:chainLength

outchain_1984 = dina(Y = Y_1984, Q = Q_1984,
                     chain_length = chainLength)

# Summarize posterior samples for g and 1-s
mgs_1984 = apply(outchain_1984$GamS[, chain_samples], 1, mean)
sgs_1984 = apply(outchain_1984$GamS[, chain_samples], 1, sd)
mss_1984 = 1 - apply(outchain_1984$SigS[, chain_samples], 1, mean)
sss_1984 = apply(outchain_1984$SigS[, chain_samples], 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[, chain_samples], 1, mean)
PI_Sd = apply(marg_PIs[, chain_samples], 1, sd)
PIoutput = cbind(PI_Est, PI_Sd)
colnames(PIoutput) = c('EST', 'SE')
rownames(PIoutput) = paste('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 
Y_sim = simcdm::sim_dina_items(Alphas, Q, ss, gs)
                    
## Execute MCMC DINA routine ----

# NOTE: This example uses a small chain length to reduce 
# computation time to illustrate the pedagogical concept.
# In a real-life scenario, increase the chain length to
# at least 5,000.

chainLength = 200
burnin = 100
                    
outchain = dina(Y_sim, Q, chain_length = chainLength)
    
## Summarize posterior samples for g and 1-s ----

chain_samples = burnin:chainLength
mGs = apply(outchain$GamS[, chain_samples], 1, mean)
sGs = apply(outchain$GamS[, chain_samples], 1, sd)
m1mSS = 1 - apply(outchain$SigS[, chain_samples], 1, mean)
s1mSS = apply(outchain$SigS[, chain_samples], 1, sd)
output = cbind(mGs, sGs, m1mSS, s1mSS)
colnames(output) = c('g Est', 'g SE', '1-s Est', '1-s SE')
rownames(output) = paste('Item', 1:J)
print(output, digits = 3)
                            
## Summarize marginal skill distribution ---- 

# Via 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)

## End(Not run)