Latent class models for nonmonotone dichotomous items

Starting from perfectly discriminating nonmonotone dichotomous items, a class of probabilistic models with or without response errors and with or without intrinsically unscalable respondents is described. All these models can be understood as simply restricted latent class analysis. Thus, the estimation and identifiability of the parameters (class sizes and item latent probabilities) as well as the chi-squared goodness-of-fit tests (Pearson and likelihood-ratio) are free of the problems. The applicability of the proposed variants of latent class models is demonstrated on real attitudinal data.This research was supported by the Kulturamt der Stadt Wien, Magistratsabteilung 7.The author wishes to thank the editor, Ivo W. Molenaar, as well as Clifford C. Clogg and the anonymous reviewers for their valuable comments on the earlier drafts of this paper.     

The effect of difficulty and chance success on item-test correlation and on test reliability

An equation is derived for predicting the effect of chance success, relative to item difficulty, on item-test correlation. The values predicted by this equation and by equations derived by Guilford and Carroll for predicting the effect of chance success on item difficulty and test reliability are compared with empirical values in an experiment which used identical test items in multiple-choice and answer-only form.Condensation of a dissertation presented in partial fulfillment of the requirements for the Ph.D. degree to the University of Chicago. Grateful acknowledgment is made to Professor Harold Gulliksen for his guidance as thesis advisor and to Professor L. L. Thurstone and Dr. D. W. Fiske of the University of Chicago who served as members of the thesis committee. The author is also indebted to Professor S. S. Wilks for review of the derivations and development of statistical tests used in the thesis, to Dr. L. R Tucker for technical advice, and to Dr. W. G. Mollenkopf for critical comments on the derivations and interpretations. The writer expresses appreciation to the Educational Testing Service for making available its technical facilities, and to the University of Chicago for the flexible administrative arrangement which made this thesis possible.     

Simple statistical methods for scalogram analysis

Simple statistical methods are developed to test whether coefficients of reproducibility, of homogeneity, or of consistency differ significantly from what can be expected if responses to different items are statistically independent. Simple methods are also developed for estimating the variance of coefficients of reproducibility when it is not assumed that responses to different items are independent. These estimates are used to test whether a coefficient differs significantly from any prescribed value, and also to obtain confidence intervals for these coefficients. The rationale for the measurement of reproducibility is also discussed.This research was carried out at the Statistical Research Center, University of Chicago, under sponsorship of the Statistics Branch, Office of Naval Research, and of the Social Science Research Committee, University of Chicago. Reproduction in whole or in part is permitted for any purpose of the United States Government. The author is indebted to Jacob Gewirtz for some very helpful comments.This paper is dedicated to Professor Jacob Marschak on the occasion of his sixtieth birthday, July 23, 1958.     

Characterizing the manifest probabilities of latent trait models

The problem of characterizing the manifest probabilities of a latent trait model is considered. The item characteristic curve is transformed to the item passing-odds curve and a corresponding transformation is made on the distribution of ability. This results in a useful expression for the manifest probabilities of any latent trait model. The result is then applied to give a characterization of the Rasch model as a log-linear model for a 2 ^J -contingency table. Partial results are also obtained for other models. The question of the identifiability of “guessing” parameters is also discussed. The research reported here is collaborative in every respect and the order of authorship is alphabetical. Dr. Cressie was a Visiting Research Scientist at ETS during the Fall of 1980. His current address is: School of Mathematical Sciences, The Flinders University of South Australia, Bedford Park SA, 5042, AUSTRALIA. The preparation of this paper was supported, in part, by the Program Statistics Research Project in the Research Statistics Group at ETS.     

Two-alternative learning situations with partial reinforcement

The comparative effects of reward and nonreward on learning are considered in connection with a two-alternative learning situation. Conditions are more general in nature than those discussed in an earlier article. In the statistical model proposed, the question of whether reward and nonreward are equivalent in their effects on learning reduces to testing a composite hypothesis on a multivariate probability distribution. An asymptotic test of this hypothesis is described, and its use is illustrated with data from psychological experiments.This paper was prepared with the partial support of the Office of Naval Research, and may be reproduced in whole or in part for any purpose of the United States government.I wish to express my gratitude to Professors J. Neyman and E. L. Scott of the University of California, Department of Statistics, for their constant assistance and encouragement throughout the research that led to this paper and during its preparation, and to Professor F. W. Irwin of the University of Pennsylvania Department of Psychology for his many helpful suggestions and comments.     

