Mixed Effects Models and Extensions in Ecology with R

Contributors......Page 0
 1.1 What Is in the Book?......Page 19
  1.1.1 To Include or Not to Include GLM and GAM......Page 21
  1.1.3 Flowchart of the Content......Page 22
 1.2 Software......Page 23
 1.4 What We Did Not Do and Why......Page 24
 1.5 How to Cite R and Associated Packages......Page 25
 1.6 Our R Programming Style......Page 26
 1.7 Getting Data into R......Page 27
  1.7.1 Data in a Package......Page 28
 2.1 Data Exploration......Page 29
  2.1.1 Cleveland Dotplots......Page 30
  2.1.2 Pairplots......Page 32
  2.1.4 xyplot from the Lattice Package......Page 33
 2.2 The Linear Regression Model......Page 35
  2.3.2 Normality......Page 37
  2.3.3 Heterogeneity......Page 38
  2.3.5 Independence......Page 39
  2.3.6 Example 1; Wedge Clam Data......Page 40
   2.3.6.1 Model Validation......Page 41
  2.3.7 Example 2; Moby's Teeth......Page 44
  2.3.8 Example 3; Nereis......Page 46
 2.4 Where to Go from Here......Page 48
 3.1 Introduction......Page 52
 3.2 Additive Modelling......Page 53
  3.2.1 GAM in gam and GAM in mgcv......Page 54
  3.2.2 GAM in gam with LOESS......Page 55
   3.2.2.1 LOESS Smoothing......Page 56
  3.2.3 GAM in mgcv with Cubic Regression Splines......Page 59
 3.3 Technical Details of GAM in mgcv......Page 61
  3.3.1 A (Little) Bit More Technical Information on Regression Splines......Page 64
  3.3.2 Smoothing Splines Alias Penalised Splines......Page 66
  3.3.3 Cross-Validation......Page 68
  3.3.5 Two More Things......Page 70
 3.4 GAM Example 1; Bioluminescent Data for Two Stations......Page 72
  3.4.1 Interaction Between a Continuous and Nominal Variable......Page 76
 3.5 GAM Example 2: Dealing with Collinearity......Page 80
 3.6 Inference      Means that it is difficult and that it can be skipped upon first reading.......Page 83
 3.7 Summary and Where to Go from Here?......Page 84
to 4  Dealing with Heterogeneity......Page 87
  4.1.1 Linear Regression Applied on Squid......Page 88
  4.1.2 The Fixed Variance Structure......Page 90
  4.1.3 The VarIdent Variance Structure......Page 91
  4.1.4 The varPower Variance Structure......Page 94
  4.1.6 The varConstPower Variance Structure......Page 96
  4.1.7 The varComb Variance Structure......Page 97
  4.1.8 Overview of All Variance Structures......Page 98
  4.1.9 Graphical Validation of the Optimal Model......Page 100
  4.2.1 Linear Regression Applied on the Benthic Biodiversity Data......Page 102
  4.2.2 GLS Applied on the Benthic Biodiversity Data......Page 105
  4.2.3 A Protocol......Page 106
  4.2.4 Application of the Protocol on the Benthic Biodiversity Data......Page 108
   4.2.4.1 Round 1 of the Backwards Selection......Page 110
   4.2.4.2 Round 2 of the Backwards Selection......Page 111
   4.2.4.3 Round 3 of the Backwards Selection......Page 112
   4.2.4.5 The Aftermath......Page 113
 5.1 Introduction......Page 117
 5.2 2Stage Analysis Method......Page 118
  5.3.1 Introduction......Page 121
  5.3.2 The Random Intercept Model......Page 122
  5.3.3 The Random Intercept and Slope Model......Page 125
  5.3.4 Random Effects Model......Page 127
 5.4 Induced Correlations......Page 128
 5.5 The Marginal Model......Page 130
 5.6 Maximum Likelihood and REML Estimation......Page 132
  5.6.1 Illustration of Difference Between ML and REML......Page 135
 5.7 Model Selection in (Additive) Mixed Effects Modelling......Page 136
  5.8.1 The Wrong Approach......Page 138
   5.8.1.1 Step 2 of the Protocol......Page 139
