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.     

Fungible Weights in Multiple Regression

Every set of alternate weights (i.e., nonleast squares weights) in a multiple regression analysis with three or more predictors is associated with an infinite class of weights. All members of a given class can be deemed fungiblebecause they yield identical SSE (sum of squared errors) and R ² values. Equations for generating fungible weights are reviewed and an example is given that illustrates how fungible weights can be profitably used to evaluate parameter sensitivity in multiple regression. The author wishes to thank Drs. Robyn Dawes, William Grove, Markus Keel, Leslie Yonce, Joe Rausch, the editor, and three anonymous reviewers for helpful comments on earlier versions of this article.     

Locating the Extrema of Fungible Regression Weights

In a multiple regression analysis with three or more predictors, every set of alternate weights belongs to an infinite class of “fungible weights” (Waller, Psychometrica, in press) that yields identical SSE (sum of squared errors) and R ² values. When the R ² using the alternate weights is a fixed value, fungible weights (a _i ) that yield the maximum or minimum cosine with an OLS weight vector (b) are called “fungible extrema.” We describe two methods for locating fungible extrema and we report R code (R Development Core Team, 2007) for one of the methods. We then describe a new approach for populating a class of fungible weights that is derived from the geometry of alternate regression weights. Finally, we illustrate how fungible weights can be profitably used to gauge parameter sensitivity in linear models by locating the fungible extrema of a regression model of executive compensation (Horton & Guerard, Commun. Stat. Simul. Comput. 14:441–448, 1985).     

Algebras and Matrices for Annotated Logics

We study the matrices, reduced matrices and algebras associated to the systems SA_T of structural annotated logics. In previous papers, these systems were proven algebraizable in the finitary case and the class of matrices analyzed here was proven to be a matrix semantics for them.We prove that the equivalent algebraic semantics associated with the systems SA_T are proper quasivarieties, we describe the reduced matrices, the subdirectly irreducible algebras and we give a general decomposition theorem. As a consequence we obtain a decision procedure for these logics.     

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.     

瑞文推理测验上的性别差异

瑞文推理测验上是否存在性别差异？与较早期的研究结果不同, 较近的一系列研究发现在瑞文推理测验上存在性别差异, 且性别差异具有发展性。当前可以解释该性别差异的观点主要有两种, 其一是污染观, 即认为瑞文推理测验中发现的性别差异实质上是由于测验中其他成分的“污染”所致, 而在它所主要代表的g因素的发展中并不存在性别差异; 其二, 较新的研究提示, “污染”可能存在, 但在g因素上仍然具有性别差异, 这种差异是由生理上的原因造成的。研究者由此提出了智力性别差异的发展理论。本文主要从儿童青少年发展的角度回顾了有关瑞文推理测验性别差异的研究, 综述并分析了关于性别差异研究结果不一致的可能原因。     

Some Properties Of Three Factor Contribution Matrices

Some properties of three matrices which have been proposed as quantitative bases for factor interpretation are discussed. The first of these contains as elements the covariances of the tests with weighted factor scores. Interpretations of the row sums and column sums of this matrix are given and its independence of the primary factor-reference vector distinction is demonstrated. Two matrices of part correlations are shown to be proportional by columns to the primary factor pattern (and thus to the reference vector structure). One of these matrices is shown to be equal to the reference vector structure.     

Matrices,primitive satisfaction and finitely based logics

We examine the notion of primitive satisfaction in logical matrices. Theorem II. 1, being the matrix counterpart of Baker's well-known result for congruently distributive varieties of algebras (cf [1], Thm. 1.5), links the notions of primitive and standard satisfaction. As a corollary we give the matrix version of Jónsson's Lemma, proved earlier in [4]. Then we investigate propositional logics with disjunction. The main result, Theorem III. 2, states a necessary and sufficient condition for such logics to be finitely based.     

A Note on Matrices for Systems of nonsense-logics

We construct a class K of algebras which are matrices of the logical system Z introduced in [4]. It is shown that algebras belonging to the class K are decomposable into disjoint subalgebras which are Boolean algebras.To the memory of Jerzy SupeckiTranslated From the Polish by Jan Zygmunt.     

