APPLICATION OF THE STOCHASTIC EM METHOD TO LATENT REGRESSION MODELS

The reporting methods used in large scale assessments such as the National Assessment of Educational Progress (NAEP) rely on a latent regression model. The first component of the model consists of a p-scale IRT measurement model that defines the response probabilities on a set of cognitive items in p scales depending on a p-dimensional latent trait variable θ = (θ1, … θp). In the second component, the conditional distribution of this latent trait variable θ is modeled by a multivariate, multiple regression on a set of predictor variables, which are usually based on student, school and teacher variables in assessments such as NAEP. In order to fit the latent regression model using the maximum (marginal) likelihood estimation technique, multivariate integrals have to be evaluated. In the computer program MGROUP used by ETS for fitting the latent regression model to data from NAEP and other sources, the integration is currently done either by numerical quadrature (for problems up to two dimensions) or by an approximation of the integral. CGROUP, the current operational version of the MGROUP program used in NAEP and other assessments since 1993, is based on Laplace approximation that may not provide fully satisfactory results, especially if the number of items per scale is small. This paper examines the application of stochastic expectation-maximization (EM) methods (where an integral is approximated by an average over a random sample) to NAEP-like settings. We present a comparison of CGROUP with a promising implementation of the stochastic EM algorithm that utilizes importance sampling. Simulation studies and real data analysis show that the stochastic EM method provides a viable alternative to CGROUP for fitting multivariate latent regression models.

[1]  J. Booth,et al.  Maximizing generalized linear mixed model likelihoods with an automated Monte Carlo EM algorithm , 1999 .

[2]  R. Mislevy.  Estimating latent distributions , 1984 .

[3]  Eugene G. Johnson,et al.  Scaling Procedures in NAEP , 1992 .

[4]  Xiao-Li Meng,et al.  Fitting Full-Information Item Factor Models and an Empirical Investigation of Bridge Sampling , 1996 .

[5]  G. C. Wei,et al.  A Monte Carlo Implementation of the EM Algorithm and the Poor Man's Data Augmentation Algorithms , 1990 .

[6]  Identifiers California,et al.  Annual Meeting of the National Council on Measurement in Education , 1998 .

[7]  A. Beaton,et al.  Chapter 1: Overview of the National Assessment of Educational Progress , 1992 .

[8]  J. Geweke,et al.  Bayesian Inference in Econometric Models Using Monte Carlo Integration , 1989 .

[9]  Irwin S. Kirsch,et al.  THE INTERNATIONAL ADULT LITERACY SURVEY (IALS): UNDERSTANDING WHAT WAS MEASURED , 2001 .

[10]  J-P Fox.  Stochastic EM for estimating the parameters of a multilevel IRT model. , 2003, The British journal of mathematical and statistical psychology.

[11]  Sik-Yum Lee,et al.  Maximum Likelihood Estimation of Nonlinear Structural Equation Models with Ignorable Missing Data , 2003 .

[12]  Peter Green,et al.  Markov chain Monte Carlo in Practice , 1996 .

[13]  N. Thomas,et al.  Asymptotic Corrections for Multivariate Posterior Moments with Factored Likelihood Functions , 1993 .

[14]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[15]  A. Winsor.  Sampling techniques. , 2000, Nursing times.

[16]  Matthias von Davier,et al.  COMPARING CONDITIONAL AND MARGINAL DIRECT ESTIMATION OF SUBGROUP DISTRIBUTIONS , 2003 .

[17]  E. Muraki,et al.  Chapter 3: Scaling Procedures in NAEP , 1992 .

[18]  Matthew S. Johnson,et al.  A BAYESIAN HIERARCHICAL MODEL FOR LARGE-SCALE EDUCATIONAL SURVEYS: AN APPLICATION TO THE NATIONAL ASSESSMENT OF EDUCATIONAL PROGRESS , 2004 .

[19]  A. Zellner.  An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests for Aggregation Bias , 1962 .

[20]  Albert E. Beaton.  Implementing the New Design: The NAEP 1983-84 Technical Report. , 1987 .

[21]  Rebecca Zwick,et al.  Overview of the National Assessment of Educational Progress , 1992 .

[22]  C. McCulloch.  Maximum Likelihood Variance Components Estimation for Binary Data , 1994 .

[23]  Matthias von Davier,et al.  A UNIFIED APPROACH TO IRT SCALE LINKING AND SCALE TRANSFORMATIONS , 2004 .

[24]  Robert J. Mislevy,et al.  Estimation of Latent Group Effects , 1985 .

[25]  R. Kass,et al.  Approximate Bayesian Inference in Conditionally Independent Hierarchical Models (Parametric Empirical Bayes Models) , 1989 .

[26]  R. Gonzalez,et al.  Random effects diagonal metric multidimensional scaling models , 2001 .

[27]  Jon Cohen,et al.  Comparison of Partially Measured Latent Traits across Nominal Subgroups , 1999 .