“Best Possible” Systematic Estimates of Communalities

At least four approaches have been used to estimate communalities that will leave an observed correlation matrix R Gramian and with minimum rank. It has long been known that the square of the observed multiple-correlation coefficient is a lower bound to any communality of a variable of R . This lower bound actually provides a “best possible” estimate in several senses. Furthermore, under certain conditions basic to the Spearman-Thurstone common-factor theory, the bound must equal the communality in the limit as the number of observed variables increases. Otherwise, this type of theory cannot hold for R .