# Stochastic Approximation EM for Exploratory Item Factor Analysis

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.

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