(4) Proposition III.— To fond the coefficient of correlation of two indices in terms of the coefficients of correlation of the four absolute measurements and their coefficients of variation.
Let /s»s and a5*/«4 be the two indices. nfSuPu = S ( t , - *") Mathematical Contributions to the Theory of Evolution. 493 W3W4 \ W i wi3 niiTn-i /„ . €2 C4 e2e4 , e42 x 1+- \ —1 —V42+ TuViVi) 1 m2 m4 14' _ . • Q / C1 «*\ / g* \ ^is»2iS \m* m4/ ’
if we neglect terms of the cubic order. /» 2 132 24 = t u*M( ri 2ViV2— r 23t?2r 3 + r34r3r 4) . Hence, finally, _____ r 18^2—Ti4r M + ra4r3r4 ^ Y ^ ^ Vi2+V83—2ri3Vit;3 ^V i+ .. '
(5) Thus we have expressed p in terms of the four coefficients of correlation and the four coefficients of variation of the absolute measurements which form the indices. We may draw the following conclusions :
(i.) The correlation between two indices will always vanish when the four absolute measurements forming the indices are quite uncorrelated,
(ii.) If two of the organs are perfectly correlated, let us say made identical: for example, the third and fourth, so that ru = 1, and r3 = r4, we find rittW — ri3t>,v3— r^ffph + vf v vi* + v3 ~ 2 r l3ViV3 v/vi2+V3 —2 ri3v2v3 (▼).
This is the coefficient of correlation between two indices with the same denominator (xfx3 and
The value of p in (v) does not vanish if the remaining organs be quite uncorrelated, i.e., ru = r13 = r^ = 0. In this case _ ____ vf P°. " / Vx+V? •f Vi +V3 (vi).
This is the measure of the spurious correlation. For the special
