Relationships between two systems of factor analysis

Considering only population values, it is shown that the complete set of factors of a correlation matrix with units in the diagonal cells may be transformed into the factors derived by factoring these correlations with communalities in the diagonal cells. When the correlations are regarded as observed values, the common factors derived as a transformation of the complete set of factors of the correlation matrix with units in the diagonal cells satisfy Lawley's requirement for a maximum likelihood solution and are a first approximation to Rao's canonical factors.     

Some necessary conditions for common-factor analysis

LetR be any correlation matrix of ordern, with unity as each main diagonal element. Common-factor analysis, in the Spearman-Thurstone sense, seeks a diagonal matrixU ² such thatG = R – U ² is Gramian and of minimum rankr. Lets ₁ be the number of latent roots ofR which are greater than or equal to unity. Then it is proved here thatr s ₁. Two further lower bounds tor are also established that are better thans ₁. Simple computing procedures are shown for all three lower bounds that avoid any calculations of latent roots. It is proved further that there are many cases where the rank of all diagonal-free submatrices inR is small, but the minimum rankr for a GramianG is nevertheless very large compared withn. Heuristic criteria are given for testing the hypothesis that a finiter exists for the infinite universe of content from which the sample ofn observed variables is selected; in many cases, the Spearman-Thurstone type of multiple common-factor structure cannot hold.This research was made possible in part by an uncommitted grant-in-aid from the Behavioral Sciences Division of the Ford Foundation.     

Some mathematical remarks concerning boundary conditions in the factorial analysis of ability

This paper is a mathematical supplement to the preceding paper by Professor Godfrey H. Thomson. It gives rigorous proofs of theorems enunciated by him and by Dr. J. Ridley Thompson, and extends them. Its basic theorem is that if a matrix of correlations is to be factorized without the aid of higher factors thans-factors (withn-s zero loadings), then the largest latent root of the matrix must not exceed the sum of thes largest communalities on the diagonal.     

Some rao-guttman relationships

An examination of the determinantal equation associated with Rao's canonical factors suggests that Guttman's best lower bound for the number of common factors corresponds to the number of positive canonical correlations when squared multiple correlations are used as the initial estimates of communality. When these initial communality estimates are used, solving Rao's determinantal equation (at the first stage) permits expressing several matrices as functions of factors that differ only in the scale of their columns; these matrices include the correlation matrix with units in the diagonal, the correlation matrix with squared multiple correlations as communality estimates, Guttman's image covariance matrix, and Guttman's anti-image covariance matrix. Further, the factor scores associated with these factors can be shown to be either identical or simply related by a scale change. Implications for practice are discussed, and a computing scheme which would lead to an exhaustive analysis of the data with several optional outputs is outlined.     

A note on the reflection of signs in the extraction of centroid factors

This note suggests that the reflection of residuals in the centroid method of factor analysis should be continued, whenever possible, after all the sums of the columns in the correlation matrix, excluding diagonal values, are positive. A criterion is given for determining whether further reflection is possible in such cases.     

Testing pattern hypotheses for covariance matrices

Maximum likelihood estimates of the free parameters, and an asymptotic likelihood-ratio test, are given for the hypothesis that one or more elements of a covariance matrix are zero, and/or that two or more of its elements are equal. The theory applies immediately to a transformation of the covariance matrix by a known nonsingular matrix. Estimation is by Newton's method, starting conveniently from a closed-form least-squares solution.Numerical illustrations include a test for equality of diagonal blocks of a covariance matrix, and estimation of quasi-simplex structures.     

Rank-reducibility of a symmetric matrix and sampling theory of minimum trace factor analysis

One of the intriguing questions of factor analysis is the extent to which one can reduce the rank of a symmetric matrix by only changing its diagonal entries. We show in this paper that the set of matrices, which can be reduced to rankr, has positive (Lebesgue) measure if and only ifr is greater or equal to the Ledermann bound. In other words the Ledermann bound is shown to bealmost surely the greatest lower bound to a reduced rank of the sample covariance matrix. Afterwards an asymptotic sampling theory of so-called minimum trace factor analysis (MTFA) is proposed. The theory is based on continuous and differential properties of functions involved in the MTFA. Convex analysis techniques are utilized to obtain conditions for differentiability of these functions.     

