Equation Derivation of the Probability Distribution of IBS Score among Full Sibling Pairs

Objective To derive the general equation of the probability distribution of identity by state （IBS） score among biological full sibling pairs by calculating STR allele frequency. Methods Based on the Mendelian genetics law and the hypothesis that parents of biological full siblings （FS） were unrelated individuals, the IBS score and corresponding probability of different genotype combinations in the offspring when unrelated individuals of different genotype combinations give birth to two offsprings were derived. Results Given f （i=1, 2, …, m） as the frequency of the ith allele of a STR locus, the probability of sharing 2 alleles （p2FS）, 1 allele （p1FS） or 0 allele （p0FS） with biological full sibling pairs on the locus can be respectively expressed as follows： (see the text). The sum of p2FS, p1FS and p0FS must be 1. As for the multiple genotyping system that contained n STR loci, IBS scores between biological full sibling pairs conform to binomial distribution： IBS~B（2n, π1）. The population rate π1, can be given by the formula： (see the text). Conclusion The alternative hypothesis in biological full sibling testing is that two appraised individuals are biological full siblings. The probability of the corresponding alternative hypothesis of any STR locus combination or IBS score can be directly calculated by the equations presented in this study, and the calculation results are the basis for explanations of the evidence.