Nonparametric Estimation of Item and Respondent Locations from Unfolding-type Items

Unlike their monotone counterparts, nonparametric unfolding response models, which assume the item response function is unimodal, have seen little attention in the psychometric literature. This paper studies the nonparametric behavior of unfolding models by building on the work of Post (1992). The paper provides rigorous justification for a class of nonparametric estimators of respondents’ latent attitudes by proving that the estimators consistently rank order the respondents. The paper also suggests an algorithm for the rank ordering of items along the attitudes scale. Finally, the methods are evaluated using simulated data. This research was supported in part by an Educational Testing Service Gulliksen Fellowship, and by the National Science Foundation, Grant DMS-97.05032. The author would like to thank Brian Junker for his help and support on this paper and Paul Holland, Steve Fienberg, and Jay Kadane for their helpful comments.     

Three-way metric unfolding via alternating weighted least squares

Three-way unfolding was developed by DeSarbo (1978) and reported in DeSarbo and Carroll (1980, 1981) as a new model to accommodate the analysis of two-mode three-way data (e.g., nonsymmetric proximities for stimulus objects collected over time) and three-mode, three-way data (e.g., subjects rendering preference judgments for various stimuli in different usage occasions or situations). This paper presents a revised objective function and new algorithm which attempt to prevent the common type of degenerate solutions encountered in typical unfolding analysis. We begin with an introduction of the problem and a review of three-way unfolding. The three-way unfolding model, weighted objective function, and new algorithm are presented. Monte Carlo work via a fractional factorial experimental design is described investigating the effect of several data and model factors on overall algorithm performance. Finally, three applications of the methodology are reported illustrating the flexibility and robustness of the procedure.We wish to thank the editor and reviewers for their insightful comments.     

Avoiding degeneracy in multidimensional unfolding by penalizing on the coefficient of variation

Multidimensional unfolding methods suffer from the degeneracy problem in almost all circumstances. Most degeneracies are easily recognized: the solutions are perfect but trivial, characterized by approximately equal distances between points from different sets. A definition of an absolutely degenerate solution is proposed, which makes clear that these solutions only occur when an intercept is present in the transformation function. Many solutions for the degeneracy problem have been proposed and tested, but with little success so far. In this paper, we offer a substantial modification of an approach initiated bythat introduced a normalization factor based on thevariance in the usual least squares loss function. Heiser unpublishedthesis, (1981) and showed that the normalization factor proposed by Kruskal and Carroll was not strong enough to avoid degeneracies. The factor proposed in the present paper, based on the coefficient of variation, discourages or penalizes nonmetric transformations of the proximities with small variation, so that the procedure steers away from solutions with small variation in the interpoint distances. An algorithm is described for minimizing the re-adjusted loss function, based on iterative majorization. The results of a simulation study are discussed, in which the optimal range of the penalty parameters is determined. Two empirical data sets are analyzed by our method, clearly showing the benefits of the proposed loss function.The authors would like to thank the editor, an associate editor, and three reviewers for their valuable comments and suggestions to improve the quality of this work.This revised article was published online in August 2005 with the PDF paginated correctly.     

Least squares metric,unidimensional unfolding

The partial derivatives of the squared error loss function for the metric unfolding problem have a unique geometry which can be exploited to produce unfolding methods with very desirable properties. This paper details a simple unidimensional unfolding method which uses the geometry of the partial derivatives to find conditional global minima; i.e., one set of points is held fixed and the global minimum is found for the other set. The two sets are then interchanged. The procedure is very robust. It converges to a minimum very quickly from a random or non-random starting configuration and is particularly useful for the analysis of large data sets with missing entries.This paper benefits from many conversations with and suggestions from Howard Rosenthal.     

An individual differences model for multidimensional scaling

A quantitative system is presented to permit the determination of separate multidimensional perceptual spaces for individuals having different viewpoints about stimulus interrelationships. The structure of individual differences in the perception of stimulus relationships is also determined to provide a framework for ascertaining the varieties of consistent individual viewpoints and their relationships with other variables.This research was supported in part by the National Institute of Mental Health, United States Public Health Service, under Research Grants M-2878 and M-4186 to Educational Testing Service, in part by Educational Testing Service, and in part by the Office of Naval Research under Contract Nonr-1834(39) and the University of Illinois. The authors wish to thank Drs. Harold Gulliksen and Douglas N. Jackson for their helpful comments and Miss Henrietta Gallagher for supervising the computations. Portions of this paper were presented at the American Psychological Association meetings in Chicago, September 1960.This paper was written while Dr. Messick was a Fellow at the Center for Advanced Study in the Behavioral Sciences.     

