The conjunctive model of hierarchical classes

This paper describes the conjunctive counterpart of De Boeck and Rosenberg's hierarchical classes model. Both the original model and its conjunctive counterpart represent the set-theoretical structure of a two-way two-mode binary matrix. However, unlike the original model, the new model represents the row-column association as a conjunctive function of a set of hypothetical binary variables. The conjunctive nature of the new model further implies that it may represent some conjunctive higher order dependencies among rows and columns. The substantive significance of the conjunctive model is illustrated with empirical applications. Finally, it is shown how conjunctive and disjunctive hierarchical classes models relate to Galois lattices, and how hierarchical classes analysis can be useful to construct lattice models of empirical data.The research reported in this paper was supported by NATO (Grant CRG.921321 to Iven Van Mechelen and Seymour Rosenberg) and by the Research Fund of Katholieke Universiteit Leuven (Grants PDM92/19 and POR93/3 to Iven Van Mechelen; Grants OT89/9 and F91/56 to Paul De Boeck).     

A relation between a between-item multidimensional IRT model and the mixture rasch model

Two generalizations of the Rasch model are compared: the between-item multidimensional model (Adams, Wilson, and Wang, 1997), and the mixture Rasch model (Mislevy & Verhelst, 1990; Rost, 1990). It is shown that the between-item multidimensional model is formally equivalent with a continuous mixture of Rasch models for which, within each class of the mixture, the item parameters are equal to the item parameters of the multidimensional model up to a shift parameter that is specific for the dimension an item belongs to in the multidimensional model. In a simulation study, the relation between both types of models also holds when the number of classes of the mixture is as small as two. The relation is illustrated with a study on verbal aggression. Frank Rijmen was supported by the Fund for Scientific Research Flanders (FWO). This research is also funded by the GOA/2000/02 granted from the KU Leuven. We would like to thank Kristof Vansteelandt for providing the data of the study on verbal aggression.     

A Speeded Item Response Model with Gradual Process Change

An item response theory model for dealing with test speededness is proposed. The model consists of two random processes, a problem solving process and a random guessing process, with the random guessing gradually taking over from the problem solving process. The involved change point and change rate are considered random parameters in order to model examinee differences in both respects. The proposed model is evaluated on simulated data and in a case study. The research reported in this paper was supported by IAP P5/24 and GOA/2005/04, both awarded to Paul De Boeck and Iven Van Mechelen, and by IAP P6/03, awarded to Iven Van Mechelen. Yuri Goegebeur’s research was supported by a grant of the Danish Natural Science Research Council.     

A taxonomy of latent structure assumptions for probability matrix decomposition models

A taxonomy of latent structure assumptions (LSAs) for probability matrix decomposition (PMD) models is proposed which includes the original PMD model (Maris, De Boeck, & Van Mechelen, 1996) as well as a three-way extension of the multiple classification latent class model (Maris, 1999). It is shown that PMD models involving different LSAs are actually restricted latent class models with latent variables that depend on some external variables. For parameter estimation a combined approach is proposed that uses both a mode-finding algorithm (EM) and a sampling-based approach (Gibbs sampling). A simulation study is conducted to investigate the extent to which information criteria, specific model checks, and checks for global goodness of fit may help to specify the basic assumptions of the different PMD models. Finally, an application is described with models involving different latent structure assumptions for data on hostile behavior in frustrating situations.Note: The research reported in this paper was partially supported by the Fund for Scientific Research-Flanders (Belgium) (project G.0207.97 awarded to Paul De Boeck and Iven Van Mechelen), and the Research Fund of K.U. Leuven (F/96/6 fellowship to Andrew Gelman, OT/96/10 project awarded to Iven Van Mechelen and GOA/2000/02 awarded to Paul De Boeck and Iven Van Mechelen). We thank Marcel Croon and Kristof Vansteelandt for commenting on an earlier draft of this paper.     

