STATISTICAL & BIOMETRIC TABLES
1 Probability Theory (i): One Random Variable 1 11 Introduction 1 12 Discrete RandomVariables, Definitions 3 121 Probability Distributions and Parameters 3 122 Independence 6 13 Six Important Discrete Probability Distributions 8 131 One Bernoulli Trial 8 132 The Binomial Distribution 8 133 The Hypergeometric Distribution 10 134 The UniformDistribution 13 135 The Geometric Distribution 14 136 The Negative Binomial and the Generalized Geometric Distributions 16 137 The Poisson Distribution 17 138 Approximations 18 14 TheMean of a Discrete RandomVariable 19 15 The Variance of a Discrete RandomVariable 21 16 General Moments of a Probability Distribution 23 17 The Probability-Generating Function 24 18 Continuous RandomVariables 26 19 The Mean, Variance, and Median of a Continuous Random Variable 28 191 Definitions 28 192 Chebyshev’s Inequality 29 110 Five Important Continuous Distributions 30 1101 The UniformDistribution 30 1102 The Normal Distribution 31 1103 The Normal Approximation to a Discrete Distribution 32 1104 The Exponential Distribution 34 1105 The Gamma Distribution 36 1106 The Beta Distribution 37 111 TheMoment-Generating Function 38 112 Events 41 1121 What Are Events? 41 1122 Complements, Unions, and Intersections 42 1123 Probabilities of Events 43 1124 Conditional Probabilities 44 1125 Independence of Events 46 113 The Memoryless Property of the Geometric and the Exponential Distributions 48 114 Entropy and Related Concepts 49 1141 Entropy 49 1142 Relative Entropy 50 1143 Scores and Support 50 115 Transformations 51 116 EmpiricalMethods 53 2 Probability Theory (ii): Many Random Variables 62 21 Introduction 62 22 The Independent Case 64 23 Generating Functions 66 231 Properties of Probability-Generating Functions 66 232 Properties of Moment-Generating Functions 68 24 The Dependent Case 70 241 Covariance and Correlation 70 242 TheMultinomial Distribution 71 243 TheMultivariate Normal Distribution 72 25 Marginal Distributions 73 26 Conditional Distributions 75 27 Expected Values of Functions of Many Random Variables 80 28 Asymptotic Distributions 83 29 Indicator Random Variables 83 291 Definitions 83 292 Example: Sequencing EST Libraries 84 210 Derived Random Variables (i): Sums, Averages, and Minima 87 2101 Sums and Averages 87 2102 The Minimum of n RandomVariables 90 211 Derived Random Variables (ii): The Maximum of n RandomVariables 92 2111 Distributional Properties: Continuous Random Variables 92 2112 Distributional Properties: Discrete Random Variables 94 2113 An Asymptotic Formula for the Distribution of Xmax 97 212 Order Statistics 99 2121 Definition 99 2122 Example: The UniformDistribution 101 2123 The SampleMedian 102 213 Transformations 103 3 Statistics (i): An Introduction to Statistical Inference 111 31 Introduction 111 32 Classical and BayesianMethods 112 33 Classical EstimationMethods 113 331 Unbiased Estimation 114 332 Confidence Intervals 114 333 Biased Estimators 116 34 Classical Hypothesis Testing 118 341 General Principles 118 342 P-Values 122 343 Power Calculations 123 35 Hypothesis Testing: Examples 124 351 Example 1 Testing for a Mean: the One-Sample Case 124 352 Example 2 The Two-Sample t-Test 125 353 Example 3 Tests on Variances 129 354 Example 4 Testing for the Parameters in a Multinomial Distribution 129 355 Example 5 Association tests 130 36 Likelihood Ratios, Information, and Support 133 37 Hypothesis Testing Using a Maximum as Test Statistic 134 371 The Normal Distribution 134 372 P-values for the Maximum of Geometric Random Variables 136 38 Nonparametric Alternatives to the One-Sample and Two- Sample t-Tests 138 381 The Two-Sample Permutation Test 139 382 TheMann–Whitney Test 142 383 The Wilcoxon Signed-Rank Test 143 384 What is Assumed in a Non-Parametric Test? 145 39 The Bayesian Approach to Hypothesis Testing 146 310 The Bayesian Approach to Estimation 148 311 Multiple Testing 149 312 Combining the Results of Several Experiments 151 4 Stochastic Processes (i): Poisson Processes and Markov Chains 155 41 The Homogeneous Poisson Process and the Poisson Distribution 155 42 The Poisson and the Binomial Distributions 158 43 The Poisson and the Gamma Distributions 159 44 The Pure-Birth Process 159 45 Introduction to FiniteMarkov Chains 161 