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).     

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).     

Models for ordinal hierarchical classes analysis

This paper proposes an ordinal generalization of the hierarchical classes model originally proposed by De Boeck and Rosenberg (1998). Any hierarchical classes model implies a decomposition of a two-way two-mode binary arrayM into two component matrices, called bundle matrices, which represent the association relation and the set-theoretical relations among the elements of both modes inM. Whereas the original model restricts the bundle matrices to be binary, the ordinal hierarchical classes model assumes that the bundles are ordinal variables with a prespecified number of values. This generalization results in a classification model with classes ordered along ordinal dimensions. The ordinal hierarchical classes model is shown to subsume Coombs and Kao's (1955) model for nonmetric factor analysis. An algorithm is described to fit the model to a given data set and is subsequently evaluated in an extensive simulation study. An application of the model to student housing data is discussed.     

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.     

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 models for three-way three-mode binary data: interrelations and model selection

Several hierarchical classes models can be considered for the modeling of three-way three-mode binary data, including the INDCLAS model (Leenen, Van Mechelen, De Boeck, and Rosenberg, 1999), the Tucker3-HICLAS model (Ceulemans, Van Mechelen, and Leenen, 2003), the Tucker2-HICLAS model (Ceulemans and Van Mechelen, 2004), and the Tucker1-HICLAS model that is introduced in this paper. Two questions then may be raised: (1) how are these models interrelated, and (2) given a specific data set, which of these models should be selected, and in which rank? In the present paper, we deal with these questions by (1) showing that the distinct hierarchical classes models for three-way three-mode binary data can be organized into a partially ordered hierarchy, and (2) by presenting model selection strategies based on extensions of the well-known scree test and on the Akaike information criterion. The latter strategies are evaluated by means of an extensive simulation study and are illustrated with an application to interpersonal emotion data. Finally, the presented hierarchy and model selection strategies are related to corresponding work by Kiers (1991) for principal component models for three-way three-mode real-valued data.     

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.     

Uniqueness of real-valued hierarchical classes models

Two novel uniqueness theorems are derived for the family of hierarchical classes (HICLAS) models, a family of structural decomposition models for N-way N-mode data that imply simultaneous hierarchically organized classifications of all modes involved in the data. The theorems generalize earlier results on binary HICLAS models to the integer- and real-valued cases. In addition, they allow for a shorter and insightful proof of a result on Boolean matrix invertibility that goes back to earlier work of Luce (1952) and Rutherford (1963).     

The transfer of contextual control over equivalence classes through equivalence classes: a possible model of social stereotyping.

In Experiment 1, subjects acquired conditional equivalence classes controlled by three male and three female names as contextual stimuli. When equivalence relations were tested using new names not used in training (three male and three female), contextual control remained intact. Thus, generalized control of the composition of conditional equivalence classes by characteristically gender-identified names was shown. A basic analysis of this finding was tested in Experiment 2. Contextual equivalence classes were established using as contextual stimuli nonrepresentational visual figures that were members of additional pretrained three-member equivalence classes. When other stimuli in the pretrained equivalence classes were used as contextual stimuli, the conditional equivalence classes remained intact. Control subjects showed that this effect depended on the equivalence relations established in pretraining. The results show that contextual control over equivalence classes can transfer through equivalence classes. The implications of this phenomenon for social stereotyping are discussed.     

Uniqueness of N-way N-mode hierarchical classes models

This paper presents two uniqueness theorems for the family of hierarchical classes models, a collection of order preserving Boolean decomposition models for binary N-way N-mode data. The theorems are compared with uniqueness results for the closely related family of N-way N-mode principal component models. It is concluded that the two-way two-mode PCA and N-way N-mode TuckerN models suffer more from a lack of identifiability than their hierarchical classes analogues, whereas the uniqueness conditions for N-way N-mode PARAFAC/CANDECOMP models are less restrictive than the ones derived for their N-way N-mode hierarchical classes counterparts.     

Predicting the extension of equivalence classes from primary generalization gradients: the merger of equivalence classes and perceptual classes.

