Some statistical and computational problems in pedigree linkage studies

The affected sibling pair test is central in nonparametric family linkage studies. This thesis introduces the multiple linkage test based on affected sibling pair (ASP) samples. The familywise p-values of such multiple linkage tests are estimated by simulating the allele sharing patterns of the siblings under both a Markov chain model and its Gaussian approximation. The importance sampling method is applied to improve both Monte Carlo simulation procedures. This thesis explores the adequacy of Gaussian approximation as it relates to the sibling pair sample size.; Further topics concerning ASP tests are discussed as well. We demonstrate the asymptotic equivalence of several commonly used allele sharing statistics to measure the sharing status of affected sibships under the null hypothesis of no linkage. An efficient computational algorithm to derive the distribution of allele sharing patterns given non-fully informative genotype information is introduced.; Since the Gaussian approximation can be viewed as a first order Edgeworth approximation of the standardized sum of total allele sharing, this thesis investigates whether higher order Edgeworth approximation will lead to a more accurate estimation of the error rate of a multiple linkage test. The Edgeworth approximation of the allele sharing distribution along one chromosome is developed and compared to its Gaussian approximation.; The multiple linkage test using allele sharing of affected sibling pairs is designated to identify a chromosome that harbors some disease gene. We will also use this test to locate a new trait on a linkage inferred chromosome. We introduce Holm's step down multiple sequential rejective procedure to conduct a linkage test along a chromosome. Given the penetrance model of the markers, we present the allele sharing distribution at a trait and a marker as a function of the genetic distance between these two.