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High-Dimensional Maximum Marginal Likelihood Item Factor Analysis by Adaptive Quadrature

Published online by Cambridge University Press:  01 January 2025

Stephen Schilling  and
R. Darrell Bock
Stephen Schilling*
Affiliation:
School of Education, University of Michigan
R. Darrell Bock
Affiliation:
Center for Health Statistics, University of Illinois at Chicago
*
Requests for reprints should be sent to Stephen Schilling, Assistant Professor, University of Michigan, School of Education, Ann Arbor, MI 48109, USA. E-mail: schillsg@umich.edu
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Abstract

Although the Bock–Aitkin likelihood-based estimation method for factor analysis of dichotomous item response data has important advantages over classical analysis of item tetrachoric correlations, a serious limitation of the method is its reliance on fixed-point Gauss-Hermite (G-H) quadrature in the solution of the likelihood equations and likelihood-ratio tests. When the number of latent dimensions is large, computational considerations require that the number of quadrature points per dimension be few. But with large numbers of items, the dispersion of the likelihood, given the response pattern, becomes so small that the likelihood cannot be accurately evaluated with the sparse fixed points in the latent space. In this paper, we demonstrate that substantial improvement in accuracy can be obtained by adapting the quadrature points to the location and dispersion of the likelihood surfaces corresponding to each distinct pattern in the data. In particular, we show that adaptive G-H quadrature, combined with mean and covariance adjustments at each iteration of an EM algorithm, produces an accurate fast-converging solution with as few as two points per dimension. Evaluations of this method with simulated data are shown to yield accurate recovery of the generating factor loadings for models of up to eight dimensions. Unlike an earlier application of adaptive Gibbs sampling to this problem by Meng and Schilling, the simulations also confirm the validity of the present method in calculating likelihood-ratio chi-square statistics for determining the number of factors required in the model. Finally, we apply the method to a sample of real data from a test of teacher qualifications.

Keywords

factor analysisitem response theorylatent variablesEM algorithmmarginal likelihood estimationGLS estimationadaptive quadraturemonte carlo integration
Type
Original Paper
Information
Psychometrika , Volume 70 , Issue 3 , September 2005 , pp. 533 - 555
Copyright
Copyright © 2005 The Psychometric Society

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