Range restrictions for product-moment correlation matrices

It is well-known that for a trivariate distribution if two correlations are fixed the remaining one is constrained. Indeed, if one correlation is fixed, then the remaining two are constrained. Both results are extended to the case of a multivariate distribution. The results are applied to some special patterned matrices.     

Fungible Correlation Matrices: A Method for Generating Nonsingular,Singular, and Improper Correlation Matrices for Monte Carlo Research

For a fixed set of standardized regression coefficients and a fixed coefficient of determination (R-squared), an infinite number of predictor correlation matrices will satisfy the implied quadratic form. I call such matrices fungible correlation matrices. In this article, I describe an algorithm for generating positive definite (PD), positive semidefinite (PSD), or indefinite (ID) fungible correlation matrices that have a random or fixed smallest eigenvalue. The underlying equations of this algorithm are reviewed from both algebraic and geometric perspectives. Two simulation studies illustrate that fungible correlation matrices can be profitably used in Monte Carlo research. The first study uses PD fungible correlation matrices to compare penalized regression algorithms. The second study uses ID fungible correlation matrices to compare matrix-smoothing algorithms. R code for generating fungible correlation matrices is presented in the supplemental materials.     

Identification of Matrices in Science and Engineering

Engineering science is a scientific discipline that from the point of view of epistemology and the philosophy of science has been somewhat neglected. When engineering science was under philosophical scrutiny it often just involved the question of whether engineering is a spin-off of pure and applied science and their methods. We, however, hold that engineering is a science governed by its own epistemology, methodology and ontology. This point is systematically argued by comparing the different sciences with respect to a particular set of characterization criteria. This revised version was published online in August 2006 with corrections to the Cover Date.     

Inspection time and everyday functioning: A longitudinal study

We tested whether level and/or changes in inspection time (IT) in an elderly sample (70–85 years at baseline) predicted future independence in everyday functioning, independent from chronological age. Sixty-eight participants were initially assessed on IT and cognitive abilities in 2003, reassessed on IT in 2005, and then completed the Everyday Problems Test, a test of independence in everyday living, in 2008. Controlling for age, lower fluid reasoning at baseline (Raven’s-subset) and slowing IT over 18 months independently predicted poorer everyday functioning in 2008, together accounting for one-third variance. This supports the proposition that IT has utility as a biomarker for less favourable ageing.     

Superior fluid intelligence in children with Asperger's disorder

Asperger's disorder is one of autistic spectrum disorders; sharing clinical features with autism, but without developmental delay in language acquisition. There have been some studies of intellectual functioning in autism so far, but very few in Asperger's disorder. In the present study, we investigated abstract reasoning ability, whose form of intelligence has been labeled fluid intelligence in the theory of Cattell [Cattell, R. B. (1963). Theory of fluid and crystallized intelligence: A critical experiment. Journal of Educational Psychology, 54, 1-22.], in children with Asperger's disorder. A test of fluid intelligence, the Raven's Standard Progressive Matrices Test, was administered to 17 children with Asperger's disorder and 17 age-, gender-, and FIQ-matched normal children. The results showed that children with Asperger's disorder outperformed on the test of fluid reasoning than typically developing children. We suggest that individuals with Asperger's disorder have higher fluid reasoning ability than normal individuals, highlighting superior fluid intelligence.     

Matrices with a given number of violations of Regular Minimality

A row (or column) of an n×n matrix complies with Regular Minimality (RM) if it has a unique minimum entry which is also a unique minimum entry in its column (respectively, row). The number of violations of RM in a matrix is defined as the number of rows (equivalently, columns) that do not comply with RM. We derive a formula for the proportion of n×n matrices with a given number of violations of RM among all n×n matrices with no tied entries. The proportion of matrices with no more than a given number of violations can be treated as the p-value of a permutation test whose null hypothesis states that all permutations of the entries of a matrix without ties are equiprobable, and the alternative hypothesis states that RM violations occur with lower probability than predicted by the null hypothesis. A matrix with ties is treated as being represented by all matrices without ties that have the same set of strict inequalities among their entries.     

Cut-Elimination and Quantification in Canonical Systems

Canonical Propositional Gentzen-type systems are systems which in addition to the standard axioms and structural rules have only pure logical rules with the sub-formula property, in which exactly one occurrence of a connective is introduced in the conclusion, and no other occurrence of any connective is mentioned anywhere else. In this paper we considerably generalize the notion of a “canonical system” to first-order languages and beyond. We extend the Propositional coherence criterion for the non-triviality of such systems to rules with unary quantifiers and show that it remains constructive. Then we provide semantics for such canonical systems using 2-valued non-deterministic matrices extended to languages with quantifiers, and prove that the following properties are equivalent for a canonical system G: (1) G admits Cut-Elimination, (2) G is coherent, and (3) G has a characteristic 2-valued non-deterministic matrix.     

Beyond Rasiowa's Algebraic Approach to Non-classical Logics

This paper reviews the impact of Rasiowa's well-known book on the evolution of algebraic logic during the last thirty or forty years. It starts with some comments on the importance and influence of this book, highlighting some of the reasons for this influence, and some of its key points, mathematically speaking, concerning the general theory of algebraic logic, a theory nowadays called Abstract Algebraic Logic. Then, a consideration of the diverse ways in which these key points can be generalized allows us to survey some issues in the development of the field in the last twenty to thirty years. The last part of the paper reviews some recent lines of research that in some way transcend Rasiowa's approach. I hope in this way to give the reader a general view of Rasiowa's key position in the evolution of Algebraic Logic during the twentieth century. This paper is an extended version of the invited talk given by the author at the conference Trends in Logic III, dedicated to the memory of A. MOSTOWSKI, H. RASIOWA and C. RAUSZER, and held in Warsaw and Ruciane-Nida from 23rd to 25th September 2005. Presented by Jacek Malinowski     

Some developments in multivariate generalizability

This article is concerned with estimation of components of maximum generalizability in multifacet experimental designs involving multiple dependent measures. Within a Type II multivariate analysis of variance framework, components of maximum generalizability are defined as those composites of the dependent measures that maximize universe score variance for persons relative to observed score variance. The coefficient of maximum generalizability, expressed as a function of variance component matrices, is shown to equal the squared canonical correlation between true and observed scores. Emphasis is placed on estimation of variance component matrices, on the distinction between generalizability- and decision-studies, and on extension to multifacet designs involving crossed and nested facets. An example of a two-facet partially nested design is provided.Appreciation is expressed to the Office of Research in Medical Education, University of Texas Medical Branch, for permitting use of their data.     

Two-matrix orthogonal rotation procedures

Examples are presented in which it is either desirable or necessary to transform two sets of orthogonal axes to simple structure positions by means of the same transformation matrix. A solution is then outlined which represents a two-matrix extension of the general orthomax orthogonal rotation criterion. In certain circumstances, oblique two-matrix solutions are possible using the procedure outlined and the Harris-Kaiser [1964] logic. Finally, an illustrative example is presented in which the preceding technique is applied in the context of an inter-battery factor analysis.The work reported herein was supported by Grant S72-1886 from the Canada Council. The author acknowledges the helpful contributions of Nancy Reid and Lawrence Ward to parts of this paper.