Some Necessary Conditions for Common-Factor Analysis

Let R be any correlation matrix of order n , with unity as each main diagonal element. Common-factor analysis, in the Spearman-Thurstone sense, seeks a diagonal matrix U 2 such that G = R − U 2 is Gramian and of minimum rank r . Let s 1 be the number of latent roots of R which are greater than or equal to unity. Then it is proved here that r ≥ s 1 . Two further lower bounds to r are also established that are better than s 1 . 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 in R is small, but the minimum rank r for a Gramian G is nevertheless very large compared with n . Heuristic criteria are given for testing the hypothesis that a finite r exists for the infinite universe of content from which the sample of n observed variables is selected; in many cases, the Spearman-Thurstone type of multiple common-factor structure cannot hold.