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.     

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.     

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).     

Estimation for the multiple factor model when data are missing

A maximum likelihood method of estimating the parameters of the multiple factor model when data are missing from the sample is presented. A Monte Carlo study compares the method with 5 heuristic methods of dealing with the problem. The present method shows some advantage in accuracy of estimation over the heuristic methods but is considerably more costly computationally.This paper is based on the author's doctoral dissertation at the Department of Psychology, University of Illinois at Urbana-Champaign. The author gratefully acknowledges the aid of Drs. Robert Bohrer, Charles Lewis, Robert Linn, Maurice Tatsuoka, and Ledyard Tucker.     

Multidimensional unfolding analyses of ranking data for groups

One probabilistic version of Coombs' unfolding model called the MMUR (Marginalization model for the Multidimensional Unfolding analysis of Ranking data) is extended to treat ranking data for groups. One favorable feature of the model is that it can both take into consideration individual differences without estimating the subject parameters and capture the differences between the groups in a systematic manner. Another advantage lies in the fact that one can see the group differences in the geometrical point configuration, since the model shows how the ideal points of the groups differ from each other in space. Four applications are provided which demonstrate that the model is useful for clarifying systematic differences in this type of data.     

Bayesian Thurstonian models for ranking data using JAGS

A Thurstonian model for ranking data assumes that observed rankings are consistent with those of a set of underlying continuous variables. This model is appealing since it renders ranking data amenable to familiar models for continuous response variables—namely, linear regression models. To date, however, the use of Thurstonian models for ranking data has been very rare in practice. One reason for this may be that inferences based on these models require specialized technical methods. These methods have been developed to address computational challenges involved in these models but are not easy to implement without considerable technical expertise and are not widely available in software packages. To address this limitation, we show that Bayesian Thurstonian models for ranking data can be very easily implemented with the JAGS software package. We provide JAGS model files for Thurstonian ranking models for general use, discuss their implementation, and illustrate their use in analyses.     

Fitting the factor analysis model

When the covariance matrix (p×P) does not satisfy the formal factor analysis model for m factors, there will be no factor matrix (p×m) such that =(-) is diagonal. The factor analysis model may then be replaced by a tautology where is regarded as the covariance matrix of a set of residual variates. These residual variates are linear combinations of discarded common factors and unique factors and are correlated. Maximum likelihood, alpha and iterated principal factor analysis are compared in terms of the manner in which is defined, a maximum determinant derivation for alpha factor analysis being given. Weighted least squares solutions using residual variances and common variances as weights are derived for comparison with the maximum likelihood and alpha solutions. It is shown that the covariance matrix defined by maximum likelihood factor analysis is Gramian, provided that all diagonal elements are nonnegative. Other methods can define a which is nonGramian even when all diagonal elements are nonnegative.A modified version of this paper forms part of a Ph.D. thesis submitted to the University of South Africa.Presently at the National Institute for Personnel Research, South Africa.     

A multistage ranking model

In this paper, we propose a (n–1)² parameter, multistage ranking model, which represents a generalization of Luce's model. We propose then×n item-rank relative frequency matrix (p-matrix) as a device for summarizing a set of rankings. As an alternative to the traditional maximum likelihood estimation, for the proposed model we suggest a method which estimates the parameters from thep-matrix. An illustrative numerical example is given. The proposed model and its differences from Luce's model are briefly discussed. We also show some specialp-matrix patterns possessed by the Thurstonian models and distance-based models.     

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.     

A general model for the analysis of multilevel data

A general model is developed for the analysis of multivariate multilevel data structures. Special cases of the model include repeated measures designs, multiple matrix samples, multilevel latent variable models, multiple time series, and variance and covariance component models.We would like to acknowledge the helpful comments of Ruth Silver. We also wish to thank the referees for helping to clarify the paper. This work was partly carried out with research funds provided by the Economic and Social Research Council (U.K.).     

