1 Basic Genetics ............................................... 1
1.1 Introduction ............................................ 1
1.2 Genes and Chromosomes ................................... 1
1.3 Meiosis ................................................. 2
1.4 Mendel's Laws ........................................... 3
1.4.1 Mendel's First Law ................................ 3
1.4.2 Mender's Second Law ............................... 4
1.5 Linkage and Mapping ..................................... 5
1.6 Interference ............................................ 8
1.7 Quantitative Genetics ................................... 8
1.7.1 Population Properties of Genes .................... 8
1.7.2 A General Quantitative Genetic Model .............. 9
1.7.3 Genetic Models for the Backcross and F2 Design ... 10
1.7.4 Epistatic Model .................................. 12
1.7.5 Heritability and Its Estimation .................. 12
1.7.6 Genetic Architecture ............................. 14
1.7.7 The Estimation of Gene Number .................... 15
1.8 Molecular Genetics ..................................... 16
1.9 SNP .................................................... 18
1.10 Exercises .............................................. 20
1.11 Note ................................................... 20
1.11.1 Modeling Unequal Genetic Effects by the Gamma
Function ........................................ 21
1.11.2 Modeling Unequal Genetic Effects by
a Geometric Series .............................. 22
2 Basic Statistics ............................................ 25
2.1 Introduction ............................................ 25
2.1.1 Populations and Models ............................ 25
2.1.2 Samples ........................................... 27
2.2 Likelihood Estimation ................................... 27
2.3 Hypothesis Testing ...................................... 31
2.3.1 The Pearson Chi-Squared Test ...................... 31
2.3.2 Likelihood Ratio Tests ............................ 33
2.3.3 Simulation-Based Approach ......................... 37
2.3.4 Bayesian Estimation ............................... 38
2.4 Exercises ............................................... 40
3 Linkage Analysis and Map Construction ....................... 43
3.1 Introduction ........................................... 43
3.2 Experimental Design .................................... 44
3.3 Mendelian Segregation .................................. 45
3.3.1 Testing Marker Segregation Patterns .............. 45
3.4 Segregation Patterns in a Full-Sib Family .............. 46
3.5 Two-Point Analysis ..................................... 49
3.5.1 Double Backcross ................................. 50
3.5.2 Double Intercross-F2 ............................. 52
3.6 Three-Point Analysis ................................... 56
3.7 Multilocus Likelihood and Locus Ordering ............... 58
3.8 Estimation with Many Loci .............................. 61
3.9 Mixture Likelihoods and Order Probabilities ............ 62
3.10 Map Functions .......................................... 63
3.10.1 Mather's Formula ................................ 64
3.10.2 The Morgan Map Function ......................... 65
3.10.3 The Haldane Map Function ........................ 65
3.10.4 The Kosambi Map Function ........................ 66
3.11 Exercises .............................................. 69
3.12 Notes: Algorithms and Software for Map Construction .... 71
4 A General Model for Linkage Analysis in Controlled
Crosses ..................................................... 77
4.1 Introduction ............................................ 77
4.2 Fully Informative Markers: A Diplotype Model ............ 78
4.2.1 Two-Point Analysis ................................ 78
4.2.2 A More General Formulation ........................ 81
4.2.3 Three-Point Analysis .............................. 82
4.2.4 A More General Formulation ........................ 86
4.3 Fully Informative Markers: A Genotype Model ............. 88
4.3.1 Parental Diplotypes ............................... 88
4.4 Joint modeling of the Linkage, Parental Diplotype, and
Gene Order .............................................. 91
4.5 Partially Informative Markers ........................... 96
4.5.1 Joint modeling of the Linkage and Parental
Diplotype ......................................... 96
4.5.2 Joint modeling of the Linkage, Parental
Diplotype, and Gene Order ......................... 98
4.6 Exercises ............................................... 99
4.6.1 One dominant marker .............................. 100
4.6.2 One dominant marker and one F2 codominant
marker ........................................... 100
4.7 Notes .................................................. 101
4.7.1 Linkage Analysis ................................. 101
4.7.2 The Diplotype Probability ........................ 102
4.7.3 Gene Order ....................................... 105
4.7.4 M-Point Analysis ................................. 106
5 Linkage Analysis with Recombinant Inbred Lines ............. 107
5.1 Introduction ........................................... 107
5.2 RILs by Selfing ........................................ 107
