Study of half-sib correlations and parent-offspring correlations under half-sib mating system

Abstract

A purely theoretical investigation entitled ‘a study of half-sib correlations and parent-offspring correlations under half-sib mating system’ was carried out with the following objectives. i) to derive the joint distribution- correlation table as well as the correlation of half-sib pairs under different generations of half-sib mating, in the case of autosomal genes, assuming single locus with two alleles. ii) to derive the joint distribution – correlation table as well as the correlation of parent-offspring pairs under different generations of half-sib mating, in the case of autosomal genes, assuming single locus with two alleles. iii) to derive the joint distribution –correlation table and the correlation between both the parents and several offspring as well as the correlation between one parent and several offspring under different generations of half-sib mating, in the case of autosomal genes, assuming single locus with two alleles. iv) to derive the joint distribution –correlation table as well as the correlation of the different ‘ half-sib pairs, viz., brother-sister, brother-brother and sister-sister and also the correlations of different parent- offspring pairs, viz., mother-daughter, mother-son and father-daughter, in the sex-linked gene case. As the generation matrix of half-sib pairs is too tedious to derive, the joint distribution of half-sib pairs both in the autosomal gene case and sex-linked gene case, was derived from first principles and the correlation was calculated from the correlation table, assuming additive genic effects and by using the product-moment correlation coefficient formula. The correlations were worked out for the first ten generations of half-sib mating. The joint distribution of parent –offspring pairs under half-sib mating system, both in the autosomal gene case and sex-linked gene case was derived by using the generation matrix approach. Here also, the correlations were worked out for the first ten generations of half-sib mating. A comparative study of the half-sib correlations and parent-offspring correlations, conducted both numerically and graphically, revealed that even though the half-sib correlations are of lower order than parent-offspring correlations, the rate of increase in the former is much higher than the rate of increase in the latter. Among the correlations of various half-sib pairs in the sex-linked gene case, the correlation of sister-sister pairs was found to be of the highest order. It was also observed that the correlation of brother-brother pairs can be easily obtained by the I.T.O method developed by Li and sacks (1954). Regarding the various parent-offspring pairs, mother-son correlations and father-daughter correlations were found to be the same and there was no correlation between father and son which is quite logical in the case of half-sib mating as the son receives his father’s y-chromosome. A comparative study of the half-sib correlations and parent-offspring correlations in the sex-linked gene case was carried out both numerically and graphically. It was noted that even though the mother-son correlations and the father-daughter correlations are of very high order ranging from 0.707 to 0.909, the sister-sister correlation exceeds them after the sixth generation. From this it is obvious that the correlation between sister-sister pairs increases much rapidly than the correlation between mother-son or father-daughter pairs. It was also noted that the correlations of all these six types of relative pairs tend to unity as the number of generations increases indefinitely. A study was also made on the correlation between both the parents and k offspring as well as that between one parent and k offspring, in the case of autosomal genes. A general term (ie. when number of offspring is k) for the correlation in each of the above cases was derived for each of the ten generation of half-sib mating. The correlation were found to increase with the number of generations and k, the number of offspring. It was also noted that the correlations in both parent case are comparatively of higher order than that of one parent case and the rate of increase in the former was much greater than the rate of increase in the latter. The numerical values of these correlations were worked out for k = 1 to 10 and also for the first ten generations of half-sib mating as given in table (1.19) and (1.22) respectively.