Remarks on the identifiability of thurstonian paired comparison models under multiple judgment

It is well-known that the representations of the Thurstonian models for difference judgment data are not unique. It has been shown that equivalence classes can be formed to provide a more meaningful partition of the covariance structures of the Thurstonian ranking models. In this paper, we examine the equivalence relations between Thurstonian covariance structure models for paired comparison data obtained under multiple judgment and discuss their implications on the general identification constraints and methods to check for parameter identifiability in restricted models.The author is indebted to Ulf Böckenholt and Albert Maydeu-Olivares for their significant comments and suggestions which led to considerable improvement in this article.     

Limited information estimation and testing of Thurstonian models for paired comparison data under multiple judgment sampling

We relate Thurstonian models for paired comparisons data to Thurstonian models for ranking data, which assign zero probabilities to all intransitive patterns. We also propose an intermediate model for paired comparisons data that assigns nonzero probabilities to all transitive patterns and to some but not all intransitive patterns.There is a close correspondence between the multidimensional normal ogive model employed in educational testing and Thurstone's model for paired comparisons data under multiple judgment sampling with minimal identification restrictions. Alike the normal ogive model, Thurstonian models have two formulations, a factor analytic and an IRT formulation. We use the factor analytic formulation to estimate this model from the first and second order marginals of the contingency table using estimators proposed by Muthén. We also propose a statistic to assess the fit of these models to the first and second order marginals of the contingency table. This is important, as a model may reproduce well the estimated thresholds and tetrachoric correlations, yet fail to reproduce the marginals of the contingency table if the assumption of multivariate normality is incorrect.A simulation study is performed to investigate the performance of three alternative limited information estimators which differ in the procedure used in their final stage: unweighted least squares (ULS), diagonally weighted least squares (DWLS), and full weighted least squares (WLS). Both the ULS and DWLS show a good performance with medium size problems and small samples, with a slight better performance of the ULS estimator.This paper is based on the author's doctoral dissertation; Ulf Böckenholt, advisor. The final stages of this research took place while the author was at the Department of Statistics and Econometrics, Universidad Carlos III de Madrid. The author is indebted to Adolfo Hernández for stimulating discussions that helped improve this paper, and to Ulf Böckenholt and the Associate Editor for a number of helpfulsuggestions to a previous draft.     

Item Response Modeling of Paired Comparison and Ranking Data

The comparative format used in ranking and paired comparisons tasks can significantly reduce the impact of uniform response biases typically associated with rating scales. Thurstone's (1927, 1931) model provides a powerful framework for modeling comparative data such as paired comparisons and rankings. Although Thurstonian models are generally presented as scaling models, that is, stimuli-centered models, they can also be used as person-centered models. In this article, we discuss how Thurstone's model for comparative data can be formulated as item response theory models so that respondents' scores on underlying dimensions can be estimated. Item parameters and latent trait scores can be readily estimated using a widely used statistical modeling program. Simulation studies show that item characteristic curves can be accurately estimated with as few as 200 observations and that latent trait scores can be recovered to a high precision. Empirical examples are given to illustrate how the model may be applied in practice and to recommend guidelines for designing ranking and paired comparisons tasks in the future.     

Remarks on the identifiability of thurstonian ranking models: Case V, case III, or neither?

It is well-known that the representations of the Thurstonian Case III and Case V models for paired comparison data are not unique. Similarly, when analyzing ranking data, other equivalent covariance structures can substitute for those given by Thurstone in these cases. That is, we may more broadly define the family of covariance structures satisfying Case III and Case V conditions. This paper introduces the notion of equivalence classes which defines a more meaningful partition of the covariance structures of the Thurstonian ranking models. In addition, the equivalence classes of Case V and Case III are completely characterized.     

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.     

Identification and Small Sample Estimation of Thurstone's Unrestricted Model for Paired Comparisons Data

The interpretation of a Thurstonian model for paired comparisons where the utilities' covariance matrix is unrestricted proved to be difficult due to the comparative nature of the data. We show that under a suitable constraint the utilities' correlation matrix can be estimated, yielding a readily interpretable solution. This set of identification constraints can recover any true utilities' covariance matrix, but it is not unique. Indeed, we show how to transform the estimated correlation matrix into alternative correlation matrices that are equally consistent with the data but may be more consistent with substantive theory. Also, we show how researchers can investigate the sample size needed to estimate a particular model by exploiting the simulation capabilities of a popular structural equation modeling statistical package.     

