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 array M into two component matrices, called bundle matrices, which represent the association relation and the set-theoretical relations among the elements of both modes in M. 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.

Test items are often evaluated and compared by contrasting the shapes of their item characteristics curves (ICC's) or surfaces. The current paper develops and applies three general (i.e., nonparametric) comparisons of the shapes of two item characteristic surfaces: (i) proportional latent odds, (ii) uniform relative difficulty, and (iii) item sensitivity. Two items may be compared in these ways while making no assumption about the shapes of item characteristic surfaces for other items, and no assumption about the dimensionality of the latent variable. Also studied is a method for comparing the relative shapes of two item characteristic curves in two examinee populations.

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.

Network analysis is an increasingly popular approach to study mental disorders in all their complexity. Multiple methods have been developed to extract networks from cross-sectional data, with these data being either continuous or binary. However, when it comes to time series data, most efforts have focused on continuous data. We therefore propose ConNEcT, a network approach for binary symptom data across time. ConNEcT allows to visualize and study the prevalence of different symptoms as well as their co-occurrence, measured by means of a contingency measure in one single network picture. ConNEcT can be complemented with a significance test that accounts for the serial dependence in the data. To illustrate the usefulness of ConNEcT, we re-analyze data from a study in which patients diagnosed with major depressive disorder weekly reported the absence or presence of eight depression symptoms. We first extract ConNEcTs for all patients that provided data during at least 104 weeks, revealing strong inter-individual differences in which symptom pairs co-occur significantly. Second, to gain insight into these differences, we apply Hierarchical Classes Analysis on the co-occurrence patterns of all patients, showing that they can be grouped into meaningful clusters. Core depression symptoms (i.e., depressed mood and/or diminished interest), cognitive problems and loss of energy seem to co-occur universally, but preoccupation with death, psychomotor problems or eating problems only co-occur with other symptoms for specific patient subgroups.

This paper proposes a multidimensional generalization of Coombs' (1964) parallelogram model for “pick any/n” data, which result from each of a number of subjects having selected a number of objects (s)he likes most from a prespecified set of n objects. In the model, persons and objects are represented in a low dimensional space defined by a set of ordinal variables with a prespecified number of categories; objects are represented as points and persons as intervals on each dimension. A conjunctive combination rule is assumed implying that a person selects an object if and only if the object is within the subject's interval on each dimension. An algorithm for fitting the model to a data set is presented and evaluated in a simulation study. The model is illustrated with data on preferences regarding holiday trips.

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.

A two-step weighted least squares estimator for multiple factor analysis of dichotomized variables is discussed. The estimator is based on the first and second order joint probabilities. Asymptotic standard errors and a model test are obtained by applying the Jackknife procedure.

In many psychological research domains stimulus-response profiles are explained by conjecturing a sequential process in which some variables mediate between stimuli and responses. Charting sequential processes is often a complex task because (1) many possible mediating variables may exist, and (2) interindividual differences may occur in the relationship between these mediating variables and the response. Recently, Ceulemans and Van Mechelen (Psychometrika 73(1):107–124, 2008) addressed these challenges by developing the CLASSI model. A major drawback of CLASSI is that it requires information about the same set of stimuli for all participants (i.e., crossed data), whereas recently a number of data gathering techniques have been proposed in which the set of stimuli differs across participants, yielding nested data. Therefore we present the CLASSI-N model, which extends the CLASSI model to nested data. A simulated annealing algorithm is proposed. The results of a simulation study are discussed as well as an application to data concerning depression.

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.

Multilevel covariance structure models have become increasingly popular in the psychometric literature in the past few years to account for population heterogeneity and complex study designs. We develop practical simulation based procedures for Bayesian inference of multilevel binary factor analysis models. We illustrate how Markov Chain Monte Carlo procedures such as Gibbs sampling and Metropolis-Hastings methods can be used to perform Bayesian inference, model checking and model comparison without the need for multidimensional numerical integration. We illustrate the proposed estimation methods using three simulation studies and an application involving student's achievement results in different areas of mathematics.

It is demonstrated in this paper that two major tests for 2 × 2 talbes are highly related from a Bayesian perspective. Although it is well-known that Fisher's exact and Pearson's chi-square tests are asymptotically equivalent, the present analysis shows that a formal similarity also exists in small samples. The key assumption that leads to the resemblance is the presence of a continuous parameter measuring association. In particular, it is shown that Pearson's probability can be obtained by integrating a two-moment approximation to the posterior distribution of the log-odds ratio. Furthermore, Pearson's chi-square test gave an excellent approximation to the actual Bayes probability in all 2×2 tables examined, except for those with extremely disproportionate marginal frequencies.

