Correlation and Application of Statistics to Problems of Heredity 21
If U for any character be the deviate of the generant from the mean of the race, then the individual endowed with such U's for all characters would represent the stirp of any family. Unfortunately Galton does not give us any method for determining the U of the generant. I think, however, if we take the character U of the generant to be that linear function of the characters of all the ancestry which gives the highest correlation R with the character in the offspring, it throws light on Galton's idea. In this case U is simply proportional to the multiple regression expression. If we make the following hypotheses, which have considerable experimental evidence in their favour, namely
(a) that the individual correlations of offspring with male and with ,female ancestors are equal,
(b) that such correlations with individual ancestors die out in a geometrical ratio, i.e. the correlations of the offspring with individual parents (father or mother), with individual grandparent (male or female), with individual great-grandparent, etc. form a series r,, r,a, r, a2, etc., where a is less than unity, then it can be demonstrated that the deviate U will be given by the formula
where h,, h,, h3, etc. are the deviates of the midparental characters in the successive grades of ancestry and y, /3 are constants, which can be found in terms of r, and a. Further, the fraternity of which U defines the stirp will
vary round U with variability a,/1-R2, where R (the "coefficient of multiple correlation") is known in terms of r, and a, or of y and 8.
The expression for U, or the deviate of the generant which defines the stirp, has been termed the Law of Ancestral Inheritance'. It is not a biological hypothesis, but the mathematical expression of statistical variates, which obey, as many measurable characters in man, certain forms of frequency distribution, these being maintained in successive generations. It can be applied with special values of y and /3 to many biological hypotheses. We are, however, not concerned to discuss these matters here, but merely to point out that in the papers we are now dealing with Galton was feeling his way upwards towards this Law of Ancestral Inheritance, though I venture to think by a faulty stairway. The somewhat complicated mathematics of multiple correlation with its repeated appeals to the geometrical notions of hyperspace remained a closed chamber to him, necessary as multiple correlation now is for many practical problems of modern statistics. As I have said there is a true generant, i.e. one in which we insert the true values of the
different ancestral midparental deviates, namely h„ h2, h3f ... as above, and a
probable generant for which we only know h, and put in probable values
* Biometrika, Vol. VIII, pp. 239-243.
t Roy. Soc. Proc. Vol. LXII, p. 386. For the fuller mathematical treatment see Biometrika, Vol. viii, pp. 239-240 and Vol. xvll, pp. 129 et seq.