Estimation of the MIRID: A program and a SAS-based approach

The MIRID CML program is a program for the estimation of the parameter values of two different componential IRT models: the Rasch—MIRID and the OPLM—MIRID (Butter, 1994; Butter, De Boeck, & Verhelst, 1998). To estimate the parameters of both models, the program uses a CML approach. The model parameters can also be estimated with a MML approach that can be implemented in PROC NLMIXED of SAS Version 8. Both the MIRID CML program and the MML SAS approach are explained and compared in a simulation study. The results showed that they did about equally well in estimating the values of the item parameters but that there were some differences in the estimation of the person parameters, as could be expected from the differential assumptions regarding the distribution of the persons. The SAS MML approach is much slower than the MIRID CML program, but it is more flexible.     

Psychometric latent response models

In this paper, some psychometric models will be presented that belong to the larger class oflatent response models (LRMs). First, LRMs are introduced by means of an application in the field ofcomponential item response theory (Embretson, 1980, 1984). Second, a general definition of LRMs (not specific for the psychometric subclass) is given. Third, some more psychometric LRMs, and examples of how they can be applied, are presented. Fourth, a method for obtaining maximum likelihood (ML) and some maximum a posteriori (MAP) estimates of the parameters of LRMs is presented. This method is then applied to theconjunctive Rasch model. Fifth and last, an application of the conjunctive Rasch model is presented. This model was applied to responses to typical verbal ability items (open synonym items).This paper presents theoretical and empirical results of a research project supported by the Research Council [Onderzoeksraad] of the University of Leuven (grant number 89-9) to Paul De Boeck and Luc Delbeke.     

Cognitive psychology meets psychometric theory: on the relation between process models for decision making and latent variable models for individual differences

This article analyzes latent variable models from a cognitive psychology perspective. We start by discussing work by Tuerlinckx and De Boeck (2005), who proved that a diffusion model for 2-choice response processes entails a 2-parameter logistic item response theory (IRT) model for individual differences in the response data. Following this line of reasoning, we discuss the appropriateness of IRT for measuring abilities and bipolar traits, such as pro versus contra attitudes. Surprisingly, if a diffusion model underlies the response processes, IRT models are appropriate for bipolar traits but not for ability tests. A reconsideration of the concept of ability that is appropriate for such situations leads to a new item response model for accuracy and speed based on the idea that ability has a natural zero point. The model implies fundamentally new ways to think about guessing, response speed, and person fit in IRT. We discuss the relation between this model and existing models as well as implications for psychology and psychometrics.     

Formulation and Application of the Hierarchical Generalized Random-Situation Random-Weight MIRID

The process-component approach has become quite popular for examining many psychological concepts. A typical example is the model with internal restrictions on item difficulty (MIRID) described by Butter (1994) Butter, R. 1994. Item response models with internal restrictions on item difficulty, Doctoral thesis Leuven, , Belgium: Katholieke Universiteit..  [Google Scholar] and Butter, De Boeck, and Verhelst (1998). This study proposes a hierarchical generalized random-situation random-weight MIRID. The proposed model is more flexible for formulating endogenous latent variables within a multilevel framework, allowing the analysis of polytomous data with complex models (e.g., including item discriminations, random situations, random weights, and heteroskedasticity). The parameters in the proposed model can be estimated using the computer program WinBUGS, which adopts Markov Chain Monte Carlo algorithms. To illustrate the application of the proposed model, a real data set about guilt is analyzed and a comparison of MIRIDs for various conditions is conducted.     

Probability matrix decomposition models

In this paper, we consider a class of models for two-way matrices with binary entries of 0 and 1. First, we considerBoolean matrix decomposition, conceptualize it as alatent response model (LRM) and, by making use of this conceptualization, generalize it to a larger class of matrix decomposition models. Second,probability matrix decomposition (PMD) models are introduced as a probabilistic version of this larger class of deterministic matrix decomposition models. Third, an algorithm for the computation of the maximum likelihood (ML) and the maximum a posteriori (MAP) estimates of the parameters of PMD models is presented. This algorithm is an EM-algorithm, and is a special case of a more general algorithm that can be used for the whole class of LRMs. And fourth, as an example, a PMD model is applied to data on decision making in psychiatric diagnosis. This paper is based on a chapter of the first author's doctoral dissertation, written at the University of Leuven and supervised by Paul De Boeck.     