In Experiment 1, 6 college students were given generalization tests using 25 line lengths as samples with a long line, a short line, and a “neither” option as comparisons. The neither option was to be used if a sample did not go with the other comparisons. Then, four-member equivalence classes were formed. Class 1 included three nonsense words and the short line. Class 2 included three other nonsense words and the long line. After repeating the generalization test for line length, additional tests were conducted using members of the equivalence classes (i.e., nonsense words and lines) as comparisons and intermediate-length lines as samples. All Class 2 comparisons were selected in the presence of the test lines that also evoked the selection of the long line in the generalization test that had been given before equivalence class formation. Class 1 yielded complementary findings. Thus, the preclass primary generalization gradient predicted which test lines acted as members of each equivalence class. Regardless of using comparisons that were nonsense words or lines, the post-class-formation gradients overlapped, showing the substitutability of class members. Experiment 2 assessed the discriminability of the intermediate-length test lines from the Class 1 (shortest) and Class 2 (longest) lines. The test lines that functioned as members of an equivalence class were discriminable from the line that was a member of the same class by training. Thus, these test lines also acted as members of a dimensionally defined class of “long” or “short” lines. Extension of an equivalence class, then, involved its merger with a dimensionally defined class, which converted a close-ended class to an open-ended class. These data suggest a means of predicting class membership in naturally occurring categories.     

Robust estimation of the hierarchical model for responses and response times

Van der Linden's (2007, Psychometrika, 72, 287) hierarchical model for responses and response times in tests has numerous applications in psychological assessment. The success of these applications requires the parameters of the model to have been estimated without bias. The data used for model fitting, however, are often contaminated, for example, by rapid guesses or lapses of attention. This distorts the parameter estimates. In the present paper, a novel estimation approach is proposed that is robust against contamination. The approach consists of two steps. In the first step, the response time model is fitted on the basis of a robust estimate of the covariance matrix. In the second step, the item response model is extended to a mixture model, which allows for a proportion of irregular responses in the data. The parameters of the mixture model are then estimated with a modified marginal maximum likelihood estimator. The modified marginal maximum likelihood estimator downweights responses of test-takers with unusual response time patterns. As a result, the estimator is resistant to several forms of data contamination. The robustness of the approach is investigated in a simulation study. An application of the estimator is demonstrated with real data.     

Transfer of a conditional ordering response through conditional equivalence classes

Eight adult humans were taught conditional discriminations in a matching-to-sample format that led to the formation of two four-member equivalence classes. When subjects were taught to select one comparison stimulus from each class in a set order, they then ordered all other members of the equivalence classes without explicit training. When the ordering response itself was brought under conditional control, conditional sequencing also transferred to all other members of the two equivalence classes. When the conditional discriminations in the matching-to-sample task were brought under higher order conditional control, the eight stimulus members were arranged into four conditional equivalence classes. Both ordering and conditional ordering transferred to all members of the four conditional equivalence classes; for some subjects this occurred without a typical test for equivalence. One hundred twenty untrained sequences emerged from eight trained sequences for all subjects. Transfer of functions through equivalence classes may contribute to a behavior-analytic approach to semantics and generative grammar.     

用于处理不努力作答的标准化残差系列方法和混合多层模型法的比较

文章采用模拟研究, 分别在混合多层模型假设满足和违背的情境下, 比较了混合多层模型方法与标准化残差系列方法在识别不努力作答和参数估计方面的表现。结果显示：(1)不存在不努力作答或其严重性低时, 各方法表现接近; (2)不努力作答严重性高时, 固定参数迭代标准化残差法普遍更优, 混合多层模型法仅在假设满足且两种作答反应时差异大的条件下表现较好。建议实际应用中优先选择固定参数迭代标准化残差法。     

The effect of exercise and childbirth classes on fear of childbirth and locus of labor pain control