评估职业兴趣的结构

Holland的职业兴趣理论对当今兴趣研究产生了广泛、深远的影响。但是目前,还没有研究考察Holland理论假设在中国的适用性。研究共检验了5个与Holland的6种兴趣类型(RiASEC)有关的假设,其中两个是由本文研究者提出。研究搜集了在中国进行的、与Holland理论有关的研究,并最终选择了9个RIASEC相关矩阵。假定顺序的随机化检验以及方差分析的结果表明,Holland的圆形假设与数据的拟合程度在5个模型中最差;多维标度法结果表明,RIASEC的空间结构图并没有很好地符合Holland的圆形假设。     

Multivariate thurstonian models

The recent development of probabilistic Thurstonian choice models for analyzing preference data has been motivated by the need to describe both inter- and intra-individual difference, the multidimensional nature of choice objects, and the effects of similarity and comparability among choice objects. A common feature of these models is that they focus on asingle preference judgment. It is customary, however, to ask subjects not only for an overall preference judgment but also for additional paired comparison responses regarding specific attributes. This paper proposes a generalization of Thurstonian probabilistic choice models for analyzing both multiple preference responses and their relationships. The approach is illustrated by modeling data from two multivariate preference experiments.I am grateful to James, Austin, Ingo Böckenholt, and anonymous referees for helpful comments on this research. This paper is based on research presented at the Meeting of the Society for the Multivariate Analysis in the Behavioral Sciences, Groningen, The Netherlands, December 1988.     

迫选式人格测验的传统计分与IRT计分模型

迫选测验的传统计分方式会产生自模式数据, 不能进行传统的信效度检验、因素分析和方差分析等。近年来研究者提出了一些基于项目反应理论的计分模型, 如瑟斯顿IRT模型和MUPP模型等, 它们可以规避自模式数据的弊端。瑟斯顿IRT模型方便进行参数估计, 模型定义灵活; 而MUPP模型的拓展性较差, 参数估计的方法有待提高。另一方面, 已有研究者基于MUPP模型开发了一些抗作假的迫选测验, 而瑟斯顿IRT模型距离这种应用还比较远。此外, 两个模型的适用性和有效性都有待更多的实证研究来检验。     

The fit between Strong-Campbell Interest Inventory General Occupational Themes and Holland's hexagonal model

Using a multidimensional scaling procedure, the present study examined the fit of Holland's RIASEC hexagon model to the internal relationships among the Strong-Campbell Interest Inventory (SCII) General Occupational Theme scales. SCII intercorrelation matrices for both sexes as reported in the SCII Manual were submitted, separately for each sex, to TORSCA 9 nonmetric scaling analysis. The Wakefield and Doughtie procedure was used to compare obtained TORSCA coordinates with expectations from Holland's hexagonal model. As a comparison, identical analyses were performed on Vocational Preference Inventory (VPI) scale intercorrelation matrices, these data having originally served as the basis for advancing the utility of the hexagon model. For females, the SCII-hexagon fit was not good, with a near reversal of the Social and Enterprising scales. For males, the SCII-hexagon fit was good. For either SCII or VPI scales, the female data met expectations from Holland's model less often than the male data. A replication study on SCII data for 305 female clients of a vocational assessment clinic confirmed the previously observed sex differences. Sex differences in the structure of vocational preferences are discussed.     

A structural examination of the Learning Experiences Questionnaire

The underlying factor structure of the Learning Experiences Questionnaire (LEQ; Schaub, 2004) was examined using data from 742 male and female college-age respondents. The LEQ items reflect a variety of learning experiences (generated based on Bandura's (1986, 1997) four sources of self-efficacy perceptions) that might occur in each of Holland's (1997) six RIASEC domains. The LEQ was expected to have a multi-dimensional factor structure, with dimensions representing both RIASEC context and learning experience type. A multi-trait multi-method framework was used to compare the relative fits of a series of models with different theoretically implied dimensionalities, estimated using confirmatory factor analysis. Results supported the proposed multi-dimensional structure. Results of follow-up tests of measurement invariance for men and women revealed that configural and partial factorial invariance held, implying that the LEQ measure functions similarly for men and women.     

Analyzing individual differences in vocational, leisure, and family interests: A multitrait-multimethod approach

This study investigates the latent structure of individual differences in vocational, leisure, and family interests. The participants consisted of 302 French adults who rated their like or dislike for activities based on Holland’s RIASEC typology and presented in three life domains: work, family and leisure. The multitrait-multimethod model proposed by Eid [Eid, M. (2000). A multitrait-multimethod model with minimal assumptions. Psychometrika 65, 241-261.] was used with the vocational interests as standard comparison, and was applied separately on each of the 6 RIASEC types. Results indicated that interests strongly intercorrelated whatever their vocational vs. leisure or family contextualization. However, since these correlations were controlled for measurement error, the results also suggested that the effects of these contextualizations of the activities could not be ignored because they accounted for a non negligible part of the variance for several activities. Theoretical and practical implications are discussed and directions for future research are presented.     

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.     

Adherence to RIASEC structure in relation to career exploration and parenting style: Longitudinal and idiothetic considerations

We explored the “idiothetic” cognitive structure of RIASEC occupational percepts in a sample of Italian middle and high school students over a one year period, examining the possible bidirectional linkages between cognitive-vocational structure, involvement in career exploration activities, and exposure to authoritative parenting style. The focus was on the extent to which individuals’ thinking deviates from the normative RIASEC circumplex structure. Results indicated that there was less stability in the occupational percepts of middle school students over time, but both student groups showed change in the direction of greater adherence to circular structure. In addition, deviation from the circular model was related to subsequent career exploration, and initial levels of career exploration and parental authoritativeness were predictive of later circular structure, especially in middle school students. The results support the importance of examining individual variation in cognitive-vocational structure in relation to career development models and interventions.     

