Communality of a Variable: Formulation by Cluster Analysis

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Abstract
The communality of a variable represents the degree of its generality across n − 1 behaviors. Domain-sampling principles provide a fundamental conception and definition of the communality. This definition may be alternatively stated in eight different ways. Three definitions lead to precise formulas that determine the true value of the communality: (i) from the k necessary and sufficient dimensions derived by iterated factoring, (ii) from the n − 1 remaining variable-domains, and (iii) from k′ multiple clusters of the n variables. Seven definitions provide approximation formulas: (i) one from the k dimensions as initially factored, (ii) one from the n − 1 remaining variables, and (iii) five from a single cluster. Rank of the matrix is not a desiratum in some definitions. Using an example designed by Guilford to illustrate multiple-factor analysis, applications of the formulas based on the three precise definitions recover the true communalities, and five approximation formulas each gives values closer than the ad hoc estimates usually employed in factor analysis.