Henry F. Kaiser

The Application of Electronic Computers to Factor Analysis

more stodgy and less exciting application of computers to psychological problems. Let me warn you about how I am going to talk today. I have not conducted a survey of available computer programs for factor analytic computations, nor have I done an analysis of the problems of the application of computers to factor analysis in any way that could be considered scientific. I am saying that I shall ask you to listen to my opinions about the applications of computers to factor

The varimax criterion for analytic rotation in factor analysis

An analytic criterion for rotation is defined. The scientific advantage of analytic criteria over subjective (graphical) rotational procedures is discussed. Carrolls criterion and the quartimax criterion are briefly reviewed; the varimax criterion is outlined in detail and contrasted both logically and numerically with the quartimax criterion. It is shown that thenormal varimax solution probably coincides closely to the application of the principle of simple structure. However, it is proposed that the ultimate criterion of a rotational procedure is factorial invariance, not simple structure—although the two notions appear to be highly related. The normal varimax criterion is shown to be a two-dimensional generalization of the classic Spearman case, i.e., it shows perfect factorial invariance for two pure clusters. An example is given of the invariance of a normal varimax solution for more than two factors. The oblique normal varimax criterion is stated. A computational outline for the orthogonal normal varimax is appended.

Computer Program for Varimax Rotation in Factor Analysis

THE purpose of this report is to outline an electronic computer program for varimax rotation in factor analysis [2]. We shall break up a varimax program into small sections and indicate in some detail the function each section is to perform, scaling problems within the section, possibly useful gimmicks, etc., without presuming to write down individual instructions. Given an arbitrary reference factor matrix (e.g., principal axes, centroids) with n tests and r factors, the varimax criterion requires that we make orthogonal rotations on this matrix such that

Sample and population score matrices and sample correlation matrices from an arbitrary population correlation matrix

A method for generating sample and population score matrices and sample correlation matrices from a given population correlation matrix is developed. An example giving the desired matrices for a population Guttman simplex correlation matrix is presented.

Formulas for component scores

General formulas for obtaining scores for individuals on components (factors derived from correlation matrices with unit communality estimates) are given. They are specialized to give Hotellings formula for principal component scores. Formulas for scores on components rotated from principal components are developed.

Scaling a simplex

A least-squares solution for scaling the variables of a Guttman simplex is developed. The procedure yields a ratio scale, two varieties of interval scale, and orders the variables. A measure of the goodness of fit of the scale to the data is suggested. An example of the application of the method is given. The problem of non-positive correlations is discussed.

A note on the Tryon-Kaiser solution for the communalities

The Tryon-Kaiser solution for the communalities is reviewed. Numerical investigation suggests that the procedure is applicable if and only if the correlation matrix has unique minimum rank communalities. This implies that this approach to the communality problem is not general enough to be of practical use.

Abacs for determination of a correlation coefficient corrected for restriction of range

Abacs approximating the product-moment correlation for both explicit and implicit selection are presented. These abacs give accuracy to within .01 of the corresponding analytic estimate.

Comparison of Factored Dimensions of Peer and Superior Ratings on a Situational-Type Rating Scale of Intellectual Aptitude

Perceptual ( P F ) The ability to explore visually possible courses of action in Foresight order to select the most effective one. Conceptual (CF) The ability to anticipate the needs or the consequences of a Foresight given problem situation. Penetration (Pe) The ability to see beyond the immediate and obvious. Experiential (EE) The ability to appraise aspects of common situations in terms Evaluation of agreement with experience. Sensitivity (SP) The abiliry to recognize practical problems. to Problems Adaptive ( A X ) The ability to change set in order to meet new requirements Flexibility imposed by changing problems. Spontaneous (SF) The ability to produce a diversity of ideas in a situation that Flexibility is relatively unrestricted. General (GR) The ability to comprehend or structure problems in preparaReasoning tion for solving them. Verbal ( V C ) Knowledge of words and their meanings. Comprehension Originality ( O r ) The ability to produce remotely associated, clever, or uncommon responses. Ideational ( I F ) The ability to call up many ideas in a situation in which Fluency there is little restriction and quality does nor count. Leadership (LR) Criterion variable within the rating scale ( 4 items). Rating

Program for Inverting a Gramian Matrix

A COMMON application of electronic computers is solving simultaneous linear equations. Fundamentally, the solution involves inverting the (square) matrix of coefficients of the unknowns. In the vast majority of uses in educational and psychological measurement, this matrix is Gramian-or real, symmetric, positive definite. (A Gramian matrix is often defined as real, symmetric, non-negative definite; however, for our problem the matrix of course must be nonsingular, or positive definite.) Examples of such matrices are the coe5cients in the normal equations for regression coefficients, in the normal equations for least-squares curve fitting, and, most generally, the matrix of coefficients for the unknowns in any statistical problem involving the general linear hypothesis, e.g., analysis of variance and covariance. It is not surprising, therefore, that the problem of inverting Gramian matrices has received major sttention in standard works on linear computations. It is not the purpose of this paper to contribute to the theory of inverting Gramian matrices; rather, we shall undertake the more modest task of outlining in some detail a particular computer program for the problem. It is hoped that the following discussion will relieve a computer programmer of puzzling over the general question of matrix inversion so that he may-in a relatively routine fashion c o d e an e5cient matrix inversion program in short order.