Exploratory longitudinal factor analysis in multiple populations

For multiple populatios, a longtidinal factor analytic model which is entirely exploratory, that is, no explicit identification constraints, is proposed. Factorial collapse and period/practice effects are allowed. An invariant and/or stationary factor pattern is permitted. This model is formulated stochastically. To implement this model a stagewise EM algorithm is developed. Finally a numerical illustration utilizing Nesselroade and Baltes' data is presented.The authors wish to thank Barbara Mellers and Henry Kaiser for their helpful comments and John Nesselroade for providing us the data for our illustration. This research wwa supported in part by a grant (No. AG03164) from the National Institute on Aging to William Meredith. Details of the derivations and a copy of the PROC MATRIX program are available upon request from the first author.     

On scaling models applied to data from several groups

Statistical methods are presented to facilitate a more complete analysis of results obtained when a scaling model is applied to data from two or more groups. These methods can be used to (a) compare the corresponding estimated latent distributions obtained using the scaling model applied to the different groups, (b) compare the corresponding estimated item reliabilities (or item response error rates) for the different groups, and (c) test whether the scaling model applied to the several groups can be replaced by a more parsimonious scaling model that includes various homogeneity constraints (i.e., constraints that describe which parameters in the model are the same for the several groups). Various kinds of scaling models are considered here in the multiple-group context.Support for this research was provided in part by the National Science Foundation, to Clogg by Grant No. SES-7823759 and to Goodman by Grant No. SES-8303838. Clogg and Goodman were Fellows at the Center for Advanced Study in the Behavioral Sciences when part of the research was done, with financial support provided in part by National Science Foundation grant BNS-8011494 to the Center. The authors are indebted to Mark P. Becker and James W. Shockey for helpful comments.     

The scientific error of Friedman's poor scholarship: A rejoinder

A recent article by Friedman published in this journal alleges that a study by Whitcher and Fisher (1979) in the Journal of Personality and Social Psychology is sexist research. An examination of his critique reveals that the facts are not as he purports. A reading of the Whitcher and Fisher (1979) study shows that the authors followed explicitly APA (1977) guidelines for nonsexist language in writing their article.In his article, Friedman makes a number of critical comments of a general nature that bear no relationship to the focus of our study (e.g., his comments about being touched by doctors; and patients' expectations about the role of doctors, both on p. 749). Unfortunately, he does not clarify which of his comments are aimed at our study and which are not. Therefore, we encourage readers unfamiliar with our research to review our original article.     

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.     

Factor analysis by generalized least squares

Aitken's generalized least squares (GLS) principle, with the inverse of the observed variance-covariance matrix as a weight matrix, is applied to estimate the factor analysis model in the exploratory (unrestricted) case. It is shown that the GLS estimates are seale free and asymptotically efficient. The estimates are computed by a rapidly converging Newton-Raphson procedure. A new technique is used to deal with Heywood cases effectively.The work on this project was done when the first author was Research Statistician at Educational Testing Service, Princeton, N. J. The second author was in part supported by a grant from the Research Committee of the University of Wisconsin Graduate School. The authors wish to thank Michael Browne for many helpful comments and Marielle van Thillo for valuable assistance in the numerical computations.     

Global optimality of the successive Maxbet algorithm

The Maxbet method is an alternative to the method of generalized canonical correlation analysis and of Procrustes analysis. Contrary to these methods, it does not maximize the inner products (covariances) between linear composites, but also takes their sums of squares (variances) into account. It is well-known that the Maxbet algorithm, which has been proven to converge monotonically, may converge to local maxima. The present paper discusses an eigenvalue criterion which is sufficient, but not necessary for global optimality. However, in two special cases, the eigenvalue criterion is shown to be necessary and sufficient for global optimality. The first case is when there are only two data sets involved; the second case is when the inner products between all variables involved are positive, regardless of the number of data sets.The authors are obliged to Henk Kiers for critical comments on a previous draft.     

Least squares metric,unidimensional scaling of multivariate linear models

The squared error loss function for the unidimensional metric scaling problem has a special geometry. It is possible to efficiently find the global minimum for every coordinate conditioned on every other coordinate being held fixed. This approach is generalized to the case in which the coordinates are polynomial functions of exogenous variables. The algorithms shown in the paper are linear in the number of parameters. They always descend and, at convergence, every coefficient of every polynomial is at its global minimum conditioned on every other parameter being held fixed. Convergence is very rapid and Monte Carlo tests show the basic procedure almost always converges to the overall global minimum.The author thanks Ivo Molenaar, three anonymous referees, and Howard Rosenthal for their many helpful comments.     