46 Transition Probabilities and the Transition Probability Matrix 162 47 Markov Chains with Absorbing States 164 48 Markov Chains with No Absorbing States 164 481 Stationary Distributions 165 482 Example 166 49 The Graphical Representation of a Markov Chain 167 410 Modeling 168 5 The Analysis of One DNA Sequence 174 51 Shotgun Sequencing 174 511 Introduction 174 512 Contigs 175 513 Anchored Contigs 179 52 Modeling DNA 183 53 Modeling Signals in DNA 185 531 Introduction 185 532 Weight Matrices: Independence 185 533 Markov Dependencies 186 534 Maximal Dependence Decomposition 187 54 Long Repeats 188 55 r-Scans 192 56 The Analysis of Patterns 196 561 Introduction 196 562 Counting conventions 197 563 Notation and assumptions 198 57 Overlaps Counted 198 571 Number of Occurrences 198 572 Approximations to the Distribution of Y1(N) 201 573 Distance Between Occurrences 203 574 Beginning at the Origin 205 58 Overlaps Not Counted 207 581 General Comments 207 582 Distance Between Recurrences 208 583 Number of Recurrences 209 59 Motifs 211 6 The Analysis of Multiple DNA or Protein Sequences 220 61 Two Sequences: Frequency Comparisons 220 62 Alignments 222 63 Simple Tests for Significant Similarity in an Alignment 223 64 Alignment Algorithms for Two Sequences 228 641 Introduction 228 642 Gapped Global Comparisons and Dynamic Programming Algorithms 230 643 Fitting One Sequence into Another Using a Linear GapModel 233 644 Local Alignments with a Linear Gap Model 234 645 Other GapModels 235 646 Limitations of the Dynamic Programming Alignment Algorithms 237 65 Protein Sequences and Substitution Matrices 238 651 Introduction 238 652 BLOSUM Substitution Matrices 239 653 PAM Substitution Matrices 244 654 A Simple Symmetric EvolutionaryMatrix 248 66 Multiple Sequences 250 7 Stochastic Processes (ii): Random Walks 257 71 Introduction 257 72 The Simple RandomWalk 259 721 Introduction 259 73 The Difference Equation Approach 260 731 Absorption Probabilities 260 732 Mean Number of Steps Taken Until the Walk Stops 261 74 TheMoment-Generating Function Approach 262 741 Introduction andWald’s identity 262 742 Absorption Probabilities 263 743 Mean Number of Steps Until the Walk Stops 264 744 An Asymptotic Case 265 75 GeneralWalks 266 76 General Walks: Asymptotic Theory 268 761 Introduction 268 762 The Renewal Theorem 268 763 UnrestrictedWalks 269 764 RestrictedWalks 271 8 Statistics (ii): Classical Estimation Theory 275 81 Introduction 275 82 Criteria for “Good” Estimators 275 83 MaximumLikelihood Estimation 277 831 The Discrete Case 277 832 The Continuous Case 279 833 Invariance Property 281 834 Asymptotic Properties 281 835 Many Parameters 285 84 OtherMethods of Estimation 285 841 Introduction 285 842 TheMethod ofMoments 285 843 Least Squares and Multiple Regression 287 85 MultivariateMethods 291 851 Introduction 291 852 Parameter Estimation 291 853 Principal Components 292 86 Bootstrap Methods: Estimation and Confidence Intervals 295 861 Introduction 295 862 The “Plug-in” Concept 295 863 Classical EstimationMethods 296 864 Bootstrap EstimationMethods 297 865 Bootstrap Confidence Intervals 299 866 Examples 299 9 Statistics (iii): Classical Hypothesis Testing Theory 304 91 Introduction 304 92 Simple Fixed-Sample-Size Tests 304 921 The Likelihood Ratio 304 922 The Neyman–Pearson Lemma 305 923 Example Sequencematching 307 93 Classical Hypothesis Testing: Composite Fixed-Sample- Size Tests 308 931 Introduction 308 932 Parameter Spaces and the Likelihood Ratio ? 308 933 Example: t-tests 309 94 The ?2 log ? Approximation 313 941 Theory 313 942 Examples 315 95 The Analysis of Variance (ANOVA) 318 951 Introduction 318 952 From t to F: Sums of Squares and the F Statistic 319 953 One-way Fixed Effects ANOVA 320 954 The Two-Way Fixed Effects ANOVA 323 955 Multi-Way Fixed Effects ANOVAs 326 956 The 2m Design, Confounding, and Fractional Replication 326 957 RandomandMixed EffectsModels 328 96 MultivariateMethods 329 961 Introduction 329 962 One-sample T2 Tests 330 963 Two-sample T2 Tests 331 964 Optimal Linear Functions: Discriminant Functions 332 97 ANOVA: the RepeatedMeasures Case 332 98 Bootstrap Methods: the Two-sample t-test 334 99 Sequential Analysis 336 991 The Sequential Probability Ratio Test 336 992 The Power Function for a Sequential