Multiple rectilinear prediction and the resolution into components

It is assumed that a battery ofn tests has been resolved into components in a common factor space ofr dimensions and a unique factor space of at mostn dimensions, wherer is much less thann. Simplified formulas for ordinary multiple and partial correlation of tests are derived directly in terms of the components. The best (in the sense of least squares) linear regression equations for predicting factor scores from test scores are derived also in terms of the components. Spearman's single factor prediction formulas emerge as special cases. The last part of the paper shows how the communality is an upper bound for multiple correlation. A necessary and sufficient condition is established for the square of the multiple correlation coefficient of testj on the remainingn—1 tests to approach the communality of testj as a limit asn increases indefinitely whiler remains constant. Limits are established for partial correlation and regression coefficients and for the prediction of factor scores.I am indebted to Professor Dunham Jackson for helpful criticism of most of this paper.     

Multiple rectilinear prediction and the resolution into components: II

Given a battery ofn tests that has already been resolved intor orthogonal common factors andn unique factors, procedures are outlined for computing the following types of linear multiple regressions directly from the factor loadings: (i) the regression of any one test on then?1 remaining tests; (ii) all then different regressions of ordern?1 for then tests, computed simultaneously; (iii) the regression of any common factor on then tests; (iv) the regressions of all the common factors on then tests computed simultaneously; (v) the regression of any unique factor on then tests; (vi) the regressions of all the unique factors on then tests, computed simultaneously. Multiple and partial correlations are then determined by ordinary formulas from the regression coefficients. A worksheet with explicit instructions is provided, with a completely worked out example. Computing these regressions directly from the factor loadings is a labor-saving device, the efficiency of which increases as the number of tests increases. The amount of labor depends essentially on the number of common factors. This is in contrast to computations based on the original test intercorrelations, where the amount of labor increases more than proportionately as the number of tests increases. The procedures evaluate formulas developed in a previous paper (2). They are based essentially on a shortened way of computing the inverse of the test intercorrelation matrix by use of the factor loadings.     

On oblique procrustes rotation

The equations involved in the rotation of an arbitrary factor matrix to a least squares fit to a specified factor structure have been known for many years. These equations, in general, cannot be solved by purely algebraic means, and an approximate solution has previously been used in practical applications.In this paper an effective iterative method for obtaining the exact solution is developed. By algebraic manipulation the set of equations is expressed in the form of one polynomial equation in one unknown. Newton's method is suggested for solving this equation. Practical applications of the procedure indicate that convergence within small tolerance limits is generally attained after few iterations.Part of this research was carried out at the National Institute for Personnel Research (South Africa). It was completed while the author was a Visiting Research Psychologist at Educational Testing Service.     

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.     

Constrained least squares estimators of oblique common factors

An expression is given for weighted least squares estimators of oblique common factors, constrained to have the same covariance matrix as the factors they estimate. It is shown that if as in exploratory factor analysis, the common factors are obtained by oblique transformation from the Lawley-Rao basis, the constrained estimators are given by the same transformation. Finally a proof of uniqueness is given.The research reported in this paper was partly supported by Natural Sciences and Engineering Research Council Grant No. A6346.     

On browne's solution for oblique procrustes rotation

Browne [1967] has given a method of solving the problem (originally stated by Mosier, [1939]) of finding a least squares fit to a specified factor structure. The problem is one of minimizing the sum of squared residuals of —FT with Diag (T'T)=I. Browne's solution involves the eigenvectors and values ofF'F and leads to an iterative solution.This paper gives a form of the solution which does not involve solution of an eigenvalue problem but does require an iteration similar to Browne's. It suggests the possible existence of a singularity, and a simple modification of Browne's computational procedure is proposed which deals with this case. A better starting value for the iteration is also proposed for which convergence is guaranteed using the ordinary Newton iteration.Part of this work was presented at the April 1969 meetings of the Psychometric Society. The anthor is indebted to Dr. Ledyard Tucker for some helpful discussions. This work was supported in part by a PHS Research Grant No. MH-10006 from the National Institute of Mental Health, and Grant No. GM-12868 from the Institute of General Medical Sciences, Public Health Service.     

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.     

On the equivalence of two oblique congruence rotation methods,and orthogonal approximations

Tucker's method of oblique congruence rotation is shown to be equivalent to a procedure by Meredith. This implies that Monte Carlo studies on congruence by Nesselroade, Baltes and Labouvie and by Korth and Tucker are highly comparable. The problem of rotating two matrices orthogonally to maximal congruence has not yet been solved. An approximate solution to this problem can be derived from Tucker's method. Even better results can be obtained from a Procrustes rotation followed by rotation to simple structure.     

