Limited information estimation and testing of Thurstonian models for preference data

Thurstonian models provide a rich representation of choice behavior that does not assume that stimuli are judged independently of each other, and they have an appealing substantive interpretation. These models can be seen as multivariate standard normal models that have been discretized using a set of thresholds and that impose certain restrictions on these thresholds and on the inter-correlations among the underlying normal variates. In this paper we provide a unified framework for modeling preference data under Thurstonian assumptions and we propose a limited information estimation and testing framework for it. Although these methods have a long tradition in psychometrics, until recently only their application to rating data has been considered. Here we shall give an overview of how these methods can be readily applied to fit not only rating data, but also paired comparison and ranking data. The limited information methods discussed here are appealing because they are extremely fast, they are able to estimate models essentially of any size, they can easily accommodate external information about the stimuli and / or respondents, and in simulations they have been found to be very robust to data sparseness.  2002 Elsevier Science B.V. All rights reserved.

[1]  Georgina Bermann.  Estimation and inference in bivariate and multivariate ordinal probit models , 1993 .

[2]  Geert De Soete,et al.  The Wandering Ideal Point Model: A Probabilistic Multidimensional Unfolding Model for Paired Comparisons Data , 1986 .

[3]  Ulf Böckenholt,et al.  Applications of Thurstonian Models to Ranking Data , 1993 .

[4]  Gail Gong,et al.  Pseudo Maximum Likelihood Estimation: Theory and Applications , 1981 .

[5]  P. Bentler,et al.  A testing method for covariance structure analysis , 1994 .

[6]  Yoshio Takane,et al.  MAXIMUM LIKELIHOOD ESTIMATION IN THE GENERALIZED CASE OF THURSTONE'S MODEL OF COMPARATIVE JUDGMENT , 1980 .

[7]  Ulrich Küsters,et al.  A note on sequential ML estimates and their asymptotic covariances , 1990 .

[8]  B. Muthén,et al.  Robust inference using weighted least squares and quadratic estimating equations in latent variable modeling with categorical and continuous outcomes , 1997 .

[9]  Bengt Muthén,et al.  Technical aspects of Muthén's liscomp approach to estimation of latent variable relations with a comprehensive measurement model , 1995 .

[10]  Willem J. Heiser,et al.  Multidimensional mapping of preference data , 1981 .

[11]  R. Tsai,et al.  Remarks on the identifiability of thurstonian ranking models: Case V, case III, or neither? , 2000 .

[12]  Clifford C. Clogg,et al.  Handbook of statistical modeling for the social and behavioral sciences , 1995 .

[13]  Ulrich Küsters,et al.  Hierarchische Mittelwert- und Kovarianzstrukturmodelle mit nichtmetrischen endogenen Variablen , 1987 .

[14]  A. Satorra,et al.  Corrections to test statistics and standard errors in covariance structure analysis. , 1994 .

[15]  Anders Christoffersson,et al.  A short note on the estimation of the asymptotic covariance matrix for polychoric correlations , 1996 .

[16]  B. Mellers,et al.  Similarity and Choice. , 1994 .

[17]  Jan de Leeuw,et al.  On the relationship between item response theory and factor analysis of discretized variables , 1987 .

[18]  L. Thurstone.  Rank order as a psycho-physical method. , 1931 .

[19]  L. Thurstone.  A law of comparative judgment. , 1994 .

[20]  Henry E. Brady.  Factor and ideal point analysis for interpersonally incomparable data , 1989 .

[21]  Ulf Böckenholt,et al.  Modeling within-subject dependencies in ordinal paired comparison data , 1997 .

[22]  M. Browne.  Asymptotically distribution-free methods for the analysis of covariance structures. , 1984, The British journal of mathematical and statistical psychology.

[23]  U. Böckenholt,et al.  Maximum likelihood estimation of factor and ideal point models for paired comparison data , 2001 .

[24]  Joseph S. Verducci,et al.  Probability Models and Statistical Analyses for Ranking Data , 1992 .

[25]  James Arbuckle,et al.  A GENERAL PROCEDURE FOR PARAMETER ESTIMATION FOR THE LAW OF COMPARATIVE JUDGEMENT , 1973 .

[26]  A. Tversky.  Elimination by aspects: A theory of choice. , 1972 .

[27]  Alan Agresti,et al.  Analysis of Ordinal Paired Comparison Data , 1992 .

[28]  Karl G. Jöreskog,et al.  On the estimation of polychoric correlations and their asymptotic covariance matrix , 1994 .

[29]  Albert Maydeu-Olivares,et al.  Thurstonian modeling of ranking data via mean and covariance structure analysis , 1999 .

[30]  B. Muthén.  Contributions to factor analysis of dichotomous variables , 1978 .

[31]  Albert Maydeu-Olivares,et al.  Limited information estimation and testing of Thurstonian models for paired comparison data under multiple judgment sampling , 2001 .

[32]  Ulf Olsson,et al.  Maximum likelihood estimation of the polychoric correlation coefficient , 1979 .

[33]  L. V. Jones,et al.  The measurement and prediction of judgment and choice. , 1970 .

[34]  P M Bentler,et al.  A two-stage estimation of structural equation models with continuous and polytomous variables. , 1995, The British journal of mathematical and statistical psychology.

[35]  Peter M. Bentler,et al.  Covariance structure analysis of ordinal ipsative data , 1998 .

[36]  G. Arminger,et al.  Specification and Estimation of Mean- and Covariance-Structure Models , 1995 .

[37]  C. Raghavendra Rao,et al.  Multivariate Analysis and Its Applications , 1984 .

[38]  J. S. Long,et al.  Testing Structural Equation Models , 1993 .

[39]  Anders Christoffersson,et al.  Factor analysis of dichotomized variables , 1975 .

[40]  Geert De Soete,et al.  A maximum likelihood method for fitting the wandering vector model , 1983 .

[41]  B. Muthén.  A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicators , 1984 .

[42]  J. Teugels.  Some representations of the multivariate Bernoulli and binomial distributions , 1990 .

[43]  U. Böckenholt,et al.  BAYESIAN ESTIMATION OF THURSTONIAN RANKING MODELS BASED ON THE GIBBS SAMPLER , 1999 .

[44]  Yoshio Takane,et al.  Analysis of Covariance Structures and Probabilistic Binary Choice Data , 1989 .

[45]  B. R. Dansie.  Normal order statistics as permutation probability models , 1986 .

[46]  Albert Maydeu-Olivares,et al.  Thurstonian Covariance and Correlation Structures for Multiple Judgment Paired Comparison Data , 2003 .