Exact tests for the rasch model via sequential importance sampling

Rasch proposed an exact conditional inference approach to testing his model but never implemented it because it involves the calculation of a complicated probability. This paper furthers Rasch’s approach by (1) providing an efficient Monte Carlo methodology for accurately approximating the required probability and (2) illustrating the usefulness of Rasch’s approach for several important testing problems through simulation studies. Our Monte Carlo methodology is shown to compare favorably to other Monte Carlo methods proposed for this problem in two respects: it is considerably faster and it provides more reliable estimates of the Monte Carlo standard error.This Research was supported in part by National Science Foundation grant DMS-0203762 and a University of Pennsylvania Research Foundation grant.The authors are grateful to Don Burdick for helpful comments. In addition, the authors wish to thank the editor, the associate editor, and the referees for their helpful suggestions.This revised article was published online in August 2005 with the PDF paginated correctly.     

Are the characteristics of narrative comments related to improvement in multirater feedback ratings over time?

Researchers have paid almost no attention to the narrative comments that typically accompany multirater feedback reports despite the fact that both anecdotal and empirical evidence suggest that feedback recipients devote considerable attention to such comments. The authors examined improvement in upward feedback ratings over a 1-year period for 176 managers as a function of (a) the number of narrative comments each manager received, (b) whether those comments were favorable (vs. unfavorable), and (c) whether the comments were behavior/task focused (vs. trait focused). The authors found that managers who received a small number of unfavorable, behavior/task-focused comments improved more than did other managers, whereas managers who received a large number of unfavorable, behavior/task-focused comments declined more than did other managers.     

Average exponent from magnitude production

When magnitude production is used to obtain a psychophysical power function for a group ofSs, the group exponent is shown to be the harmonic mean of individual-S exponents under the conditions usually employed in empirical research. The conditions are (a) the same numbers are presented to eachS, (b) responses to each number are pooled overSs by taking geometric means, and (c) the function is fit by the method of least squares in which regression is of the logarithm ofSs' responses upon the logarithm of numbers presented byE.This paper was supported in part by grant no. APA-151 from the National Research Council of Canada. The author wishes to thank William W. Rozeboom for his helpful comments on this paper.     

A general solution of the weighted orthonormal procrustes problem

A general solution for weighted orthonormal Procrustes problem is offered in terms of the least squares criterion. For the two-demensional case. this solution always gives the global minimum; for the general case, an algorithm is proposed that must converge, although not necessarily to the global minimum. In general, the algorithm yields a solution for the problem of how to fit one matrix to another under the condition that the dimensions of the latter matrix first are allowed to be transformed orthonormally and then weighted differentially, which is the task encountered in fitting analogues of the IDIOSCAL and INDSCAL models to a set of configurations.The authors are grateful to the Editor and the anonymous reviewers for their helpful comments on an earlier draft of this paper.     

Direct versus indirect response-reinforcer relationships in teaching autistic children

One possibility suggested by the literature for maximizing the efficiency of behavior modification procedures concerns the relationship between target behaviors and their reinforcers. Therefore, in this experiment three severely autistic children were taught a total of six new target behaviors (in a multiple baseline design) employing two different response-reinforcer relationships: (1) those where the target behaviors were a direct part of the response chain required to procure a reinforcer (e.g., opening the lid of a container to obtain a food reward inside the container); and (2) those where the target behavior was an indirect part of the chain leading to the reinforcer (e.g., the therapist handing the child a food reward after the child had opened the lid of an empty container). In all cases, the results showed rapid acquisition only when the target behavior was a direct part of the chain leading to the reinforcer. The results are discussed in terms of several possible conceptualizations concerning efficient reinforcement contingencies, and in terms of their implications for teaching autistic children.This investigation was supported by U.S. Public Health Service Research Grants MH 28210 and MH 28231, and by U.S. Office of Education Research Grant G007802084 from the Bureau for the Education of the Handicapped. Portions of the results described in this article were discussed in a presentation at the 13th Annual AABT Convention in San Francisco, 1979. The authors are particularly grateful to Andrew Egel, Roberta Elman, Janis Costello, and Karen Britten, for their helpful comments, and to Jody Fisher and Gina Richman, for their assitance in the data collection.     

Profiles of Counseling Pioneers: A Visit With Kathleen and Gilbert Wrenn

The authors have had the honor of knowing Gilbert and Kathleen Wrenn for 20 years; this story is about them. As Gilbert stated, “I am not a lone star, but part of a constellation… ”(Wachowiak & Aubrey, 1976, p. 76). The authors are privileged to know that for Gilbert, Kathleen (his partner of 72 years in 1996) is the center of that constellation. This article briefly captures pieces of the journey of 72+ years, on which Gilbert and Kathleen, each at 96 years of age, are continuing to travel together.