Some properties of the communality in multiple factor theory

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