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

Concepts and models from psychometrics

The concepts and methods of psychometrics originated under trait and behavioral psychology, with relatively simple data, used mainly for purposes of prediction and selection. Ideas emerged over that nevertheless hold value for the new psychological perspectives, contexts of use, and forms of data and analytic tools we are now seeing. In this chapter we review some fundamental models and ideas from psychometrics that are be profitably reconceived, extended, and augmented in in the new world of assessment. Methods we address include classical test theory, generalizability theory, item response theory, latent class models, cognitive diagnosis models, factor analysis, hierarchical models, and Bayesian networks. Key concepts are these: (1) The essential nature of psychometric models (observations, constructs, latent variables, and probability-based reasoning). (2) The interplay of design and discovery in assessment. (3) Understanding the measurement issues of validity, reliability, comparability, generalizability, and fairness as social values that pertain even as forms of data, analysis, context, and purpose evolve

Does adaptive testing violate local independence?

adaptive testing, conditional independence, item response theory (IRT), local independence,