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.     

Facilitating Local Implementation of Special Education Program Evaluation

The results of a project assessing the applicability of four program evaluation models is described. Survey data from districts documented an interest in local-level special education evaluation models. A training institute for local staff was therefore established to provide staff with competency in four evaluation models. Subsequently, the models were replicated in eight local sites from which representatives had expressed an interest in using the models. All sites completed evaluation planning and implementation activities. Findings suggest that each of the four models in the project can be used with minimal levels of outside assistance and can be readily adapted to local needs. However, each model places somewhat different demands on local staff; and some models may be more appropriate for particular evaluation purposes.     

Minimax d-optimal designs for item response theory models

Various different item response theory (IRT) models can be used in educational and psychological measurement to analyze test data. One of the major drawbacks of these models is that efficient parameter estimation can only be achieved with very large data sets. Therefore, it is often worthwhile to search for designs of the test data that in some way will optimize the parameter estimates. The results from the statistical theory on optimal design can be applied for efficient estimation of the parameters.A major problem in finding an optimal design for IRT models is that the designs are only optimal for a given set of parameters, that is, they are locally optimal. Locally optimal designs can be constructed with a sequential design procedure. In this paper minimax designs are proposed for IRT models to overcome the problem of local optimality. Minimax designs are compared to sequentially constructed designs for the two parameter logistic model and the results show that minimax design can be nearly as efficient as sequentially constructed designs.     

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.     

A nonlinear mixed model framework for item response theory

Mixed models take the dependency between observations based on the same cluster into account by introducing 1 or more random effects. Common item response theory (IRT) models introduce latent person variables to model the dependence between responses of the same participant. Assuming a distribution for the latent variables, these IRT models are formally equivalent with nonlinear mixed models. It is shown how a variety of IRT models can be formulated as particular instances of nonlinear mixed models. The unifying framework offers the advantage that relations between different IRT models become explicit and that it is rather straightforward to see how existing IRT models can be adapted and extended. The approach is illustrated with a self-report study on anger.     

The emulation theory of representation: motor control, imagery, and perception

The emulation theory of representation is developed and explored as a framework that can revealingly synthesize a wide variety of representational functions of the brain. The framework is based on constructs from control theory (forward models) and signal processing (Kalman filters). The idea is that in addition to simply engaging with the body and environment, the brain constructs neural circuits that act as models of the body and environment. During overt sensorimotor engagement, these models are driven by efference copies in parallel with the body and environment, in order to provide expectations of the sensory feedback, and to enhance and process sensory information. These models can also be run off-line in order to produce imagery, estimate outcomes of different actions, and evaluate and develop motor plans. The framework is initially developed within the context of motor control, where it has been shown that inner models running in parallel with the body can reduce the effects of feedback delay problems. The same mechanisms can account for motor imagery as the off-line driving of the emulator via efference copies. The framework is extended to account for visual imagery as the off-line driving of an emulator of the motor-visual loop. I also show how such systems can provide for amodal spatial imagery. Perception, including visual perception, results from such models being used to form expectations of, and to interpret, sensory input. I close by briefly outlining other cognitive functions that might also be synthesized within this framework, including reasoning, theory of mind phenomena, and language.     

Mediation analysis with structural equation models: Combining theory,design, and statistics

Statistical tests of indirect effects can hardly distinguish between genuine and spurious mediation effects. The present research demonstrates, however, that mediation analysis can be improved by combining a significance test of the indirect effect with assessing the fit of causal models. Testing only the indirect effect can be misleading, because significant results may also be obtained when the underlying causal model is different from the mediation model. We use simulated data to demonstrate that additionally assessing the fit of causal models with structural equation models can be used to exclude subsets of models that are incompatible with the observed data. The results suggest that combining structural equation modeling with appropriate research design and theoretically stringent mediation analysis can improve scientific insights. Finally, we discuss limitations of the structural equation modeling approach, and we emphasize the importance of non‐statistical methods for scientific discovery.     

Nonlinear Bayesian analysis for single case designs

Several authors have suggested the use of multilevel models for the analysis of data from single case designs. Multilevel models are a logical approach to analyzing such data, and deal well with the possible different time points and treatment phases for different subjects. However, they are limited in several ways that are addressed by Bayesian methods. For small samples Bayesian methods fully take into account uncertainty in random effects when estimating fixed effects; the computational methods now in use can fit complex models that represent accurately the behavior being modeled; groups of parameters can be more accurately estimated with shrinkage methods; prior information can be included; and interpretation is more straightforward. The computer programs for Bayesian analysis allow many (nonstandard) nonlinear models to be fit; an example using floor and ceiling effects is discussed here.     

