Further Application of the Principles of Direct Rotation in Factor Analysis

Abstract

Essentially, factor analysis presents a statistical method of summarizing a matrix of intercorrelations in such a way as to describe each of the variables in terms of a limited num ber of assumed factors. An important end product of the analysis, therefore, is a set of linear equations describing each of the variables in terms of the assumed factors. This set of equations, which may be regarded as multiple regression equations, will be called a factor pattern, following Holzinger. 1 The analysis of a correlation matrix, with communalities in the diagonals, may be made in terms of either correlated or uncorrelated common factors. The latter solution is called an orthogonal one, and is illustrated by such well-known types of solution as the bi-factor, the centroid, and the principal-factor solution. A solution in terms of correlated common factors is called an ob lique solution. It is this type of solution that Thurstone emphasizes in his recent text. 2 The procedure for computing an oblique sol ution that has been followed most commonly is that of first calculating an orthogonal solution, such as the centroid, and then rotating this in itial orthogonal solution to the desired oblique solution. In such a procedure the calculation of the initial orthogonal solution serves prim arily as a method of estimating the number of common factors, or minimum rank of the given correlation matrix, R, and of determining the communalities. Any initial solution that yields the same number of common factors and the same communalities would serve equally well as this intermediate stage in developing the de sired oblique solution. It, therefore, is evident that the initial orthogonal solution might be re placed by an initial oblique solution in this pro cedure. This opens the possibility of combin ing certain earlier developments in factor meth od to yield a modified procedure for arriving at the desired oblique solution. Holzinger has shown explicitly that the de sired oblique solution with m common factors may be calculated quite simply if the variables whose correlations make up the matrix can be grouped into exactly m distinct clusters. 3 This method is essentially one of sectioning the cor relation matrix into groups of variables of ap proximate unit rank, and then passing axes through these groups, or clusters. The com putation provides for determining the correla tion of each variable with each of these axes (the structure matrix, S) and the intercorrela tions, , of the axes or factors. 4 As Holzinger shows, the oblique pattern, P, is given by: