On admissible communalities in factor analysis

A set of inequalities on communalities developed by Darroch [1] is used to provide some rules for determining admissible sets of communalities in factor analysis.     

Maximum likelihood estimation and factor analysis

Fisher's method of maximum likelihood is applied to the problem of estimation in factor analysis, as initiated by Lawley, and found to lead to a generalization of the Eckart matrix approximation problem. The solution of this in a special case is applied to show how test fallability enters into factor determination, it being noted that the method of communalities underestimates the number of factors.Dr. George Brown of Princeton University has independently made the same suggestion in some unpublished work.     

Obtaining maximum-likelihood estimates of factor loadings and communalities using an easily implemented iterative computer procedure

This paper considers the problem of computing estimates of factor loadings, specific variances, and communalities for a factor analytic model. The equations for maximum-likelihood estimators are discussed. Iterative formulas are developed to solve the maximum-likelihood equations and a simple and efficient method of implementation on a digital computer is described. Use of the iterative formulas and computing techniques for other estimators of factor loadings and communalities is also considered to provide a very general approach for this aspect of factor analysis.     

A simple orthogonal multiple factor approximation procedure

This paper describes a simple orthogonal multiple factor approximation procedure that involves no inversion of the sign of negative residuals, the estimation of only as many communalities as there are factors, and none or only a few minor rotations of the axes in an attempt to obtain a meaningful solution. It also suggests a technique for the estimation of those communalities that must be estimated. The factor loadings obtained by means of this procedure, which we shall designate as the pre-selection procedure, are affected by the order in which the factors are obtained, showing a reduction in variance accounted for by each successive factor, as is characteristic of the centroid, bi-factor, and principal factor solutions. The entire procedure takes considerably less time than that involved in the orthodox centroid method alone.     

A numerical approach to the approximate and the exact minimum rank of a covariance matrix

A concept of approximate minimum rank for a covariance matrix is defined, which contains the (exact) minimum rank as a special case. A computational procedure to evaluate the approximate minimum rank is offered. The procedure yields those proper communalities for which the unexplained common variance, ignored in low-rank factor analysis, is minimized. The procedure also permits a numerical determination of the exact minimum rank of a covariance matrix, within limits of computational accuracy. A set of 180 covariance matrices with known or bounded minimum rank was analyzed. The procedure was successful throughout in recovering the desired rank.The authors are obliged to Paul Bekker for stimulating and helpful comments.     

An algebraic solution for the communalities

Factorial analysis begins with ann ×n correlation matrixR, whose principal diagonal entries are unknown. If the common test space of the battery is under investigation, the communality of each test is entered in the appropriate diagonal cell. This value is the portion of the test's variance shared with others in the battery. The communalities must be so estimated thatR will maintain the rank determined by its side entries, after the former have been inserted. Previous methods of estimating the communalities have involved a certain arbitrariness, since they depended on selecting test subgroups or parts of the data inR. A theory is presented showing that this difficulty can be avoided in principle. In its present form, the theory is not offered as a practical computing procedure. The basis of the new method lies in the Cayley-Hamilton theorem: Any square matrix satisfies its own characteristic equation.     

Clergy dissatisfaction: denominational hierarchy as a salient factor

The author contends that there is a perceived lack of support from denominational hierarchy, and that the impact of this lack on clergy has been overlooked in the literature. The factor is important because it will underpin the success of any strategies attempted to address the cited stressors. As well, no treatment strategy for those who struggle with these issues has been defined which seems to ignore that this unique population requires unique considerations.     

A definitive,large-sample factoring of personality structure in objective measures,as a basis for the high school objective-analytic battery

Eighty-two subtests chosen from a series of preceding researches as the best marker variables for 12 previously checked and interpreted personality factors (U.I. 16, 17, 19, 20, 21, 23–26, 28, 32, and 33) were administered to 2,522 boys of high school age. The ensuing factor analysis, aimed to check and precision the factor patterns, to lead to the definitive HSOA battery, and to yield scores for further age and genetic analyses, observed the following procedural requirements: (1) Scree test for number of factors, (2) iterated communalities, (3) blind automated rotation, carried further by rotoplot to a maximum simple structure, (4) a test of simple structure significance, and (5) matching of patterns for invariance against other studies. As before in this domain, 19–20 factors were found, and the 10 for which definite markers had been made available matched very well with former patterns. Although a small minority of tests failed to load as hypothesized, this results in some weakness of factor estimate only on U.I. 32, Exvia-Invia. For on all 10 factors for the planned HSOA battery, with this possible exception, the validities of weighted battery scores reached acceptable levels.     

