Note: Intro -- 1 Introduction -- 1.1 What Is in the Book? -- 1.1.1 To Include or Not to Include GLM and GAM -- 1.1.2 Case Studies -- 1.1.3 Flowchart of the Content -- 1.2 Software -- 1.3 How to Use This Book If You Are an Instructor -- 1.4 What We Did Not Do and Why -- 1.5 How to Cite R and Associated Packages -- 1.6 Our R Programming Style -- 1.7 Getting Data into R -- 1.7.1 Data in a Package -- 2 Limitations of Linear Regression Applied on Ecological Data -- 2.1 Data Exploration -- 2.1.1 Cleveland Dotplots -- 2.1.2 Pairplots -- 2.1.3 Boxplots -- 2.1.4 xyplot from the Lattice Package -- 2.2 The Linear Regression Model -- 2.3 Violating the Assumptions -- Exception or Rule? -- 2.3.1 Introduction -- 2.3.2 Normality -- 2.3.3 Heterogeneity -- 2.3.4 Fixed X -- 2.3.5 Independence -- 2.3.6 Example 1 -- Wedge Clam Data -- 2.3.6.1 Model Validation -- 2.3.7 Example 2 -- Moby's Teeth -- 2.3.8 Example 3 -- Nereis -- 2.3.9 Example 4 -- Pelagic Bioluminescence -- 2.4 Where to Go from Here -- 3 Things Are Not Always Linear -- Additive Modelling -- 3.1 Introduction -- 3.2 Additive Modelling -- 3.2.1 GAM in gam and GAM in mgcv -- 3.2.2 GAM in gam with LOESS -- 3.2.2.1 LOESS Smoothing -- 3.2.3 GAM in mgcv with Cubic Regression Splines -- 3.3 Technical Details of GAM in mgcv -- 3.3.1 A (Little) Bit More Technical Information on Regression Splines -- 3.3.2 Smoothing Splines Alias Penalised Splines -- 3.3.3 Cross-Validation -- 3.3.4 Additive Models with Multiple Explanatory Variables -- 3.3.5 Two More Things -- 3.4 GAM Example 1 -- Bioluminescent Data for Two Stations -- 3.4.1 Interaction Between a Continuous and Nominal Variable -- 3.5 GAM Example 2: Dealing with Collinearity -- 3.6 Inference Means that it is difficult and that it can be skipped upon first reading. -- 3.7 Summary and Where to Go from Here? -- 4 Dealing with Heterogeneity. , 4.1 Dealing with Heterogeneity -- 4.1.1 Linear Regression Applied on Squid -- 4.1.2 The Fixed Variance Structure -- 4.1.3 The VarIdent Variance Structure -- 4.1.4 The varPower Variance Structure -- 4.1.5 The varExp Variance Structure -- 4.1.6 The varConstPower Variance Structure -- 4.1.7 The varComb Variance Structure -- 4.1.8 Overview of All Variance Structures -- 4.1.8.1 Which One to Choose? -- 4.1.9 Graphical Validation of the Optimal Model -- 4.2 Benthic Biodiversity Experiment -- 4.2.1 Linear Regression Applied on the Benthic Biodiversity Data -- 4.2.2 GLS Applied on the Benthic Biodiversity Data -- 4.2.3 A Protocol -- 4.2.4 Application of the Protocol on the Benthic Biodiversity Data -- 4.2.4.1 Round 1 of the Backwards Selection -- 4.2.4.2 Round 2 of the Backwards Selection -- 4.2.4.3 Round 3 of the Backwards Selection -- 4.2.4.4 Round 4 of the Backwards Selection -- 4.2.4.5 The Aftermath -- 5 Mixed Effects Modelling for Nested Data -- 5.1 Introduction -- 5.2 2Stage Analysis Method -- 5.3 The Linear Mixed Effects Model -- 5.3.1 Introduction -- 5.3.2 The Random Intercept Model -- 5.3.3 The Random Intercept and Slope Model -- 5.3.4 Random Effects Model -- 5.4 Induced Correlations -- 5.4.1 Intraclass Correlation Coefficient -- 5.5 The Marginal Model -- 