A generalization of the bush-mosteller model with some significance tests

A generalization of the Bush-Mosteller learning model is proposed in connection with two-alternative learning situations with continuous reinforcement. The problem is to test the hypothesis that reward and non-reward are equally effective in promoting learning. Statistically this reduces to testing a hypothesis about the value of a single parameter, while a set of other parameters remains unspecified. The test presented has the property of being asymptotically locally most powerful among all tests of the same size and asymptotically similar. The application of the test is illustrated.I would like to express my gratitude to Professors J. Neyman and E. L. Scott, Department of Statistics, University of California, for their constant assistance and encouragement throughout the research that led to this paper and during its preparation, and to Professor F. W. Irwin, Department of Psychology, University of Pennsylvania, for his many helpful suggestions and comments.     

Explaining change in psychology: The road not taken

I wish to express sincere thanks to the following persons who have given me the benefit of their comments and criticisms: Anthony S. Bryk, Renford Bamreough, Simon Blackburn, David Bloor, Jeremy Butterfield, Edward Craig, Jerry A. Fodor, Robert P. Gordon, P.M.S. Hacker, David Hansen, D.W. Hamlyn, Philip W. Jackson, Anthony Kenny, Peggy J. Miller, and Robert J. Richards. In addition, I gratefully acknowledge the support I received from a Spencer Seed Grant, University of Chicago, which allowed me to go to England in the summer of 1985 and discuss a version of this work with colleagues. Finally, I owe thanks to Benton Center for Curriculum and Instruction, University of Chicago, for various support services.     

Psychometrics in Psychological Research: Role Model or Partner in Science?

This is a reaction to Borsboom's (2006) discussion paper on the issue that psychology takes so little notice of the modern developments in psychometrics, in particular, latent variable methods. Contrary to Borsboom, it is argued that latent variables are summaries of interesting data properties, that construct validation should involve studying nomological networks, that psychological research slowly but definitely will incorporate latent variable methods, and that the role of psychometrics in psychology is that of partner, not role model. Requests for reprints should be sent to Klaas Sijtsma, Department of Methodology and Statistics, FSW, Tilburg University, PO Box 90153, 5000 LE, Tilburg, The Netherlands.     

On the relations among regular, equal unique variances, and image factor analysis models

We investigate under what conditions the matrix of factor loadings from the factor analysis model with equal unique variances will give a good approximation to the matrix of factor loadings from the regular factor analysis model. We show that the two models will give similar matrices of factor loadings if Schneeweiss' condition, that the difference between the largest and the smallest value of unique variances is small relative to the sizes of the column sums of squared factor loadings, holds. Furthermore, we generalize our results and discus the conditions under which the matrix of factor loadings from the regular factor analysis model will be well approximated by the matrix of factor loadings from Jöreskog's image factor analysis model. Especially, we discuss Guttman's condition (i.e., the number of variables increases without limit) for the two models to agree, in relation with the condition we have shown, and conclude that Schneeweiss' condition is a generalization of Guttman's condition. Some implications for practice are discussed.Kentaro Hayashi is a visiting Assistant Professor, Department of Mathematics, Bucknell University, Lewisburg PA 17837, and Peter M. Bentler is Professor, Departments of Psychology and Statistics, University of California, Los Angeles CA 90095-1563. (Emails: Khayashi@bucknell.edu, bentler@ucla.edu) Parts of this paper were discussed in a session on Factor Analysis (J. ten Berge, Chair) at the IFCS-98 International Conference, Rome, July, 1998. This work was supported by National Institute on Drug Abuse grant DA 01070. The authors thank Professors Hans Schneeweiss and Ke-Hai Yuan, and four anonymous referees, for their invaluable comments which led to an improved version of this paper.     

Maximum Likelihood Methods in Treating Outliers and Symmetrically Heavy-Tailed Distributions for Nonlinear Structural Equation Models with Missing Data

By means of more than a dozen user friendly packages, structural equation models (SEMs) are widely used in behavioral, education, social, and psychological research. As the underlying theory and methods in these packages are vulnerable to outliers and distributions with longer-than-normal tails, a fundamental problem in the field is the development of robust methods to reduce the influence of outliers and the distributional deviation in the analysis. In this paper we develop a maximum likelihood (ML) approach that is robust to outliers and symmetrically heavy-tailed distributions for analyzing nonlinear SEMs with ignorable missing data. The analytic strategy is to incorporate a general class of distributions into the latent variables and the error measurements in the measurement and structural equations. A Monte Carlo EM (MCEM) algorithm is constructed to obtain the ML estimates, and a path sampling procedure is implemented to compute the observed-data log-likelihood and then the Bayesian information criterion for model comparison. The proposed methodologies are illustrated with simulation studies and an example. The research described herein was fully supported by a grant (CUHK 4243/03H) from the Rearch Grants Council of the Hong Kong Special Administration Region. The authors are thankful to the Editor, the Associate Editor, and anonymous reviewers for valuable comments which improve the paper significantly, and are grateful to ICPSR and the relevant funding agency for allowing the use of their data. Requests for reprints should be sent to S. Y. Lee, Department of Statistics, The Chinese University of Hong Kong, Shatin, N. T., Hong Kong.     

