A stochastic multidimensional scaling procedure for the spatial representation of three-mode, three-way pick any/ J data

This paper presents a new stochastic multidimensional scaling procedure for the analysis of three-mode, three-way pick any/ J data. The method provides either a vector or ideal-point model to represent the structure in such data, as well as “floating” model specifications (e.g., different vectors or ideal points for different choice settings), and various reparameterization options that allow the coordinates of ideal points, vectors, or stimuli to be functions of specified background variables. A maximum likelihood procedure is utilized to estimate a joint space of row and column objects, as well as a set of weights depicting the third mode of the data. An algorithm using a conjugate gradient method with automatic restarts is developed to estimate the parameters of the models. A series of Monte Carlo analyses are carried out to investigate the performance of the algorithm under diverse data and model specification conditions, examine the statistical properties of the associated test statistic, and test the robustness of the procedure to departures from the independence assumptions. Finally, a consumer psychology application assessing the impact of situational influences on consumers' choice behavior is discussed.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45749/1/11336_2005_Article_BF02294486.pd

A spatial interaction model for deriving joint space maps of bundle compositions and market segments from pick-any/J data: An application to new product options

We propose an approach for deriving joint space maps of bundle compositions and market segments from three-way (e.g., consumers x product options/benefits/features x usage situations/scenarios/time periods) pick-any/J data. The proposed latent structure multidimensional scaling procedure simultaneously extracts market segment and product option positions in a joint space map such that the closer a product option is to a particlar segment, the higher the likelihood of its being chosen by that segment. A segment-level threshold parameter is estimated that spatially delineates the bundle of product options that are predicted to be chosen by each segment. Estimates of the probability of each consumer belonging to the derived segments are simultaneously obtained. Explicit treatment of product and consumer characteristics are allowed via optional model reparameterizations of the product option locations and segment memberships. We illustrate the use of the proposed approach using an actual commercial application involving pick-any/J data gathered by a major hi-tech firm for some 23 advanced technological options for new automobiles.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47207/1/11002_2004_Article_BF00434905.pd

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.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45748/1/11336_2005_Article_BF02294463.pd

A stochastic multidimensional scaling vector threshold model for the spatial representation of “pick any/ n ” data

This paper presents a new stochastic multidimensional scaling vector threshold model designed to analyze “pick any/ n ” choice data (e.g., consumers rendering buy/no buy decisions concerning a number of actual products). A maximum likelihood procedure is formulated to estimate a joint space of both individuals (represented as vectors) and stimuli (represented as points). The relevant psychometric literature concerning the spatial treatment of such binary choice data is reviewed. The nonlinear probit type model is described, as well as the conjugate gradient procedure used to estimate parameters. Results of Monte Carlo analyses investigating the performance of this methodology with synthetic choice data sets are presented. An application concerning consumer choices for eleven competitive brands of soft drinks is discussed. Finally, directions for future research are presented in terms of further applications and generalizing the model to accommodate three-way choice data.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45743/1/11336_2005_Article_BF02294452.pd

A simulated annealing methodology for clusterwise linear regression

In many regression applications, users are often faced with difficulties due to nonlinear relationships, heterogeneous subjects, or time series which are best represented by splines. In such applications, two or more regression functions are often necessary to best summarize the underlying structure of the data. Unfortunately, in most cases, it is not known a priori which subset of observations should be approximated with which specific regression function. This paper presents a methodology which simultaneously clusters observations into a preset number of groups and estimates the corresponding regression functions' coefficients, all to optimize a common objective function. We describe the problem and discuss related procedures. A new simulated annealing-based methodology is described as well as program options to accommodate overlapping or nonoverlapping clustering, replications per subject, univariate or multivariate dependent variables, and constraints imposed on cluster membership. Extensive Monte Carlo analyses are reported which investigate the overall performance of the methodology. A consumer psychology application is provided concerning a conjoint analysis investigation of consumer satisfaction determinants. Finally, other applications and extensions of the methodology are discussed.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45745/1/11336_2005_Article_BF02296405.pd