Studying Dimensions of God Representation: Choosing Closed or Open-Ended Research Questions

God representation was studied by means of a questionnaire and by means of an open-ended question in a representative sample of students at the Katholieke Universiteit Leuven, Belgium. Data were analyzed for the two approaches by means of factor analysis and cluster analysis. Comparison of the two methods indicates that the answer structure of the two data sets is similar, but that the item list gave a somewhat better prediction of the answer to the open-ended question than vice versa.     

Tucker3 hierarchical classes analysis

This paper presents a new model for binary three-way three-mode data, called Tucker3 hierarchical classes model (Tucker3-HICLAS). This new model generalizes Leenen, Van Mechelen, De Boeck, and Rosenberg's (1999) individual differences hierarchical classes model (INDCLAS). Like the INDCLAS model, the Tucker3-HICLAS model includes a hierarchical classification of the elements of each mode, and a linking structure among the three hierarchies. Unlike INDCLAS, Tucker3-HICLAS (a) does not restrict the hierarchical classifications of the three modes to have the same rank, and (b) allows for more complex linking structures among the three hierarchies. An algorithm to fit the Tucker3-HICLAS model is described and evaluated in an extensive simulation study. An application of the model to hostility data is discussed.The first author is a Research Assistant of the Fund for Scientific Research-Flanders (Belgium). The research reported in this paper was partially supported by the Research Council of K.U. Leuven (GOA/2000/02). We are grateful to Kristof Vansteelandt for providing us with an interesting data set.     

The Effect of Forgiveness on Marital Satisfaction in Relation to Marital Stability

The present study examined the association between forgiveness and marital satisfaction in relation to marital stability. It was a comparative study between first-married and remarried adults involving 787 respondents from the Flanders region in Belgium. The current study showed that, although in the overall forgiveness, there was no significant difference between the first-married and remarried, the first-married significantly differed in two subscales of Enright Forgiveness Inventory (EFI) from the remarried. A significant difference in the marital satisfaction was found between the first-married and remarried adults. The present study also reported a significant positive correlation between forgiveness and general-life adjustment.This study was carried out at the Institute for the Family and Sexuality Sciences, Katholieke Universiteit Leuven, as part of docotral research.     

Causal inferences with group based trajectory models

A central theme of research on human development and psychopathology is whether a therapeutic intervention or a turning-point event, such as a family break-up, alters the trajectory of the behavior under study. This paper lays out and applies a method for using observational longitudinal data to make more confident causal inferences about the impact of such events on developmental trajectories. The method draws upon two distinct lines of research: work on the use of finite mixture modeling to analyze developmental trajectories and work on propensity scores. The essence of the method is to use the posterior probabilities of trajectory group membership from a finite mixture modeling framework, to create balance on lagged outcomes and other covariates established prior to t for the purpose of inferring the impact of first-time treatment at t on the outcome of interest. The approach is demonstrated with an analysis of the impact of gang membership on violent delinquency based on data from a large longitudinal study conducted in Montreal. The research has been supported by the National Science Foundation (NSF) (SES-99113700) and the National Institute of Mental Health (RO1 MH65611-01A2). It also made heavy use of data collected with the support from Québec’s CQRS and FCAR funding agencies, Canada’s NHRDP and SSHRC funding agencies, and the Molson Foundation. We thank Stephen Fienberg, Susan Murphy, Paul Rosenbaum, the editor, Paul De Boeck, and two anonymous reviewers for their insightful suggestions.     

Fruits and vegetables categorized: an application of the generalized context model

In the study reported in this paper, we investigated the categorization of well-known and novel food items in the categories fruits and vegetables. Predictions based on Nosofsky’s (1984, 1986) generalized context model (GCM), on a multiplicative-similarity prototype model, and on an instantiation model as applied in Storms, De Boeck, and Ruts (2001) were compared. Despite suggestions in the literature that prototype models predict categorization from large categories better than exemplar models do, our results showed that the exemplar-based GCM yielded clearly better predictions than did a (multiplicative-similarity) prototype model.     

