New lower and upper bounds for communality in factor analysis

We derive several relationships between communalities and the eigenvalues for ap ×p correlation matrix under the usual factor analysis model. For suitable choices ofj, _j (), where _j () is thej-th largest eigenvalue of , provides either a lower or an upper bound to the communalities for some of the variables. We show that for at least one variable, 1 - _p () improves on the use of squared mulitiple correlation coefficient as a lower bound.This research was done while the second author was at Tokyo Institute of Technology.     

A new rotational method in factor analysis

A new rotational method, particularly appropriate when a positive manifold can be assumed, is presented in theory and in computational detail. It is applied to a five-dimensional factorial matrix in which a simple configuration is known to exist. The new technique is much more efficient in yielding the simple configuration than are the methods formerly used. Landahl's transformations can be combined with this method to advantage.     

Some properties of the communality in multiple factor theory

Several theorems concerning properties of the communaltiy of a test in the Thurstone multiple factor theory are established. The following theorems are applicable to a battery ofn tests which are describable in terms ofr common factors, with orthogonal reference vectors.1. The communality of a testj is equal to the square of the multiple correlation of testj with ther reference vectors.2. The communality of a testj is equal to the square of the multiple correlation of testj with ther reference vectors and then—1 remaining tests. Corollary: The square of the multiple correlation of a testj with then—1 remaining tests is equal to or less than the communality of testj. It cannot exceed the communality.3. The square of the multiple correlation of a testj with then—1 remaining tests equals the communality of testj if the group of tests containsr statistically independent ests teach with a communality of unity.4. With correlation coefficients corrected for attenuation, when the number of tests increases indefinitely while the rank of the correlational matrix remains unchanged, the communality of a testj equals the square of the multiple correlation of testj with then—1 remaining tests.5. With raw correlation coefficients, it is shown in a special case that the square of the multiple correlation of a testj with then—1 remaining tests approaches the communality of testj as a limit when the number of tests increases indefinitely while the rank of correlational matrix remains the same. This has not yet been proved for the general case.The author wishes to express his appreciation of the encouragement and assistance given him by Dr. L. L. Thurstone.     

A new algorithm for the least-squares solution in factor analysis

A new algorithm to obtain the least-squares or MINRES solution in common factor analysis is presented. It is based on the up-and-down Marquardt algorithm developed by the present authors for a general nonlinear least-squares problem. Experiments with some numerical models and some empirical data sets showed that the algorithm worked nicely and that SMC (Squared Multiple Correlation) performed best among four sets of initial values for common variances but that the solution might sometimes be very sensitive to fluctuations in the sample covariance matrix.Numerical computation was made on a NEAC S-1000 computer in the Computer Center, Osaka University.     

A simple method of factor analysis

A simple method for extracting correlated factors simultaneously is described. The method is based on the idea that the centroid pattern coefficients for the sections of unit rank of the complete matrix may be interpreted as structure values for the entire matrix. Only the routine centroid average process is required.     

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 comparison of the LISREL and EQS programs for obtaining parameter estimates in confirmatory factor analysis studies

Recently, researchers in psychology have been provided with the techniques for confirming factor analytic structures. Many software packages are now available that provide the researcher with the ability to test factor structures and measurement models with some precision. However, the literature fails to provide the researcher with any comparison of the available software. This paper reviews the basic premise of confirmatory factor analysis and describes the use of two of the more popular software packages (e.g., LISREL and EQS) for obtaining parameter estimates. The two programs are then compared for similarity in estimates and ease of use.     

NFACOM: A new program for relating solutions in exploratory factor analysis

A new version of the program FACOM (comparison of exploratory factor analysis solutions) is presented. The new version (NFACOM) greatly extends the possibilities of the previous versions. It is now possible to use congruence as well as least squares comparisons in both orthogonal and oblique situations. What is more, n solutions, not just two, can be compared. Recently developed algorithms have been implemented; others, which have proved not to be theoretically well founded, have been omitted.     

A new estimator of the uniqueness in factor analysis

A closed form estimator of the uniqueness (unique variance) in factor analysis is proposed. It has analytically desirable properties—consistency, asymptotic normality and scale invariance. The estimation procedure is given through the application to the two sets of Emmett's data and Holzinger and Swineford's data. The new estimator is shown to lead to values rather close to the maximum likelihood estimator.     

A method for obtaining psychophysical estimates of movement costs

Although researchers generally accept the proposition that movement costs are taken into account in the planning of voluntary movements, there is no established psychophysical method for estimating such costs. The authors tested and introduce a possible method. Participants were given every possible pair of tasks from a set of tasks that varied along one or more dimensions. In each trial, they indicated which task was easier to perform. The authors used participants' probability of preferring any given task over others to estimate the judged relative cost (JRC) of that task, and they used the JRCs of all the tasks to quantify the relation between task preference and task properties. The method was applied in 3 experiments in which university students (Experiment 1, N = 20; Experiment 2, N = 40; Experiment 3, N = 20) chose which spatial target to move to in carrying an object from one position to another. The data were consistent with the hypothesis that the JRC of moving an object increases with the degree to which the object must be translated and rotated. The simplicity of the method encourages its application to a wide range of questions about motor behavior.     

