The abilities of man : their nature and measurement / [Charles Edward Spearman].

  • Charles Spearman
Date:
1927
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Licence: In copyright

Credit: The abilities of man : their nature and measurement / [Charles Edward Spearman]. Source: Wellcome Collection.

    433/472 (page 419)
    known formula for partial correlations. Let rap,g denote the correlation that a would have with p if the influence of the general factor g were eliminated. Then, by Yule’s formula, ?ap. <7 — ?ap Yo<7 rPff But bv definition of r, av’ (I -rl0)h{i-rla)h so that r r cip. g = 0, — fat) • Y. VV> and similarly, rbv=rbg . r.pg. Hence rag/rl>g =raJrhv> and in the same way = raJrbQ, which gives us at once the above equation (4).* Subsequently, the only change made in the criterion has been the obvious conversion of (4) into the more convenient “ tetrad equation ” (see p. 73), namely, ^av^bq ~ ^bjfaq ~~ (5) 3. Reversibility of proof.—The preceding demonstration had been to the effect that, when every variable could be divided into the two factors g and s, then the criterion (in whichever of its forms) would necessarily be satisfied. There remained the far more difficult reverse problem, namely, as to whether, when the criterion was satisfied, then every variable would necessarily be divisible into the said two factors. This problem was first solved, and affirmatively, by Garnett for the case of “ normal ” frequency distribution of the variables.'}' Another solution, this time covering all frequency distributions whatsoever, so long as the number of variables is large, was given by the present writer.J Finally, a complete solution of the problem, including all manner of distribution and any number of variables, was given by the present writer, as follows : § From (4) or (5) we may readily get ^xv ^-xz* vz> (^) where \xz is constant whilst y takes all values except % or z. Our question, then, is tantamount to asking whether, on assum- ing (6), each of the variables involved, say a, can be reduced to the form a-fan + $a> (7) where 1. fa, fb, etc., are constant for all particular values of a, b, etc. 2. t] is an element common to all the variables. 3. Sa, Sb, etc., are uncorrelated with 4. Sa, Sb, etc., are uncorrelated with each other. * Hart and Spearman, Brit. J. Psych. 1912, v. p. 58. f Proc. Royal Society, A. lxxxxvi. 1919. f Psych. Rev. 1920, pp. 167-8. § Proc. Royal Society, A. ci. 1922.