Structural Equation Modeling of Paired-Comparison and Ranking Data.

Abstract

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.Keywords: Lisrel, preference data, choice models, random utility models, factor analysisThe methods of ranking and paired comparison play an essential role in the measurement of preferences, attitudes, and values. In a ranking task, respondents are presented with a set of alternatives (which, in the literature, are also referred to as options, stimuli, or items) and are asked to order them from most to least preferred. In a paired-comparison task, respondents are presented with pairs selected from the set of available alternatives and are instructed to select the more preferred alternative from each pair. The popularity of paired-comparison and ranking methods can be traced back to three reasons. First, by asking for a comparison of choice alternatives, these methods impose minimal constraints on the response behavior of a respondent. Especially when differences between choice alternatives are small, these methods are likely to provide more information about individual preferences than is obtainable by rating methods. Second, internal consistency checks are available that facilitate the identification of respondents who do not have well-defined preferences, values, or attitudes. If respondents can be shown to be consistent in their judgments, one can have much greater confidence in the obtained measurements and the predictive value of the derived scales in further applications. Third, paired-comparison and ranking data provide a rich source of information about the effects of individual differences and perceived similarity relationships among choice alternatives. This article presents two applications that illustrate this important feature.Because of their versatility, the methods of paired comparison and rankings are used in a wide range of studies.

How IRT can solve problems of ipsative data in forced-choice questionnaires.

“…In turn, the Thurstonian factor model is a factor-analytic model embedded within the Law of Comparative Judgment. Although the latter was introduced by Thurstone as early as 1927, embedding a factor-analytic structure within it (and hence developing an IRT counterpart) was only possible very recently, for computational reasons (see Maydeu-Olivares, 1999, 2001Maydeu-Olivares & Böckenholt, 2005;Tsai & Böckenholt, 2001).…”

“…Thurstone ( Thurstonian factor analytic models (or Thurstonian factor models) are less well known than classical models such as Case V because their estimation has only recently become feasible (Maydeu-Olivares & Böckenholt, 2005;Tsai & Böckenholt, 2001). Thurstonian factor models are similar to second-order factor analysis models with binary outcomes.…”

“…The latent utilities (first-order factors), in turn, depend on psychological attributes (second-order factors). Maydeu-Olivares and Böckenholt (2005) show how to embed these models within a familiar SEM framework so that they can be easily estimated and tested.…”

“…The latent utilities (first-order factors), in turn, depend on psychological attributes (second-order factors). Maydeu-Olivares and Böckenholt (2005) show how to embed these models within a familiar SEM framework so that they can be easily estimated and tested. The Thurstonian IRT model is a first-order model that links the binary outcomes to the traits directly, by substituting the latent utilities with linear functions describing their relationships with the underlying traits .…”

Thurstonian Scaling of Compositional Questionnaire Data

“…Conceptualizing the observed variables in compositional questionnaires as the differences of item utilities provides a straightforward connection to established models of choice dataThurstonian factor models (Maydeu-Olivares & Böckenholt, 2005). These models have been applied to forced-choice questionnaires (Brown & Maydeu-Olivares, 2011a;, where the observed comparative judgments are binary (prefer item i or item k).…”