A marginalization model for the multidimensional unfolding analysis of ranking data

A marginalization model for the multidimensional unfolding analysis of ranking data is presented. A subject samples one of a number of random points that are multivariate normally distributed. The subject perceives the distances from the point to all the stimulus points fixed in the same multidimensional space. The distances are error perturbed in this perception process. He/she produces a ranking dependent on these error-perturbed distances. The marginal probability of a ranking is obtained according to this ranking model and by integrating out the subject (ideal point) parameters, assuming the above distribution. One advantage of the model is that the individual differences are captured using the posterior probabilities of subject points. Three sets of ranking data are analyzed by the model.     

Influence analysis of ranking data

This paper develops diagnostic measures to identify those observations in Thurstonian models for ranking data which unduly influence parameter estimates that are obtained by the partition maximum likelihood approach of Chan and Bentler (1998). Diagnostic measures are constructed by employing the local influence approach that uses geometric techniques to assess the effect of small perturbations on a postulated statistical model. Very little additional effort is required to compute the proposed diagnostic measures, because all of the necessary building blocks are readily available after a usual fit of the model. The work described in this paper was partially supported by the grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (RGC Ref. No. CUHK4186/98P and RGC Direct Grant ID2060178). The authors are grateful to the Editor and four anonymous referees for their helpful comments.     

Bayesian analysis of order-statistics models for ranking data

In this paper, a class of probability models for ranking data, the order-statistics models, is investigated. We extend the usual normal order-statistics model into one where the underlying random variables follow a multivariate normal distribution. Bayesian approach and the Gibbs sampling technique are used for parameter estimation. In addition, methods to assess the adequacy of model fit are introduced. Robustness of the model is studied by considering a multivariate-t distribution. The proposed method is applied to analyze the presidential election data of the American Psychological Association (APA).The author is grateful to K. Lam, K.F. Lam, the Editor, an associate editor, and three reviewers for their valuable comments and suggestions. This research was substantially supported by the CRCG grant 335/017/0015 of the University of Hong Kong and a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU 7169/98H). Upon completion of this paper, I became aware that similar work had been done independently by K.G. Yao and U. Böckenholt (1999).     

A Thurstonian Analysis of Preference Change

This paper presents a Thurstonian model for the analysis of preference change. Preferences are expressed in the form of rankings, possibly with ties. A vector-autoregression framework is used to investigate relationships between past and current rankings. It is shown that this approach yields a parsimonious and easily interpretable representation of individual preference differences in time-dependent ranking data. A detailed analysis of the 1992 National Election Study illustrates the proposed approach.     

A latent trait model for dichotomous choice data

The PARELLA model is a probabilistic parallelogram model that can be used for the measurement of latent attitudes or latent preferences. The data analyzed are the dichotomous responses of persons to stimuli, with a one (zero) indicating agreement (disagreement) with the content of the stimulus. The model provides a unidimensional representation of persons and items. The response probabilities are a function of the distance between person and stimulus: the smaller the distance, the larger the probability that a person will agree with the content of the stimulus. An estimation procedure based on expectation maximization and marginal maximum likelihood is developed and the quality of the resulting parameter estimates evaluated.I gratefully acknowledge Ivo Molenaar and Wijbrandt van Schuur for their advice and encouragement during the course of the investigation, Derk-Jan Kiewiet who constructed the program for the ML estimator for the person parameter and Anne Boomsma, Wendy Post, Tom Snijders, and David Thissen for their comments on smaller aspects of the investigation.     

Estimating Optimal Weights for Compound Scores: A Multidimensional IRT Approach

A method is proposed for constructing indices as linear functions of variables such that the reliability of the compound score is maximized. Reliability is defined in the framework of latent variable modeling [i.e., item response theory (IRT)] and optimal weights of the components of the index are found by maximizing the posterior variance relative to the total latent variable variance. Three methods for estimating the weights are proposed. The first is a likelihood-based approach, that is, marginal maximum likelihood (MML). The other two are Bayesian approaches based on Markov chain Monte Carlo (MCMC) computational methods. One is based on an augmented Gibbs sampler specifically targeted at IRT, and the other is based on a general purpose Gibbs sampler such as implemented in OpenBugs and Jags. Simulation studies are presented to demonstrate the procedure and to compare the three methods. Results are very similar, so practitioners may be suggested the use of the easily accessible latter method. A real-data set pertaining to the 28-joint Disease Activity Score is used to show how the methods can be applied in a complex measurement situation with multiple time points and mixed data formats.     

A stochastic multidimensional unfolding approach for representing phased decision outcomes

This paper presents a stochastic multidimensional unfolding (MDU) procedure to spatially represent individual differences in phased or sequential decision processes. The specific application or scenario to be discussed involves the area of consumer psychology where consumers form judgments sequentially in their awareness, consideration, and choice set compositions in a phased or sequential manner as more information about the alternative brands in a designated product/service class are collected. A brief review of the consumer psychology literature on these nested congnitive sets as stages in phased decision making is provided. The technical details of the proposed model, maximum likelihood estimation framework, and algorithm are then discussed. A small scale Monte Carlo analysis is presented to demonstrate estimation proficiency and the appropriateness of the proposed model selection heuristic. An application of the methodology to capture awareness, consideration, and choice sets in graduate school applicants is presented. Finally, directions for future research and other potential applications are given.     

