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iz 3 



linear regressions throughout, and is the 

 .assumption justified? For example, has 

 .any adequate test been made to show that 

 the regression of, say, "Cancellation" on 

 "Opposites" is really linear? There is 

 •nothing in Spearman's book to tell us. 



To return to our g and s, however, there 

 are other remarks which seem pertinent. 

 First, we should observe that the validity 

 of the decomposition of variables into a 

 (g, s) system does not depend in the least 

 on whether there truly exist any physical 

 or mental realities corresponding to g 

 land s. They need be nothing more than 

 a convenient mathematical fiction; and 

 their values will be just as determinate. 

 The best proof of such a statement is a 

 •demonstration that a system of variables 

 can be devised which can be produced 

 both by a (g, s) system and by a completely 

 -different system. 



Suppose four variables, having the in- 

 tercorrelations shown: 





A 



B 



C 



D 



A 





0.895 



0.694 



0.578 



B 



0.895 





0.633 



0.517 



C 



0.694 



0.633 





0.408 



D 



0.578 



0.517 



0.408 





In this table the tetrad differences are all 

 izero, and accordingly we should be able 

 to split the variables into a (g, s) system; 

 which we can do, according to Spearman's 

 •equations, with the following results: 



m ax — 0.994£z + O.Io6jaa: 



mbx = 0.899^ + 0.438.^ 

 m cx = Q.ioxgx + o.-jz.xs ex 

 m dx = o.578gs + o.ii^Sdx 



But we can equally well suppose these 

 correlations to have arisen from the fol- 

 lowing system 



max — Vlx -\~ Vix 



mbx = Vix + Vix + Vix 



m c .x = Vix + Vix + V Zx + Vgx 



mdx = Vix + Vix + Vix + fs* + f6» 



in which v%, v 2 , etc., are independent 

 variables (having, in this case, equal 

 standard deviations). And, as may be 

 inferred, an infinite number of other de- 

 compositions are possible. It is to be 

 observed also that any system of correla- 

 tions can be represented by such a system 

 of variables, whether or not the tetrad 

 difference is zero. 



This possibility is, of course, not over- 

 looked by Spearman. He points out, 

 first, that the (g, s) system differs from our 

 00 system in being unique; there is one 

 and only one (g, s) system which will 

 satisfy the conditions. Second, he points 

 out that a relation can be set up between 

 the (g, j) system and such a 00 system, 

 whereby we can determine the (g, s) 

 values from the 00 values. Finally, he 

 urges that the (g, s) system is simpler and 

 requires fewer assumptions, and accord- 

 ingly should be given preference. With 

 this question, however, we need not con- 

 cern ourselves. We have merely pointed 

 out that there is no necessary physical 

 reality corresponding to the mathematical 

 analysis. An analogy would be the de- 

 composition of a velocity into compo- 

 nent velocities which add by the parallelo- 

 gram law, or the decomposition of an 

 alternating current into components in 

 quadrature with each other — one of the 

 most useful devices in the theory of al- 

 ternating current circuits. Unless and un- 

 til we have evidence of the existence of g 

 and s from some other source than the 

 tetrad equation, we must regard them 

 fundamentally as mathematical expres- 

 sions and not as physically existing quan- 

 tities. 



Obviously, however, this should not 

 prevent our using g and s if they are use- 

 ful to us . The fact that the decomposition 

 of velocities is an artifice of mathematics 

 does not make it less useful. And, in 

 Spearman's view, there are grounds for 



