  The stochastic approximation EM algorithm (SAEM) is described for the
estimation of item and person parameters given test data coded as dichotomous
or ordinal variables. The method hinges upon the eigenanalysis of missing
variables sampled as augmented data; the augmented data approach was introduced
by Albert's seminal work applying Gibbs sampling to Item Response Theory in
1992. Similar to maximum likelihood factor analysis, the factor structure in
this Bayesian approach depends only on sufficient statistics, which are
computed from the missing latent data. A second feature of the SAEM algorithm
is the use of the Robbins-Monro procedure for establishing convergence.
Contrary to Expectation Maximization methods where costly integrals must be
calculated, this method is well-suited for highly multidimensional data, and an
annealing method is implemented to prevent convergence to a local maximum
likelihood. Multiple calculations of errors applied within this framework of
Markov Chain Monte Carlo are presented to delineate the uncertainty of
parameter estimates. Given the nature of EFA (exploratory factor analysis), an
algorithm is formalized leveraging the Tracy-Widom distribution for the
retention of factors extracted from an eigenanalysis of the sufficient
statistic of the covariance of the augmented data matrix. Simulation conditions
of dichotomous and polytomous data, from one to ten dimensions of factor
loadings, are used to assess statistical accuracy and to gauge computational
time of the EFA approach of this IRT-specific implementation of the SAEM
algorithm. Finally, three applications of this methodology are also reported
that demonstrate the effectiveness of the method for enabling timely analyses
as well as substantive interpretations when this method is applied to real
data.