Covariance structure analysis of ordinal ipsative data

Data are ipsative if they are subject to a constant-sum constraint for each individual. In the present study, ordinal ipsative data (OID) are defined as the ordinal rankings across a vector of variables. It is assumed that OID are the manifestations of their underlying nonipsative vector y, which are difficult to observe directly. A two-stage estimation procedure is suggested for the analysis of structural equation models with OID. In the first stage, the partition maximum likelihood (PML) method and the generalized least squares (GLS) method are proposed for estimating the means and the covariance matrix of A_cy, where A_c is a known contrast matrix. Based on the joint asymptotic distribution of the first stage estimator and an appropriate weight matrix, the generalized least squares method is used to estimate the structural parameters in the second stage. A goodness-of-fit statistic is given for testing the hypothesized covariance structure. Simulation results show that the proposed method works properly when a sufficiently large sample is available.This research was supported by National Institute on Drug Abuse Grants DA01070 and DA10017. The authors are indebted to Dr. Lee Cooper, Dr. Eric Holman, Dr. Thomas Wickens for their valuable suggestions on this study, and Dr. Fanny Cheung for allowing us to use her CPAI data set in this article. The authors would also like to acknowledge the helpful comments from the editor and the two anonymous reviewers.     

On the relationship between item response theory and factor analysis of discretized variables

Equivalence of marginal likelihood of the two-parameter normal ogive model in item response theory (IRT) and factor analysis of dichotomized variables (FA) was formally proved. The basic result on the dichotomous variables was extended to multicategory cases, both ordered and unordered categorical data. Pair comparison data arising from multiple-judgment sampling were discussed as a special case of the unordered categorical data. A taxonomy of data for the IRT and FA models was also attempted.The work reported in this paper has been supported by Grant A6394 to the first author from the Natural Sciences and Engineering Research Council of Canada.     

Thurstonian modeling of ranking data via mean and covariance structure analysis

Although Thurstonian models provide an attractive representation of choice behavior, they have not been extensively used in ranking applications since only recently efficient estimation methods for these models have been developed. These, however, require the use of special-purpose estimation programs, which limits their applicability. Here we introduce a formulation of Thurstonian ranking models that turns an idiosyncratic estimation problem into an estimation problem involving mean and covariance structures with dichotomous indicators. Well-known standard solutions for the latter can be readily applied to this specific problem, and as a result any Thurstonian model for ranking data can be fitted using existing general purpose software for mean and covariance structure analysis. Although the most popular programs for covariance structure analysis (e.g., LISREL and EQS) cannot be presently used to estimate Thurstonian ranking models, other programs such as MECOSA already exist that can be straightforwardly used to estimate these models.This paper is based on the author's doctoral dissertation. Ulf Böckenholt was my advisor. The author is indebted to Ulf Böckenholt for his comments on a previous version of this paper and to Gerhard Arminger for his extensive support on the use of MECOSA. The final stages of this research took place while the author was at the Department of Statistics and Econometrics, Universidad Carlos III de Madrid. Conversations with my colleague there, Adolfo Hernández, helped to greatly improve this paper.     

Multidimensional scaling models for reaction times and same-different judgments

A method for joint analysis of reaction times and same-different judgments is discussed. A set of stimuli is assumed to have some parametric representation which uniquely defines dissimilarities between the stimuli. Those dissimilarities are then related to the observed reaction times and same-different judgments through a model of psychological processes. Three representation models of dissimilarities are considered, the Minkowski power distance model, the linear model, and Tversky's feature matching model. Maximum likelihood estimation procedures are developed and implemented in the form of a FORTRAN program. An example is given to illustrate the kind of analyses that can be performed by the proposed method.The work reported in this paper is supported by Grant A6394 to the first author from the Natural Sciences and Engineering Research Council of Canada. Portions of this study have been presented at the Psychometric Society meeting in Chapel Hill, N.C., in May, 1981. We thank Tony Marley, Jim Ramsay and anonymous reviewers for their helpful comments. MAXRT, a computer program which performs the computations described in this paper may be obtained by writing to the first author.     

A parametric procedure for ultrametric tree estimation from conditional rank order proximity data

The psychometric and classification literatures have illustrated the fact that a wide class of discrete or network models (e.g., hierarchical or ultrametric trees) for the analysis of ordinal proximity data are plagued by potential degenerate solutions if estimated using traditional nonmetric procedures (i.e., procedures which optimize a STRESS-based criteria of fit and whose solutions are invariant under a monotone transformation of the input data). This paper proposes a new parametric, maximum likelihood based procedure for estimating ultrametric trees for the analysis of conditional rank order proximity data. We present the technical aspects of the model and the estimation algorithm. Some preliminary Monte Carlo results are discussed. A consumer psychology application is provided examining the similarity of fifteen types of snack/breakfast items. Finally, some directions for future research are provided.     

Goodness of fit tests for paired comparison models

A goodness of fit test presented by Andersen is shown to be incorrect. The correct test is described and a re-analysis of Andersen's data is provided.     

Marginal maximum likelihood estimation of item response theory (IRT) equating coefficients for the common-examinee design

A method of estimating item response theory (IRT) equating coefficients by the common-examinee design with the assumption of the two-parameter logistic model is provided. The method uses the marginal maximum likelihood estimation, in which individual ability parameters in a common-examinee group are numerically integrated out. The abilities of the common examinees are assumed to follow a normal distribution but with an unknown mean and standard deviation on one of the two tests to be equated. The distribution parameters are jointly estimated with the equating coefficients. Further, the asymptotic standard errors of the estimates of the equating coefficients and the parameters for the ability distribution are given. Numerical examples are provided to show the accuracy of the method.     

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 the Bock-Aitkin procedure—From an EM algorithm perspective

The relationship between the EM algorithm and the Bock-Aitkin procedure is described with a continuous distribution of ability (latent trait) from an EM-algorithm perspective. Previous work has been restricted to the discrete case from a probit-analysis perspective. The author is grateful to Bradley A. Hanson for valuable discussion and comments. Thanks also go to Terry A. Ackerman, Meichu Fan, Subrata Kundu, and Robert K. Tsutakawa for their help and encouragement in this study.