复杂决策规则下MIRT的分类准确性和分类一致性

介绍多维项目反应理论模型下分类准确性和分类一致性指标, 采用蒙特卡罗方法实现复杂决策规则下指标计算, 并从数学上证明分类准确性指标两类估计量在均匀先验和相同决策规则条件下依概率收敛于同一真值。研究结果表明：分类准确性指标可以比较准确地评价分类结果的准确性; 分类一致性指标可以较好地评价分类结果的重测一致性; 在一定条件下, 基于能力量尺的指标优于基于原始总分的指标; 纵使测验维度增加, 估计精度仍比较好; 随着测验长度和维度间相关增加, 分类准确性和分类一致性更高。指标可以用来评价标准参照测验或计算机分类测验的多种决策规则下分类信度和效度。     

Latent roots of random data correlation matrices with squared multiple correlations on the diagonal: A monte carlo study

In order to make the parallel analysis criterion for determining the number of factors easy to use, regression equations for predicting the logarithms of the latent roots of random correlation matrices, with squared multiple correlations on the diagonal, are presented. The correlation matrices were derived from distributions of normally distributed random numbers. The independent variables are log (N–1) and log {[n(n–1)/2]–[(i–1)n]}, whereN is the number of observations;n, the number of variables; andi, the ordinal position of the eigenvalue. The results were excellent, with multiple correlation coefficients ranging from .9948 to .9992.This research was supported by the Office of Naval Research under Contract N00014-67-A-0305-0012, Lloyd G. Humphreys, principal investigator, and by the Department of Computer Science of which Richard G. Montanelli, Jr., is a member.     

Properties and applications of gramian factoring

The Gramian factorizationG of a GramianR is square and symmetric and has no negative characteristic roots. It is shown to be that square factorization that is, in the least-squares sense, most isomorphic toR, most like a scalarK, and most highly traced, and to be the necessary and sufficient relation between the oblique vectors of an oblique transformation and the orthogonal vectors of the least-squares orthogonal counterpart. A slightly modified Gramian factorization is shown to be the factorization that is most isomorphic to a specified diagonalD, and to be the main part of an iterative procedure for obtaining simplimax, a square factor matrix with simple structure maximized in the sense of having the largest sum of squared diagonal loadings. Several published applications of Gramian factoring are cited.     

On the least-squares orthogonalization of an oblique transformation

After proving a special case of a theorem stated by Eckart and Young, namely, that an oblique transformationG is the product of two different orthogonal transformations and an intervening diagonal, this note shows that the best fitting orthogonal approximation toG is obtained simply by replacing the intervening diagonal by the identity matrix. This result is shown to be identical with two earlier orthogonalizing procedures whenG is of full rank. A multiplicity of solutions is shown for the case of a singularG.I am grateful to J. J. Mellinger for pointing out a flaw in a previous version of this paper.Opinions expressed herein are those of the author, not necessarily those of the Army.     

Computing Cohen’s kappa coefficients using SPSS MATRIX

This short paper proposes a general computing strategy to compute Kappa coefficients using the SPSS MATRIX routine. The method is based on the following rationale. If the contingency table is considered as a square matrix, then the observed proportions of agreement lie in the main diagonal’s cells, and their sum equals the trace of the matrix, whereas the proportions of agreement expected by chance are the joint product of marginals. The generalization to weighted kappa, which requires an additional square matrix of disagreement weights, both matrices having the same order, becomes possible by the use of the Hadamard product-that is, the elementwise direct product of two matrices.     

Approximate uniqueness estimates for singular correlation matrices

The residual variance (one minus the squared multiple correlation) is often used as an approximation to the uniqueness in factor analysis. An upper bound approximation to the residual variance is given for the case when the correlation matrix is singular. The approximation is computationally simpler than the exact solution, especially since it can be applied routinely without prior knowledge as to the singularity or nonsingularity of the correlation matrix.     

,SOME CHARACTERISTICS OF LATENT SCORES

A number of multivariate psychometric models hypothesize that a data matrix of observed, scores equals the sum of two mutually orthogonal latent matrices. Relationships among latent and observed scores are investigated psychometrically using the concept of the general inverse. A number of previously unrecognized characteristics of the latent scores are thus brought to light.     

