High-dimensional maximum marginal likelihood item factor analysis by adaptive quadrature

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 upto 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.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43596/1/11336_2003_Article_1141.pd

Item Response Modeling of Forced-Choice Questionnaires

Multidimensional forced-choice formats can significantly reduce the impact of numerous response biases typically associated with rating scales. However, if scored with classical methodology, these questionnaires produce ipsative data, which lead to distorted scale relationships and make comparisons between individuals problematic. This research demonstrates how item response theory (IRT) modeling may be applied to overcome these problems. A multidimensional IRT model based on Thurstone’s framework for comparative data is introduced, which is suitable for use with any forced-choice questionnaire composed of items fitting the dominance response model, with any number of measured traits, and any block sizes (i.e., pairs, triplets, quads, etc.). Thurstonian IRT models are normal ogive models with structured factor loadings, structured uniquenesses, and structured local dependencies. These models can be straightforwardly estimated using structural equation modeling (SEM) software Mplus. A number of simulation studies are performed to investigate how latent traits are recovered under various forced-choice designs and provide guidelines for optimal questionnaire design. An empirical application is given to illustrate how the model may be applied in practice. It is concluded that when the recommended design guidelines are met, scores estimated from forced-choice questionnaires with the proposed methodology reproduce the latent traits well

Thurstonian modeling of ranking data via mean and covariance structure analysis

GMM, GLS, WLS estimation, UMD, ULS, EWMD estimation, permutation data, random utility models, structural equations models,