Thompson, Elizabeth A
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NSF-CBMS regional conference series in probability and statistics, 6, page i--169. Institute of Mathematical Statistics, (2000)

1 Genes, Pedigrees and Genetic Models 1 1.1 DNA, alleles, loci, genotypes, and phenotypes ............ . 1 1.2 Mendel's laws and meiosis indicators. 3 1.3 Pedigrees: the conditional independence structure. 4 1.4 Models, parameters, and inferences . . . . . . . . . . . . . . . . . . 7 2 Likelihood, Estimation and Testing 11 2.1 Likelihood and log-likelihood ...................... . 11 2.2 Estimation, information, and testing . . . . . . . . . . . . . . . . . . 13 2.3 Population allele frequencies ........ .. . . .. . . .. . . . . 16 2.4 The EM algorithm; general formulation . . . . . . . . . . . . . . . . 20 2.5 Gene counting and the ABO blood types ...... . . . . . . . . . 22 2.6 EM estimation for quantitative trait data ...... . . . . . . . . . 25 3 Gene Identity by Descent 3.1 Kinship and inbreeding coefficients ....... . . . . . . . . . . . . 29 3.2 Methods of computation .......... . .. .. . .. .. . .. . 30 3.3 Data on inbred individuals . . . . . . . . . . . . . . . . . . . . . . . 32 3.4 Multi-gamete kinship and gene ibd .................. . 34 3.5 Patterns of gene ibd in pairs of individuals ...... . . . . . . . . 36 3.6 Observations on related individuals ....... . . . . . . . . . . . 39 3.7 Monte Carlo estimation of expectations ...... . . . . . . . . . . 44 3.8 Reduction of Monte Carlo variance ....... . . . . . . . . . . . 46 4 Genetic Linkage 4.1 Linkage and recombination: genetic distance ...... . . . . . . . 49 4.2 Haplotypes, linkage, and association . . . . . . . . . . . . . . . . . . 51 4.3 Lod scores for two-locus linkage analysis ...... . . . . . . . . . 53 4.4 Power, information and Elods ..................... . 55 4.5 Two-locus kinship and gene identity ....... . . . . . . . . . . . 59 This content downloaded from 128.223.96.12 on Wed, 13 Dec 2017 17:45:42 UTC All use subject to http://about.jstor.org/termsCONTENTS 4.6 Homozygosity mapping with a single marker .... ......... 61 4.7 Meiosis at multiple linked loci ....... . . . . . . . . . . . . . . 64 4.8 Multi-locus kinship and gene identity ...... . . . . . . . . . . . 65 5 Models for Meiosis 69 5.1 The meiosis process ........... ... .. .. .. .. .. .. . 69 5.2 From chromatids to crossovers ........ . . . . . . .. . . . . . 71 5.3 From chiasmata to recombination patterns ...... . . . . . . . . 72 5.4 The chiasmata avoidance process ....... . . . . . . . . . . . . . 73 5.5 Chromatid interference .......... . .. .. .. . .. .. . . . 75 5.6 Count-location models for chiasmata ...... . . . . . . . . . . . 76 5.7 Renewal process models of chiasma formation ..... . . . . . . . 77 6 Likelihoods on Pedigrees 6.1 The Baum algorithm and "Peeling" .......... . . . . . . . . . . 81 6.2 Exact likelihoods for multiple markers ...... . . . . . . . . . . . 83 6.3 Computations on large but simple pedigrees ..... . . . . . . . . 84 6.4 Example of peeling a zero-loop pedigree ...... . . . . . . . . . . 86 6.5 Computations on complex pedigrees . . . . . . . . . . . . . . . . . . 90 6.6 Models with Gaussian random effects ...... . . . . . . . . . . . 91 7 Monte Carlo Estimates on Pedigrees 93 7.1 Baum algorithm for conditional probabilities ..... . . . . . . . . 93 7.2 An EM algorithm for map estimation ...... . . . . . . . . . . . 95 7.3 Importance sampling for likelihoods ....... . . . . . . . . . . . 96 7.4 Risk probabilities and reverse peeling ...... . . . . . . . . . . . 97 7.5 Elods and SIMLINK ........... ... .. .. .. .. .. .. . 99 7.6 Sequential imputation .......... . .. . .. .. . .. .. . . . 100 8 Markov chain Monte Carlo on Pedigrees 103 8.1 Simulation conditional on data: MCMC ............... . 103 8.2 Single-site updating methods . . . . . . . . . . . . . . . . . . . . . . 107 8.3 Combining exact computation and Monte Carlo ..... . . . . . . 109 8.4 Tightly-linked loci: the M-sampler .................. . 111 9 Likelihood Ratios for Genetic Analysis 115 9.1 Monte Carlo likelihood ratio estimation . . . . . . . . . . . . . . . . 115 9.2 Monte Carlo relative likelihood surfaces . . . . . . . . . . . . . . . . 116 9.3 Monte Carlo EM for the mixed model ...... . . . . . . . . . . . 118 9.4 Likelihood estimators for complex models ...... . . . . . . . . . 120 9.5 Likelihood estimation of gene locations ...... . . . . . . . . . . 123 9.6 Marker ibd and complete-data log-likelihoods . . . . . . . . . . . . . 125 10 Case studies using the M- and LM-samnplers 129 10.1 Background to a study ......................... 129 10.2 Conditional gene ibd probabilities ................... 131 10.3 Likelihoods and log-likelihoods ..................... 133 10.4 Gene ibd in a smaller example . . . . . . . . . . . . . . . . . . . . . 135 10.5 MCMC lod score estimation . . . . . . . . . . . . . . . . . . . . . . 137 10.6 Better MCMC lod scores ....................... . 140 11 Other Monte Carlo Likelihoods in Genetics 147 11.1 Improving pedigree samplers . . . . . . . . . . . . . . . . . . . . . . 147 11.2 Interference by Metropolis-Hastings ................. . 149 11.3 Inference of typing or pedigree error ................. . 154 11.4 Other Monte-Carlo procedures for linkage analysis ..... . . . . . 156 11.5 Monte-Carlo likelihoods in population genetics . . . . . . . . . . . . 156
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