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114 6. Applications of Identity CoefficientsEarl NL, Heyman A, Clark CM, Roses AD (1991) Linkage studies infamilial Alzheimer disease: evidence for chromosome 19 linkage. AmerJ Hum Genet 48:1034–1050[10] Risch N (1990) Linkage strategies for genetically complex traits. I.Multilocus models. Amer J Hum Genet 46:22–228[11] Thompson EA (1986) Pedigree Analysis in Human Genetics. JohnsHopkins University Press, Baltimore[12] Weeks DE, Lange K (1988) The affected-pedigree-member method oflinkage analysis. Amer J Hum Genet 42:315–326[13] Weeks DE, Lange K (1992) A multilocus extension of the affectedpedigree-membermethod of linkage analysis. Amer J Hum Genet50:859–868
7Computation of MendelianLikelihoods7.1 IntroductionRigorous analysis of human pedigree data is a vital concern in genetic epidemiology,human gene mapping, and genetic counseling. In this chapterwe investigate efficient algorithms for likelihood computation on pedigreedata, placing particular stress on the pioneering algorithm of Elston andStewart [8]. It is no accident that their research coincided with the introductionof modern computing. To analyze human pedigree data is tedious,if not impossible, without computers. Pedigrees lack symmetry, andall simple closed-form solutions in mathematics depend on symmetry. Theachievement of Elston and Stewart [8] was to recognize that closed-form solutionsare less relevant than good algorithms. However, the Elston-Stewartalgorithm is not the end of the story. Evaluation of pedigree likelihoods remainsa subject sorely in need of further theoretical improvement. Linkagecalculations alone are among the most demanding computational tasks inmodern biology.7.2 Mendelian ModelsBesides the raw materials of pedigree structure and observed phenotypes,a genetic model is a prerequisite for likelihood calculation. At its mostelementary level, a model postulates the number of loci necessary to explainthe phenotypes. Mendelian models, as opposed to polygenic models, involveonly a finite number of loci. For purposes of discussion, it is convenient touse the term “genotype” when discussing the multilocus, ordered genotypesof an underlying model. Because ordered genotypes preserve phase, theyare preferable to unordered genotypes for theoretical and computationalpurposes. Of course, observed genotypes are always unordered.Any Mendelian model revolves around the three crucial notions of priors,penetrances, and transmission probabilities [8]. Prior probabilitiespertain only to founders. If G is a possible genotype for a founder,then in the absence of other knowledge, Prior(G) is the probability thatthe founder carries genotype G. Almost all models postulate that priorprobabilities conform to Hardy-Weinberg and linkage equilibrium.Penetrance functions specify the likelihood of an observed phenotype X
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Applied ProbabilityKenneth LangeSpr
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Springer Texts in StatisticsAlfrd:
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Kenneth IangeDepartment of Biomathe
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viPrefacebringing up their children
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Preface to the First EditionWhen I
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A Few Words about SoftwarePrefacexi
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ContentsPreface to the Second Editi
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8.4 Multivariate Traits . . . . . .
