Bayes modal estimation in item response models

This article describes a Bayesian framework for estimation in item response models, with two-stage prior distributions on both item and examinee populations. Strategies for point and interval estimation are discussed, and a general procedure based on the EM algorithm is presented. Details are given for implementation under one-, two-, and three-parameter binary logistic IRT models. Novel features include minimally restrictive assumptions about examinee distributions and the exploitation of dependence among item parameters in a population of interest. Improved estimation in a moderately small sample is demonstrated with simulated data.This research was supported by a grant from the Spencer Foundation, Chicago, IL. Comments and suggestions on earlier drafts by Charles Lewis, Frederic Lord, Rosenbaum, James Ramsey, Hiroshi Watanabe, the editor, and two anonymous referees are gratefully acknowledged.     

On the robustness of maximum likelihood scaling for violations of the error model

A Monte Carlo study was conducted to investigate the robustness of the assumed error distribution in maximum likelihood estimation models for multidimensional scaling. Data sets generated according to the lognormal, the normal, and the rectangular distribution were analysed with the log-normal error model in Ramsay's MULTISCALE program package. The results show that violations of the assumed error distribution have virtually no effect on the estimated distance parameters. In a comparison among several dimensionality tests, the corrected version of thex ² test, as proposed by Ramsay, yielded the best results, and turned out to be quite robust against violations of the error model.     

Optimum examinee samples for item parameter estimation in item response theory: A multi-objective programming approach

This paper proposes a multi-objective programming method for determining samples of examinees needed for estimating the parameters of a group of items. In the numerical experiments, optimum samples are compared to uniformly and normally distributed samples. The results show that the samples usually recommended in the literature are well suited for estimating the difficulty parameters. Furthermore, they are also adequate for estimating the discrimination parameters in the three-parameter model, butnot for the guessing parameters.     

Specifying optimum examinees for item parameter estimation in item response theory

Information functions are used to find the optimum ability levels and maximum contributions to information for estimating item parameters in three commonly used logistic item response models. For the three and two parameter logistic models, examinees who contribute maximally to the estimation of item difficulty contribute little to the estimation of item discrimination. This suggests that in applications that depend heavily upon the veracity of individual item parameter estimates (e.g. adaptive testing or text construction), better item calibration results may be obtained (for fixed sample sizes) from examinee calibration samples in which ability is widely dispersed.This work was supported by Contract No. N00014-83-C-0457, project designation NR 150-520, from Cognitive Science Program, Cognitive and Neural Sciences Division, Office of Naval Research and Educational Testing Service through the Program Research Planning Council. Reproduction in whole or in part is permitted for any purpose of the United States Government. The author wishes to acknowledge the invaluable assistance of Maxine B. Kingston in carrying out this study, and to thank Charles Lewis for his many insightful comments on earlier drafts of this paper.     

More on EM for ML factor analysis

We address several issues that are raised by Bentler and Tanaka's [1983] discussion of Rubin and Thayer [1982]. Our conclusions are: standard methods do not completely monitor the possible existence of multiple local maxima; summarizing inferential precision by the standard output based on second derivatives of the log likelihood at a maximum can be inappropriate, even if there exists a unique local maximum; EM and LISREL can be viewed as complementary, albeit not entirely adequate, tools for factor analysis.This work was partially supported by the Program Statistics Research Project at Educational Testing Service.     

Probabilistic multidimensional scaling: Complete and incomplete data

Simple procedures are described for obtaining maximum likelihood estimates of the location and uncertainty parameters of the Hefner model. This model is a probabilistic, multidimensional scaling model, which assigns a multivariate normal distribution to each stimulus point. It is shown that for such a model, standard nonmetric and metric algorithms are not appropriate. A procedure is also described for constructing incomplete data sets, by taking into consideration the degree of familiarity the subject has for each stimulus. Maximum likelihood estimates are developed both for complete and incomplete data sets. This research was supported by National Science Grant No. SOC76-20517. The first author would especially like to express his gratitude to the Netherlands Institute for Advanced Study for its very substantial help with this research.     

On Green's best linear composites with a specified structure,and oblique estimates of factor scores

Green solved the problem of least-squares estimation of several criteria subject to the constraint that the estimates have an arbitrary fixed covariance or correlation matrix. In the present paper an omission in Green's proof is discussed and resolved. Furthermore, it is shown that some recently published solutions for estimating oblique factor scores are special cases of Green's solution for the case of fixed covariance matrices.     

Pseudo maximum likelihood estimation and a test for misspecification in mean and covariance structure models

Using the theory of pseudo maximum likelihood estimation the asymptotic covariance matrix of maximum likelihood estimates for mean and covariance structure models is given for the case where the variables are not multivariate normal. This asymptotic covariance matrix is consistently estimated without the computation of the empirical fourth order moment matrix. Using quasi-maximum likelihood theory a Hausman misspecification test is developed. This test is sensitive to misspecification caused by errors that are correlated with the independent variables. This misspecification cannot be detected by the test statistics currently used in covariance structure analysis.For helpful comments on a previous draft of the paper we are indebted to Kenneth A. Bollen, Ulrich L. Küsters, Michael E. Sobel and the anonymous reviewers of Psychometrika. For partial research support, the first author wishes to thank the Department of Sociology at the University of Arizona, where he was a visiting professor during the fall semester 1987.     

Power of the likelihood ratio test in covariance structure analysis

A procedure for computing the power of the likelihood ratio test used in the context of covariance structure analysis is derived. The procedure uses statistics associated with the standard output of the computer programs commonly used and assumes that a specific alternative value of the parameter vector is specified. Using the noncentral Chi-square distribution, the power of the test is approximated by the asymptotic one for a sequence of local alternatives. The procedure is illustrated by an example. A Monte Carlo experiment also shows how good the approximation is for a specific case.This research was made possible by a grant from the Dutch Organization for Advancement of Pure Research (ZWO). The authors also like to acknowledge the helpful comments and suggestions from the editor and anonymous reviewers.