On the Measurement of Perceived Consumer Risk

The role of perceived risk in consumer behavior has been studied extensively by academic researchers. This paper introduces a methodology for the measurement of the effects of product features, marketing mix components, and individual differences on perceived consumer risk based on theoretical foundations in the literature. A conjoint-type model based on paired comparison judgments is estimated to provide attribute weights. A modification of a stochastic multidimensional scaling-based vector model is then used to measure and summarize individual consumer differences with respect to the impact of brand attributes and marketing mix components on latent levels of perceived consumer risk. An illustration is provided using students’ risk perceptions of sports cars.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/75462/1/j.1540-5915.1991.tb00372.x.pd

Latent class metric conjoint analysis

A latent class methodology for conjoint analysis is proposed, which simultaneously estimates market segment membership and part-worth utilities for each derived market segment using mixtures of multivariate conditional normal distributions. An E-M algorithm to estimate the parameters of these mixtures is briefly discussed. Finally, an application of the methodology to a commercial study (pretest) examining the design of a remote automobile entry device is presented.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47104/1/11002_2004_Article_BF00994135.pd

Tscale: A new multidimensional scaling procedure based on tversky's contrast model

Tversky's contrast model of proximity was initially formulated to account for the observed violations of the metric axioms often found in empirical proximity data. This set-theoretic approach models the similarity/dissimilarity between any two stimuli as a linear (or ratio) combination of measures of the common and distinctive features of the two stimuli. This paper proposes a new spatial multidimensional scaling (MDS) procedure called TSCALE based on Tversky's linear contrast model for the analysis of generally asymmetric three-way, two-mode proximity data. We first review the basic structure of Tversky's conceptual contrast model. A brief discussion of alternative MDS procedures to accommodate asymmetric proximity data is also provided. The technical details of the TSCALE procedure are given, as well as the program options that allow for the estimation of a number of different model specifications. The nonlinear estimation framework is discussed, as are the results of a modest Monte Carlo analysis. Two consumer psychology applications are provided: one involving perceptions of fast-food restaurants and the other regarding perceptions of various competitive brands of cola soft-drinks. Finally, other applications and directions for future research are mentioned.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45750/1/11336_2005_Article_BF02294658.pd

Bayesian inference for finite mixtures of generalized linear models with random effects

We present an hierarchical Bayes approach to modeling parameter heterogeneity in generalized linear models. The model assumes that there are relevant subpopulations and that within each subpopulation the individual-level regression coefficients have a multivariate normal distribution. However, class membership is not known a priori, so the heterogeneity in the regression coefficients becomes a finite mixture of normal distributions. This approach combines the flexibility of semiparametric, latent class models that assume common parameters for each sub-population and the parsimony of random effects models that assume normal distributions for the regression parameters. The number of subpopulations is selected to maximize the posterior probability of the model being true. Simulations are presented which document the performance of the methodology for synthetic data with known heterogeneity and number of sub-populations. An application is presented concerning preferences for various aspects of personal computers.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45757/1/11336_2005_Article_BF02294188.pd

Simultaneous multidimensional unfolding and cluster analysis: An investigation of strategic groups

This paper develops a maximum likelihood based methodology for simultaneously performing multidimensional unfolding and cluster analysis on two-way dominance or profile data. This new procedure utilizes mixtures of multivariate conditional normal distributions to estimate a joint space of stimulus coordinates and K ideal points, one for each cluster or group, in a T -dimensional space. The conditional mixture, maximum likelihood methodology is introduced together with an E-M algorithm utilized for parameter estimation. A marketing strategy application is provided with an analysis of PIMS data for a set of firms drawn from the same competitive industry to determine strategic groups, while simultaneously depicting strategy-performance relationships.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47056/1/11002_2004_Article_BF00436033.pd