Decision Matrices and Time in Tactical Course of Action Analysis

Tactical decision-making doctrine incorporates rules that conform to a classical additive weighting model. However, decision-making research has increasingly focused on naturalistic strategies. An experiment was conducted to investigate classical and natural approaches to tactical analysis. Teams of U.S. Army officers used either unspecified or structured procedures to analyze courses of action. Structure was imposed by providing workbooks for individual task steps. Completion times and additive weighted matrices were included. Paper-and-pencil and computer versions of materials were used. Teams from both structured groups had better justification scores than those with unspecified procedures. Computer support provided no additional advantage. The more detailed consideration of battle events provided the desired influence, whereas the classical decision rules did not seem to. Early conclusions did not result in better or poorer performance. The findings suggest modified rules for tactical analysis that emphasize detailed consideration of events and flexibility in the timing of decisions.     

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.     

The induction of solution rules in Raven's Progressive Matrices Test

In a passive task participants recall material in the format in which it was presented, whereas in an active task they transform or manipulate the material. Experiment 1 describes an active visuo-spatial task that is vulnerable to the effects of ageing, and Experiment 2 shows that it is more sensitive to the effects of ageing than a passive task carried out concurrently using the same material. In Experiment 3, similar results were obtained using active and passive verbal tasks carried out concurrently using the same material. Finally, these findings were replicated in Experiment 4 using active and passive verbal and visuo-spatial tasks that were carried out in separate experimental conditions. It is concluded that effects of ageing show an earlier onset in tasks requiring active processing than in tasks requiring the passive storage of information.     

Normative data for Raven's Coloured Progressive Matrices scale in Yemen

Results are reported for a standardization sample of 986 6- to 1-yr.-olds for the Coloured Progressive Matrices in Yemen. Younger children performed better than older children relative to British norms, and there was no significant sex difference in means or variability. In relation to a British IQ of 100 (SD=15), the sample obtained an average IQ of approximately 81.     

Modeling Noisy Data with Differential Equations Using Observed and Expected Matrices

Complex intraindividual variability observed in psychology may be well described using differential equations. It is difficult, however, to apply differential equation models in psychological contexts, as time series are frequently short, poorly sampled, and have large proportions of measurement and dynamic error. Furthermore, current methods for differential equation modeling usually consider data that are atypical of many psychological applications. Using embedded and observed data matrices, a statistical approach to differential equation modeling is presented. This approach appears robust to many characteristics common to psychological time series.     

A Tabu-Search Heuristic for Deterministic Two-Mode Blockmodeling of Binary Network Matrices

Two-mode binary data matrices arise in a variety of social network contexts, such as the attendance or non-attendance of individuals at events, the participation or lack of participation of groups in projects, and the votes of judges on cases. A popular method for analyzing such data is two-mode blockmodeling based on structural equivalence, where the goal is to identify partitions for the row and column objects such that the clusters of the row and column objects form blocks that are either complete (all 1s) or null (all 0s) to the greatest extent possible. Multiple restarts of an object relocation heuristic that seeks to minimize the number of inconsistencies (i.e., 1s in null blocks and 0s in complete blocks) with ideal block structure is the predominant approach for tackling this problem. As an alternative, we propose a fast and effective implementation of tabu search. Computational comparisons across a set of 48 large network matrices revealed that the new tabu-search heuristic always provided objective function values that were better than those of the relocation heuristic when the two methods were constrained to the same amount of computation time.     

Information Matrices and Standard Errors for MLEs of Item Parameters in IRT

The paper clarifies the relationship among several information matrices for the maximum likelihood estimates (MLEs) of item parameters. It shows that the process of calculating the observed information matrix also generates a related matrix that is the middle piece of a sandwich-type covariance matrix. Monte Carlo results indicate that standard errors (SEs) based on the observed information matrix are robust to many, but not all, conditions of model/distribution misspecifications. SEs based on the sandwich-type covariance matrix perform most consistently across conditions. Results also suggest that SEs based on other matrices are either not consistent or perform not as robust as those based on the sandwich-type covariance matrix or the observed information matrix.     

The reliability and validity of Raven's Standard Progressive Matrices for Irish apprentices

The Standard Progressive Matrices (SPM) is a 'test of a person's capacity at the time of the test to apprehend meaningless figures presented for his observation, see the relations between them, conceive the nature of the figure completing each system of relations presented and, by so doing, develop a systematic method of reasoning' (Raven et al., 1983, p. 2). It was developed in Britain in the mid-1930s (see Penrose and Raven, 1936) as a non-verbal measure of Spearman's 'g' factor, eduction of relations among abstract items.