The Derivation of Joint Distribution and Correlation between Relatives by the Use of Stochastic Matrices

Abstract

The absolute frequencies of the various genotypic parent-offspring and sib-pair combinations with respect to one pair of genes, autosomal or sex-linked, are well known to geneticists, for they are of practical value in certain types of studies in human heredity. These frequencies are, however, obtained by a rather long process, even for the simple case of full sibs. The procedure of obtaining the frequencies of unclenephew or first cousinl combinations is entirely too tedious evenly with the help of matrix notations (Hogben, 1933). The purpose of the present communication is three-fold. The first is to give a simple procedure of finding the frequencies of the various genotype combinations of near relatives by using matrices of conditional probabilities. The second purpose is to express such matrices of conditional probabilities of relatives in the form of a linear function of some basic matrices. The third is to deduce the correlations between relatives from such linear functions of the basic matrices. The meaning of these statements will be made clear in later sections. For many years there existed two apparently very different methods of obtaining the genotypic correlations between relatives. One is a straightforward but long procedure by which the frequencies of the various combinations of the stated relatives in the general population are first found and then the is calculated from such a correlation table. On the other hand, they may be obtained by the method of path coefficients, developed by Wright (1921 and later). The latter method gives the required coefficient almost instantly once the relationship is specified, but does not give us any information about the frequencies of the various combinations of the relatives in the population. Moreover, one has to be familiar with the mathematical theorems concerning the path coefficients before he can use them to derive the required correlations. The method of stochastic process and its final reduction to some basic matrices, as described in the following pages, will give us both the frequencies of relative-pairs and their in a very simple manner. It is hoped that the present method will bridge the gap between the two existing procedures.