Generalized Linear Models with Examples in R

Description

Preface......Page 3 Contents......Page 6 1.2 Conventions for Describing Data......Page 16 1.3 Plotting Data......Page 20 1.4 Coding for Factors......Page 25 1.5 Statistical Models Describe Both Random and Systematic Features of Data......Page 26 1.6 Regression Models......Page 27 1.7 Interpreting Regression Models......Page 31 1.8 All Models Are Wrong, but Some Are Useful......Page 32 1.9 The Purpose of a Statistical Model Affects How It Is Developed......Page 33 1.10 Accuracy vs Parsimony......Page 34 1.11 Experiments vs Observational Studies: Causality vs Association......Page 36 1.12 Data Collection and Generalizability......Page 37 1.13 Using R for Statistical Modelling......Page 38 1.14 Summary......Page 39 Problems......Page 40 References......Page 44 2.2 Linear Regression Models Defined......Page 46 2.3.1 Least-Squares Estimation......Page 50 2.3.2 Coefficient Estimates......Page 51 2.3.3 Estimating the Variance σ2......Page 53 2.3.5 Standard Errors of Fitted Values......Page 54 2.4.1 Coefficient Estimates......Page 55 2.4.3 Standard Errors......Page 57 to* 2.5.1 Matrix Notation......Page 58 to* 2.5.2 Coefficient Estimates......Page 59 to* 2.5.3 Estimating the Variance σ2......Page 61 to* 2.5.5 Estimating the Variance of Fitted Values......Page 62 2.6 Fitting Linear Regression Models Using R......Page 63 2.7 Interpreting the Regression Coefficients......Page 67 2.8.2 The Distribution of j......Page 68 2.8.3 Hypothesis Tests for βj......Page 69 2.8.4 Confidence Intervals for βj......Page 70 2.8.5 Confidence Intervals for μ......Page 71 2.9 Analysis of Variance for Regression Models......Page 73 2.10.1 Analysis of Variance to Compare Two NestedModels......Page 76 2.10.2 Sequential Analysis of Variance......Page 78 2.10.3 Parallel and Independent Regressions......Page 81 2.11 Choosing Between Non-nested Models: AIC and BIC......Page 85 2.12.1 Adding and Dropping Variables......Page 87 2.12.2 Automated Methods for Model Selection......Page 88 2.13 Case Study......Page 91 2.14 Using R for Fitting Linear Regression Models......Page 94 2.15 Summary......Page 97 Problems......Page 98 References......Page 105 3.1 Introduction and Overview......Page 107 3.2.3 Constant Variance......Page 108 3.2.4 Independence......Page 109 3.2.7 Approximations and Consequences......Page 110 3.3 Residuals for Normal Linear Regression Models......Page 111 3.4.1 Leverage and Extreme Covariate Values......Page 112 to* 3.4.2 The Leverages Using Matrix Algebra......Page 114 3.5.1 Plot Residuals Against xj: Linearity......Page 115 3.5.2 Partial Residual Plots......Page 116 3.5.3 Plot Residuals Against : Constant Variance......Page 118 3.5.4 Q–Q Plots and Normality......Page 119 3.5.5 Lag Plots and Dependence over Time......Page 120 3.6.1 Introduction......Page 122 3.6.2 Outliers and Studentized Residuals......Page 123 3.6.3 Influential Observations......Page 124 3.8 Remedies: Fixing Identified Problems......Page 129 3.9.1 Symmetry, Constraints and the Ladder of Powers......Page 130 3.9.2 Variance-Stabilizing Transformations......Page 131 3.9.3 Box–Cox Transformations......Page 134 3.10 Simple Transformations of Covariates......Page 135 3.11 Polynomial Trends......Page 141 3.12 Regression Splines......Page 145 3.13 Fixing Identified Outliers......Page 148 3.14 Collinearity......Page 149 3.15.1 Case Study 1......Page 152 3.15.2 Case Study 2......Page 155 3.16 Using R for Diagnostic Analysis of Linear RegressionModels......Page 160 3.17 Summary......Page 161 Problems......Page 163 References......Page 176 4.2.1 When Linear Models Are a Poor Choice......Page 179 4.2.2 Binary Outcomes and Binomial Counts......Page 180 4.2.3 Unrestricted Counts: Poisson or Negative Binomial......Page 182 4.2.4 Continuous Positive Observations......Page 183 4.3 Generalizing the Normal Linear Model......Page 185 4.4 The Idea of Likelihood Estimation......Page 186 4.5.1 Score Equations......Page 190 4.5.2 Information: Observed and Expected......Page 191 4.5.3 Standard Errors of Parameters......Page 193 4.6.1 Score