Religiosity and Marital Satisfaction

This study examined the association between religiosity and marital satisfaction among first-married and remarried adults. Seven hundred and eighty-seven heterosexually married adults from the Flanders region in Belgium completed the Maudsley Marital Questionnaire (MMQ) and a four-item religiosity scale, measuring marital satisfaction and religiosity respectively. This study found the effect of gender and marital status statistically significant (p < .0001) on religiosity. For marital satisfaction, the effect of gender and marital status statistically significant only for MMQ-S (p < .0001) and MMQ-M (p < .0001) respectively. Religiosity had a significant positive correlation (r = .19, p < .0001) with sexual-adjustment problems (MMQ-S). The ultimate aim of this study was to inform marital and relational therapists the value of a possible association religiosity has on marital satisfaction.This study was conducted at the Institute for the Family and Sexuality Sciences, Katholieke Universiteit Leuven, as part of doctoral research.     

Tucker2 hierarchical classes analysis

This paper presents a new hierarchical classes model, called Tucker2-HICLAS, for binary three-way three-mode data. As any three-way hierarchical classes model, the Tucker2-HICLAS model includes a representation of the association relation among the three modes and a hierarchical classification of the elements of each mode. A distinctive feature of the Tucker2-HICLAS model, being closely related to the Tucker3-HICLAS model (Ceulemans, Van Mechelen & Leenen, 2003), is that one of the three modes is minimally reduced and, hence, that the differences among the association patterns of the elements of this mode are maximally retained in the model. Moreover, as compared to Tucker3-HICLAS, Tucker2-HICLAS implies three rather than four different types of parameters and as such is simpler to interpret. Two types of Tucker2-HICLAS models are distinguished: a disjunctive and a conjunctive type. An algorithm for fitting the Tucker2-HICLAS model is described and evaluated in a simulation study. The model is illustrated with longitudinal data on interpersonal emotions. The first author is a Researcher of the Fund for Scientific Research—Flanders (Belgium). The research reported in this paper was partially supported by the Research Council of K.U. Leuven (GOA/2000/02). The authors are grateful to Iwin Leenen for the fruitful discussions.     

Hierarchical classes: Model and data analysis

A discrete, categorical model and a corresponding data-analysis method are presented for two-way two-mode (objects × attributes) data arrays with 0, 1 entries. The model contains the following two basic components: a set-theoretical formulation of the relations among objects and attributes; a Boolean decomposition of the matrix. The set-theoretical formulation defines a subset of the possible decompositions as consistent with it. A general method for graphically representing the set-theoretical decomposition is described. The data-analysis algorithm, dubbed HICLAS, aims at recovering the underlying structure in a data matrix by minimizing the discrepancies between the data and the recovered structure. HICLAS is evaluated with a simulation study and two empirical applications.This research was supported in part by a grant from the Belgian NSF (NFWO) to Paul De Boeck and in part by NSF Grant BNS-83-01027 to Seymour Rosenberg. We thank Iven Van Mechelen for clarifying several aspects of the Boolean algebraic formulation of the model and Phipps Arabie for his comments on an earlier draft.     

Indclas: A three-way hierarchical classes model

A three-way three-mode extension of De Boeck and Rosenberg's (1988) two-way two-mode hierarchical classes model is presented for the analysis of individual differences in binary object × attribute arrays. In line with the two-way hierarchical classes model, the three-way extension represents both the association relation among the three modes and the set-theoretical relations among the elements of each model. An algorithm for fitting the model is presented and evaluated in a simulation study. The model is illustrated with data on psychiatric diagnosis. Finally, the relation between the model and extant models for three-way data is discussed.The research reported in this paper was partially supported by NATO (Grant CRG.921321 to Iven Van Mechelen and Seymour Rosenberg).     

A generalized longitudinal mixture IRT model for measuring differential growth in learning environments

This article describes a generalized longitudinal mixture item response theory (IRT) model that allows for detecting latent group differences in item response data obtained from electronic learning (e-learning) environments or other learning environments that result in large numbers of items. The described model can be viewed as a combination of a longitudinal Rasch model, a mixture Rasch model, and a random-item IRT model, and it includes some features of the explanatory IRT modeling framework. The model assumes the possible presence of latent classes in item response patterns, due to initial person-level differences before learning takes place, to latent class-specific learning trajectories, or to a combination of both. Moreover, it allows for differential item functioning over the classes. A Bayesian model estimation procedure is described, and the results of a simulation study are presented that indicate that the parameters are recovered well, particularly for conditions with large item sample sizes. The model is also illustrated with an empirical sample data set from a Web-based e-learning environment.