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Some Properties of the Communality in Multiple Factor Theory

Published online by Cambridge University Press:  01 January 2025

Merrill Roff
Merrill Roff*
Affiliation:
Indiana University, Bloomington, Indiana
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Abstract

Several theorems concerning properties of the communaltiy of a test in the Thurstone multiple factor theory are established. The following theorems are applicable to a battery of n tests which are describable in terms of r common factors, with orthogonal reference vectors.

1. The communality of a test j is equal to the square of the multiple correlation of test j with the r reference vectors.

2. The communality of a test j is equal to the square of the multiple correlation of test j with the r reference vectors and the n—1 remaining tests.

Corollary: The square of the multiple correlation of a test j with the n—1 remaining tests is equal to or less than the communality of test j. It cannot exceed the communality.

3. The square of the multiple correlation of a test j with the n—1 remaining tests equals the communality of test j if the group of tests contains r statistically independent ests teach with a communality of unity.

4. With correlation coefficients corrected for attenuation, when the number of tests increases indefinitely while the rank of the correlational matrix remains unchanged, the communality of a test j equals the square of the multiple correlation of test j with the n—1 remaining tests.

5. With raw correlation coefficients, it is shown in a special case that the square of the multiple correlation of a test j with the n—1 remaining tests approaches the communality of test j as a limit when the number of tests increases indefinitely while the rank of correlational matrix remains the same. This has not yet been proved for the general case.

Type
Original Paper
Information
Psychometrika , Volume 1 , Issue 2 , June 1936 , pp. 1 - 6
Copyright
Copyright © 1936 The Psychometric Society

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Footnotes

*

The author wishes to express his appreciation of the encouragement and assistance given him by Dr. L. L. Thurstone.

References

Holzinger, K. J.Statistical methods for students in education. New York: Ginn & Co. Pp. vii + 372.Google Scholar
Thurstone, L. L.The vectors of the mind. Chicago: University of Chicago Press. Pp. xv + 266.CrossRefGoogle Scholar