Simple Structure Detection Through Bayesian Exploratory Multidimensional IRT Models

In modern validity theory, a major concern is the construct validity of a test, which is commonly assessed through confirmatory or exploratory factor analysis. In the framework of Bayesian exploratory Multidimensional Item Response Theory (MIRT) models, we discuss two methods aimed at investigating the underlying structure of a test, in order to verify if the latent model adheres to a chosen simple factorial structure. This purpose is achieved without imposing hard constraints on the discrimination parameter matrix to address the rotational indeterminacy. The first approach prescribes a 2-step procedure. The parameter estimates are obtained through an unconstrained MCMC sampler. The simple structure is, then, inspected with a post-processing step based on the Consensus Simple Target Rotation technique. In the second approach, both rotational invariance and simple structure retrieval are addressed within the MCMC sampling scheme, by introducing a sparsity-inducing prior on the discrimination parameters. Through simulation as well as real-world studies, we demonstrate that the proposed methods are able to correctly infer the underlying sparse structure and to retrieve interpretable solutions.     

The varimax criterion for analytic rotation in factor analysis

An analytic criterion for rotation is defined. The scientific advantage of analytic criteria over subjective (graphical) rotational procedures is discussed. Carroll's criterion and the quartimax criterion are briefly reviewed; the varimax criterion is outlined in detail and contrasted both logically and numerically with the quartimax criterion. It is shown that thenormal varimax solution probably coincides closely to the application of the principle of simple structure. However, it is proposed that the ultimate criterion of a rotational procedure is factorial invariance, not simple structure—although the two notions appear to be highly related. The normal varimax criterion is shown to be a two-dimensional generalization of the classic Spearman case, i.e., it shows perfect factorial invariance for two pure clusters. An example is given of the invariance of a normal varimax solution for more than two factors. The oblique normal varimax criterion is stated. A computational outline for the orthogonal normal varimax is appended.Part of the material in this paper is from the writer's Ph.D. thesis. I am indebted to my committee, Professors F. T. Tyler, R. C. Tryon, and H. D. Carter, chairman, for many helpful suggestions and criticisms. Dr. John Caffrey suggested the namevarimax, and wrote the original IBM 602A computer program for this criterion.I am also indebted to the staff of the University of California Computer Center for help in programming the procedures described in the paper for their IBM 701 electronic computer. Since their installation is partially supported by a grant from the National Science Foundation, the assistance of this agency is acknowledged.     

Personality described by adjectives: The generalizability of the Big Five to the Italian lexical context

Two studies have been performed in the frame of the Big Five model to describe personality. In the first study, the most useful adjectives for describing personality have been selected, trying to adopt a procedure as objective and empirically driven as possible. The resulting pool of adjectives (n = 492) has been administered to a sample of 274 subjects to verify the emergence of the Big Five in the Italian context. In the second study the pool of original adjectives has been reduced to 260 adjectives selecting the most representative terms (with regard to the factorial structure that has emerged). This pool of 260 adjectives has been administered to a sample of 862 subjects, together with the NEO-PI and the BFQ to facilitate the interpretation of the resulting factorial structure. Results showed the emergence of an Italian Big Five factorial structure that resembles the ‘canonical’ Big Five, although some of the factors, viz. Agreeableness and Emotional Stability, emerged as ‘blended’ dimensions.     

Techniques for rotating two or more loading matrices to optimal agreement and simple structure: A comparison and some technical details

Matrices of factor loadings are often rotated to simple structure. When more than one loading matrix is available for the same variables, the loading matrices can be compared after rotating them all (separately) to simple structure. An alternative procedure is to rotate them to optimal agreement, and then compare them. In the present paper techniques are described that combine these two procedures. Specifically, five techniques that combine the ideals of rotation to optimal agreement and rotation to simple structure are compared on the basis of contrived and empirical data. For the contrived data, it is assessed to what extent the rotations recover the underlying common structure. For both the contrived and the empirical data it is studied to what extent the techniques give well matching rotated matrices, to what extent these have a simple structure, and to what extent the most prominent parts of the different loading matrices agree. It was found that the simple procedure of combining a Generalized Procrustes Analysis (GPA) with Varimax on the mean of the matched loading matrices performs very well on all criteria, and, for most purposes, offers an attractive compromise of rotation to agreement and simple structure. In addition to this comparison, some technical improvements are proposed for Bloxom's rotation to simple structure and maximum similarity.This research has been made possible by a fellowship from the Royal Netherlands Academy of Arts and Sciences to the author. The author is obliged to René van der Heijden for assistance in programming the procedures in the simulation study reported in this paper, and to Jos ten Berge, three anonymous reviewers and an associate editor for helpful comments on an earlier version of this paper.     

A simple test for heterogeneity of variance in complex factorial designs

A simple procedure for testing heterogeneity of variance is developed which generalizes readily to complex, multi-factor experimental designs. Monte Carlo Studies indicate that the Z-variance test statistic presented here yields results equivalent to other familiar tests for heterogeneity of variance in simple one-way designs where comparisons are feasible. The primary advantage of the Z-variance test is in the analysis of factorial effects on sample variances in more complex designs. An example involving a three-way factorial design is presented.     