Exact Interval Estimation, Power Calculation, and Sample Size Determination in Normal Correlation Analysis

This paper considers the problem of analysis of correlation coefficients from a multivariate normal population. A unified theorem is derived for the regression model with normally distributed explanatory variables and the general results are employed to provide useful expressions for the distributions of simple, multiple, and partial-multiple correlation coefficients. The inversion principle and monotonicity property of the proposed formulations are used to describe alternative approaches to the exact interval estimation, power calculation, and sample size determination for correlation coefficients. The author thanks the referees for their constructive comments and helpful suggestions and especially the associate editor for drawing attention to several critical results which led to substantial improvements of the exposition. The work for this paper was initiated while the author was visiting the Department of Statistics, Stanford University. This research was partially supported by National Science Council Grant NSC-94-2118-M-009-004. Request for reprints should be sent to Gwowen Shieh, Department of Management Science, National Chiao Tung University, Hsinchu, Taiwan 30050, ROC.     

Aggregate item response analysis

A stochastic postulate is given for the multiple-item, successive-intervals scaling of populations. The logistic equivalent of this postulate provides an aggregate item response model in which a unidimensional submodel may be nested. This reduction provides a subtractive conjoint measurement of several items and stimuli on the same latent scale. Generalized-least-squares methods are used to estimate and test the multiple-item model, and its unidimensional reduction, on aggregate survey responses. The entire procedure is illustrated with an analysis of semantic-differential attitude data. This analysis exhibits an item selection procedure that is applicable to various social constructs.The authors dedicate this paper to the memory and contributions of Clyde Coombs.The programming and data analyses for the present paper were carried out by José Ventura of the Department of Industrial and Systems Engineering, and Jerry Meiten of the Department of Statistics, University of Florida.The study was also supported by the College of Business Administration, University of Florida, and the Faculty of Social Sciences, Hebrew University of Jerusalem.     

Gennclus: New models for general nonhierarchical clustering analysis

A general class of nonhierarchical clustering models and associated algorithms for fitting them are presented. These (metric) clustering models generalize the Shepard-Arabie Additive Clusters model in allowing for: (1). either overlapping or nonoverlapping clusters; (2). either symmetric (one-way clustering) or nonsymmetric (two-way clustering) proximities (input data); and, (3). either symmetric or diagonal weights. The GENNCLUS algorithms utilize alternating least-squares methods combining ordinary and constrained least-squares, nonlinear constrained mathematical programming, and combinatorial optimization techniques in estimating model parameters. In addition to developing the mathematical bases of these models, a comprehensive set of Monte Carlo simulations of the different models is reported. Two applications concerning brand-switching data and celebrity-brand proximities are discussed. Finally, extensions to three-way models, nonmetric analyses, and other model specifications are provided.Wayne S. DeSarbo is a Member of Technical Staff in the Mathematics and Statistics Research Center at Bell Laboratories in Murray Hill, N.J. I wish to thank R. Gnanadesikan, J. D. Carroll, and P. Arabie for their comments on a previous draft of this paper. I also wish to acknowledge the computer assistance provided by Linda Clark. Finally, I wish to thank the reviewers and editor for their very complete reviews and comments.     

Likelihood inference on the underlying structure of IRT models

The assumptions underlying item response theory (IRT) models may be expressed as a set of equality and inequality constraints on the parameters of a latent class model. It is well known that the same assumptions imply that the parameters of the manifest distribution have to satisfy a more complicated set of inequality constraints which, however, are necessary but not sufficient. In this paper, we describe how the theory for likelihood-based inference under equality and inequality constraints may be used to test the underlying assumptions of IRT models. It turns out that the analysis based directly on the latent structure is simpler and more flexible. In particular, we indicate how several interesting extensions of the Rasch model may be obtained by partial relaxation of the basic constraints. An application to a data set provided by Educational Testing Service is used to illustrate the approach.We thank Dr. Gorman and Dr. Rogers of the Educational Testing Service for providing the data analyzed in Section 4. We also thank three reviewers for comments and suggestions.This revised article was published online in August 2005 with the PDF paginated correctly.     

