Hostname: page-component-745bb68f8f-lrblm Total loading time: 0 Render date: 2025-01-08T09:42:04.357Z Has data issue: false hasContentIssue false
  • English
  • Français

Multiple Rectilinear Prediction and the Resolution into Components

Published online by Cambridge University Press:  01 January 2025

Louis Guttman
Louis Guttman*
Affiliation:
Department of Sociology, University of Minnesota
Get access Rights & Permissions [Opens in a new window]

Abstract

It is assumed that a battery of n tests has been resolved into components in a common factor space of r dimensions and a unique factor space of at most n dimensions, where r is much less than n. Simplified formulas for ordinary multiple and partial correlation of tests are derived directly in terms of the components. The best (in the sense of least squares) linear regression equations for predicting factor scores from test scores are derived also in terms of the components. Spearman's “single factor” prediction formulas emerge as special cases. The last part of the paper shows how the communality is an upper bound for multiple correlation. A necessary and sufficient condition is established for the square of the multiple correlation coefficient of test j on the remaining n−1 tests to approach the communality of test j as a limit asn increases indefinitely while r remains constant. Limits are established for partial correlation and regression coefficients and for the prediction of factor scores.

Type
Original Paper
Information
Psychometrika , Volume 5 , Issue 2 , June 1940 , pp. 75 - 99
Copyright
Copyright © 1940 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

*

I am indebted to Professor Dunham Jackson for helpful criticism of most of this paper.

References

Dwyer, P. S. The contribution of an orthogonal multiple factor solution to multiple correlation. Psychometrika, 1939, 4, 163–172.CrossRefGoogle Scholar
Guttman, Loui. A note on the derivation of formulae for multiple and partial correlation. Annals math. Statistics, 1938, 9, 305–308.CrossRefGoogle Scholar
Harman, H. H. Systems of regression equations for estimation of factors. J. educ. Psychol., 1938, 29, 431–441.CrossRefGoogle Scholar
Holzinger, K. J. Estimation of factors. Chapter III,Student Manual of Factor Analysis. Chicago: Department of Education. 1937, 29–41.Google Scholar
Jackson, Dunha. The trigonometry of correlation. Amer. math. Monthly, 1924, 31, 275–280.CrossRefGoogle Scholar
Ledermann, Walte. Shortened method of estimation of mental factors by regression. Nature, 1938, 141, 650–650.CrossRefGoogle Scholar
Ledermann, Walte. On a shortened method of estimation of mental factors by regression. Psychometrika, 1939, 4, 109–116.CrossRefGoogle Scholar
Roff, Merril. Some properties of the communality in multiple factor theory. Psychometrika, 1936, 1, 1–6.CrossRefGoogle Scholar
Spearman, C. The abilities of man, appendix, New York: Macmillan, 1927.Google Scholar
Thomson, Godfre. Some points of mathematical technique in the factorial analysis of ability. J. educ. Psychol., 1936, 27, 37–54.CrossRefGoogle Scholar
Thomson, Godfre. Methods of estimating mental factors. Nature, 1938, 141, 246–246.CrossRefGoogle Scholar
Thurstone, L. L. The vectors of mind, Chicago: Univ. Chicago Press, 1935.Google Scholar
Thurstone, L. L. Primary mental abilities. Psychometric Monographs Number 1, Chicago: Univ. Chicago Press, 1938.Google Scholar