COMPUTATION OF THE FULL LIKELIHOOD FUNCTION FOR ESTIMATING VARIANCE AT A QUANTITATIVE TRAIT LOCUS

The proportion of alleles identical by descent (IBD) determines the ge netic covariance between relatives, and thus is crucial in estimating genetic variances of quantitative trait loci (QTL). However, IBD propo rtions at QTL are unobservable and must be inferred from marker inform ation. The conventional method of QTL variance analysis maximizes the likelihood function by replacing the missing IBDs by their conditional expectations (the expectation method), while in fact the full likelih ood function should take into account the conditional distribution of IBDs (the distribution method). The distribution method for families o f more than two sibs has not been obvious because there are n(n - 1)/2 IBD variables in a family of size n, forming an n X n symmetrical mat rix. In this paper, I use four binary variables, where each indicates the event that an allele from one of the four grandparents has passed to the individual. The IBD proportion between any two sibs is then exp ressed as a function of the indicators. Subsequently, the joint distri bution of the IBD matrix is derived from the distribution of the indic ator variables. Given the joint distribution of the unknown IBDs, a me thod to compute the full likelihood function is developed for families of arbitrary sizes.