Equations......Page 194 4.6.2 Information: Observed and Expected......Page 196 to* 4.7.2 Score Equations......Page 197 to* 4.7.3 Information: Observed and Expected......Page 198 to* 4.8 Fisher Scoring for Computing MLEs......Page 200 4.9.2 Properties of MLEs for One Parameter......Page 203 to* 4.9.3 Properties of MLEs for Many Parameters......Page 204 4.10.1 Introduction......Page 205 to* 4.10.2 Global Tests......Page 208 to* 4.10.3 Tests About Subsets of Parameters......Page 210 4.10.4 Tests About One Parameter in a Set of Parameters......Page 211 4.10.5 Comparing the Three Methods......Page 213 4.11.2 Confidence Intervals for Single Parameters......Page 214 4.12 Comparing Non-nested Models: The AIC and BIC......Page 216 to* 4.14 Appendix: R Code to Fit Models to the Quilpie RainfallData......Page 218 Problems......Page 220 References......Page 223 5.2 The Two Components of Generalized Linear Models......Page 224 5.3.2 Definition of EDMs......Page 225 5.3.3 Generating Functions......Page 227 5.3.4 The Moment Generating and Cumulant Functions for EDMs......Page 228 5.3.5 The Mean and Variance of an EDM......Page 229 5.3.6 The Variance Function......Page 230 5.4.1 TheUnitDevianceandtheDispersionModelForm......Page 231 5.4.2 The Saddlepoint Approximation......Page 236 5.4.3 The Distribution of the Unit Deviance......Page 237 5.4.4 Accuracy of the Saddlepoint Approximation......Page 238 5.4.5 Accuracy of the χ21 Distribution for the UnitDeviance......Page 239 5.5.2 Offsets......Page 242 5.6 Generalized Linear Models Defined......Page 243 5.7 The Total Deviance......Page 244 5.8 Regression Transformations Approximate GLMs......Page 245 5.9 Summary......Page 247 Problems......Page 248 References......Page 253 6.2.1 Differentiating the Probability Function......Page 255 6.2.2 Score Equations and Information for β......Page 256 6.3 Computing Estimates of β......Page 257 6.4 The Residual Deviance......Page 260 to* 6.6 Estimation of β: Matrix Formulation......Page 262 6.8.1 Introduction......Page 264 6.8.3 Modified Profile Log-Likelihood Estimator of φ......Page 265 6.8.4 Mean Deviance Estimator of φ......Page 266 6.8.6 Which Estimator of φ Is Best?......Page 267 6.9 Using R to Fit GLMs......Page 269 6.10 Summary......Page 271 Problems......Page 273 References......Page 274 7.2.1 Wald Tests for Single Regression Coefficients......Page 276 7.2.2 Confidence Intervals for Individual Coefficients......Page 277 7.2.3 Confidence Intervals for μ......Page 278 7.2.4 Likelihood Ratio Tests to Compare Nested Models: χ2 Tests......Page 280 7.2.5 Analysis of Deviance Tables to Compare Nested Models......Page 281 7.2.6 Score Tests......Page 282 to* 7.2.7 Score Tests Using Matrices......Page 283 7.3 Large Sample Asymptotics......Page 284 7.4.1 The Idea of Goodness-of-Fit......Page 285 7.4.3 Pearson Goodness-of-Fit Test......Page 286 7.5 Small Dispersion Asymptotics......Page 287 7.6.1 Wald Tests for Single Regression Coefficients......Page 289 7.6.2 Confidence Intervals for Individual Coefficients......Page 291 to* 7.6.3 Confidence Intervals for μ......Page 292 7.6.4 Likelihood Ratio Tests to Compare Nested Models: F-Tests......Page 293 7.6.5 Analysis of Deviance Tables to Compare Nested Models......Page 295 7.6.6 Score Tests......Page 297 7.7 Comparing Wald, Score and Likelihood Ratio Tests......Page 298 7.8 Choosing Between Non-nested GLMs: AIC and BIC......Page 299 7.9 Automated Methods for Model Selection......Page 300 7.10 Using R to Perform Tests......Page 301 7.11 Summary......Page 303 Problems......Page 304 References......Page 307 8.2 Assumptions of GLMs......Page 308 8.3.1 Response Residuals Are Insufficient for GLMs......Page 309 8.3.2 Pearson Residuals......Page 310 8.3.4 Quantile Residuals......Page 311 8.3.4.1 Quantile Residuals: Continuous Response......Page 312 8.3.4.2 Quantile Residuals: Discrete Response......Page 313 to* 8.4.2 The Hat Matrix......Page 315 8.5 Leverage Standardized Residuals for GLMs......Page 316 8.7.1 Introduction......Page 317 8.7.3 Plots to Check the Systematic Component......Page 318 8.7.4 Plots to Check the