Non-Graphical Solutions for Cattell’s Scree Test

Abstract

Most of the strategies that have been proposed to determine the number of components that account for the most variation in a principal components analysis of a correlation matrix rely on the analysis of the eigenvalues and on numerical solutions. The Cattell's scree test is a graphical strategy with a nonnumerical solution to determine the number of components to retain. Like Kaiser's rule, this test is one of the most frequently used strategies for determining the number of components to retain. However, the graphical nature of the scree test does not definitively establish the number of components to retain. To circumvent this issue, some numerical solutions are proposed, one in the spirit of Cattell's work and dealing with the scree part of the eigenvalues plot, and one focusing on the elbow part of this plot. A simulation study compares the efficiency of these solutions to those of other previously proposed methods. Extensions to factor analysis are possible and may be particularly useful with many low-dimensional components. Several strategies have been proposed to determine the num- ber of components that account for the most variation in a principal components analysis of a correlation matrix. Most of these rely on the analysis of the eigenvalues of the corre- lation matrix and on numerical solutions. For example, Kaiser's eigenvalue greater than one rule (Guttman, 1954; Kaiser, 1960), parallel analysis (Buja & Eyuboglu, 1992; Horn, 1965; Hoyle & Duvall, 2004), or hypothesis signifi- cance tests, like Bartlett's test (1950), make use of numerical criteria for comparison or statistical significance criteria. Independently of these numerical solutions, Cattell (1966) proposed the scree test, a graphical strategy to determine the number of components to retain. Along with the Kaiser's rule, the scree test is probably the most used strategy and it is included in almost all statistical software dealing with principal components analysis. Unfortunately, it is generally recognized that the graphical nature of the Cattell's scree test does not enable clear decision-making about the number of components to retain. The previously proposed non-graphical solutions for