This study sought to track changes in intensity of fear of childbirth and locus of labor pain control in women attending an exercise program for pregnant women or traditional childbirth classes and to identify the predictors of these changes. The study was longitudinal/non-experimental in nature and run on 109 healthy primigravidae aged from 22 to 37, including 62 women participating in an exercise program for pregnant women and 47 women attending traditional childbirth classes. The following assessment tools were used: two scales developed by the present authors – the Fear of Childbirth Scale and the Control of Birth Pain Scale, three standardized psychological inventories for the big five personality traits (NEO Five Factors Inventory), trait anxiety (State–Trait Anxiety Inventory) and dispositional optimism (Life Oriented Test–Revised) and a questionnaire concerning socioeconomic status, health status, activities during pregnancy, relations with partners and expectations about childbirth. Fear of childbirth significantly decreased in women participating in the exercise program for pregnant women but not in women attending traditional childbirth classes. Several significant predictors of post-intervention fear of childbirth emerged: dispositional optimism and self-rated health (negative) and strength of the belief that childbirth pain depends on chance (positive).     

Decomposition of a Rasch partial credit item into independent binary and indecomposable trinary items

For each Rasch (Masters) partial credit item, there exists a set of independent Rasch binary and indecomposable trinary items for which the sum of the scores and the partial credit score have identical probability density functions. If each indecomposable trinary item is further expressed as the sum of two binary items, then the binary items are positively dependent and cannot be both of the Rasch type. This paper was written while the author was working with Steve Ferrara and Hillary Michaels on some technical aspects of the Maryland School Performance Assessment Program. The author had been puzzled by the fact that most MSPAP assessment items have three or less score categories. With a psychometric justification now being apparent, this paper is dedicated to both of them.     

Testing for transitive class containment as a feature of hierarchical classification

Three experiments investigated responding consistent with transitive class containment, a feature of hierarchical classification. Experiment 1 replicated key components of a preliminary attempt to model hierarchical classification (Griffee & Dougher, 2002) and tested for responding consistent with transitive class containment. Only 2 out of 5 participants showed the expected pattern. Experiment 2 tested whether repeated exposures to the Experiment 1 protocol would give rise to the expected pattern more reliably. None of 3 novel participants demonstrated the pattern. In Experiment 3, physically similar stimuli used in Experiments 1 and 2 were replaced across testing cycles by arbitrary stimuli. Transitive-class-containment-consistent responding was observed in all 3 novel participants. Implications, limitations and future research are discussed.     

Psychological distress: A hierarchical factor model of the Multiple Affect Adjective Check List (MAACL)

Recent investigations of the structure of psychological distress have indicated that hierarchical models can accommodate both unitary and multifaceted conceptions of distress. The present study tested the hierarchical framework suggested by Zuckerman, Lubin, and Rinck (1983) for the Multiple Affect Adjective Check List (MAACL), a commonly used measure of psychological distress. One- and two-factor models were estimated using maximum-likelihood methods. Results indicated that the two-factor solution, with correlated positive and negative affect factors, provided a significantly better fit to the data than did the omnibus one-factor solution. These results provide further support for hierarchical models of distress.     

The Local Minima Problem in Hierarchical Classes Analysis: An Evaluation of a Simulated Annealing Algorithm and Various Multistart Procedures

Hierarchical classes models are quasi-order retaining Boolean decomposition models for N-way N-mode binary data. To fit these models to data, rationally started alternating least squares (or, equivalently, alternating least absolute deviations) algorithms have been proposed. Extensive simulation studies showed that these algorithms succeed quite well in recovering the underlying truth but frequently end in a local minimum. In this paper we evaluate whether or not this local minimum problem can be mitigated by means of two common strategies for avoiding local minima in combinatorial data analysis: simulated annealing (SA) and use of a multistart procedure. In particular, we propose a generic SA algorithm for hierarchical classes analysis and three different types of random starts. The effectiveness of the SA algorithm and the random starts is evaluated by reanalyzing data sets of previous simulation studies. The reported results support the use of the proposed SA algorithm in combination with a random multistart procedure, regardless of the properties of the data set under study. Eva Ceulemans is a post-doctoral fellow of the Fund for Scientific Research Flanders (Belgium). Iwin Leenen is a post-doctoral researcher of the Spanish Ministerio de Educación y Ciencia (programa Ramón y Cajal). The research reported in this paper was partially supported by the Research Council of K.U. Leuven (GOA/05/04).