  The RRUM is a model that is frequently seen in language assessment studies.
The objective of this research is to advance an MCMC algorithm for the Bayesian
RRUM. The algorithm starts with estimating correlated attributes. Using a
saturated model and a binary decimal conversion, the algorithm transforms
possible attribute patterns to a Multinomial distribution. Along with the
likelihood of an attribute pattern, a Dirichlet distribution is used as the
prior to sample from the posterior. The Dirichlet distribution is constructed
using Gamma distributions. Correlated attributes of examinees are generated
using the inverse transform sampling. Model parameters are estimated using the
Metropolis within Gibbs sampler sequentially. Two simulation studies are
conducted to evaluate the performance of the algorithm. The first simulation
uses a complete and balanced Q-matrix that measures 5 attributes. Comprised of
28 items and 9 attributes, the Q-matrix for the second simulation is incomplete
and imbalanced. The empirical study uses the ECPE data obtained from the CDM R
package. Parameter estimates from the MCMC algorithm and from the CDM R package
are presented and compared. The algorithm developed in this research is
implemented in R.