Standard errors for the class of orthomax-rotated factor loadings: Some matrix results

The partial derivative matrices of the class of orthomax-rotated factor loadings with respect to the unrotated maximum likelihood factor loadings are derived. The reported results are useful for obtaining standard errors of the orthomax-rotated factor loadings, with or without row normalization (standardization) of the initial factor loading matrix for rotation. Using a numerical example, we verify our analytic formulas by comparing the obtained standard error estimates with that from some existing methods. Some advantages of the current approach are discussed.Authorship is determined by alphabetical order. The authors contributed equally to the research. Kentaro Hayashi is now at the Department of Mathematics, Bucknell University, Lewisburg, PA 17837 (email: khayashi@Bucknell.edu). Yiu-Fai Yung is now at the SAS Institute, Inc., SAS Campus Drive, Cary, NC 27513 (email: yiyung@wnt.sas.com).Part of the research was completed while Yiu-Fai Yung was a visiting scholar at the Department of Psychology, the Ohio State University. The visit was supported in part by grant N4856118101 from the NIMH and the Mason and Linda Stephenson Travel Award from the Department of Psychology, University of North Carolina at Chapel Hill. The authors are grateful to Michael Browne who suggested some relevant references and provided valuable comments on the research, and to Robert Cudeck who provided the FAS program for the numerical comparison. The expert comments by the reviewers are deeply appreciated.     

The guttman-harris uniqueness estimates

It is shown that, under very general conditions, uniqueness estimates proposed independently by Guttman [1957] and by Harris [1963] provide tighter upper bounds on the unknown uniqueness values of factor analysis than do existing estimates.The Illinois Department of Mental Health Drug Abuse Program is co-sponsored by the State of Illinois and the Department of Psychiatry, Division of the Biological Sciences and Pritzker School of Medicine, University of Chicago. Send reprint requests to Drug Abuse Programs—Research, East Pavilion, Museum of Science and Industry, 57th Street and South Lake Shore Drive, Chicago, Illinois 60637.     

Some models for individual-group comparisons and group behavior

In problem solving situations, it has been suggested the superiority of groups over individuals is due simply to the fact that groups consist of several individuals. In this paper, the Lorge and Solomon [1955] approach to such situations is reexamined using the method of maximum likelihood. Extensions to trichotomous response situations are also presented, and the resulting models are applied to data gathered by Staub [1970]. The partitioning of the likelihood ratio goodness-of-fit statistic is then discussed in order that the effects of additional variables on the response variate can be assessed. Finally, the small sample behavior of the likelihood ratio statistic is examined.This research was supported in part by Research Contract No. NSF GS-2818 from the Division of Social Sciences of the National Science Foundation to the Department of Statistics, The University of Chicago.Now at the University of Minnesota.     

Bayesian Semiparametric Structural Equation Models with Latent Variables

Structural equation models (SEMs) with latent variables are widely useful for sparse covariance structure modeling and for inferring relationships among latent variables. Bayesian SEMs are appealing in allowing for the incorporation of prior information and in providing exact posterior distributions of unknowns, including the latent variables. In this article, we propose a broad class of semiparametric Bayesian SEMs, which allow mixed categorical and continuous manifest variables while also allowing the latent variables to have unknown distributions. In order to include typical identifiability restrictions on the latent variable distributions, we rely on centered Dirichlet process (CDP) and CDP mixture (CDPM) models. The CDP will induce a latent class model with an unknown number of classes, while the CDPM will induce a latent trait model with unknown densities for the latent traits. A simple and efficient Markov chain Monte Carlo algorithm is developed for posterior computation, and the methods are illustrated using simulated examples, and several applications.     

Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm

Maximum likelihood estimation of item parameters in the marginal distribution, integrating over the distribution of ability, becomes practical when computing procedures based on an EM algorithm are used. By characterizing the ability distribution empirically, arbitrary assumptions about its form are avoided. The Em procedure is shown to apply to general item-response models lacking simple sufficient statistics for ability. This includes models with more than one latent dimension.Supported in part by NSF grant BNS 7912417 to the University of Chicago and by SSRC (UK) grant HR6132 to the University of Lancaster.We are indebted to Mark Reiser and Robert Gibbons for computer programming. David Thissen clarified a number of points in an earlier draft.