Hostname: page-component-5f745c7db-j9pcf Total loading time: 0 Render date: 2025-01-06T07:49:08.735Z Has data issue: true hasContentIssue false
  • English
  • Français

“Best Possible” Systematic Estimates of Communalities

Published online by Cambridge University Press:  01 January 2025

Louis Guttman
Louis Guttman*
Affiliation:
The Israel Institute of Applied Social Research
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Keywords

Public PolicyStatistical TheoryVariable IncreaseSystematic EstimateMinimum Rank
Type
Original Paper
Information
Psychometrika , Volume 21 , Issue 3 , September 1956 , pp. 273 - 285
Copyright
Copyright © 1956 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

This research was facilitated by a grant from the Lucius N. Littauer Foundation to the American Committee for Social Research in Israel in order to promote methodological work of the Israel Institute of Applied Social Research.

References

Golub, G. H. On the number of significant factors as determined by the method of maximum likelihood. (Stencilled) Digital Computer Laboratory, Univ. of Illinois, 1954.Google Scholar
Guttman, L.. Multiple rectilinear prediction and the resolution into components. Psychometrika, 1940, 5, 75–99CrossRefGoogle Scholar
Guttman, L.. Image theory for the structure of quantitative variates. Psychometrika, 1953, 18, 277–296CrossRefGoogle Scholar
Guttman, L.. Some necessary conditions for common-factor analysis. Psychometrika, 1954, 19, 149–161CrossRefGoogle Scholar
Guttman, L.. A new approach to factor analysis: the radex. In Lazarsfeld, Paul F. (Eds.), Mathematical thinking in the social sciences, Glencoe, Ill.: Free Press, 1954Google Scholar
Guttman, L.. The determinacy of factor score matrices, with implications for five other basic problems of common-factor theory. Brit. J. stat. Psychol., 1955, 8, 65–81CrossRefGoogle Scholar
Rao, C. R.. Estimation and tests of significance in factor analysis. Psychometrika, 1955, 20, 93–111CrossRefGoogle Scholar
Thurstone, L. L.. Multiple-factor analysis, Chicago: Univ. Chicago Press, 1947Google Scholar