Remarkable properties for diagnostics and inference of ranking data modelling

The Plackett-Luce model (PL) for ranked data assumes the forward order of the ranking process. This hypothesis postulates that the ranking process of the items is carried out by sequentially assigning the positions from the top (most liked) to the bottom (least liked) alternative. This assumption has been recently relaxed with the Extended Plackett-Luce model (EPL) through the introduction of the discrete reference order parameter, describing the rank attribution path. By starting from two formal properties of the EPL, the former related to the inverse ordering of the item probabilities at the first and last stage of the ranking process and the latter well-known as independence of irrelevant alternatives (or Luce's choice axiom), we derive novel diagnostic tools for testing the appropriateness of the EPL assumption as the actual sampling distribution of the observed rankings. These diagnostic tools can help uncovering possible idiosyncratic paths in the sequential choice process. Besides contributing to fill the gap of goodness-of-fit methods for the family of multistage models, we also show how one of the two statistics can be conveniently exploited to construct a heuristic method, that surrogates the maximum likelihood approach for inferring the underlying reference order parameter. The relative performance of the proposals, compared with more conventional approaches, is illustrated by means of extensive simulation studies.     

An individual difference model for the multidimensional analysis of preference data

A model for the analysis of paired comparison data is presented which combines features of the BTL-model with features of the Unfolding model. The model is metric, mathematically tractable, and has an exact algebraic solution. Since it is multidimensional and allows for individual differences, it is thought to be more realistic for some choice situations than either the Thurstone model or the BTL-model. No claim is made that the present model will be appropriate for all conceivable choice situations. Rather, it is argued that the fact that it is explicitly falsifiable is a point in its favor.This work was supported by a grant of the United States Public Health Service (1 R03 MH19139 01 MSM) and a David Ross (XR) grant of the Purdue Research Foundation (PRF 2132), both of which are gratefully acknowledged. We also owe thanks to Dr. Terry Cooper, whose careful scrutiny of an earlier draft of this paper allowed us to reduce the number of errors from n to n-1.     

A dynamic factor model for the analysis of multivariate time series

As a method to ascertain the structure of intra-individual variation,P-technique has met difficulties in the handling of a lagged covariance structure. A new statistical technique, coined dynamic factor analysis, is proposed, which accounts for the entire lagged covariance function of an arbitrary second order stationary time series. Moreover, dynamic factor analysis is shown to be applicable to a relatively short stretch of observations and therefore is considered worthwhile for psychological research. At several places the argumentation is clarified through the use of examples.I would like to thank WM. van der Molen, G. J. Mellenbergh and L. H. M. Oppenheimer, who provided valuable ideas that led to this formulation.     

Mean performance and the factor analysis of learning data

The difference in factor structure resulting from the factorization of correlations, covariances, and cross products is discussed. Factoring cross products has the advantage of retaining information on both means and variances; this method of factoring learning data is recommended. The conditions under which factoring covariances and cross products lead to the same essential structure are investigated.     

Model specification search using a genetic algorithm with factor reordering for a simple structure factor analysis model

Abstract:  Many techniques for automated model specification search based on numerical indices have been proposed, but no single decisive method has yet been determined. In the present article, the performance and features of the model specification search method using a genetic algorithm (GA) were verified. A GA is a robust and simple metaheuristic algorithm with great searching power. While there has already been some research applying metaheuristics to the model fitting task, we focus here on the search for a simple structure factor analysis model and propose a customized algorithm for dealing with certain problems specific to that situation. First, implementation of model specification search using a GA with factor reordering for a simple structure factor analysis is proposed. Then, through a simulation study using generated data with a known true structure and through example analysis using real data, the effectiveness and applicability of the proposed method were demonstrated.     

Structural equation modeling of paired-comparison and ranking data

L. L. Thurstone's (1927) model provides a powerful framework for modeling individual differences in choice behavior. An overview of Thurstonian models for comparative data is provided, including the classical Case V and Case III models as well as more general choice models with unrestricted and factor-analytic covariance structures. A flow chart summarizes the model selection process. The authors show how to embed these models within a more familiar structural equation modeling (SEM) framework. The different special cases of Thurstone's model can be estimated with a popular SEM statistical package, including factor analysis models for paired comparisons and rankings. Only minor modifications are needed to accommodate both types of data. As a result, complex models for comparative judgments can be both estimated and tested efficiently.