   5.8.1.2 Step 3 of the Protocol......Page 140
   5.8.1.3 Step 4 of the Protocol......Page 142
   5.8.2.3 Step 3 of the Protocol......Page 143
 5.9 Model Validation......Page 144
 5.10 Begging Behaviour of Nestling Barn Owls......Page 145
  5.10.1 Step 1 of the Protocol: Linear Regression......Page 146
  5.10.3 Step 3 of the Protocol: Choose a Variance Structure......Page 148
  5.10.5 Step 5 of the Protocol: Compare New Model with Old Model......Page 149
  5.10.6 Step 6 of the Protocol: Everything Ok?......Page 150
  5.10.7 Steps 7 and 8 of the Protocol: The Optimal Fixed Structure......Page 151
  5.10.8 Step 9 of the Protocol: Refit with REML and Validate the Model......Page 153
  5.10.10 Sorry, We are Not Done Yet......Page 155
 6.1 Temporal Correlation and Linear Regression......Page 159
  6.1.1 ARMA Error Structures......Page 166
 6.2 Linear Regression Model and Multivariate Time Series......Page 168
 6.3 Owl Sibling Negotiation Data......Page 173
 7.1 Tools to Detect Violation of Independence......Page 177
 7.2 Adding Spatial Correlation Structures to the Model......Page 182
 7.3 Revisiting the Hawaiian Birds......Page 186
  7.4.1 Moby......Page 188
  7.4.2 All Whales......Page 190
 7.5 Spatial Correlation due to a Missing Covariate......Page 193
  7.6.1 Description of the Data......Page 198
  7.6.2 Data Exploration......Page 199
  7.6.3 Linear Regression......Page 200
   7.6.4.1 Step 2 of the Protocol: Refit with gls......Page 202
   7.6.4.2 Step 3 of the Protocol: Choose an Appropriate Variance Structure......Page 203
   7.6.4.4 Steps 7--8 of the Protocol: Find the Optimal Fixed Structure......Page 204
   7.6.4.5 Step 9 of the Protocol: Refit with REML......Page 205
  7.6.5 Why All the Fuss?......Page 206
 8.1 Introduction......Page 208
 8.2 The Normal Distribution......Page 209
 8.3 The Poisson Distribution......Page 211
 8.4 The Negative Binomial Distribution......Page 213
 8.5 The Gamma Distribution......Page 215
 8.6 The Bernoulli and Binomial Distributions......Page 217
 8.7 The Natural Exponential Family......Page 219
 8.8 Zero Truncated Distributions for Count Data......Page 220
 9.1 Introduction......Page 224
 9.2 Gaussian Linear Regression as a GLM......Page 225
 9.3 Introducing Poisson GLM with an Artificial Example......Page 226
 9.4 Likelihood Criterion......Page 227
  9.5.1 Introduction......Page 230
  9.5.2 R Code and Results......Page 231
  9.5.3 Deviance......Page 232
  9.5.4 Sketching the Fitted Values......Page 233
 9.6 Model Selection in a GLM......Page 234
  9.6.2 R Code and Output......Page 235
  9.6.3 Options for Finding the Optimal Model......Page 236
  9.6.4 The Drop1 Command......Page 237
 9.7 Overdispersion......Page 238
  9.7.2 Causes and Solutions for Overdispersion......Page 239
  9.7.3 Quick Fix: Dealing with Overdispersion in a Poisson GLM......Page 240
  9.7.4 R Code and Numerical Output......Page 241
  9.7.5 Model Selection in Quasi-Poisson......Page 242
 9.8 Model Validation in a Poisson GLM......Page 243
  9.8.2 Deviance Residuals......Page 244
  9.8.4 What to Plot?......Page 245
 9.9 Illustration of Model Validation in Quasi-Poisson GLM......Page 246
  9.10.1 Introduction......Page 248
  9.10.2 Results......Page 251
 9.11 GAM......Page 253
  9.11.1 Distribution of larval Sea Lice Around Scottish Fish Farms......Page 254
 10.1 Introduction......Page 259
  10.2.1 Tuberculosis in Wild Boar......Page 260
   10.2.1.1 R Code, Results and Fitted Values......Page 263