不同定义平行测验等值的群体不变性

群体不变性是等值的一个重要假设,即对不同的考生子群体等值函数一致。本研究对不同平行测验定义下线性等值的群体不变性进行了理论分析和模拟研究,模拟研究REMSD指标通过六种不同加权方式计算。结果显示,严格平行测验在信度较低时REMSD指标更大;子群体均值差异和信度差异对REMSD的影响存在明显的交互作用;REMSD指标在期望权重等权下的最大,在分数权重采用子群体比例加权最小。最后对结果进行了讨论,对REMSD权重使用及进一步研究给出了建议。     

Some relations between guttman's principal components of scale analysis and other psychometric theory

Guttman's principal components for the weighting system are the item scoring weights that maximize the generalized Kuder-Richardson reliability coefficient. The principal component for any item is effectively the same as the factor loading of the item divided by the item standard deviation, the factor loadings being obtained from an ordinary factor analysis of the item intercorrelation matrix.     

The role of processes and contents in human memory: An item response theory approach

This study investigated the specificities and communalities of visuospatial and verbal memory contents, as well as short-term memory and working memory (STM and WM) processes with the use of Rasch models as an alternative to the previous studies based on the classical test theory. A sample of 547 undergraduate students executed four computerised tasks, each consisting of verbal-numeric WM, visuospatial WM, verbal-numeric STM, and visuospatial STM content. Confirmatory factor analyses indicated that visuospatial and verbal-numeric memory are distinct, but correlated variables. Findings also support a domain-general view of WM capacity distinct from domain-specific storage. With the use of Rasch models, our results confirm previous experimental, psychometric, and neuropsychological studies, highlighting that memory span tasks can be divided into separate subsystems for content and processes. This study also shows that better results are obtained when models with person parameter estimates (provided from Rasch models) are adopted, rather than summed raw scores.     

A comparison of the thurstone method of multiple factors with the hotelling method of principal components

The factors found by the Hotelling method of principal components present the same necessity for rotation as those found by the Thurstone method of multiple factors.The Thurstone method of multiple factors is here found superior to the Hotelling method of principal components, and is suggested as the method of choice for all problems requiring the determination of elemental components from a group of related variables, since it was found on every point of comparison that, when applied to a correlation matrix of unknown communalities, the Thurstone method was more accurate than the Hotelling method, and also required much less time.     

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.     

A set of inequalities in factor analysis

Inequalities relating the communalities to the multiple-correlation coefficients are derived. They are stronger than the well-known inequalities which have played an important role in factor analysis for the past thirty years. Necessary and sufficient conditions for equality are obtained.     

Resolution of the heywood case in the minres solution

In the course of developing the minres method of factor analysis the troublesome situation of communalities greater than one arose. This problem—referred to as the generalized Heywood case—is resolved in this paper by means of a process of minimizing the sum of squares of off-diagonal residuals. The resulting solution is superior to the otherwise very efficient original minres method without requiring additional computing time.Both authors were with the System Development Corporation when this work was done.     

Control of social desirability in personality inventories: Principal-factor deletion

Rational and statistical techniques for controlling social desirability (SD) in scales derived from self-report inventories are reviewed. The principal-factor deletion technique is explained in detail. It is applied during factor analysis when derived content scales are expected to be tainted with SD. If one principal factor is found to represent social desirability, the implicated factor is dropped and the communalities adjusted before the remaining factors are rotated to a desired simple structure criterion. Subscales derived from resulting factors are free of SD bias. Principal-factor deletion is compared with other approaches including rational techniques, covariate procedures, and target rotation.     

An improvement of the Harman-Fukuda-method for the minres solution in factor analysis

The method proposed by Harman and Fukuda to treat the so-called Heywood case in the minres method in factor analysis i.e., the case where the resulting communalities are greater than one, involves the frequent solution of eigenvalue problems. A simple method to treat this problem requiring less computing time and enjoying higher numerical stability is described in this paper.     

Separation of data as a principle in factor analysis

Two systems of factor analysis—factoring correlations with units in the diagonal cells and factoring correlations with communalities in the diagonal cells—are considered in relation to the commonly used statistical procedure of separating a set of data (scores) into two or more parts. It is shown that both systems of factor analysis imply the separation of the observed data into two orthogonal parts. The matrices used to achieve the separation differ for the two systems of factor analysis.     

Internal consistency of tests: Analyses old and new

A coefficient derived from communalities of test parts has been proposed as greatest lower bound to Guttman's immediate retest reliability. The communalities have at times been calculated from covariances between itemsets, which tends to underestimate appreciably. When items are experimentally independent, a consistent estimate of the greatest defensible internal-consistency coefficient is obtained by factoring item covariances. In samples of modest size, this analysis capitalizes on chance; an estimate subject to less upward bias is suggested. For estimating alternate-forms reliability, communality-based coefficients are less appropriate than stratified alpha.I thank Edward Haertel for comments and suggestions, and Andrew Comrey for data.