Sampling genotypes in large pedigrees with loops

Markov chain Monte Carlo (MCMC) methods have been proposed to overcome comp utational problems in linkage and segregation analyses. This approach invol ves sampling genotypes at the marker and trait loci. Scalar-Gibbs is easy t o implement, and it is widely used in genetics. However, the Markov chain t hat corresponds to scalar-Gibbs may not be irreducible when the marker locu s has more than two alleles, and even when the chain is irreducible, mixing has been observed to be slow. These problems do not arise if the genotypes are sampled jointly from the entire pedigree. This paper proposes a method to jointly sample genotypes. The method combines the Elston-Stewart algori thm and iterative peeling, and is called the ESIP sampler. For a hypothetic al pedigree, genotype probabilities are estimated from samples obtained usi ng ESIP and also scalar-Gibbs. Approximate probabilities were also obtained by iterative peeling. Comparisons of these with exact genotypic probabilit ies obtained by the Elston-Stewart algorithm showed that ESIP and iterative peeling yielded genotypic probabilities that were very close to the exact values. Nevertheless, estimated probabilities from scalar-Gibbs with a chai n of length 235000, including a burn-in of 200000 steps, were less accurate than probabilities estimated using ESIP with a chain of length 10000, with a burn-in of 5000 steps. The effective chain size (ECS) was estimated from the last 25000 elements of the chain of length 125000. For one of the ESIP samplers, the ECS ranged from 21579 to 22741, while for the scalar-Gibbs s ampler, the ECS ranged from 64 to 671. Genotype probabilities were also est imated for a large real pedigree consisting of 3223 individuals. For this p edigree, it is not feasible to obtain exact genotype probabilities by the E lston-Stewart algorithm. ESIP and iterative peeling yielded very similar re sults. However, results from scalar-Gibbs were less accurate.