Some necessary conditions for common-factor analysis (original) (raw)

Abstract

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.

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  1. The Israel Institute of Applied Social Research, Israel
    Louis Guttman

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

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Guttman, L. Some necessary conditions for common-factor analysis.Psychometrika 19, 149–161 (1954). https://doi.org/10.1007/BF02289162

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