Metropolis-Hastings Robbins-Monro Algorithm for Confirmatory Item Factor Analysis

Abstract

Item factor analysis (IFA), already well established in educational measurement, is increasingly applied to psychological measurement in research settings. However, high-dimensional confirmatory IFA remains a numerical challenge. The current research extends the Metropolis-Hastings Robbins-Monro (MH-RM) algorithm, initially proposed for exploratory IFA, to the case of maximum likelihood estimation under user-defined linear restrictions for confirmatory IFA. MH-RM naturally integrates concepts such as the missing data formulation, data augmentation, the Metropolis algorithm, and stochastic approximation. In a limited simulation study, the accuracy of the MH-RM algorithm is checked against the standard Bock-Aitkin expectation-maximization (EM) algorithm. To demonstrate the efficiency and flexibility of the MH-RM algorithm, it is applied to the IFA of real data from pediatric quality-of-life (QOL) research in comparison with adaptive quadrature-based EM algorithm. The particular data set required a confirmatory item factor model with eight factors and a variety of equality and fixing constraints to implement the hypothesized factor pattern. MH-RM converged in less than 3 minutes to the maximum likelihood solution while the EM algorithm spent well over 4 hourrs.