Some Theory and Applications of Confirmatory Second-Order Factor Analysis

This paper shows how confirmatory factor analysis can be used to test second- (and higher-) order factor models in the areas of the structure of abilities, allometry, and the separation of specific and error variance estimates. In the latter area, an idea of Joreskog's is extended to include a new conceptualization of the estimation of validity. Second-order models are placed in a hierarchy of factor analysis models, which shows how the fit of various models can be compared. The concept of discriminability is introduced to describe a situation in which two models may both be identified, and yet the goodness-of-fit for both will be the same. This problem can usually be avoided by careful design of a study. Several examples are discussed.     

Complexity of categorical syllogisms: An integration of two metrics

The complexity of categorical syllogisms was assessed using the relational complexity metric, which is based on the number of entities that are related in a single cognitive representation. This was compared with number of mental models in an experiment in which adult participants solved all 64 syllogisms. Both metrics accounted for similarly large proportions of the variance, showing that complexity depends on the number of categories that are related in a representation of the combined premises, whether represented in multiple mental models, or by a single model. This obviates the difficulty with mental models theory due to equivocal evidence for construction of more than one mental model. The “no valid conclusion” response was used for complex syllogisms that had valid conclusions. The results are interpreted as showing that the relational complexity metric can be applied to syllogistic reasoning, and can be integrated with mental models theory, which together account for a wide range of cognitive performances.     

Additive and multiplicative models for gamma distributed random variables,and their application as psychometric models for response times

A class of models for gamma distributed random variables is presented. These models are shown to be more flexible than the classical linear models with respect to the structure that can be imposed on the expected value. In particular, both additive, multiplicative, and combined additive-multiplicative models can be formulated. As a special case, a class of psychometric models for reaction times is presented, together with their psychological interpretation. By means of a comparison with existing models, this class of models is shown to offer some possibilities that are not available in existing methods. Parameter estimation by means of maximum likelihood (ML) is shown to have some attractive properties, since the models belong to the exponential family. Then, the results of a simulation study of the bias in the ML estimates are presented. Finally, the application of these models is illustrated by an analysis of the data from a mental rotation experiment. This analysis is preceded by an evaluation of the appropriateness of the gamma distribution for these data.     

Detection of differential item functioning in Rasch models by boosting techniques

Methods for the identification of differential item functioning (DIF) in Rasch models are typically restricted to the case of two subgroups. A boosting algorithm is proposed that is able to handle the more general setting where DIF can be induced by several covariates at the same time. The covariates can be both continuous and (multi‐)categorical, and interactions between covariates can also be considered. The method works for a general parametric model for DIF in Rasch models. Since the boosting algorithm selects variables automatically, it is able to detect the items which induce DIF. It is demonstrated that boosting competes well with traditional methods in the case of subgroups. The method is illustrated by an extensive simulation study and an application to real data.     

Cognitive and neural modeling of dynamics of trust in competitive trustees

Trust dynamics can be modeled in relation to experiences. In this paper two models to represent human trust dynamics are introduced, namely a model on a cognitive level and a neural model. These models include a number of parameters, providing the possibility to express certain relations between trustees. The behavior of each of the models is further analyzed by means of simulation experiments and formal verification techniques. Thereafter, both models have been compared to see whether they can produce patterns that are comparable. As each of the models has its own specific set of parameters, with values that depend on the type of person modeled, such a comparison is non-trivial. To address this, a special comparison approach is introduced, based on mutual mirroring of the models in each other. More specifically, for a given parameter values set for one model, by an automated parameter estimation procedure the most optimal values for the parameter values of the other model are determined in order to show the same behavior. Roughly spoken the results are that the models can mirror each other up to an accuracy of around 90%.     

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.     

An application of a biased version of the Bradley-Terry-Luce model to professional basketball results

Some extensions of the Bradley-Terry-Luce model for paired comparisons are presented and used to assess the existence of a home court advantage for professional basketball teams. It is shown how maximum likelihood estimates for these models can be obtained from a simple iterative scaling algorithm. Won-loss records for the regular season are used to make predictions for playoff series. The models are extended to include possible differences among divisions. These models can be applied in many other situations to incorporate group effects and order effects into analysis of paired comparisons.     