Some new results on factor indeterminacy

Some relations between maximum likelihood factor analysis and factor indeterminacy are discussed. Bounds are derived for the minimum average correlation between equivalent sets of correlated factors which depend on the latent roots of the factor intercorrelation matrix . Empirical examples are presented to illustrate some of the theory and indicate the extent to which it can be expected to be relevant in practice.     

Indeterminacy Problems in the Lisrel Model

The latent variables and errors of the Lisrel model are indeterminate even when the parameters of the model are perfectly identified. The reason for the indeterminacy is that the Lisrel model gives a solution in terms of estimation of latent variables by means of observed variables. The indeterminacy is relevant also in practice; the minimum correlation between equivalent latent variables, is often negative in empirical examples. The degree of indeterminacy of the latent variables depends on the data. The average minimum correlation is a linear combination of the eigenvalues of the correlation matrix of solutions and it is always included in weak bounds which depend on the same eigenvalues.     

A rationale and test for the number of factors in factor analysis

It is suggested that if Guttman's latent-root-one lower bound estimate for the rank of a correlation matrix is accepted as a psychometric upper bound, following the proofs and arguments of Kaiser and Dickman, then the rank for a sample matrix should be estimated by subtracting out the component in the latent roots which can be attributed to sampling error, and least-squares capitalization on this error, in the calculation of the correlations and the roots. A procedure based on the generation of random variables is given for estimating the component which needs to be subtracted.I wish to acknowledge the valuable help given by J. Jaspers and L. G. Humphreys in the development of the ideas presented in this paper.     

Graph-theoretic representations for proximity matrices through strongly-anti-Robinson or circular strongly-anti-Robinson matrices

There are various optimization strategies for approximating, through the minimization of a least-squares loss function, a given symmetric proximity matrix by a sum of matrices each subject to some collection of order constraints on its entries. We extend these approaches to include components in the approximating sum that satisfy what are called the strongly-anti-Robinson (SAR) or circular strongly-anti-Robinson (CSAR) restrictions. A matrix that is SAR or CSAR is representable by a particular graph-theoretic structure, where each matrix entry is reproducible from certain minimum path lengths in the graph. One published proximity matrix is used extensively to illustrate the types of approximation that ensue when the SAR or CSAR constraints are imposed.The authors are indebted to Boris Mirkin who first noted in a personal communication to one of us (LH, April 22, 1996) that the optimization method for fitting anti-Robinson matrices in Hubert and Arabie (1994) should be extendable to the fitting of strongly anti-Robinson matrices as well.     

On the oblique rotation of a factor matrix to a specified pattern

This paper presents a procedure for rotating an arbitrary factor matrix to maximum similarity with a specified factor pattern. The sum of squared distances between specified vectors and rotated vectors in oblique Euclidian space is minimized. An example of the application of the procedure is given.This research was supported in part by the National Institute of Child Health and Human Development, Research Grant 1 PO1 HDO1762.The names of the authors are given in alphabetical order. Their contributions to the paper are equal.     

Statistical inference of minimum rank factor analysis

For any given number of factors, Minimum Rank Factor Analysis yields optimal communalities for an observed covariance matrix in the sense that the unexplained common variance with that number of factors is minimized, subject to the constraint that both the diagonal matrix of unique variances and the observed covariance matrix minus that diagonal matrix are positive semidefinite. As a result, it becomes possible to distinguish the explained common variance from the total common variance. The percentage of explained common variance is similar in meaning to the percentage of explained observed variance in Principal Component Analysis, but typically the former is much closer to 100 than the latter. So far, no statistical theory of MRFA has been developed. The present paper is a first start. It yields closed-form expressions for the asymptotic bias of the explained common variance, or, more precisely, of the unexplained common variance, under the assumption of multivariate normality. Also, the asymptotic variance of this bias is derived, and also the asymptotic covariance matrix of the unique variances that define a MRFA solution. The presented asymptotic statistical inference is based on a recently developed perturbation theory of semidefinite programming. A numerical example is also offered to demonstrate the accuracy of the expressions.This work was supported, in part, by grant DMS-0073770 from the National Science Foundation.