One might argue, that, in intelligence tests, many factors play a role in the realization of the final test score. In that case g, what ever it might mean, must be a composite of more than one underlying factor. The question arises, how, at the same time, the correlation matrix could still be uni-factorial. According to Spearman the solution for this problem could be as follows:
"There are, however, certain special cases where g does admit of resolution into a plurality of sub-factors ... say, for example, ability and zeal. If in all tests the respective influences of these two always remained in any constant ratio, then both could quite well enter into g together; for the tetrad equation would still be able to hold." (Spearman, 1927, p. 93).For example, in the case of three latent sub-factors (e.g. ability, zeal and practice) one might have:
Suppose, however, the following equalities hold:
Then the ratios , with p = 1, 2, 3 and q = 1, 2, 3 are all constant across tests.
One may write:
The term between brackets represents one factor The weights are independent of the tests. The factor is identifiable, the factors , and , however, are not. One could consider this idea of a plurality of underlying factors as a rather contrived explanation. However, in the next section an examlple will be discussed, in which many unidentifiable factors play a role, while a unifactorial solution still holds. The example is taken from the game of golf. Everybody will agree, that proficiency in the game of golf is determined by many factors. However, it will be shown that factor analysis always results in one factor. It was on purpose, that the example was taken from a different domain than that of mental ability testing. Sometimes the pros and cons of a method are better demonstrated, when the method is appled in a domain, which is more easy to understand.