The first principal component as a basis for examining the unidimensionality of a set of variables

Abstract

A method of generating structural parameters using a second-order function of the first principal component, and termed PC('2), was compared with maximum likelihood factor analysis and principal components analysis in a Monte Carlo study of 4200 one- and two-dimensional simulated correlation matrices. The study investigated three types of factor structure, three levels of reliability, and three ranges of factor loadings in four- and eight-variable matrices. The levels of factor structure that were included were the unidimensional case, the two-factor case where the first factor accounted for three times the variance of the second factor, and the two-factor case in which both factors were of approximately equal influence. Correlation matrices were created using a high range, a low range, and a wide range of fact loadings. The reliabilities of the variables in the matrices were held constant at either a high (.90), moderate (.80), or low (.60) level. The low level of reliability was nested under the low range of factor loadings. The loadings produced by PC('2), maximum likelihood, and principal components were compared with expected loadings in the population. A normed index of fit was computed on the residential covariances to determine how well these three methods of analysis could distinguish unidimensional correlation matrices from two-dimensional matrices. Particularly in unidimensional correlation matrices (the case of primary interest), the loadings estimated with both the PC('2) method and maximum likelihood analysis were close to the expected values, but as anticipated, the principal components loadings were consistently higher than the expected values. The results of the study suggested that PC('2) was an alternative to maximum likelihood analysis in estimating structural parameters for a unidimensional set of measures, and that either method was superior to the first principal component for that purpose.