Random Component......Page 322 8.8.2 Outliers and Studentized Residuals......Page 323 8.8.3 Influential Observations......Page 324 8.9 Remedies: Fixing Identified Problems......Page 326 8.10 Quasi-Likelihood and Extended Quasi-Likelihood......Page 329 8.11 Collinearity......Page 332 8.12 Case Study......Page 333 8.13 Using R for Diagnostic Analysis of GLMs......Page 336 8.14 Summary......Page 337 Problems......Page 338 References......Page 341 9.2 Modelling Proportions......Page 343 9.3 Link Functions......Page 346 9.4 Tolerance Distributions and the Probit Link......Page 348 9.5 Odds, Odds Ratios and the Logit Link......Page 350 9.6 Median Effective Dose, ED50......Page 353 9.7 The Complementary Log-Log Link in Assay Analysis......Page 354 9.8 Overdispersion......Page 357 9.9 When Wald Tests Fail......Page 361 9.11 Case Study......Page 364 9.13 Summary......Page 370 Problems......Page 371 References......Page 377 10.2 Summary of Poisson GLMs......Page 380 10.3 Modelling Rates......Page 382 10.4.2 Two Dimensional Tables: Systematic Component......Page 387 10.4.3.2 No Marginal Totals Are Fixed......Page 389 10.4.3.3 The Grand Total Is Fixed......Page 392 10.4.3.4 The Column (or Row) Totals Are Fixed......Page 393 10.4.4.1 Introduction......Page 394 10.4.4.2 Mutual Independence......Page 395 10.4.4.4 Conditional Independence......Page 396 10.4.4.5 Uniform Association......Page 397 10.4.5 Simpson's Paradox......Page 398 10.4.6 Equivalence of Binomial and Poisson GLMs......Page 401 10.4.7 Higher-Order Tables......Page 402 10.4.8 Structural Zeros in Contingency Tables......Page 404 10.5.1 Overdispersion for Poisson GLMs......Page 406 10.5.2 Negative Binomial GLMs......Page 408 10.5.3 Quasi-Poisson Models......Page 411 10.6 Case Study......Page 413 10.8 Summary......Page 420 Problems......Page 421 References......Page 431 11.2 Modelling Positive Continuous Data......Page 434 11.3 The Gamma Distribution......Page 436 11.4 The Inverse Gaussian Distribution......Page 440 11.5 Link Functions......Page 442 11.6.1 Estimating φ for the Gamma Distribution......Page 445 11.6.2 Estimating φ for the Inverse Gaussian Distribution......Page 448 11.7.1 Case Study 1......Page 449 11.7.2 Case Study 2......Page 451 11.9 Summary......Page 454 Problems......Page 455 References......Page 463 12.2.1 Introducing Tweedie Distributions......Page 466 12.2.2 The Structure of Tweedie EDMs......Page 469 12.2.3 Tweedie EDMs for Positive Continuous Data......Page 470 12.2.4 Tweedie EDMs for Positive Continuous Data with Exact Zeros......Page 472 12.3.1 Introduction......Page 473 12.3.2 Estimation of the Index Parameter ξ......Page 474 12.3.3 Fitting Tweedie GLMs......Page 478 12.4.1 Case Study 1......Page 482 12.4.2 Case Study 2......Page 484 12.5 Using R to Fit Tweedie GLMs......Page 487 12.6 Summary......Page 488 Problems......Page 489 References......Page 497 Problems......Page 500 References......Page 509 A.2.1 Introduction to R......Page 511 A.2.4 Downloading and Installing R Packages......Page 512 A.2.5 Using R Packages......Page 513 A.3.1 Basic Use of R as an Advanced Calculator......Page 514 A.3.4 Variable Names in R......Page 516 A.3.5 Working with Vectors in R......Page 517 A.3.6 Loading Data into R......Page 519 A.3.7 Working with Data Frames in R......Page 521 A.3.8 Using Functions in R......Page 522 A.3.9 Basic Statistical Functions in R......Page 523 A.3.10 Basic Plotting in R......Page 524 A.3.11 Writing Functions in R......Page 526 to* A.3.12 Matrix Arithmetic in R......Page 528 References......Page 531 GLMsData package......Page 533 References......Page 535 Solutions from Chap.1......Page 536 Solutions from Chap.2......Page 537 Solutions from Chap.3......Page 539 Solutions from Chap.4......Page 541 Solutions from Chap.5......Page 543 Solutions from Chap.7......Page 544 Solutions from Chap.9......Page 546 Solutions from Chap.10......Page 548 Solutions from Chap.11......Page 551 Solutions from Chap.12......Page 554 Solutions from Chap.13......Page 555 References......Page 557 Data sets......Page 558 R commands......Page 560 Index......Page 564