A simulation study on the sampling distribution of the linkage disequilibrium

The aim of this study is to determine the distribution of the quadratic forms related to the linkage disequilibrium coefficients calculated from the samples drawn from populations with non-zero linkage disequilibrium coefficient. It is known that the average of coefficient of linkage disequilibrium (D) calculated from samples drawn from populations in linkage equilibrium is zero (µD = 0) and statistic with equation number 1.2 in the text, which is a quadratic form, shows a chi-square distribution. It is studied the distribution of three different quadratic statistics one is 1.2 calculated from samples of size n = 10, 20, 50 generated through simulation from populations with linkage disequilibrium coefficients of μD = 0 and μD = 0,15 for a model of two alleles, two loci. The results show that the first quadratic form (statistic number 1.2) which is much known statistics in literature, has found to be fitted noncentral chi square distribution with 3 df in the samples drawn from the populations of μD = 0,15. The second quadratic form (equation number 3.1) in the methods was stable in being fitted to the central chi square distributions of 1 df in the existence of linkage disequilibrium. The third quadratic form (equation number 3. 2) has a distribution fitted to chi square with 3 df. The distribution of three different quadratic statistics , one of which is a quadratic form numbered 3.8 in the text, calculated from samples of size n=100 generated through simulation from populations with linkage disequilibrium coefficients of μD=0 and μD≠0 and two alleles in one locus and three alleles in the other were studied. As a result, it was found that this (3.8 statistic number) quadratic form, calculated from the samples drawn from the population with a linkage disequilibrium coefficient of zero, showed a central Chi-Square distribution with 2 df (central). For the statistics calculated from samples drawn from non-equilibrium populations, a suitable distribution could not be found among those tried, some other distributions should be tested.