, Volume 19, Issue 2, pp 149-161

Some necessary conditions for common-factor analysis

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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 matrixU 2 such thatG = R − U 2 is Gramian and of minimum rankr. Lets 1 be the number of latent roots ofR which are greater than or equal to unity. Then it is proved here thatrs 1. Two further lower bounds tor are also established that are better thans 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 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.

This research was made possible in part by an uncommitted grant-in-aid from the Behavioral Sciences Division of the Ford Foundation.