Big five factors and facets and the prediction of behavior

The authors compared the Big 5 factors of personality with the facets or traits of personality that constitute those factors on their ability to predict 40 behavior criteria. Both the broad factors and the narrow facets predicted substantial numbers of criteria, but the latter did noticeably better in that regard, even when the number of facet predictors was limited to the number of factor predictors. Moreover, the criterion variance accounted for by the personality facets often included large portions not predicted by the personality factors. The narrow facets, therefore, were able to substantially increase the maximum prediction achieved by the broad factors. The results of this study are interpreted as supporting a more detailed approach to personality assessment, one that goes beyond the measurement of the Big 5 factors alone.     

On the prediction of confusion matrices from similarity judgments

Three observers viewed visual representations of eight complex sounds in both a pairwise similarity-judgment task and an identification task. A multidimensional scaling procedure applied to the similarity judgments yielded a three-dimensional perceptual space and the relative positions of the stimuli in that space. A probabilistic decision model based on weighted interstimulus distances served to predict well the confusion matrices of the identification task. Three conditions of the identification task, calling for identification of different subsets of the eight stimuli, led the observers to vary the weights they placed on the dimensions; they apparently adjusted the weights to maximize the percent correct identification. An additional group of 14 subjects, participating only in the similarity-judgment task, manifested the same three dimensions as the observers (corresponding to the locus of low-frequency energy, the locus of midfrequency energy, and visual contrast), and also a fourth dimension (corresponding to the periodicity, or waxing and waning, of the sound). Although not evident in the scaling analysis for the three observers, our utilization of the additional dimension increased significantly the variance accounted for in their identification responses. The overall accuracy of the predictions from a perceptual space to identification responses supplies a substantial validation of the use of multidimensional scaling procedures to reveal perceptual structure in demonstrating the ability of that structure to account for behavior in an independent task. The empirical success of this approach, furthermore, suggests a relatively simple and practical means of predicting, and possibly enhancing, identification performance for a given set of visual or auditory stimuli.     

Mediatory cue proximity and mental subnormality in the perception of rectilinear dot progressions

This study investigated the relationship between illusory displacement of sequentially presented light stimuli and their proximity to mediatory cues. Twenty-four retardates, 24 equal CA normals, and 24 equal MA normals were tested to determine the relative stabilizing effects of four different frame conditions on these groups. Illusory displacement decreased from least to most proximate cue condition, was least for the MA group, and was most for retardates It was concluded that interstimulus mediation referents, rather than restricted eye movement, facilitated veridical perception of successive light positions. nt]mis|The author extends sincere appreciation to Dr. Ralph Osborne, superintendent of the Hamilton Township school system, Mr. George Murphy, principal of Hamilton High School West, and Mrs. Isabel Robinson, principal of Kuser School, for their generous cooperation in providing students and testing space for this experiment.     

Sources of optical information useful for perception of speed of rectilinear self-motion

Magnitude judgments of the speed of self-motion were examined. The principal independent variables were edge rate, global optical flow rate, and the type of texture (grid or dot). Results indicated that edge rate and global optical flow rate had additive effects on magnitude judgments, with edge rate accounting for a larger portion of the variance. Effects were independent of texture type. Secondary variables examined were viewing condition and task load. Attempts were made to control the availability of flatness cues. Evidence indicates that the effectiveness of global optical flow rate varied with the control of flatness cues. A secondary running auditory Sternberg task was used to prevent edge counting; the presence of this task did not reduce the effect of edge rate. These results replicate and extend previous work by D. H. Owen and colleagues.     

Assessment of the school readiness section of the early detection inventory: Preschool prediction across situational factors

The efficiency of (a) a school readiness test, (b) mental age, and (c) specific sensory and perceptual skills was tested as predictors of word recognition at the end of the first school year. Five hundred and eigth children in four schools with three significantly different socioeconomic backgrounds were tested in kindergarten. Two schools were in the same middle-class area: one was a traditional classroom school, the other was open space in structure. When the variables were examined by means of stepwise regression analysis and partial correlations, the school readiness test appeared as the most efficient predictor in each of the four schools, as well as in the total group. Mental age and auditory skills ranked either second or third in importance depending on the school situation.     

A comparison of graphic and analytic solutions for both oblique and orthogonal simple structures for factors of employee morale

Solutions for simple structure obtained from empirical data through the use of the Oblimax, Quartimax, and Varimax procedures are matched with independently determined graphic solutions for the same data. The relative merits of the two analytic orthogonal solutions (Quartimax and Varimax) are discussed. Factors of employee morale are interpreted.     

On Green's best linear composites with a specified structure,and oblique estimates of factor scores

Green solved the problem of least-squares estimation of several criteria subject to the constraint that the estimates have an arbitrary fixed covariance or correlation matrix. In the present paper an omission in Green's proof is discussed and resolved. Furthermore, it is shown that some recently published solutions for estimating oblique factor scores are special cases of Green's solution for the case of fixed covariance matrices.