5.6 Maximum Likelihood and REML Estimation -- 5.6.1 Illustration of Difference Between ML and REML -- 5.7 Model Selection in (Additive) Mixed Effects Modelling -- 5.8 RIKZ Data: Good Versus Bad Model Selection -- 5.8.1 The Wrong Approach -- 5.8.1.1 Step 2 of the Protocol -- 5.8.1.2 Step 3 of the Protocol -- 5.8.1.3 Step 4 of the Protocol -- 5.8.2 The Good Approach -- 5.8.2.1 Step 1 of the Protocol -- 5.8.2.2 Step 2 of the Protocol -- 5.8.2.3 Step 3 of the Protocol -- 5.8.2.4 Step 4 of the Protocol -- 5.9 Model Validation -- 5.10 Begging Behaviour of Nestling Barn Owls. , 5.10.1 Step 1 of the Protocol: Linear Regression -- 5.10.2 Step 2 of the Protocol: Fit the Model with GLS -- 5.10.3 Step 3 of the Protocol: Choose a Variance Structure -- 5.10.4 Step 4: Fit the Model -- 5.10.5 Step 5 of the Protocol: Compare New Model with Old Model -- 5.10.6 Step 6 of the Protocol: Everything Ok? -- 5.10.7 Steps 7 and 8 of the Protocol: The Optimal Fixed Structure -- 5.10.8 Step 9 of the Protocol: Refit with REML and Validate the Model -- 5.10.9 Step 10 of the Protocol -- 5.10.10 Sorry, We are Not Done Yet -- 6 Violation of Independence -- Part I -- 6.1 Temporal Correlation and Linear Regression -- 6.1.1 ARMA Error Structures -- 6.2 Linear Regression Model and Multivariate Time Series -- 6.3 Owl Sibling Negotiation Data -- 7 Violation of Independence -- Part II -- 7.1 Tools to Detect Violation of Independence -- 7.2 Adding Spatial Correlation Structures to the Model -- 7.3 Revisiting the Hawaiian Birds -- 7.4 Nitrogen Isotope Ratios in Whales -- 7.4.1 Moby -- 7.4.2 All Whales -- 7.5 Spatial Correlation due to a Missing Covariate -- 7.6 Short Godwits Time Series -- 7.6.1 Description of the Data -- 7.6.2 Data Exploration -- 7.6.3 Linear Regression -- 7.6.4 Protocol Time -- 7.6.4.1 Step 2 of the Protocol: Refit with gls -- 7.6.4.2 Step 3 of the Protocol: Choose an Appropriate Variance Structure -- 7.6.4.3 Steps 4--6 of the Protocol: Find the Optimal Random Structure -- 7.6.4.4 Steps 7--8 of the Protocol: Find the Optimal Fixed Structure -- 7.6.4.5 Step 9 of the Protocol: Refit with REML -- 7.6.5 Why All the Fuss? -- 8 Meet the Exponential Family -- 8.1 Introduction -- 8.2 The Normal Distribution -- 8.3 The Poisson Distribution -- 8.3.1 Preparation for the Offset in GLM -- 8.4 The Negative Binomial Distribution -- 8.5 The Gamma Distribution -- 8.6 The Bernoulli and Binomial Distributions -- 8.7 The Natural Exponential Family. , 8.7.1 Which Distribution to Select? -- 8.8 Zero Truncated Distributions for Count Data -- 9 GLM and GAM for Count Data -- 9.1 Introduction -- 9.2 Gaussian Linear Regression as a GLM -- 9.3 Introducing Poisson GLM with an Artificial Example -- 9.4 Likelihood Criterion -- 9.5 Introducing the Poisson GLM with a Real Example -- 9.5.1 Introduction -- 9.5.2 R Code and Results -- 9.5.3 Deviance -- 9.5.4 Sketching the Fitted Values -- 9.6 Model Selection in a GLM -- 9.6.1 Introduction -- 9.6.2 R Code and Output -- 9.6.3 Options for Finding the Optimal Model -- 9.6.4 The Drop1 Command -- 9.6.5 Two Ways of Using the Anova Command -- 9.6.6 Results -- 9.7 Overdispersion -- 9.7.1 