5.2.1 Two-Point Analysis ............................... 107
5.2.2 Three-Point Analysis ............................. 111
5.3 RILs by Sibling Mating ................................. 117
5.3.1 Two-point Analysis ............................... 117
5.3.2 Three-point Analysis ............................. 118
5.4 Bias Reduction ......................................... 118
5.4.1 RILs by Selfing .................................. 118
5.4.2 RILs by Sibling Mating ........................... 119
5.5 Multiway RILs .......................................... 120
5.6 Exercises .............................................. 120
5.7 Note ................................................... 122
6 Linkage Analysis for Distorted and Misclassified Markers ... 123
6.1 Introduction ........................................... 123
6.2 Gametic Differential Viability ......................... 123
6.2.1 One-Gene Model ................................... 123
6.2.2 Two-Gene Model ................................... 127
6.2.3 Simulation ....................................... 130
6.3 Zygotic Differential Viability ......................... 132
6.3.1 One-Gene Model ................................... 132
6.3.2 Two-Gene Model ................................... 133
6.3.3 Simulation ....................................... 134
6.4 Misclassification ...................................... 134
6.4.1 One-gene Model ................................... 134
6.4.2 Two-Gene Model ................................... 138
6.5 Simulation ............................................. 141
6.6 Exercises .............................................. 142
7 Special Considerations in Linkage Analysis .................. 145
7.1 Introduction ........................................... 145
7.2 Linkage Analysis with a Complicated Pedigree ........... 145
7.2.1 A Nuclear Family ................................. 145
7.2.2 Multipoint Estimation of Identical-By-Descent
Sharing .......................................... 149
7.2.3 A Complex Pedigree ............................... 150
7.3 Information Analysis of Dominant Markers ............... 160
7.3.1 Introduction ..................................... 160
7.3.2 Segregation Analysis ............................. 161
7.3.3 Linkage Analysis ................................. 165
7.4 Exercises .............................................. 168
8 Marker Analysis of Phenotypes .............................. 171
8.1 Introduction ........................................... 171
8.2 QTL Regression Model ................................... 172
8.3 Analysis at the Marker ................................. 174
8.3.1 Two-Sample t Test ................................ 174
8.3.2 Analysis of Variance ............................. 176
8.3.3 Genetic Analysis ................................. 179
8.4 Moving Away from the Marker ............................ 181
8.4.1 Likelihood ....................................... 181
8.5 Power Calculation ...................................... 184
8.6 Marker Interaction Analysis ............................ 188
8.6.1 ANOVA ............................................ 188
8.6.2 Genetic Analysis ................................. 192
8.7 Whole-Genome Marker Analysis ........................... 195
8.8 Exercises .............................................. 198
9 The Structure of QTL Mapping ............................... 203
9.1 Introduction ........................................... 203
9.2 The Mixture Model ...................................... 204
9.2.1 Formulation ...................................... 204
9.2.2 Structure, Setting, and Estimation ............... 206
9.3 Population Genetic Structure of the Mixture Model ...... 207
9.3.1 Backcross / F2 ................................... 207
9.3.2 Outbred Crosses .................................. 208
9.3.3 Recombinant Inbred Lines ......................... 208
9.3.4 Natural Populations .............................. 208
9.4 Quantitative Genetic Structure of the Mixture Model .... 208
9.4.1 Additive-Dominance Model ......................... 209
9.4.2 Additive-Dominance-Epistasis Model ............... 210
9.4.3 Multiplicative-Epistatic Model ................... 211
9.4.4 Mechanistic Model ................................ 212
9.5 Experimental Setting of the Mixture Model .............. 213
9.6 Estimation in the Mixture Model ........................ 214
9.7 Computational Algorithms for the Mixture Model ......... 216
9.7.1 EM Algorithm ..................................... 216
9.7.2 Monte Carlo EM ................................... 216
9.7.3 Stochastic EM .................................... 217
9.7.4 An EM Algorithm / Newton-Raphson Hybrid .......... 217
9.7.5 Some Cautions .................................... 218
9.7.6 Bayesian Methods ................................. 218
9.7.7 Estimating the Number of Components in
a Mixture Model .................................. 220
9.8 Exercises .............................................. 221
10 Interval Mapping with Regression Analysis .................. 223
10.1 Introduction .......................................... 223
10.2 Linear Regression Model ............................... 224
