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 ofn tests which are describable in terms ofr common factors, with orthogonal reference vectors.
1. The communality of a testj is equal to the square of the multiple correlation of testj with ther reference vectors.
2. The communality of a testj is equal to the square of the multiple correlation of testj with ther reference vectors and then—1 remaining tests.
Corollary: The square of the multiple correlation of a testj with then—1 remaining tests is equal to or less than the communality of testj. It cannot exceed the communality.
3. The square of the multiple correlation of a testj with then—1 remaining tests equals the communality of testj if the group of tests containsr 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 testj equals the square of the multiple correlation of testj with then—1 remaining tests.
5. With raw correlation coefficients, it is shown in a special case that the square of the multiple correlation of a testj with then—1 remaining tests approaches the communality of testj 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.
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References
Holzinger, K. J. Statistical methods for students in education. New York: Ginn & Co. Pp. vii + 372.
Thurstone, L. L. The vectors of the mind. Chicago: University of Chicago Press. Pp. xv + 266.
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The author wishes to express his appreciation of the encouragement and assistance given him by Dr. L. L. Thurstone.
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Roff, M. Some properties of the communality in multiple factor theory. Psychometrika 1, 1–6 (1936). https://doi.org/10.1007/BF02287999
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DOI: https://doi.org/10.1007/BF02287999