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.     

A note on the computation of the second-order derivatives of the elementary symmetric functions in the rasch model

Jansen (1984) gave explicit formulas for the computation of the second-order derivatives of the elementary symmetric functions. But they are only applicable to those pairs of items which have unequal parameters. It is shown here that for equal parameters similar explicit formulas do exist, too, facilitating the application of the Newton-Raphson procedure to estimate the parameters in the Rasch model and related models according to the conditional maximum likelihood principle.The author wishes to thank the reviewers for their helpful comments on an earlier draft of this paper.     

Minimum rank and minimum trace of covariance matrices

This paper considers some mathematical aspects of minimum trace factor analysis (MTFA). The uniqueness of an optimal point of MTFA is proved and necessary and sufficient conditions for a point x to be optimal are established. Finally, some results about the connection between MTFA and the classical minimum rank factor analysis will be presented.     

Investigation into Combinatory Systems with Dual Combinators

Combinatory logic is known to be related to substructural logics. Algebraic considerations of the latter, in particular, algebraic considerations of two distinct implications (, ), led to the introduction of dual combinators in Dunn & Meyer 1997. Dual combinators are "mirror images" of the usual combinators and as such do not constitute an interesting subject of investigation by themselves. However, when combined with the usual combinators (e.g., in order to recover associativity in a sequent calculus), the whole system exhibits new features. A dual combinatory system with weak equality typically lacks the Church-Rosser property, and in general it is inconsistent. In many subsystems terms "unexpectedly" turn out to be weakly equivalent. The paper is a preliminary attempt to investigate some of these issues, as well as, briefly compare function application in symmetric -calculus (cf. Barbanera & Berardi 1996) and dual combinatory logic.     

Weak Consistency for Ensuring Priority Vectors Reliability

Consistency and transitivity are important and leading research topics in the study of decision‐making in terms of pairwise comparison matrices. In this paper, we search for conditions that, in case of inconsistency, guarantee ordinal compatibility between ordinal ranking (actual ranking) derived from a transitive matrix and cardinal rankings provided by the most usual priority vectors proposed in the scientific literature. We provide the notion of weak consistency; it is a condition weaker than consistency and stronger than transitivity and ensures that vectors associated with a matrix, by means of a strictly increasing synthesis functional, provide a preference order, on the related set of decision elements, equal to the actual ranking. This notion extends, to the case in which the decision‐maker can be indifferent between two or more alternatives/criteria, weak consistency introduced in previous papers under constraint of no indifference. Finally, we introduce an order relation on the rows of the matrix, that is, a simple order if and only if weak consistency is satisfied; this simple order allows us to easily determine the actual ranking on the set of decision elements. Copyright © 2015 John Wiley & Sons, Ltd.     

Age-related Changes in Memory and Fluid Reasoning in a Sample of Healthy Old People

ABSTRACT

Participants in the Healthy Old People in Edinburgh (HOPE) study (N = 398) were assessed on Raven's Progressive Matrices and Logical Memory on up to three occasions. Covariates included education, social class, disease and medication status, blood pressure and study outcome. Raven's score declined linearly with age, whereas decline in Logical Memory was accelerating. There was significant variation in individuals' rates of decline for Ravens but not Logical Memory. Slope–intercept covariances were not significant. Those who later developed dementia already exhibited lower scores, more so for Logical Memory than Raven's. Death and study attrition were related to performance, again greater for Logical Memory. Conclusions: The HOPE approach of progressive screening is a feasible and practical method for studying healthy cognitive ageing. As predicted for an initially healthy sample, rates of decline were relatively homogeneous. The hypothesis of differential decline was not supported, nor was a strict interpretation of the hypothesis that cognitive ageing is entirely pathology driven.     

A General Approach for Fitting Pure Exploratory Bifactor Models

This article proposes a procedure for fitting a pure exploratory bifactor solution in which the general factor is orthogonal to the group factors, but the loadings on the group factors can satisfy any orthogonal or oblique rotation criterion. The proposal combines orthogonal Procrustes rotations with analytical rotations and consists of a sequence of four steps. The basic input is a semispecified target matrix that can be (a) defined by the user, (b) obtained by using Schmid-Leiman orthogonalization, or (c) automatically built from a conventional unrestricted solution based on a prescribed number of factors. The relevance of the proposal and its advantages over existing procedures is discussed and assessed via simulation. Its feasibility in practice is illustrated with two empirical examples in the personality domain.     

Graph coloring, minimum-diameter partitioning, and the analysis of confusion matrices

It is well known that minimum-diameter partitioning of symmetric dissimilarity matrices can be framed within the context of coloring the vertices of a graph. Although confusion data are typically represented in the form of asymmetric similarity matrices, they are also amenable to a graph-coloring perspective. In this paper, we propose the integration of the minimum-diameter partitioning method with a neighborhood-based coloring approach for analyzing digraphs corresponding to confusion data. This procedure is capable of producing minimum-diameter partitions with the added desirable property that vertices with the same color have similar in-neighborhoods (i.e., directed edges entering the vertex) and out-neighborhoods (i.e., directed edges exiting the vertex) for the digraph corresponding to the minimum partition diameter.     

The effects of process and outcome accountability on judgment process and performance

This article challenges the view that it is always better to hold decision makers accountable for their decision process rather than their decision outcomes. In three multiple-cue judgment studies, the authors show that process accountability, relative to outcome accountability, consistently improves judgment quality in relatively simple elemental tasks. However, this performance advantage of process accountability does not generalize to more complex configural tasks. This is because process accountability improves an analytical process based on cue abstraction, while it does not change a holistic process based on exemplar memory. Cue abstraction is only effective in elemental tasks (in which outcomes are a linear additive combination of cues) but not in configural tasks (in which outcomes depend on interactions between the cues). In addition, Studies 2 and 3 show that the extent to which process and outcome accountability affect judgment quality depends on individual differences in analytical intelligence and rational thinking style.     

Symmetric Propositions and Logical Quantifiers

Symmetric propositions over domain and signature are characterized following Zermelo, and a correlation of such propositions with logical type- quantifiers over is described. Boolean algebras of symmetric propositions over and Σ are shown to be isomorphic to algebras of logical type- quantifiers over . This last result may provide empirical support for Tarski’s claim that logical terms over fixed domain are all and only those invariant under domain permutations.