Often problems result in the collection of coupled data, which consist of different N-way N-mode data blocks that have one or more modes in common. To reveal the structure underlying such data, an integrated modeling strategy, with a single set of parameters for the common mode(s), that is estimated based on the information in all data blocks, may be most appropriate. Such a strategy implies a global model, consisting of different N-way N-mode submodels, and a global loss function that is a (weighted) sum of the partial loss functions associated with the different submodels. In this paper, such a global model for an integrated analysis of a three-way three-mode binary data array and a two-way two-mode binary data matrix that have one mode in common is presented. A simulated annealing algorithm to estimate the model parameters is described and evaluated in a simulation study. An application of the model to real psychological data is discussed.

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.

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.

Studies of agreement commonly occur in psychiatric research. For example, researchers are often interested in the agreement among radiologists in their review of brain scans of elderly patients with dementia or in the agreement among multiple informant reports of psychopathology in children. In this paper, we consider the agreement between two raters when rating a dichotomous outcome (e.g., presence or absence of psychopathology). In particular, we consider logistic regression models that allow agreement to depend on both rater- and subject-level covariates. Logistic regression has been proposed as a simple method for identifying covariates that are predictive of agreement (Coughlin et al., 1992). However, this approach is problematic since it does not take account of agreement due to chance alone. As a result, a spurious association between the probability (or odds) of agreement and a covariate could arise due entirely to chance agreement. That is, if the prevalence of the dichotomous outcome varies among subgroups of the population, then covariates that identify the subgroups may appear to be predictive of agreement. In this paper we propose a modification to the standard logistic regression model in order to take proper account of chance agreement. An attractive feature of the proposed method is that it can be easily implemented using existing statistical software for logistic regression. The proposed method is motivated by data from the Connecticut Child Study (Zahner et al., 1992) on the agreement among parent and teacher reports of psychopathology in children. In this study, parents and teachers provide dichotomous assessments of a child's psychopathology and it is of interest to examine whether agreement among the parent and teacher reports is related to the age and gender of the child and to the time elapsed between parent and teacher assessments of the child.

The problem of choosing the correct number of clusters is as old as cluster analysis itself. A number of authors have suggested various indexes to facilitate this crucial decision. One of the most extensive comparative studies of indexes was conducted by Milligan and Cooper (1985). The present piece of work pursues the same goal under different conditions. In contrast to Milligan and Cooper's work, the emphasis here is on high-dimensional empirical binary data. Binary artificial data sets are constructed to reflect features typically encountered in real-world data situations in the field of marketing research. The simulation includes 162 binary data sets that are clustered by two different algorithms and lead to recommendations on the number of clusters for each index under consideration. Index results are evaluated and their performance is compared and analyzed.

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.

The authors introduce subset conjunction as a classification rule by which an acceptable alternative must satisfy some minimum number of criteria. The rule subsumes conjunctive and disjunctive decision strategies as special cases.

Subset conjunction can be represented in a binary-response model, for example, in a logistic regression, using only main effects or only interaction effects. This results in a confounding of the main and interaction effects when there is little or no response error. With greater response error, a logistic regression, even if it gives a good fit to data, can produce parameter estimates that do not reflect the underlying decision process. The authors propose a model in which the binary classification of alternatives into acceptable/unacceptable categories is based on a probabilistic implementation of a subset-conjunctive process. The satisfaction of decision criteria biases the odds toward one outcome or the other. The authors then describe a two-stage choice model in which a (possibly large) set of alternatives is first reduced using a subset-conjunctive rule, after which an alternative is selected from this reduced set of items. They describe methods for estimating the unobserved consideration probabilities from classification and choice data, and illustrate the use of the models for cancer diagnosis and consumer choice. They report the results of simulations investigating estimation accuracy, incidence of local optima, and model fit.

Advances in computer technology have made possible a greater sophistication in the statistical analysis of pedigree data, however this is not necessarily manifest by fitting more comprehensive causative models. Planned twin and family studies measure numerous explanatory variables, including perhaps genetic and DNA marker information status on all pedigree members, and the cohabitation of all pairs of individuals. A statistical analysis should examine the contribution of these measured factors on individual means, and in explaining the variation and covariation between individuals, concurrently with the postulated effect of unmeasured factors such as polygenes. We present two models that meet this requirement: the Multivariate Normal Model for Pedigree Analysis for quantitative traits, and a Log-Linear Model for Binary Pedigree Data. For both models, important issues are examination of fit, detection of outlier pedigrees and outlier individuals, and critical examination of the model assumptions. Procedures for fulfilling these needs and examples of modelling are discussed.