   10.2.1.2 General Comments......Page 265
  10.2.2 Parasites in Cod......Page 266
 10.3 GLM for Proportional Data......Page 268
 10.4 GAM for AbsencePresence Data......Page 271
 10.5 Where to Go from Here?......Page 273
 11.1 Introduction......Page 274
   11.2.1.1 Mathematics for the Zero-Truncated Poisson Model......Page 276
  11.2.2 Illustration of Poisson and NB Truncated Models......Page 278
 11.3 Too Many Zeros......Page 282
  11.3.1 Sources of Zeros......Page 283
  11.3.3 Two-Part Models Versus Mixture Models, and Hippos......Page 284
  11.4.1 Mathematics of the ZIP and ZINB......Page 287
   11.4.1.2 Summary......Page 290
  11.4.2 Example of ZIP and ZINB Models......Page 291
   11.4.2.1 Model Validation......Page 297
 11.5 ZAP and ZANB Models, Alias Hurdle Models......Page 299
  11.5.1 Mathematics of the ZAP and ZANB......Page 300
  11.5.2 Example of ZAP and ZANB......Page 301
 11.6 Comparing Poisson, Quasi-Poisson, NB, ZIP, ZINB, ZAP and ZANB GLMs......Page 303
  11.6.0 Option 4: Hypothesis Tests -- Poisson Versus NB......Page 304
 11.7 Flowchart and Where to Go from Here......Page 305
  12.1.1 The California Bird Data......Page 307
  12.1.2 The Owl Data......Page 311
  12.1.3 The Deer Data......Page 312
  12.2.1 Introduction......Page 314
  12.2.2 Step 1 of the GEE: Systematic Component and Link Function......Page 315
  12.2.4 Step 3 of the GEE: The Association Structure......Page 316
  12.2.4 Option 1: The Unstructured Correlation......Page 317
  12.2.4 Option 2: AR-1 Correlation......Page 318
  12.2.4 Option 3: Exchangeable Correlation......Page 319
  12.2.4 Option 4: Another Correlation Structure -- Stationary Correlation......Page 320
 12.3 Why All the Fuss?......Page 321
  12.3.1 A Bit of Maths......Page 322
 12.5 Examples of GEE......Page 325
  12.5.1 A GEE for the California Birds......Page 326
  12.5.2 A GEE for the Owls......Page 328
  12.5.3 A GEE for the Deer Data......Page 331
 12.6 Concluding Remarks......Page 332
to 13  GLMM and GAMM......Page 334
 13.1 Setting the Scene for Binomial GLMM......Page 335
  13.2.1 Deer Data......Page 338
   13.2.1.1 Comparison of Results......Page 343
  13.2.2 The Owl Data Revisited......Page 344
 13.3 The Underlying Mathematics in GLMM......Page 350
 14.1 Introduction......Page 353
  14.1.1 Explanatory Variables......Page 354
 14.2 Data Exploration......Page 355
 14.3 Trends and Auto-correlation......Page 360
 14.4 Using Ice Extent as an Explanatory Variable......Page 362
 14.5 SOI and Differences Between Arrival and Laying Dates......Page 364
 14.6 Discussion......Page 369
 14.7 What to Report in a Paper......Page 371
 15.1 Introduction......Page 372
 15.2 Data Exploration......Page 374
 15.3 Estimation of Trends for the Bird Data......Page 376
  15.3.1 Model Validation......Page 377
  15.3.2 Failed Approach 1......Page 381
  15.3.3 Failed Approach 2......Page 382
 15.4 Dealing with Independence......Page 383
 15.5 To Transform or Not to Transform......Page 386
 15.6 Birds and Explanatory Variables......Page 387
 15.7 Conclusions......Page 389
 15.8 What to Write in a Paper......Page 390
  16.1.1 Roadkills......Page 391
 16.2 Data Exploration......Page 393
 16.3 GAM......Page 397
 16.4 Understanding What the Negative Binomial is Doing......Page 402
 16.5 GAMM: Adding Spatial Correlation......Page 403
 16.7 What to Write in a Paper......Page 405
 17.1 Biological Introduction......Page 406
 17.2 The Data and Underlying Questions......Page 407