The fit of Holland’s RIASEC model to US occupations

Holland’s [Holland, J. L. (1959). A theory of occupational choice. Journal of Counseling Psychology, 6, 35-45; Holland, J. L. (1997). Making vocational choices: A theory of vocational personalities and work environments (3rd ed.). Odessa, FL: Psychological Assessment Resources, Inc.] RIASEC types were initially developed using a restricted range of occupational titles. Holland’s type classification system has been extended to encompass the full range of occupations in the US, using both statistical and expert rating methods. However, the extent that Holland’s classification model is sufficient to represent the full range of occupational interests has not been examined. Multidimensional scaling (MDS) was used to analyze college students’ (266 men, 572 women) interests in occupations representing approximately 85% of the US labor market. A two-dimensional MDS solution of the full set of occupations did not fit Holland’s model, but limiting the analysis to occupations used in Holland-based measures produced the expected RIASEC structure. In comparison, a three-dimensional solution included Prediger’s [Prediger, D. J. (1982). Dimensions underlying Holland’s hexagon: Missing link between interests and occupations? Journal of Vocational Behavior, 21, 259-287] dimensions (Things/People and Data/Ideas) consistent with Holland’s model, but also included prestige and sex-type dimensions that were not orthogonal to Prediger’s dimensions. These results demonstrate that the RIASEC types are not sufficient to represent the full range of occupational interests and are confounded with prestige and sex-type.     

Individuals and environments: Linking ability and skill ratings with interests

Holland's (1997) theory of corresponding person and work environment structures was evaluated by comparing the integration of individual and occupational ratings of interests, abilities, and skills. Occupational ratings were obtained from the U.S. Department of Labor's O*NET database (U.S. Department of Labor, 2007). College students (494 women, 526 men) provided self-ratings of their interests, abilities, and skills. Property vector fitting was used to embed ability and skill ratings into the Realistic, Investigative, Artistic, Social, Enterprising, and Conventional (RIASEC) interest structure, and bootstrapping was used to generate confidence intervals for the angles of the vectors and the magnitude of their fit to the Holland model. Across the individual and occupational ratings, 18 of 45 (40%) ability vectors and 41 of 48 (85%) skill vectors were fit into the RIASEC model. No significant gender differences were found in the integration of self-rated abilities and skills into the RIASEC circumplex; however, some differences were found between individual and environmental ratings. Obtained results highlight the potential utility and limitations of using Holland's model for representing both individual and occupational data in a common structure.     

Structure of interest in a Caribbean sample: Application of the Personal Globe Inventory

The structural validity of the Personal Globe Inventory (PGI, Tracey, 2002) as applied to a Caribbean sample of university students in Jamaica (N = 103) and Trinidad (N = 118) was examined. The fit of the data to Holland's six-type RIASEC, the PGI eight-type model, and the PGI spherical structure was examined using the randomized test of hypothesized order relations. The results demonstrated that the circular structure (i.e. RIASEC and the eight-type model), unlike the spherical structure, fits the Caribbean data well. No structural differences were found between gender and nationality. Further examination of the cultural perception of prestige in a Caribbean sample is warranted. Overall, the results support the structural validity of the PGI in Caribbean applications.     

The HEXACO and five-factor models of personality in relation to RIASEC vocational interests

The current study extended the empirical research on the overlap of vocational interests and personality by (a) testing hypothesized relations between RIASEC interests and the personality dimensions of the HEXACO model, and (b) exploring the HEXACO personality model's predictive advantage over the five-factor model (FFM) in capturing RIASEC interests. Results of correlations based on a sample of 437 college students (157 men and 280 women) revealed support for eight of 21 hypothesized HEXACO–RIASEC relations for men and nine of 21 for women. Results of multiple regressions revealed that HEXACO accounted for more variance in RIASEC interests than did the FFM, and that HEXACO accounted for significant incremental variance beyond that explained by the FFM in four of six RIASEC interests for men and in five of six RIASEC interests for women.     

Structure of vocational interests for diverse groups on the 2005 strong interest inventory

This study was designed to examine the structure of vocational interests in a diverse sample of individuals who completed the 2005 revision of the Strong Interest Inventory. We examined the fit of three racial/ethnic groups (African American, Caucasian, and Latino/a), both genders, and three levels of professional status (GRS participant, student, and working adult; N = 136, 219) with a less-stringent circular RIASEC ordering as well as a more stringent, equidistant RIASEC ordering. Results from circular unidimensional scaling analyses revealed a circular RIASEC order for all the samples with the exception of African American females in the General Representative Sample and Latinos in the adult sample. Additionally, results indicated gender differences in the adult and student samples suggesting a better fit with Holland's model for males than females.     

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.