On the determination of the chief correlations between collaterals in the case of a simple Mendelian population mating at random

Abstract

(1) In a paper communicated to the Royal Society in April, 1909, Prof. Pearson obtained the gametic correlations between the offspring and the ancestry in each grade in a simple Mendelian population mating at random. By a “simple Mendelian population” for a given character, he understood one which started with any definite ratio of dominant individuals (AA) to recedents ( aa ). These mating at random give rise to a population which may be written in the form p 2 (AA) + 2 pq (A a ) + q 2 ( aa ), and of which, without selection, the proportions of dominants, recedents, and hybrids are known to remain constant, with continued random mating, during successive generations. Prof. Pearson found that, both in the case of gametic and somatic characters, the ancestral correlations diminished in a geometrical progression, and thus obeyed the fundamental principle of the Law of Ancestral Heredity, as deduced from observations on man and other living forms. (2) It is difficult to believe that the characters dealt with in the case of Mendelian investigations on animals can be largely affected by environment, but it is easy to allow for this influence by the method of partial correlation. If, in an investigation on any given character, the subscript 1 denote offspring, 2 the ancestor in any generation, 3 the offspring’s environment, and 4 the ancestor’s environment, then the correlation between 1 and 2 for constant 3 and 4 is given by 34 ρ 12 = r 12 (1 - r 34 2 ) - r 13 r 23 - r 14 r 24 + r 34 ( r 14 r 23 + r 13 r 24 )/{1 - r 23 2 - r 34 2 - r 24 2 + 2 r 23 r 34 r 24 } ½ {1 - r 13 2 - r 14 2 - r 34 2 + 2 r 13 r 34 r 14 } ½ .