Kinetic Models of (M-R)-Systems

Kinetic models using enzyme kinetics are developed for the three ways that Louie proved that Rosen’s minimal (M-R)-System can be closed to efficient cause; i.e., how the “replication” component can itself be entailed from within the system. The kinetic models are developed using the techniques of network thermodynamics. As a demonstration, each model is simulated using a SPICE circuit simulator using arbitrarily chosen rate constants. The models are built from SPICE sub-circuits representing the key terms in the chemical rate equations. The models include the addition of an ad hoc semi-permeable membrane so the system can achieve steady state fluxes and also to illustrate the need for all the efficient cause agents to be continually replaced. Comments are made about exactly what is being simulated.     

An integrative review of process dissociation and related models in social cognition

This chapter reviews the use of formal dual process models in social psychology, with a focus on the process dissociation model and related multinomial models. The utility of the models is illustrated using studies of social and affective influences on memory, judgement and decision making, and social attitudes and stereotypes. We then compare and contrast the process dissociation model with other approaches, including implicit and explicit tests, signal detection theory, and multinomial models. Finally we show how several recently proposed multinomial models can be integrated into a single family of models, of which process dissociation is a specific instance. We describe how these process models can be used as both theoretical and measurement tools to answer questions about the role of automatic and controlled processes in social behaviour.     

Distinguishing Between Latent Classes and Continuous Factors with Categorical Outcomes: Class Invariance of Parameters of Factor Mixture Models

Factor mixture models are latent variable models with categorical and continuous latent variables that can be used as a model-based approach to clustering. A previous article covered the results of a simulation study showing that in the absence of model violations, it is usually possible to choose the correct model when fitting a series of models with different numbers of classes and factors within class. The response format in the first study was limited to normally distributed outcomes. This article has 2 main goals, first, to replicate parts of the first study with 5-point Likert scale and binary outcomes, and second, to address the issue of testing class invariance of thresholds and loadings. Testing for class invariance of parameters is important in the context of measurement invariance and when using mixture models to approximate nonnormal distributions. Results show that it is possible to discriminate between latent class models and factor models even if responses are categorical. Comparing models with and without class-specific parameters can lead to incorrectly accepting parameter invariance if the compared models differ substantially with respect to the number of estimated parameters. The simulation study is complemented with an illustration of a factor mixture analysis of 10 binary depression items obtained from a female subsample of the Virginia Twin Registry.     

Commentary on reconceptualizing personality disorder categories using trait dimensions

The ability of trait models of personality to account for categories of personality disorder is examined with particular reference to the five-factor approach. Although dimensional models are consistent with the evidence, and trait models can accommodate personality disorder categories, it is not clear that such models are directly applicable to clinical use. The lower-order or facet structure, in particular, needs further development to capture clinical concepts. It is also suggested that trait models do not account for all aspects of personality disorder-attention also needs to be paid to the organizing and integrative functions of personality.     

Geometric representation of association between categories

Categories can be counted, rated, or ranked, but they cannot be measured. Likewise, persons or individuals can be counted, rated, or ranked, but they cannot be measured either. Nevertheless, psychology has realized early on that it can take an indirect road to measurement: What can be measured is the strength of association between categories in samples or populations, and what can be quantitatively compared are counts, ratings, or rankings made under different circumstances, or originating from different persons. The strong demand for quantitative analysis of categorical data has thus created a variety of statistical methods, with substantial contributions from psychometrics and sociometrics. What is the common basis of these methods dealing with categories? The basic element they share is that the sample space has a special geometry, in which categories (or persons) are point masses forming a simplex, while distributions of counts or profiles of ratings are centers of gravity, which are also point masses. Rankings form a discrete subset in the interior of the simplex, known as the permutation polytope, and paired comparisons form another subset on the edges of the simplex. Distances between point masses form the basic tool of analysis. The paper gives some history of major concepts, which naturally leads to a new concept: the shadow point. It is then shown how loglinear models, Luce and Rasch models, unfolding models, correspondence analysis and homogeneity analysis, forced classification and classification trees, as well as other models and methods, fit into this particular geometrical framework.