Some necessary conditions for common-factor analysis

LetR be any correlation matrix of ordern, with unity as each main diagonal element. Common-factor analysis, in the Spearman-Thurstone sense, seeks a diagonal matrixU2 such thatG = R − U2 is Gramian and of minimum rankr. Lets1 be the number of latent roots ofR which are greater than or equal to unity. Then it is proved here thatr ≧s1. Two further lower bounds tor are also established that are better thans1. Simple computing procedures are shown for all three lower bounds that avoid any calculations of latent roots. It is proved further that there are many cases where the rank of all diagonal-free submatrices inR is small, but the minimum rankr for a GramianG is nevertheless very large compared withn. Heuristic criteria are given for testing the hypothesis that a finiter exists for the infinite universe of content from which the sample ofn observed variables is selected; in many cases, the Spearman-Thurstone type of multiple common-factor structure cannot hold.