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