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.     

Enhancing Interpretability in Factor Analysis by Means of Mathematical Optimization

Abstract

Exploratory Factor Analysis (EFA) is a widely used statistical technique to discover the structure of latent unobserved variables, called factors, from a set of observed variables. EFA exploits the property of rotation invariance of the factor model to enhance factors’ interpretability by building a sparse loading matrix. In this paper, we propose an optimization-based procedure to give meaning to the factors arising in EFA by means of an additional set of variables, called explanatory variables, which may include in particular the set of observed variables. A goodness-of-fit criterion is introduced which quantifies the quality of the interpretation given this way. Our methodology also exploits the rotational invariance of EFA to obtain the best orthogonal rotation of the factors, in terms of the goodness-of-fit, but making them match to some of the explanatory variables, thus going beyond traditional rotation methods. Therefore, our approach allows the analyst to interpret the factors not only in terms of the observed variables, but in terms of a broader set of variables. Our experimental results demonstrate how our approach enhances interpretability in EFA, first in an empirical dataset, concerning volumes of reservoirs in California, and second in a synthetic data example.     

Mathematical Studies of the Information in the Stimulus-Response Matrix

This paper considers the information transmitted in absolute judgments as encoded in a stimulus-response matrix (e.g., see Garner and Hake, 1951). When transmitted information is plotted against the number of stimulus categories in the matrix, one obtains a curve that increases monotonically toward a plateau, which is the maximum information transmittable per stimulus for the particular range of stimuli employed. We demonstrate that although the maximum information transmitted is an attribute of the stimulus continuum itself, the shape of the curve is an empirical property of the stimulus-response matrix, which is determined, in part, by maintaining a constant stimulus category width. Therefore, in principle, each curve of information transmitted vs number of stimulus categories can be determined by a single point: the rightmost point on the graph. Copyright 2001 Academic Press.     

Some Theory and Applications of Confirmatory Second-Order Factor Analysis

This paper shows how confirmatory factor analysis can be used to test second- (and higher-) order factor models in the areas of the structure of abilities, allometry, and the separation of specific and error variance estimates. In the latter area, an idea of Joreskog's is extended to include a new conceptualization of the estimation of validity. Second-order models are placed in a hierarchy of factor analysis models, which shows how the fit of various models can be compared. The concept of discriminability is introduced to describe a situation in which two models may both be identified, and yet the goodness-of-fit for both will be the same. This problem can usually be avoided by careful design of a study. Several examples are discussed.     

Psychometric Properties and Confirmatory Factor Analysis of the UCLA Loneliness Scale

This study investigated the psychometric properties of the UCLA Loneliness Scale. Test-retest and internal consistency reliability were found to be highly acceptable, and support for the concurrent related validity of the scale was also indicated. Confirmatory factor analysis provided support for the unidimensionality of the scale. Evidence was found that the scale is not multivariate normal and in fact appears to be bimodal. Implications point to the need to reconsider the nature of the loneliness construct and to develop conceptually or theoretically based models for testing.     

Some Properties of Orthologics

In this paper, we present three main results on orthologics. Firstly, we give a sufficient condition for an orthologic to have variable separation property and show that the orthomodular logic has this property. Secondly, we show that the class of modular orthologics has an infinite descending chain. Finally we show that there exists a continuum of orthologics.     

Brief Report: Some Possible Uses Of Factor Analysis In Multivariate Studies

Factor analysis is used in item selection in the hopes of producing a small number of factors each of which will represent a unidimensional sub- scale. If item analysis has been successful in producing truly independent subscales, it might be hoped that the number of factors would equal the number of subscales and that each factor would be highly defined by a single subscale. Factor analysis when used in studies of organization, is not assumed to produce factors that represent unidimensional scales. Rather, factor analysis is used to reveal various substructures that exist within an organization. If several variables are loaded on a single factor, the variables can be regarded as nodes of interaction between measured dimensions of organization.     

Clustered Common Factor Exploration in Factor Analysis

Psychometrika - The factor analysis (FA) model does not permit unique estimation of the common and unique factor scores. This weakness is notorious as the factor indeterminacy in FA. Luckily, some...     

Some Versions of the Number Problem Have No Solution

This article addresses Taruek’s much discussed Number Problem from a non-consequentialist point of view. I argue that some versions of the Number Problem have no solution, meaning that no alternative is at least as choice-worthy as the others, and that the best way to behave in light of such moral indeterminacy is to let chance make the decision. I contrast my proposal with F M Kamm’s nonconsequentialist argument for saving the greatest number, the Argument for Best Outcomes, which I argue does not follow from the premises it is based on.     