10.3 Interval Mapping in the Backcross ..................... 224
10.3.1 Conditional Probabilities ...................... 224
10.3.2 Conditional Regression Model ................... 226
10.3.3 Estimation and Test ............................ 228
10.4 Interval Mapping in an F2 ............................. 230
10.5 Remarks ............................................... 233
10.6 Exercises ............................................. 234
11 Interval Mapping by Maximum Likelihood Approach ............ 237
11.1 Introduction .......................................... 237
11.2 QTL Interval Mapping in a Backcross ................... 238
11.2.1 The Likelihood ................................. 238
11.2.2 Maximizing the Likelihood ...................... 240
11.3 Hypothesis Testing .................................... 246
11.3.1 Model for Incorporating Double Recombination ... 252
11.3.2 Model for Incorporating Interference ........... 252
11.4 QTL Interval Mapping in an F2 ......................... 254
11.4.1 No Double Recombination ........................ 254
11.4.2 Independence ................................... 257
11.4.3 Interference ................................... 260
11.4.4 Testing Hypotheses ............................. 262
11.5 Factors That Affect QTL Detection ..................... 262
11.6 Procedures for QTL Mapping ............................ 263
11.6.1 The Number of QTLs ............................. 263
11.6.2 Locations of Individual QTLs ................... 266
11.7 Exercises ............................................. 267
12 Threshold and Precision Analysis ........................... 269
12.1 Introduction .......................................... 269
12.2 Threshold Determination ............................... 270
12.2.1 Background ..................................... 270
12.2.2 Analytical Approximations ...................... 271
12.2.3 Simulation Studies ............................. 275
12.2.4 Permutation Tests .............................. 275
12.2.5 A Quick Approach ............................... 276
12.2.6 A Score Statistic .............................. 277
12.3 Precision of Parameter Estimation ..................... 279
12.3.1 Asymptotic Variance-Covariance Matrix .......... 279
12.3.2 Simulation Studies ............................. 282
12.4 Confidence Intervals for the QTL Location ............. 283
12.5 Exercises ............................................. 285
13 Composite QTL Mapping ...................................... 287
13.1 Introduction .......................................... 287
13.2 Composite Interval Mapping for a Backcross ............ 288
13.2.1 The Likelihood ................................. 288
13.2.2 Maximizing the Likelihood ...................... 289
13.2.3 Hypothesis Testing ............................. 290
13.3 Composite Interval Mapping for an F2 .................. 291
13.4 A Statistical Justification of Composite Interval
Mapping ............................................... 293
13.4.1 Conditional Marker (Co)variances ............... 293
13.4.2 Conditional QTL Variance ....................... 295
13.4.3 Marker Selection ............................... 298
13.5 Comparisons Between Composite Interval Mapping and
Interval Mapping ...................................... 298
13.6 Multiple Interval Mapping ............................. 301
13.7 Exercises ............................................. 302
14 QTL Mapping in Outbred Pedigrees ........................... 303
14.1 Introduction .......................................... 303
14.2 A Fixed-Effect Model for a Full-Sib Family ............ 304
14.2.1 Introduction ................................... 304
14.2.2 A Mixture Model for a Parental Diplotype ....... 304
14.2.3 Quantitative Genetic Model ..................... 308
14.2.4 Likelihood Analysis ............................ 309
14.2.5 Fitting Marker Phenotypes ...................... 312
14.2.6 Hypothesis Tests ............................... 313
14.2.7 The Influence of Linkage Phases ................ 316
14.3 Random-Effect Mapping Model for a Complicated
Pedigree .............................................. 317
14.3.1 Introduction ................................... 317
14.3.2 Statistical Model .............................. 319
14.3.3 IBD at a QTL ................................... 320
14.3.4 The Likelihood ................................. 321
14.3.5 Hypothesis Testing ............................. 323
14.4 Exercises ............................................. 325
A General Statistical Results and Algorithms ................. 331
A.1 Likelihood Asymptotics ................................. 331
A.2 General Form of the EM Algorithm ....................... 332
B R Programs ................................................. 335
B.1 Chapter 2 .............................................. 335
B.2 Chapter 8 .............................................. 337
C References ................................................. 343
Author Index ............................................... 355
Subject Index .............................................. 361
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