An empirical verification of the Wherry-Gaylord iterative factor analysis procedure

A comparison of the Wherry-Gaylord iterative factor analysis procedure and the Thurstone multiple-group analysis of sub-tests shows that the two methods result in the same factors. The Wherry-Gaylord method has the advantage of giving factor loadings for items. The number of iterations needed can be reduced by doing a factor analysis of sub-tests, re-grouping sub-tests according to factors, and using each group as a starting point for iterations.This research was carried out under Contract No. WSW-2503, between the Department of the Army and Ohio State University. This paper is based on the final report PRS No. 827 under that contract. The opinions expressed herein regarding matters relating to the Department of the Army are those of the authors and are not necessarily official.     

A feasible method for standard errors of estimate in maximum likelihood factor analysis

A jackknife-like procedure is developed for producing standard errors of estimate in maximum likelihood factor analysis. Unlike earlier methods based on information theory, the procedure developed is computationally feasible on larger problems. Unlike earlier methods based on the jackknife, the present procedure is not plagued by the factor alignment problem, the Heywood case problem, or the necessity to jackknife by groups. Standard errors may be produced for rotated and unrotated loading estimates using either orthogonal or oblique rotation as well as for estimates of unique factor variances and common factor correlations. The total cost for larger problems is a small multiple of the square of the number of variables times the number of observations used in the analysis. Examples are given to demonstrate the feasibility of the method.The research done by R. I. Jennrich was supported in part by NSF Grant MCS 77-02121. The research done by D. B. Clarkson was supported in part by NSERC Grant A3109.     

A new dynamic systems method for the analysis of early socioemotional development

This study reports on a new dynamic systems method for studying infant socioemotional development, using conventional statistical techniques to portray dynamic systems constructs. State space grids were constructed from two ordinal variables, distress intensity and attention to mother, and hypothetical attractors were identified as grid cells with high cumulative duration of behavior. Attractor and state space characteristics were operationalized and tested, first to assess the utility of the method and second to reconceptualize and extend conventional developmental hypotheses. The basin strength and relaxation time of hypothetical attractors demonstrated their ‘attractiveness’ and predicted consistency in attractor location across sessions. Developmental changes and individual continuities in the organization of behavior were also revealed, in ways that would be inaccessible to conventional research methods.     

A new method for solving simultaneous linear equations associated with multivariate analysis

A method is outlined whereby the equations resulting from multivariate analyses may be solved. Only the upper or lower triangular coefficient matrix need be employed. All solutions involving any combination(s) of the variables represented in the coefficient matrix are readily obtainable without permutation of the matrix. All solutions are given immediately without the need for back solving. One variable is added to or subtracted from the regression equation with each application of the method. The method may be applied as repeatedly as needed for any particular solution.     

More efficient parameter estimates for factor analysis of ordinal variables by ridge generalized least squares

Data in psychology are often collected using Likert‐type scales, and it has been shown that factor analysis of Likert‐type data is better performed on the polychoric correlation matrix than on the product‐moment covariance matrix, especially when the distributions of the observed variables are skewed. In theory, factor analysis of the polychoric correlation matrix is best conducted using generalized least squares with an asymptotically correct weight matrix (AGLS). However, simulation studies showed that both least squares (LS) and diagonally weighted least squares (DWLS) perform better than AGLS, and thus LS or DWLS is routinely used in practice. In either LS or DWLS, the associations among the polychoric correlation coefficients are completely ignored. To mend such a gap between statistical theory and empirical work, this paper proposes new methods, called ridge GLS, for factor analysis of ordinal data. Monte Carlo results show that, for a wide range of sample sizes, ridge GLS methods yield uniformly more accurate parameter estimates than existing methods (LS, DWLS, AGLS). A real‐data example indicates that estimates by ridge GLS are 9–20% more efficient than those by existing methods. Rescaled and adjusted test statistics as well as sandwich‐type standard errors following the ridge GLS methods also perform reasonably well.     

A generalized simplex for factor analysis

By a simplex is meant a set of statistical variables whose interrleations reveal a simple order pattern. For the case of quantitative variables, an order model was analyzed previously which allowed only for positive correlations among the variables and a limited type of gradient among the correlation coefficients. The present paper analyzes a more general model and shows how it is more appropriate to empirical data. Among the novel features emerging from the analysis are: (a) the factoring implied of the correlation matrix; (b) the use of a non-Euclidean distance function; and (c) the possible underlying psychological theories.Read at the International Congress of Psychology, Montreal, June 7–12, 1954. This research was facilitated in part by an uncommitted grant-in-aid to the writer from the Behavioral Sciences Division of the Ford Foundation.     

An inter-battery method of factor analysis

The inter-battery method of factor analysis was devised to provide information relevant to the stability of factors over different selections of tests. Two batteries of tests, postulated to depend on the same common factors, but not parallel tests, are given to one sample of individuals. Factors are determined from the correlation of the tests in one battery with the tests in the other battery. These factors are only those that are common to the two batteries. No communality estimates are required. A statistical test is provided for judging the minimum number of factors involved. Rotation of axes is carried out independently for the two batteries. A final step provides the correlation between factors determined by scores on the tests in the two batteries. The correlations between corresponding factors are taken as factor reliability coefficients.This research was jointly supported by Princeton University and the Office of Naval Research under contract N6onr-270-20 and the National Science Foundation under grant NSF G-642; Harold Gulliksen, principal investigator. The preparation of this paper and the accompanying material has been aided by the Educational Testing Service. The author is grateful to Professors Harold Gulliksen and Samuel S. Wilks for their many most helpful comments and suggestions.