The simultaneous maximization of congruence for two or more matrices under orthogonal rotation

While a rotation procedure currently exists to maximize simultaneously Tucker's coefficient of congruence between corresponding factors of two factor matrices under orthogonal rotation of one factor matrix, only approximate solutions are known for the generalized case where two or more matrices are rotated. A generalization and modification of the existing rotation procedure to simultaneously maximize the congruence is described. An example using four data matrices, comparing the generalized congruence maximization procedure with alternative rotation procedures, is presented. The results show a marked improvement of the obtained congruence using the generalized congruence maximization procedure compared to other procedures, without a significant loss of success with respect to the least squares criterion. A computer program written by the author to perform the rotations is briefly discussed.     

A monotonically convergent algorithm for orthogonal congruence rotation

Brokken has proposed a method for orthogonal rotation of one matrix such that its columns have a maximal sum of congruences with the columns of a target matrix. This method employs an algorithm for which convergence from every starting point is not guaranteed. In the present paper, an iterative majorization algorithm is proposed which is guaranteed to converge from every starting point. Specifically, it is proven that the function value converges monotonically, and that the difference between subsequent iterates converges to zero. In addition to the better convergence properties, another advantage of the present algorithm over Brokken's one is that it is easier to program. The algorithms are compared on 80 simulated data sets, and it turned out that the new algorithm performed well in all cases, whereas Brokken's algorithm failed in almost half the cases. The derivation of the algorithm is given in full detail because it involves a series of inequalities that can be of use to derive similar algorithms in different contexts.This research has been made possible by a fellowship from the Royal Netherlands Academy of Arts and Sciences to the first author. The authors are obliged to Willem J. Heiser and Jos M. F. ten Berge for useful comments on an earlier version of this paper.     

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.     

The equivalence of two methods of parameter estimation for the rasch model

Two methods of estimating parameters in the Rasch model are compared. It is shown that estimates for a certain loglinear model for the score × item × response table are equivalent to the unconditional maximum likelihood estimates for the Rasch model.     

A general rotation criterion and its use in orthogonal rotation

Measures of test parsimony and factor parsimony are defined. Minimizing their weighted sum produces a general rotation criterion for either oblique or orthogonal rotation. The quartimax, varimax and equamax criteria are special cases of the expression. Two new criteria are developed. One of these, the parsimax criterion, apparently gives excellent results. It is argued that one of the most important factors bearing on the choice of a rotation criterion for a particular problem is the amount of information available on the number of factors that should be rotated. This research was supported by the National Research Council of Canada research grant 291-13 to Dr. G. A. Ferguson.     

A note on identification in the oblique and orthogonal factor analysis models

Conditions for removing the indeterminancy due to rotation are given for both the oblique and orthogonal factor analysis models. The conditions indicate why published counterexamples to conditions discussed by Jöreskog are not identifiable.The author would like to thank Gordon Bechtel and the reviewers for their comments and suggestions.     

Independent detectors for expansion and rotation, and for orthogonal components of deformation.

It is well known that optic flow--the smooth transformation of the retinal image experienced by a moving observer--contains valuable information about the three-dimensional layout of the environment. From psychophysical and neurophysiological experiments, specialised mechanisms responsive to components of optic flow (sometimes called complex motion) such as expansion and rotation have been inferred. However, it remains unclear (a) whether the visual system has mechanisms for processing the component of deformation and (b) whether there are multiple mechanisms that function independently from each other. Here, we investigate these issues using random-dot patterns and a forced-choice subthreshold summation technique. In experiment 1, we manipulated the size of a test region that was permitted to contain signal and found substantial spatial summation for signal components of translation, expansion, rotation, and deformation embedded in noise. In experiment 2, little or no summation was found for the superposition of orthogonal pairs of complex motion patterns (eg expansion and rotation), consistent with probability summation between pairs of independent detectors. Our results suggest that optic-flow components are detected by mechanisms that are specialised for particular patterns of complex motion.     

