Gibbs sampler to estimate the rRUM

Obtains samples from posterior distributon for the reduced Reparametrized Unified Model (rRUM).

rrum(
  Y,
  Q,
  chain_length = 10000L,
  as = 1,
  bs = 1,
  ag = 1,
  bg = 1,
  delta0 = rep(1, 2^ncol(Q))
)

Arguments

Y

A matrix with N rows and J columns, where N reperesnts the number of individuals and J the number of items. Y indicates the indviduals' responses to each of the items.

Q

A matrix with J rows and K columns indicating which attributes are required to answer each of the items. An entry of 1 indicates attribute k is required to answer item j. An entry of one indicates attribute k is not required.

chain_length

A numeric indicating the number of iterations of Gibbs sampler to be run. Default is set to 10000.

as

A numeric, parameter for the prior distribution of pistar. High values as encourage higher values of pistar and lower values of rstar.

bs

A numeric, parameter for the prior distribution of pistar. High values as encourage lower values of pistar and higher values of rstar.

ag

A numeric, parameter for the prior distribution of rstar. High values as encourage higher values of rstar.

bg

A numeric, parameter for the prior distribution of pistar. High values as encourage lower values of rstar.

delta0

A vector, parameters for the Dirichlet prior on pi.

Value

A list that contains

• PISTAR: A matrix where each column represents one draw from the posterior distribution of pistar.

• RSTAR: A \(J x K x chain_length\) array where J reperesents the number of items, and K represents the number of attributes. Each slice represents one draw from the posterior distribution of rstar.

• PI: A matrix where each column reperesents one draw from the posterior distribution of pi.

• ALPHA: An \(N x K x chain_length\) array where N reperesents the number of individuals, and K represents the number of attributes. Each slice represents one draw from the posterior distribution of alpha.

References

Culpepper, S. A. & Hudson, A. (In Press). An improved strategy for Bayesian estimation of the reduced reparameterized unified model. Applied Psychological Measurement.

Hudson, A., Culpepper, S. A., & Douglas, J. (2016, July). Bayesian estimation of the generalized NIDA model with Gibbs sampling. Paper presented at the annual International Meeting of the Psychometric Society, Asheville, North Carolina.

See also

simcdm::sim_rrum_items()

Author

Steven Andrew Culpepper, Aaron Hudson, and James Joseph Balamuta

Examples

# Set seed for reproducibility
set.seed(217)

## Define Simulation Parameters

N = 1000 # Number of Individuals
J = 6    # Number of Items
K = 2    # Number of Attributes

# Matrix where rows represent attribute classes
As = attribute_classes(K) 

# Latent Class probabilities
pis = c(.1, .2, .3, .4) 

# Q Matrix
Q = rbind(c(1, 0),
          c(0, 1),
          c(1, 0),
          c(0, 1),
          c(1, 1),
          c(1, 1)
    )
    
# The probabiliies of answering each item correctly for individuals 
# who do not lack any required attribute
pistar = rep(.9, J)

# Penalties for failing to have each of the required attributes
rstar  = .5 * Q

# Randomized alpha profiles
alpha  = As[sample(1:(K ^ 2), N, replace = TRUE, pis),]

# Simulate data
rrum_items = simcdm::sim_rrum_items(Q, rstar, pistar, alpha)

if (FALSE) {
# Note: This portion of the code is computationally intensive.

# Recover simulation parameters with Gibbs Sampler
Gibbs.out = rrum(rrum_items, Q)

# Iterations to be discarded from chain as burnin
burnin = 1:5000 

# Calculate summarizes of posterior distributions
rstar.mean  = with(Gibbs.out, apply(RSTAR[,,-burnin], c(1, 2), mean))
pistar.mean = with(Gibbs.out, apply(PISTAR[,-burnin], 1, mean))
pis.mean    = with(Gibbs.out, apply(PI[,-burnin], 1 ,mean))
}