  17.3.1 Approach 1......Page 409
  17.3.2 Approach 2......Page 414
  17.3.3 Approach 3......Page 415
 17.4 Estimating Common Patterns Using Additive Mixed Modelling......Page 416
  17.4.1 One Smoothing Curve for All Stations......Page 417
  17.4.2 Four Smoothers; One for Each Month......Page 421
  17.4.3 Smoothing Curves for Groups Based on Geographical Distances......Page 424
  17.4.4 Smoothing Curves for Groups Based on Source Correlations......Page 425
 17.5 Choosing the Best Model......Page 426
 17.6 Discussion......Page 427
 17.7 What to Write in a Paper......Page 428
 18.1 Introduction......Page 430
  18.1.1 Biological Background of the Project......Page 431
 18.2 Data Exploration......Page 433
 18.3 A Statistical Data Analysis Strategy for DIN......Page 436
 18.4 Results for Temperature......Page 446
 18.5 Results for DIAT1......Page 447
 18.6 Comparing Phytoplankton and Environmental Trends......Page 449
 18.7 Conclusions......Page 451
 18.8 What to Write in a Paper......Page 453
 19.1 Introduction......Page 454
 19.2 Data Exploration......Page 455
 19.3 Analysis of the Data......Page 457
  19.3.0 Step 1 of the Protocol......Page 458
  19.3.0 Steps 2--6 of the Protocol......Page 459
  19.3.0 Steps 7 and 8 of the Protocol......Page 461
  19.3.0 Step 9 of the Protocol......Page 462
 19.5 What to Write in a Paper......Page 465
 20.1 Introduction......Page 466
 20.2 Data Exploration......Page 467
 20.3 Data Analysis......Page 469
  20.3.1 Intraclass Correlations......Page 473
 20.4 Discussion......Page 474
 20.5 What to Write in a Paper......Page 475
 21.1 Introduction......Page 476
 21.2 The Data......Page 478
  21.3.1 Collinearity......Page 480
  21.3.2 Spatial Auto-correlation......Page 486
 21.4 Generalised Linear Mixed Effects Modelling......Page 488
  21.4.1 Model Selection......Page 490
  21.4.2 Model Adequacy......Page 494
 21.5 Discussion......Page 497
 21.6 What to Write in a Paper......Page 499
 22.1 Introduction......Page 500
 22.2 Data Exploration......Page 502
 22.3 GLM Results Assuming Independence......Page 504
 22.4 GEE Results......Page 506
 22.5 GLMM Results......Page 507
 22.6 Discussion......Page 508
 22.7 What to Write in a Paper......Page 509
 23.1 Introduction......Page 510
 23.2 Preliminary Results......Page 511
 23.3 GLM......Page 514
  23.3.1 Validation......Page 516
  23.4.1 Theory Behind Bayesian Statistics......Page 517
  23.4.2 Markov Chain Monte Carlo Techniques......Page 518
 23.5 Fitting the Poisson Model in BRugs......Page 519
  23.5.1 Code in R......Page 520
  23.5.2 Model Code......Page 521
  23.5.3 Initialising the Chains......Page 522
  23.5.4 Summarising the Posterior Distributions......Page 524
  23.5.5 Inference......Page 525
 23.6 Poisson Model with Random Effects......Page 527
 23.7 Poisson Model with Random Effects and Auto-correlation......Page 530
 23.8 Negative Binomial Distribution with Auto-correlated Random Effects......Page 532
 23.9 Conclusions......Page 535
 A.2 Data Exploration......Page 537
  A.2.1 Step 1: Outliers......Page 538
  A.2.2 Step 2: Collinearity......Page 539
 A.3 Linear Regression......Page 542
  A.3.1 Model Selection......Page 546
  A.3.2 Model Validation......Page 548
  A.3.3 Model Interpretation......Page 549
 A.4 Additive Modelling......Page 552
 A.6 Information Theory and Multi-model Inference......Page 556
 A.7 Maximum Likelihood Estimation in Linear Regression Context......Page 558
References......Page 559
Index......Page 568