Introduction -- 9.7.2 Causes and Solutions for Overdispersion -- 9.7.3 Quick Fix: Dealing with Overdispersion in a Poisson GLM -- 9.7.4 R Code and Numerical Output -- 9.7.5 Model Selection in Quasi-Poisson -- 9.8 Model Validation in a Poisson GLM -- 9.8.1 Pearson Residuals -- 9.8.2 Deviance Residuals -- 9.8.3 Which One to Use? -- 9.8.4 What to Plot? -- 9.9 Illustration of Model Validation in Quasi-Poisson GLM -- 9.10 Negative Binomial GLM -- 9.10.1 Introduction -- 9.10.2 Results -- 9.11 GAM -- 9.11.1 Distribution of larval Sea Lice Around Scottish Fish Farms -- 10 GLM and GAM for Absence--Presence and Proportional Data -- 10.1 Introduction -- 10.2 GLM for AbsencePresence Data -- 10.2.1 Tuberculosis in Wild Boar -- 10.2.1.1 R Code, Results and Fitted Values -- 10.2.1.2 General Comments -- 10.2.2 Parasites in Cod -- 10.3 GLM for Proportional Data -- 10.4 GAM for AbsencePresence Data -- 10.5 Where to Go from Here? -- 11 Zero-Truncated and Zero-Inflated Models for Count Data -- 11.1 Introduction -- 11.2 Zero-Truncated Data -- 11.2.1 The Underlying Mathematics for Truncated Models -- 11.2.1.1 Mathematics for the Zero-Truncated Poisson Model. , 11.2.1.2 Mathematics for the Negative Binomial Truncated Model -- 11.2.1.3 Summary -- 11.2.2 Illustration of Poisson and NB Truncated Models -- 11.3 Too Many Zeros -- 11.3.1 Sources of Zeros -- 11.3.2 Sources of Zeros for the Cod Parasite Data -- 11.3.3 Two-Part Models Versus Mixture Models, and Hippos -- 11.4 ZIP and ZINB Models -- 11.4.1 Mathematics of the ZIP and ZINB -- 11.4.1.1 The Mean and the Variance in ZIP and ZINB Models -- 11.4.1.2 Summary -- 11.4.2 Example of ZIP and ZINB Models -- 11.4.2.1 Model Validation -- 11.4.2.2 Model Interpretation -- 11.5 ZAP and ZANB Models, Alias Hurdle Models -- 11.5.1 Mathematics of the ZAP and ZANB -- 11.5.2 Example of ZAP and ZANB -- 11.6 Comparing Poisson, Quasi-Poisson, NB, ZIP, ZINB, ZAP and ZANB GLMs -- 11.6.0 Option 1: Common Sense -- 11.6.0 Option 2: Model Validation -- 11.6.0 Option 3: Information Criteria -- 11.6.0 Option 4: Hypothesis Tests -- Poisson Versus NB -- 11.6.0 Option 5: Compare Observed and Fitted Values -- 11.7 Flowchart and Where to Go from Here -- 12 Generalised Estimation Equations -- 12.1 GLM: Ignoring the Dependence Structure -- 12.1.1 The California Bird Data -- 12.1.2 The Owl Data -- 12.1.3 The Deer Data -- 12.2 Specifying the GEE -- 12.2.1 Introduction -- 12.2.2 Step 1 of the GEE: Systematic Component and Link Function -- 12.2.3 Step 2 of the GEE: The Variance -- 12.2.4 Step 3 of the GEE: The Association Structure -- 12.2.4 Option 1: The Unstructured Correlation -- 12.2.4 Option 2: AR-1 Correlation -- 12.2.4 Option 3: Exchangeable Correlation -- 12.2.4 Option 4: Another Correlation Structure -- Stationary Correlation -- 12.3 Why All the Fuss? -- 12.3.1 A Bit of Maths -- 12.4 Association for Binary Data -- 12.5 Examples of GEE -- 12.5.1 A GEE for the California Birds -- 12.5.2 A GEE for the Owls -- 12.5.3 A GEE for the Deer Data -- 12.6 Concluding Remarks -- 13 GLMM and GAMM. , 13.1 Setting the Scene for Binomial GLMM.

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