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Correlation and Application of Statistics to Problems of Heredity 23

understands, for having determined the midparental contribution to be 2 from

either series, he now writes* of the values of 1 and 1

n m

"These values differ but slightly from J, and their mean is closely 71, so that we may fairly accept that result. Hence the influence, pure and simple, of the midparent may be taken as 1, of the midgrandparent J, of the midgreatgrandparent g and so on. That of the individual parent would be I, of the individual grandparent IL, of an individual in the next generation I and so on."

Thus Galton reaches his Separate Contribution of each Ancestor to the Heritage of the Child, a principle which is often spoken of as his Law of Ancestral Heredity. In reaching it he apparently drops his 1 series altogether

n

and follows his 1 series with its geometrical system of taxation. This is m

distinctly more in keeping with the expression for the generant deviate U above, which runs in a geometrical series. If we assume all the ancestors to

have the same deviation h, we have U=1 y h, and, if the offspring value

-18

might in such a uniform breed be also taken as h, it follows that y = 1 -,8. Hence if we take the first midparents' contribution to be 2, i.e. y = 2, with Galton, it follows that R =1, and our series is Galton's geometrical series with his radix value, a half. But I venture to think it was inspiration rather than correct reasoning which led him to a geometrical series for U.

On the other hand his multiple regression coefficients 2,1, a, ... suffice to determine what the correlations between an individual ancestor in any generation and the offspring ought to be. They take the values for parents •3,

for grandparents 2 x •3, for great-grandparents 1 x •3 and so on. Galton found

his midparental regression - and took his parental to be 2t. This is not so far from •3, that we could say it confutes Galton's Ancestral Law. But we find Galton taking the grandparental regression and therefore the correlation

the great-grandparental -- and so on. These values form a series a, a', a3, ... for the individual ancestral •correlations and lead to y = 1, 8 = 0, or to the generant U being solely determined by the parents, the higher ancestry contributing nothing to the generant\$. Hence it follows that Galton's Ancestral Law is not in keeping with the values he has taken for his individual ancestral correlations. The reasoning by which he has reached one or the other is defective. As I have said Galton's guess at a geometrical relation for the coefficients of U was an inspiration, but his idea that a grandson is the son of a son and so his regression (and with a stable population his correlation) must be 2 x 2 = 2 is fallacious. Regression coefficients cannot be obtained from each other in this manner.

* Roy. Soc. Proc. Vol. LXII, p. 62.

t This will be equal to the correlation, for the variabilities of both variates are taken to be the same.

\$ See Phil. Tran8. Vol. 187, A, p. 306, 1896.