THEORY OF STATISTICS. 
of Chap. IV. § 6, and indicates a source of fallacies similar to 
those there discussed. 
20. We have seen (§ 9) that r,, is the correlation between x, , 
and x,,, and that we might determine the value of this partial 
correlation by drawing up the actual correlation table for the two 
residuals in question. Suppose, however, that instead of drawing 
up a single table we drew up a series of tables for values of x, 
and z,, associated with values of x, lying within successive 
class-intervals of its range. In general the value of r,,, would 
not be the same (or approximately the same) for all such tables, 
but would exhibit some systematic change as the value of x, 
increased. Hence 7, should be regarded, in general, as of the 
nature of an average correlation: the cases in which it measures 
the correlation between x,, and x,, for every value of x, (cf. 
Chap. XVI.) are probably exceptional. The process for deter- 
mining partial associations (¢f. Chap. IV.) is, it will be remembered, 
thorough and complete, as we always obtain the actual tables 
exhibiting the association between, say, 4 and B in the universe 
of C’s and the universe of y's: that these two associations may 
differ materially, is illustrated by Example i. of Chap. IV. 
(pp. 45-6). It might sometimes serve as a useful check on 
partial-correlation work to reclassify the observations by the 
fundamental methods of that chapter. For the general case an 
extension of the method of the “ correlation-ratio ” (Chap. X., § 20) 
might be useful, though exceedingly laborious. It is actually 
employed in the paper cited in ref. 7 and the theory more fully 
developed in ref. 8. 
REFERENCES. 
The preceding chapter is written from the standpoint of refs. 3 and 4, and with the 
notation and method of ref. 5. The theory of correlation for several variables was 
developed by Edgeworth and Pearson (refs. 1 and 2) from the standpoint of the normal ” 
distribution of frequency (cf. Chap. XVL.). 
Theory. 
(1) EpGEWoRTH, F. Y., “On Correlated Averages,” Phil. Mag., 5th Series, vol. xXxiv., 
1892, p. 194. 
(2) PEARSON, KARL, ‘ Regression, Heredity, and Panmixia,” Phil. Trans. Roy. Soc., 
Series A, vol. clxxxvii., 1896, p. 253. 
(3) YULE, G. U., *“On the Significance of Bravais’ Formulae for Regression, etc., in the 
case of Skew Correlation,” Proc. Roy. Soc., vol. Ix., 1897, p. 477. 
(4) YULE, G. U., “On the Theory of Correlation,” Jour. Roy. Stat. Soc., vol. 1x., 1897, 
p. 812. 
(5) YULE, G. U., “On the Theory of Correlation for any number of Variables treated 
by a New System of Notation,” Proc. Roy. Soc., Series A, vol. 1xxix., 1907, p. 182. 
(6) HOOKER, R. H., and G. U. YULE, ‘Note on Estimating the Relative Influence of 
Two Variables upon a Third,” Jour. Roy. Stat. Soc., vol. Ixix., 1906, p. 197. 
(7) BROWN, J. W., M. GREENWOOD, and FRANCES Woob, “A Study of Index-Corre- 
lations,” Jour. Roy. Stat. Soc., vol. 1xxvii., 1614, pp. 317-46. (The partial or 
“solid ” correlation-ratio is used.) : " ; 
(8) ISSERLIS, L., “On the Partial Correlation-Ratio, Pt. I. Theoretical,” Biometrika, 
vol. x., 1914, pp. 391-411. 
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