Test 338 993 TheMean Sample Size 340 994 The Effect of Boundary Overshoot 342 10 BLAST 345 101 Introduction 345 102 The Comparison of Two Aligned Sequences 346 1021 Introduction 346 1022 The BLAST RandomWalk 346 1023 Parameter Calculations 347 1024 The Choice of a Score 349 1025 Bounds and Approximations for the BLAST PValue 351 1026 The Normalized and the Bit Scores 354 1027 The Number of High-Scoring Excursions 355 1028 The Karlin–Altschul Sum Statistic 356 103 The Comparison of Two Unaligned Sequences 358 1031 Introduction 358 1032 Theoretical and Empirical Background 359 1033 Edge Effects 361 1034 Multiple Testing 363 104 The Comparison of a Query Sequence Against a Database 364 105 Printouts 365 1051 Example 366 1052 AMore Complicated Example 369 106 MinimumSignificance Lengths 372 1061 A Correct Choice of n 372 1062 An Incorrect Choice of n 374 107 BLAST: A Parametric or a Non-parametric Test? 375 108 Gapped BLAST and PSI BLAST 376 1081 Gapped BLAST 376 1082 PSI BLAST 378 109 Relation to Sequential Analysis 380 11 Stochastic Processes (iii): Markov Chains 384 111 Introduction 384 112 Markov Chains with No Absorbing States 385 1121 Introduction 385 1122 Convergence to the Stationary Distribution 385 1123 Stationary Distributions: A Numerical Example 386 1124 Reversibility and Detailed Balance 386 113 Higher-OrderMarkov Dependence 389 1131 Testing for Higher-Order Markov Dependence 389 1132 Testing for a Uniform Stationary Distribution 389 114 Patterns in Sequences with First-Order Markov Dependence 390 115 Markov ChainMonte Carlo 392 1151 The Hastings–Metropolis Algorithm 392 1152 Gibbs Sampling 393 1153 Simulated Annealing 395 116 Markov Chains with Absorbing States 398 1161 Theory 398 1162 Examples 399 117 Continuous-TimeMarkov Chains 403 1171 Definitions 403 1172 Time-Dependent Solutions 404 1173 The Stationary Distribution 404 1174 Detailed Balance 405 1175 Exponential Holding Times 405 1176 The Embedded Chain 406 12 Hidden Markov Models 409 121 What is a HiddenMarkovModel? 409 122 Three Algorithms 411 1221 The Forward and Backward Algorithms 411 1222 The Viterbi Algorithm 413 1223 The Estimation Algorithms 414 123 Applications 417 1231 Modeling Protein Families 417 1232 Multiple Sequence Alignments 419 1233 Pfam 421 1234 Gene Finding 422 13 Gene Expression, Microarrays, and Multiple Testing 430 131 Introduction 430 1311 Introduction toMicroarrays 430 1312 Microarray Data 432 1313 Sources of Bias and Variation 438 132 The Statistical Analysis of Microarray Data: One Gene 443 1321 Introduction 443 1322 Determining Whether a Gene is Expressed 443 1323 Testing for Differential Expression 444 133 Differential Expression – Multiple Genes 449 1331 Introduction 449 1332 Ranked lists 449 1333 The Choice of Statistic 450 1334 Confidence Measures 451 1335 The Family-Wise Error Rate (FWER) 454 1336 The False Discovery Rate (FDR) 459 1337 The ANOVA Approach:Many Genes 467 1338 Comparing Two Groups by Discriminant Analysis 470 134 Principal Components andMicroarrays 471 135 ClusteringMethods 471 1351 Hierarchical Clustering 471 1352 Other Forms of Clustering 472 14 Evolutionary Models 475 141 Models of Nucleotide Substitution 475 142 Discrete-TimeModels 476 1421 The Jukes–CantorModel 476 1422 The KimuraModels 478 1423 Further Generalizations of the Kimura Models 479 1424 The FelsensteinModels 480 1425 The HKYModel 482 1426 OtherModels 483 1427 The Reversibility Criterion 483 1428 The Simple Symmetric Amino AcidModel 484 143 Continuous-TimeModels 484 1431 The Continuous-Time Jukes–Cantor Model 485 1432 The Continuous-Time KimuraModel 489 1433 The Continuous-Time FelsensteinModel 492 1434 The Continuous-Time HKYModel 493 1435 Continuous-Time Amino AcidModel 493 1436 A Remark About Bias 494 15 Phylogenetic Tree Estimation 497 151 Introduction 497 152 Distances 499 153 Tree Reconstruction: The Ultrametric Case 501 154 Tree Reconstruction: the Neighbor-Joining Approach 505 155 Inferred Distances 509 156 Tree Reconstruction: Parsimony 511 157 Tree Estimation:MaximumLikelihood 512 158 Example 517 159 Modeling, Estimation, and Hypothesis Testing 522 1591 Estimation and Hypothesis Testing 522 1592 Bootstrapping and Phylogenies 523 1593 Assumptions and Problems 528 1594 Phylogenetic Models and Hypothesis Testing 531