Mathematical Proof Theory in the Light of Ordinal Analysis

We give an overview of recent results in ordinal analysis. Therefore,we discuss the different frameworks used in mathematical proof-theory, namely subsystem of analysis including reversemathematics, Kripke–Platek set theory, explicitmathematics, theories of inductive definitions,constructive set theory, and Martin-Löfs typetheory.     

Ignoring Clustering in Confirmatory Factor Analysis: Some Consequences for Model Fit and Standardized Parameter Estimates

In many situations, researchers collect multilevel (clustered or nested) data yet analyze the data either ignoring the clustering (disaggregation) or averaging the micro-level units within each cluster and analyzing the aggregated data at the macro level (aggregation). In this study we investigate the effects of ignoring the nested nature of data in confirmatory factor analysis (CFA). The bias incurred by ignoring clustering is examined in terms of model fit and standardized parameter estimates, which are usually of interest to researchers who use CFA. We find that the disaggregation approach increases model misfit, especially when the intraclass correlation (ICC) is high, whereas the aggregation approach results in accurate detection of model misfit in the macro level. Standardized parameter estimates from the disaggregation and aggregation approaches are deviated toward the values of the macro- and micro-level standardized parameter estimates, respectively. The degree of deviation depends on ICC and cluster size, particularly for the aggregation method. The standard errors of standardized parameter estimates from the disaggregation approach depend on the macro-level item communalities. Those from the aggregation approach underestimate the standard errors in multilevel CFA (MCFA), especially when ICC is low. Thus, we conclude that MCFA or an alternative approach should be used if possible.     

探索性因素分析因子抽取方法的比较

探索性因素分析决定因子抽取的方法主要有Bartlett法、K1原则、碎石检验法、Aaker原则、PA、MAP等六种,通过对样本1的395名大学生的学习过程问卷调查获得真实数据,运用这六种方法进行因素分析分别抽取7、4、2、4、3、2个因子;应用样本2的383名大学生的问卷调查数据进行验证性因素分析,结果显示,碎石检验方法与MAP方法抽取的二因素模型更理想。研究表明,因子抽取需兼顾“简约性原则”与“完备性原则”,同时要根据一定的理论建构、专业知识和经验来决定因子数。     

论因素分析方法的整合

传统的因素分析是探索性的因素分析(Exploratory Factor Analysis,以下简称EFA),现代统计方法增加了验证性的因素分析(Confirmatory Factor Analysis,以下简称CFA)。对二者的关系及其内部的诸多问题进行分析是非常必要的。     

Analysis and Decomposition in Frege and Russell

Michael Dummett has long argued that Frege is committed to recognizing a distinction between two sorts of analysis of propositional contents: 'analysis', which reveals the entities that one must grasp in order to apprehend a given propositional content; and 'decomposition', which is used in recognizing the validity of certain inferences. Whereas any propositional content admits of a unique ultimate 'analysis' into simple constituents, it also admits of distinct 'decompositions', no one of which is ultimately privileged over the others. I argue that although Russell accepts this distinction between analysis and decomposition, Frege does not. In particular, I consider claims which Dummett makes regarding how 'analysis' and 'decomposition' are related to two different models Frege at least suggests in discussing the composition of thoughts, the part/whole model and the function/argument model; and I argue that in each case, while Russell accepts views which Dummett attributes to Frege, Frege does not.     

广义量词的相关性质研究

本文是笔者对S.Peters与D.Westerst(?)hl([7])的成果的拓展研究。首先介绍相关的概念。其次,笔者详细证明了类型为〈1,1〉的广义量词的对称性与单调性的关系定理。然后,笔者给出了该类量词的余对称性、余相交性和余驻留性定义,接着笔者提出并证明了关于这三个性质的四个定理,而且还详细证明了余对称性与单调性的关系定理。最后,笔者探讨了具有(余)驻留性和(余)对称性的〈1,1〉类型的广义量词的数字三角形的特点。由于〈1,1〉类型的广义量词在自然语言中普遍存在,所以,本文的研究对广义量词理论的发展具有一定的理论价值,对自然语言的计算机信息处理也具有一定的实践指导意义。     

Some Closure Properties of Finite Definitions

There is no known syntactic characterization of the class of finite definitions in terms of a set of basic definitions and a set of basic operators under which the class is closed. Furthermore, it is known that the basic propositional operators do not preserve finiteness. In this paper I survey these problems and explore operators that do preserve finiteness. I also show that every definition that uses only unary predicate symbols and equality is bound to be finite.