Factor simplicity index and transformations

A scale-invariant index of factorial simplicity is proposed as a summary statistic for principal components and factor analysis. The index ranges from zero to one, and attains its maximum when all variables are simple rather than factorially complex. A factor scale-free oblique transformation method is developed to maximize the index. In addition, a new orthogonal rotation procedure is developed. These factor transformation methods are implemented using rapidly convergent computer programs. Observed results indicate that the procedures produce meaningfully simple factor pattern solutions.This investigation was supported in part by a Research Scientist Development Award (K02-DA00017) and research grants (MH24149 and DA01070) from the U. S. Public Health Service. The assistance of Andrew L. Comrey, Henry F. Kaiser, Bonnie Barron, Marion Hee, and several anonymous reviewers is gratefully acknowledged.     

Studentized maximum root procedures for coherent analyses of two-factor fixed-effects designs

The authors provide generalizations of R. J. Boik's (1993) studentized maximum root (SMR) procedure that allow for simultaneous inference on families of product contrasts including simple effect contrasts and differences among simple effect contrasts in coherent analyses of data from 2-factor fixed-effects designs. Unlike the F-based simultaneous test procedures (STPs) proposed by M. A. Betz and K. R. Gabriel (1978) for coherent analyses allowing for inferences on all factorial contrasts, SMR STPs are designed for analyses where each contrast of interest is a product contrast, which is usually the case in analyses of data from factorial experiments. When both factors have more than 2 levels, SMR STPs always provide more power and precision than F STPs for inferences on product contrasts.     

On the Concept of the Scale

The theory of linear arrays provides a definition of linear order from the reflexive, symmetric, but non-transitive relation of matching. However, a distance function is not generally available for the elements of a linear array. Given the original intended interpretation of the matching predicate as holding between phenomenal qualia, this result presents an apparent contradiction to the existence of human practices, specifically the tradition of musical practice described by common-practice music theory, that involve precise judgments of phenomenal distance. This paper resolves that seeming contradiction by constructing a notion of the musical scale out of the theory of linear arrays. First, the theory of linear arrays is briefly reviewed. Then, criteria of adequacy for the construction are adduced from common-practice music theory, the theory of measure and the semiotic theory of notation. The construction is presented along with results indicating its fulfillment of the criteria of adequacy. Finally, the relations between several variables introduced in the course of the construction are investigated.     

Factorial invariance in multilevel confirmatory factor analysis

This paper presents a procedure to test factorial invariance in multilevel confirmatory factor analysis. When the group membership is at level 2, multilevel factorial invariance can be tested by a simple extension of the standard procedure. However level‐1 group membership raises problems which cannot be appropriately handled by the standard procedure, because the dependency between members of different level‐1 groups is not appropriately taken into account. The procedure presented in this article provides a solution to this problem. This paper also shows Muthén's maximum likelihood (MUML) estimation for testing multilevel factorial invariance across level‐1 groups as a viable alternative to maximum likelihood estimation. Testing multilevel factorial invariance across level‐2 groups and testing multilevel factorial invariance across level‐1 groups are illustrated using empirical examples. SAS macro and Mplus syntax are provided.     

A convex approach for multicriteria decision making in hierarchical systems

In this paper hierarchical multicriteria optimization problems are addressed in a convex programming framework. It is assumed that the criteria are aggregated into a nonlinear function, which renders the problem nonseparable, in general. The projection of the problem onto the criteria space is used to obtain an equivalent separable problem, solved through a relaxation procedure implemented on basis of a multilevel structure. At the upper level of the structure, the decision making process involves the solution of a multicriteria problem formulated in the criteria space. The solution encountered at the upper level originates a lower level parametric optimization problem with separable structure that can be treated by standard coordination-decomposition techniques. The convergence of the overall procedure is ensured. The paper includes an application of the approach proposed for the control of dynamic systems with linear quadratic structure. © 1998 John Wiley & Sons, Ltd.     

Regional Analysis of Self‐Reported Personality Disorder Criteria

ABSTRACT Building on the theoretical work of Louis Guttman, we propose that the core problem facing research into the multidimensional structure of the personality disorders is not the identification of factorial simple structure but rather detailed characterization of the multivariate configuration of the diagnostic criteria. Dimensions rotated to orthogonal or oblique simple structure are but one way out of many to characterize a multivariate map, and their current near universal application represents a choice for a very particular set of interpretive advantages and disadvantages. We use multidimensional scaling and regional interpretation to investigate the structure of 78 self‐reported personality disorder criteria from a large sample of military recruits and college students. Results suggest that the criteria have a three‐dimensional radex structure that conforms only loosely to the 10 existing personality disorder (PD) categories. Regional interpretation in three dimensions elucidates several important aspects of PDs and their interrelationships.     

