Paul S. Dwyer

The determination of the factor loadings of a given test from the known factor loadings of other tests

A technique is indicated by which approximations to the factor loadings of a new test may be obtained if factor loadings of a given group of tests and the correlations of the new test with the other tests are known. The technique is applicable to any orthogonal system and is especially adapted to cases in which Σajiajk = 0 wheni ≠k. Application is also made to the simultaneous determination of the factor weights of a group of tests in which no additional common factor is present. The technique is useful in adding tests to a completed factorial solution and in using factorial solutions involving errors to give results which are approximately correct.

Some Applications of Matrix Derivatives in Multivariate Analysis

Abstract It is claimed that the reasons for using matrices of derivatives, in appropriate situations, are as compelling as those for using matrices. This paper provides basic material for such use. Different types of matrix derivatives are defined and illustrated. Simple and easy techniques are then derived and are shown to be applicable to a considerable collection of matrix functions. Applications are made to such problems as establishing matrix integrals from scalar ones, determining maximum likelihood estimates for complex likelihood functions, optimizing matrix functions when there are matrices of side conditions, and evaluating the Jacobians of certain classes of transformations. The emphasis is on simplicity of derivation and on breadth of application.

Basic Instructions in Statistical Computations

The varied instructional programs of a large university Training in the various programs at Michigan Computational equipment at Michigan The nature of statistical computations The use of digital machines The control of computational error The use of direct or iterative methods The accumulation of inherent error General remarks with reference to a suitable program of basic instruction in statistical computations

Multivariate Maxima and Minima with Matrix Derivatives

Abstract The purpose of this paper is the presentation of formulae for obtaining matrix derivatives of the second order to use in making tests for maxima and minima. The theory of such second order derivatives is presented. These formulae require the rearrangment of the parameter elements in vector form and the transformed results feature Kronecker products, which have certain desirable properties. Application is made to several types of problems.

SOLUTION OF THE PERSONNEL CLASSIFICATION PROBLEM WITH THE METHOD OF OPTIMAL REGIONS

The personnel classification problem is identified mathematically with other problems in the social and biological sciences. This mathematical problem is shown to be a special case of the general mathematical problem of linear programming. It is proposed here that the personnel classification problem may be solved directly by methods particularly appropriate to it as well as by the simplex method, which is a standard method for solving the general linear programming problem. The method of optimal regions is derived and illustrated in this paper.

The evaluation of multiple and partial correlation coefficients from the factorial matrix

This paper shows how to compute multiple correlation coefficients, partial correlation coefficients, and regression coefficients from the factorial matrix. Special emphasis is given to computation technique and to approximation formulas. The method is extremely flexible in application since it may be applied to any subset of the original set of observed variables. It is also extremely useful when many of these coefficients are desired.

The contribution of an orthogonal multiple factor solution to multiple correlation

A method is indicated by which multiple factor analysis may be used in determining a number,r, and then in selectingr “predicting” variables out ofn variables so that each of the remainingn-r variables may be predicted almost as well from ther variables as it could be predicted from all then—1 variables.

The solution of simultaneous equations

This paper is an attempt to integrate the various methods which have been developed for the numerical solution of simultaneous linear equations. It is demonstrated that many of the common methods, including the Doolittle Method, are variations of the method of “single division.” The most useful variation of this method, in case symmetry is present, appears to be the Abbreviated Doolittle method. The method of multiplication and subtraction likewise can be abbreviated in various ways of which the most satisfactory form appears to be the new Compact method. These methods are then applied to such problems as the solution of related equations, the solution of groups of equations, and the evaluation of the inverse of a matrix.

The detailed method of optimal regions

The detailed method of optimal regions is an extended form of the method of optimal regions which has been found effective in solving the personnel classification problem when the number of job categories is small. The automatic determination of the successive values of thevi, made possible by the more exact techniques of the detailed method, provide easier solutions for the more complex problems and provide solutions, which, for the most part, can be mechanized. In a sense the detailed method of optimal regions is more than a detailed form of the method of optimal regions. It is essentially a method of transformations by which the original matrix is reduced to a matrix from which the solution is easily obtained.

Computation with Multiple k-Statistics

Abstract This paper discusses the computation of multiple k-statistics and provides techniques for seminvariant k … through weight 8. After a review of pertinent background material, specific techniques for calculation through weight 4 are presented for desk calculation with application to moment formulae involving the mean and variance. Then the various methods of calculating seminvariant k … through weight 8 are examined, Basic formulae needed for the recommended methods are presented in tabular form. The general problem, of moments and estimates of moments of k-statistics is then treated. A machine program is available through weight 8. Specific references are given throughout so the reader who is unfamiliar with this material may examine earlier results.