The orthogonal approximation of an oblique structure in factor analysis

A procedure is derived for obtaining an orthogonal transformation which most nearly transforms one given matrix into another given matrix, according to some least-squares criterion of fit. From this procedure, three analytic methods are derived for obtaining an orthogonal factor matrix which closely approximates a given oblique factor matrix. The case is considered of approximating a specified subset of oblique vectors by orthogonal vectors.Part of this research was carried out while the author was a psychometric fellow at the Educational Testing Service, Princeton, New Jersey.     

A technique for facilitating the rotation of factor axes,based on an equivalence between persons and tests

A technique is outlined which may facilitate the rotation of factor axes to a meaningful position. It is based on certain relationships between the results of test and person factor analysis, and consists essentially of supplementing the test factor space with tests which are thetest-equivalents of persons or groups of persons. These persons may be, for instance, well-known types in the domain being investigated, or even freaks. The ways in which these persons may be selected and used to determine the final rotated position of the factor axes is discussed.     

On the equivalence of superordinate concepts

Psychological studies of superordinates have generally treated them as equivalent. However, many languages distinguish mass superordinates (e.g., clothing) from count superordinates (e.g., vehicle). In the present paper, experimental evidence is presented which suggests that the two types of superordinates are conceptually distinct as well. One study showed that the members of mass superordinates more often co-occur. A second study showed that people more often interact with the members of mass superordinates in temporal proximity whereas people primarily interact with single members of count superordinates on a specific occasion. Also, properties that characterize an individual are a more salient aspect of count superordinates. These findings imply that mass superordinates refer to unindividuated groups of objects, united by spatial and functional contiguity. Two other studies supported this hypothesis by showing that the class inclusion relation between a single object and a category is stronger for count superordinates. Taken together, the findings suggest that mass superordinates are not true taxonomic categories. We relate the findings to previous views of superordinates and to the count/mass distinction in general.     

A simple graphical method for orthogonal rotation of axes

A simple method of making orthogonal rotations by projecting coordinates from one plot to another is presented. An easily constructed apparatus for rapid and accurate work is described in detail. By the technique proposed 12 to 15 rotations of pairs of axes involving 20 test variables can be completed per hour by a practiced technician.     

Functional equivalence of the two sides of the human tongue

Taste sensitivity to sucrose and NaCl of the left and right side of the human tongue was compared. The results did not show a significant difference when the data of the whole experimental group were taken together. Analysis of the individual results showed that there was one exception: A female subject showed a quality-specific difference between left and right. A speculative explanation is offered for this result.     

On the rectilinear prediction of oblique factors

The general problem of estimating correlated or uncorrelated factors is treated. It is specifically indicated wherein the prediction of oblique factors differs from that of orthogonal factors. A shortened method of estimation of correlated factors is developed.     

Factors influencing two types of congruence in multirater judgments

In two studies of 360° judgments, it was shown that congruence measured in terms of intercorrelations between ratings by different people (congruence‐r) and in terms of the discrepancy between those ratings (congruence‐d) were empirically independent of each other. Congruence‐r was greatest, and congruence‐d was smallest, between judgments made by a target person and his or her supervisor, rather than by subordinates or peers; the differential observability of rated behaviors predicted congruence‐r but not congruence‐d. Variations in both forms of congruence were found to arise from the source of a judgment, and from a target person's age, gender, cognitive ability, and certain personality attributes. Over‐ and under‐rating associated with a personality or another third factor was shown to derive from substantially different correlations between such a factor and the two separate judgments under consideration.     

A further comparison of oblique factor transformation methods

Four prominent oblique transformation techniques—promax, the Harris-Kaiser procedure, biquartimin, and direct oblimin—are examined and compared. Additionally, two newly-developed procedures, falling into the category designated as Case III by Harris and Kaiser [1964], are presented and included in the comparisons. The techniques are compared in light of their freedom from bias in the interfactor correlations, and their ability to yield clear simple structures, over many data sets—some constructed and some real—varying widely in terms of number of variables and factors, factorial complexity, and clarity of the hyperplanes. Results are discussed, and implications for practice are noted.