The judgement of behaviour as aggressive and sanctionable

The influence of the factors norm deviation, intent, and injury, on the judgement of a critical act as aggressive and sanctionable was tested using a 2 × 2 × 2 factorial design. Scenarios depicting aggressive interactions represented the experimental conditions. 859 school children rated these scenarios on bipolar rating scales. The results show that norm deviation, intent, and injury, are important criteria for the interpretation of aggressive behaviour. Judging an action as sanctionable proved to be more sensitive for the variation of factors than the interpretation of behaviour as aggressive. In addition, an analysis of how the scenarios representing the experimental conditions were subjectively interpreted by the subjects was carried out. This showed that all combinations of the three definition criteria are psychologically meaningful. However, preferences were observed for specific interpretation patterns.     

Taking the U.S. Patent Office Criteria Seriously: A Quantitative Three-Criterion Creativity Definition and Its Implications

Although creativity has recently attracted considerable theoretical and empirical research, researchers have yet to reach a consensus on how best to define the phenomenon. To help establish a consensus, a definition is proposed that is based on the three criteria used by the United States Patent Office to evaluate applications for patent protection. The modified version uses the criteria of novelty, utility, and surprise. Moreover, creativity assessments based on these three criteria are quantitative and multiplicative rather than qualitative or additive. This three-criterion definition then leads to four implications regarding (a) the limitations to domain-specific expertise, (b) the varieties of comparable creativities, (c) the contrast between subjective and objective evaluations, and (d) the place of blind variation and selective retention in the creative process. These implications prove that adding the third criterion has critical consequences for understanding the phenomenon. Creativity is not only treated with superior sophistication, but also paradoxes that appear using the most common two-criterion definition readily disappear when the third criterion is included in the analysis. Hence, the conceptual differences between two- and three-criterion definitions are not trivial.     

The A Priori and the Empirical in Theories of Emotion: Smedslund's “Conceptual Analysis” of Emotion

Abstract

This paper examines the suggestion made by Smedslund that the empirical laws of emotion proposed by Frijda are a priori necessities, derivable from a definition of emotion Smedslund proposes. First, a distinction is drawn between a causal and non-causal reading of Smedslund's own definition of emotion. The suggestion is made that although on the causal reading the definition might provide sufficient conditions for being in an emotional state, causal regularities are empirical, so this reading would not meet his own criteria for an a priori definition. Moreover, showing that Frijda's laws are derivable from the definition thus understood would not undermine their empirical status. The non-causal reading is then examined and questions are raised about whether it can provide necessary and sufficient conditions for a state to be a state of emotion. Finally, the suggestion is made that the central issue in the question of whether or not a theory of emotion is wholly empirical turns not on whether it can be derived from a definition of emotion, but, rather, on the extent to which the explanations proposed of the regularities governing emotions must appeal to a priori rationality constraints.     

Rank‐based permutation approaches for non‐parametric factorial designs

Inference methods for null hypotheses formulated in terms of distribution functions in general non‐parametric factorial designs are studied. The methods can be applied to continuous, ordinal or even ordered categorical data in a unified way, and are based only on ranks. In this set‐up Wald‐type statistics and ANOVA‐type statistics are the current state of the art. The first method is asymptotically exact but a rather liberal statistical testing procedure for small to moderate sample size, while the latter is only an approximation which does not possess the correct asymptotic α level under the null. To bridge these gaps, a novel permutation approach is proposed which can be seen as a flexible generalization of the Kruskal–Wallis test to all kinds of factorial designs with independent observations. It is proven that the permutation principle is asymptotically correct while keeping its finite exactness property when data are exchangeable. The results of extensive simulation studies foster these theoretical findings. A real data set exemplifies its applicability.     

Éléments de validité de contenu des questionnaires d’intérêts Inforizon et exploration de dimensions sous-jacentes à l’organisation de facteurs d’intérêts

This study concerns the confirmatory factorial analysis test of the correspondence between a theory of the definition of the interests proposed in a list of factors and the responses from 5304 subjects in two self-evaluation questionnaires. The questionnaires were provided by a professional advice and support IT tool, Inforizon. It allows the user to explore professional worlds, according to his particular interests. The principles at work are designed to overcome difficulties regularly encountered in the use of interest scores. The empirical structure of the responses to the questionnaires appears to be compatible with the definition model. A second stage is the exploratory research of the explanatory and parsimonious dimensions of the approximately circular organisation of the interest factors.     

Multiobjective Blockmodeling for Social Network Analysis

To date, most methods for direct blockmodeling of social network data have focused on the optimization of a single objective function. However, there are a variety of social network applications where it is advantageous to consider two or more objectives simultaneously. These applications can broadly be placed into two categories: (1) simultaneous optimization of multiple criteria for fitting a blockmodel based on a single network matrix and (2) simultaneous optimization of multiple criteria for fitting a blockmodel based on two or more network matrices, where the matrices being fit can take the form of multiple indicators for an underlying relationship, or multiple matrices for a set of objects measured at two or more different points in time. A multiobjective tabu search procedure is proposed for estimating the set of Pareto efficient blockmodels. This procedure is used in three examples that demonstrate possible applications of the multiobjective blockmodeling paradigm.