The wandering ideal point model: A probabilistic multidimensional unfolding model for paired comparisons data

In analogy with the wandering vector model, a probabilistic multidimensional unfolding model is proposed for representing paired comparisons data. It is shown that contrary to other stochastic multidimensional unfolding models, the present model does not imply strong stochastic transitivity, only moderate stochastic transitivity. A maximum likelihood parameter estimation procedure is developed and an illustrative application is presented.     

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.     

A Quasi-Metric Approach to Multidimensional Unfolding for Reducing the Occurrence of Degenerate Solutions

In multidimensional unfolding (MDU), one typically deals with two-way, two-mode dominance data in estimating a joint space representation of row and column objects in a derived Euclidean space. Unfortunately, most unfolding procedures, especially nonmetric ones, are prone to yielding degenerate solutions where the two sets of points (row and column objects) are disjointed or separated in the derived joint space, providing very little insight as to the structure of the input data. We present a new approach to multidimensional unfolding which reduces the occurrence of degenerate solutions. We first describe the technical details of the proposed method. We then conduct a Monte Carlo simulation to demonstrate the superior performance of the proposed model compared to two non-metric procedures, namely, ALSCAL and KYST. Finally, we evaluate the performance of alternative models in two applications. The first application deals with student rank-order preferences (nonmetric data) for attending various graduate business (MBA) programs. Here, we compare the performance of our model with those of KYST and ALSCAL. The second application concerns student preference ratings (metric data) for a number of popular brands of analgesics. Here, we compare the performance of the proposed model with those of two metric procedures, namely, SMACOF-3 and GENFOLD 3. Finally, we provide some directions for future research.     

Multidimensional Unfolding by Nonmetric Multidimensional Scaling of Spearman Distances in the Extended Permutation Polytope

A multidimensional unfolding technique that is not prone to degenerate solutions and is based on multidimensional scaling of a complete data matrix is proposed: distance information about the unfolding data and about the distances both among judges and among objects is included in the complete matrix. The latter information is derived from the permutation polytope supplemented with the objects, called the preference sphere. In this sphere, distances are measured that are closely related to Spearman's rank correlation and that are comparable among each other so that an unconditional approach is reasonable. In two simulation studies, it is shown that the proposed technique leads to acceptable recovery of given preference structures. A major practical advantage of this unfolding technique is its relatively easy implementation in existing software for multidimensional scaling.     

Discrete Choice Models for Ordinal Response Variables: A Generalization of the Stereotype Model

In this paper I present a class of discrete choice models for ordinal response variables based on a generalization of the stereotype model. The stereotype model can be derived and generalized as a random utility model for ordered alternatives. Random utility models can be specified to account for heteroscedastic and correlated utilities. In the case of the generalized stereotype model this includes category-specific random effects due to individual differences in response style. But unlike standard random utility models the generalized stereotype model is better suited for ordinal response variables and can be interpreted as a kind of unidimensional unfolding model. This paper discusses the specification, interpretation, identification, and estimation of generalized stereotype models. Two applications are provided for illustration. This paper benefited significantly from the comments and suggestions of the editor, associate editor, and three anonymous reviewers. It is dedicated to my late colleague, peer, and friend Bradley D. Crouch.     

Multidimensional successive categories scaling: A maximum likelihood method

A single-step maximum likelihood estimation procedure is developed for multidimensional scaling of dissimilarity data measured on rating scales. The procedure can fit the euclidian distance model to the data under various assumptions about category widths and under two distributional assumptions. The scoring algorithm for parameter estimation has been developed and implemented in the form of a computer program. Practical uses of the method are demonstrated with an emphasis on various advantages of the method as a statistical procedure.The research reported here was partly supported by Grant A6394 to the author by Natural Sciences and Engineering Research Council of Canada. Portions of this research were presented at the Psychometric Society meeting in Uppsala, Sweden, in June, 1978. MAXSCAL-2.1, a program to perform the computations discussed in this paper may be obtained from the author. Thanks are due to Jim Ramsay for his helpful comments.     

A stochastic multidimensional scaling procedure for the empirical determination of convex indifference curves for preference/choice analysis

The vast majority of existing multidimensional scaling (MDS) procedures devised for the analysis of paired comparison preference/choice judgments are typically based on either scalar product (i.e., vector) or unfolding (i.e., ideal-point) models. Such methods tend to ignore many of the essential components of microeconomic theory including convex indifference curves, constrained utility maximization, demand functions, et cetera. This paper presents a new stochastic MDS procedure called MICROSCALE that attempts to operationalize many of these traditional microeconomic concepts. First, we briefly review several existing MDS models that operate on paired comparisons data, noting the particular nature of the utility functions implied by each class of models. These utility assumptions are then directly contrasted to those of microeconomic theory. The new maximum likelihood based procedure, MICROSCALE, is presented, as well as the technical details of the estimation procedure. The results of a Monte Carlo analysis investigating the performance of the algorithm as a number of model, data, and error factors are experimentally manipulated are provided. Finally, an illustration in consumer psychology concerning a convenience sample of thirty consumers providing paired comparisons judgments for some fourteen brands of over-the-counter analgesics is discussed.     

A New Heterogeneous Multidimensional Unfolding Procedure

A variety of joint space multidimensional scaling (MDS) methods have been utilized for the spatial analysis of two- or three-way dominance data involving subjects’ preferences, choices, considerations, intentions, etc. so as to provide a parsimonious spatial depiction of the underlying relevant dimensions, attributes, stimuli, and/or subjects’ utility structures in the same joint space representation. We demonstrate that care must be taken with respect to a key assumption in existent joint space MDS models such that all estimated dimensions are utilized by each and every subject in the sample, as this assumption can lead to serious distortions with respect to the derived joint spaces. We develop a new Bayesian dimension selection methodology for the multidimensional unfolding model which accommodates heterogeneity with respect to such dimensional utilization at the individual subject level for the analysis of two or three-way dominance data. A consumer psychology application regarding the preference for Over-the-Counter (OTC) analgesics is provided. We conclude by discussing the practical implications of the results, as well as directions for future research.     