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15.2 Review of Diffusion Processes
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2 1. Basic Principles of Population
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10 1. Basic Principles of Populatio
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22 2. Counting Methods and the EM A
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40 3. Newton’s Method and Scoring
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60 4. Hypothesis Testing and Catego
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166 8. The Polygenic Model[2] Barnh
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168 8. The Polygenic Model[28] Ott
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170 9. Descent Graph Methods9.2 Rev
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172 9. Descent Graph MethodsConvers
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174 9. Descent Graph Methodsis also
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176 9. Descent Graph Methods❧121/
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178 9. Descent Graph MethodsC D E F
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180 9. Descent Graph Methodsgenetic
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❛ ❛ ❛182 9. Descent Graph Met
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184 9. Descent Graph Methodsgraph C
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186 9. Descent Graph Methodsprocess
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188 9. Descent Graph Methods9.11 Th
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190 9. Descent Graph MethodsThe las
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192 9. Descent Graph Methodssimples
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194 9. Descent Graph Methodsand the
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196 9. Descent Graph Methodslocus t
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198 9. Descent Graph Methods11. For
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200 9. Descent Graph Methods[15] Ke
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10Molecular Phylogeny10.1 Introduct
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10. Molecular Phylogeny 205==(2n
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10. Molecular Phylogeny 207✁❆
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10. Molecular Phylogeny 20910.4 Rev
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10.5 A Nucleotide Substitution Mode
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10. Molecular Phylogeny 213satisfyi
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10. Molecular Phylogeny 215most eco
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10. Molecular Phylogeny 217TABLE 10
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10. Molecular Phylogeny 219reversib
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10. Molecular Phylogeny 221so forth
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10. Molecular Phylogeny 223TABLE 10
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10. Let A and B be the 2 × 2 real
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10. Molecular Phylogeny 227this rev
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10. Molecular Phylogeny 229[14] Lak
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232 11. Radiation Hybrid Mappingon
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234 11. Radiation Hybrid Mapping1 2
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236 11. Radiation Hybrid Mappingpro
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238 11. Radiation Hybrid Mappingpar
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240 11. Radiation Hybrid Mappingtha
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242 11. Radiation Hybrid Mappingvie
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244 11. Radiation Hybrid MappingSim
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246 11. Radiation Hybrid Mappingthe
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248 11. Radiation Hybrid Mappingpri
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250 11. Radiation Hybrid MappingInt
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252 11. Radiation Hybrid MappingSho
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254 11. Radiation Hybrid Mapping[4]
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12Models of Recombination12.1 Intro
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12. Models of Recombination 259dist
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= 1 212. Models of Recombination 26
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12. Models of Recombination 263]+ 2
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12. Models of Recombination 265lies
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12. Models of Recombination 267left
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12. Models of Recombination 269matr
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12. Models of Recombination 271In t
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12.7 Application to Drosophila Data
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12. Models of Recombination 275If G
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12. Models of Recombination 2779. I
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12. Models of Recombination 279[12]
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13Sequence Analysis13.1 Introductio
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13. Sequence Analysis 283using the
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13. Sequence Analysis 285simpler re
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13. Sequence Analysis 287run entire
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13. Sequence Analysis 28913.5 Paral
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13. Sequence Analysis 291gap score
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13. Sequence Analysis 293alignment
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p R s mF (s) =p R s m +(1− s)Q(s)
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13. Sequence Analysis 297[8] Sankof
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300 14. Poisson Approximationin I [
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302 14. Poisson Approximationgene i
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304 14. Poisson Approximationestima
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306 14. Poisson Approximationapprox
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308 14. Poisson ApproximationIt fol
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310 14. Poisson ApproximationThe Ch
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312 14. Poisson Approximationof mat
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314 14. Poisson Approximation10. Co
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316 14. Poisson Approximation[10] G
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318 15. Diffusion Processesrather t
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320 15. Diffusion Processes∫ ∞=
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322 15. Diffusion Processeslation g
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324 15. Diffusion ProcessesIt follo
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326 15. Diffusion Processesfor X 0
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328 15. Diffusion Processes=≈=k 4
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330 15. Diffusion Processesc i+1,k
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332 15. Diffusion Processesc ij→i
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334 15. Diffusion Processesgrown ex
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336 15. Diffusion Processesand the
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338 15. Diffusion Processes11. Supp
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Appendix A: MolecularGenetics in Br
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A.2 From Gene to ProteinAppendix A:
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A.3 Manipulating DNAAppendix A: Mol
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Appendix A: Molecular Genetics in B
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Appendix A: Molecular Genetics in B
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Appendix B: The NormalDistributionB
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matrix I, and characteristic functi
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IndexABO locus, see Blood group loc
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Index 357Coupling × repulsion mati
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Index 359preorder traversal, 208roo
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Index 361relation to disease risk,
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Index 363Meiosis indicators, 189MEN
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Index 365Pulley principle, see Fels
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Index 367Stationary distribution, s
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