Internal multidimensional unfolding about a single-ideal—A probabilistic solution

A solution is presented for an internal multidimensional unfolding problem in which all the judgments of a rectangular proximity matrix are a function of a single-ideal object. The solution is obtained by showing that when real and ideal objects are represented by normal distributions in a multidimensional Euclidean space, a vector of distances among a single-ideal and multiple real objects follows a multivariate quadratic form in normal variables distribution. An approximation to the vector's probability density function (PDF) is developed which allows maximum likelihood (ML) solutions to be estimated. Under dependent sampling, the likelihood function contains information about the parametric distances among real object pairs, permitting the estimation of single-ideal solutions and leading to more robust multiple-ideal solutions. Tests for single- vs. multiple-ideal solutions and dependent vs. independent sampling are given. Properties of the proposed model and parameter recovery are explored. Empirical illustrations are also provided.     

A procedure for ordering object pairs consistent with the multidimensional unfolding model

A procedure for ordering object (stimulus) pairs based on individual preference ratings is described. The basic assumption is that individual responses are consistent with a nonmetric multidimensional unfolding model. The method requires data where a numerical response is independently generated for each individual-object pair. In conjunction with a nonmetric multidimensional scaling procedure, it provides a vehicle for recovering meaningful object configurations.The author wishes to thank Jack Hoadley, Larry Mayer, Sheldon Newhouse, Stuart Rabinowitz, Forrest Young, and three anonymous reviewers for their useful suggestions.     

中介效应检验程序及其应用

讨论了中介变量以及相关概念、中介效应的估计；比较了检验中介效应的主要方法；提出了一个检验程序，它包含了依次检验和Sobel检验。该程序检验的第一类和第二类错误率之和通常比单一检验方法小，既可以做部分中介检验，也可以做完全中介检验。作为示范例子，引入中介变量研究学生行为对同伴关系的影响。     

Inferential procedures for correlation coefficients corrected for attenuation

A model and computational procedure based on classical test score theory are presented for determination of a correlation coefficient corrected for attenuation due to unreliability. Next, variance-covariance expressions for the sample estimates defined earlier are derived, based on application of the delta method. Results of a Monte Carlo study are presented in which the adequacy of the derived expressions was assessed for a large number of data forms and potential hypotheses encountered in the behavioral sciences. It is shown that, based on the proposed procedures, confidence intervals for single coefficients are reasonably precise. Two-sample hypothesis tests, for both independent and dependent samples, are also accurate. However, for hypothesis tests involving a larger number of coefficients than two—both independent and dependent—the proposed procedures require largens for adequate precision. Results of a preliminary power analysis reveal no serious loss in efficiency resulting from correction for attenuation. Implications for practice are discussed.Support for the research reported in this article was provided by the Natural Sciences and Engineering Research Council of Canada. The authors acknowledge with thanks the constructive comments of the editor and three anonymous reviewers.     

The Mixed Effects Trend Vector Model

Maximum likelihood estimation of mixed effect baseline category logit models for multinomial longitudinal data can be prohibitive due to the integral dimension of the random effects distribution. We propose to use multidimensional unfolding methodology to reduce the dimensionality of the problem. As a by-product, readily interpretable graphical displays representing change are obtained. The methodology can be applied to both nominal and ordinal response variables. Relationships to standard statistical models for multinomial data are presented. Several empirical examples are given to show the merits of the proposed modeling framework.     

Nonmetric maximum likelihood multidimensional scaling from directional rankings of similarities

A maximum likelihood estimation procedure is developed for multidimensional scaling when (dis)similarity measures are taken by ranking procedures such as the method of conditional rank orders or the method of triadic combinations. The central feature of these procedures may be termed directionality of ranking processes. That is, rank orderings are performed in a prescribed order by successive first choices. Those data have conventionally been analyzed by Shepard-Kruskal type of nonmetric multidimensional scaling procedures. We propose, as a more appropriate alternative, a maximum likelihood method specifically designed for this type of data. A broader perspective on the present approach is given, which encompasses a wide variety of experimental methods for collecting dissimilarity data including pair comparison methods (such as the method of tetrads) and the pick-M method of similarities. An example is given to illustrate various advantages of nonmetric maximum likelihood multidimensional scaling as a statistical method. At the moment the approach is limited to the case of one-mode two-way proximity data, but could be extended in a relatively straightforward way to two-mode two-way, two-mode three-way or even three-mode three-way data, under the assumption of such models as INDSCAL or the two or three-way unfolding models.The first author's work was supported partly by the Natural Sciences and Engineering Research Council of Canada, grant number A6394. Portions of this research were done while the first author was at Bell Laboratories. MAXSCAL-4.1, a program to perform the computations described in this paper can be obtained by writing to: Computing Information Service, Attention: Ms. Carole Scheiderman, Bell Laboratories, 600 Mountain Ave., Murray Hill, N.J. 07974. Thanks are due to Yukio Inukai, who generously let us use his stimuli in our experiment, and to Jim Ramsay for his helpful comments on an earlier draft of this paper. Confidence regions in Figures 2 and 3 were drawn by the program written by Jim Ramsay. We are also indebted to anonymous reviewers for their suggestions.     

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.