1 INTRODUCTION TO GENERALIZED LINEAR MODELS 1 1.1 LINEAR REGRESSION APPLIED ON FISHERIES DATA 1 1.2 POISSON GLM 7 1.2.1 Poisson distribution for count data 7 1.2.2 Predictor function 8 1.2.3 Linking the mean and the predictor function 9 1.2.4 Maximum likelihood to estimate the parameters 9 1.2.5 Application of Poisson GLM on the fisheries data 11 1.2.6 Overdispersion 19 1.2.7 Adding covariates 23 1.2.8 Using the offset 24 1.3 NEGATIVE BINOMIAL GLM 26 1.3.1 Negative binomial distribution for count data 26 1.3.2 Example of Negative binomial regression 28 1.3.3 Heterogeneous Negative binomial regression 34 1.3.4 A note on modelling under-dispersion 36 1.4 BINOMIAL GLM FOR BINARY DATA 36 1.4.1 Parasites in honeybee larvae 36 1.4.2 Visualizing the data 36 1.4.3 Defining the three steps of a binomial GLM 38 1.4.4 Results for the bee data 40 1.4.5 Likelihood function for a binomial GLM 42 1.4.6 Other link functions 42 1.5 BINOMIAL GLM FOR PROPORTIONAL DATA 43 1.5.1 Binomial distribution 43 1.5.2 Predictor function 45 1.5.3 Link function 45 1.5.4 Fitting the model in R 45 1.6 OTHER DISTRIBUTIONS 47 2 GENERALIZED LINEAR MODELLING APPLIED TO RED SQUIRREL DATA 49 2.1 RED SQUIRRELS 49 2.2 IMPORTING THE DATA 50 2.3 DATA EXPLORATION 51 2.3.1 Outliers 51 2.3.2 Collinearity 52 2.3.3 Relationships 54 2.4 FITTING THE POISSON GLM IN R 55 2.4.1 Specifying the model 55 2.4.2 Execute the Poisson GLM in R 55 2.4.3 Model validation 57 2.5 FITTING THE NEGATIVE BINOMIAL GLM IN R 60 2.5.1 Using the glm.nb function 60 2.5.2 Heterogeneous negative binomial GLM 63 2.6 BAYESIAN APPROACH - RUNNING THE POISSON GLM 66 2.6.1 Obtaining and installing JAGS 66 2.6.2 Specifying the data for JAGS 67 2.6.3 Specifying the model for JAGS 68 2.6.4 Specifying the initial values 69 2.6.5 Parameters to store 69 2.6.6 Running JAGS via R 69 2.6.7 Generalizing the JAGS modelling code 72 2.7 ASSESSING MIXING OF CHAINS 74 2.7.1 Assess mixing of chains if R2jags is used 74 2.8 MODEL VALIDATION 76 2.8.1 Checking for overdispersion 76 2.8.2 Obtaining Pearson residuals 77 2.9 APPLYING A NEGATIVE BINOMIAL GLM IN JAGS 79 2.10 MIXING OF CHAINS 82 2.11 MODEL VALIDATION 83 2.12 MODEL INTERPRETATION 84 2.13 DISCUSSION 87 2.14 WHAT TO PRESENT IN A PAPER 87 3 GLM APPLIED TO PRESENCE-ABSENCE POLYCHAETA DATA 89 3.1 MARINE BENTHIC DATA 89 3.2 IMPORTING THE DATA AND HOUSEKEEPING 90 3.3 DATA EXPLORATION 91 3.4 BINARY GLM; A FREQUENTIST APPROACH 94 3.4.1 Specifying the distribution and link function 94 3.4.2 Specifying the predictor function 95 3.4.3 Running the glm function 96 3.4.4 Results of the glm function 96 3.4.5 Model selection 97 3.4.6 Results of the optimal model 100 3.4.7 Model validation 101 3.4.8 Visualizing the model 102 3.5 FITTING A BERNOULLI GLM IN JAGS 103 3.5.1 Specifying the data for JAGS 103 3.5.2 JAGS modelling code 104 3.5.3 Initial values and parameters to save 105 3.5.4 Running JAGS from R 105 3.5.5 JAGS results presented within R 106 3.6 MODEL SELECTION USING AIC, DIC AND BIC IN JAGS 107 3.7 MODEL INTERPRETATION 110 3.8 DISCUSSION 113 3.9 WHAT TO PRESENT IN A PAPER 114 4 INTRODUCTION TO MIXED EFFECTS MODELS 115 4.1 SPIDERS 115 4.2 LINEAR REGRESSION APPLIED ON THE SPIDER DATA 115 4.3 LINEAR MIXED EFFECTS MODELS 118 4.3.1 Model formulation and interpretation 118 4.3.2 Fitting a linear mixed effects model using lmer 119 4.3.3 Analysis using lmer 122 4.4 FITTING A LINEAR MIXED EFFECTS MODEL IN JAGS 128 4.5 USING A VARIABLE AS A FIXED OR RANDOM TERM? 131 4.6 RANDOM INTERCEPT AND SLOPE MODEL 131 4.7 GENERALIZED LINEAR MIXED EFFECTS MODELS 132 5 GLMM APPLIED ON HONEYBEE POLLINATION DATA 133 5.1 HONEYBEES AND DANDELION POLLEN 133 5.2 DATA DESCRIPTION AND IMPORTING THE DATA 134 5.3 DATA EXPLORATION 135 5.4 BUILDING UP A MODEL 136 5.5 POISSON GLMM USING GLMER 137 5.6 POISSON GLMM USING JAGS 140 5.6.1 Data for JAGS 140 5.6.2 JAGS modelling code 141 5.6.3 Likelihood 142 5.6.4 Priors 142 5.6.5 Initial values 144 5.6.6 Parameters to save 144 5.6.7 Executing JAGS and obtaining results 145 5.7 NEGATIVE BINOMIAL GLMM USING GLMMADMB 146 5.8 NEGATIVE BINOMIAL GLMM USING JAGS 147 5.8.1 Data for JAGS 147 5.8.2 JAGS modelling code 147 5.8.3 Initial values 148 5.8.4 Parameters to save 149 5.8.5 Executing JAGS and obtaining results 149 5.8.6 Mixing of chains 150 5.8.7 Model validation 150 5.8.8 Model interpretation 152 5.9 GLMM WITH AUTO-REGRESSIVE CORRELATION 154 5.9.1 Simulate temporal correlated counts 155 5.9.2 JAGS to estimate the Poisson GLM with AR correlation 158 5.9.3 Multiple Poisson time series 161 5.9.4 Poisson GLMM with AR correlation 161 5.10 WHAT TO PRESENT IN A PAPER 164 6 GLMM FOR STRICTLY POSITIVE DATA: BIOMASS OF RAINFOREST TREES 165 6.1 RAINFOREST TREE SPECIES 165 6.2 IMPORTING THE DATA AND HOUSEKEEPING 167 6.3 DATA EXPLORATION 168 6.3.1 Outliers 168 6.3.2 Collinearity 169 6.3.3 Relationships 170 6.4 MULTIPLE LINEAR REGRESSION: A FREQUENTIST APPROACH 173 6.5 GAMMA GLM USING A FREQUENTIST APPROACH 175 6.5.1. Formulating the gamma GLM 175 6.5.2 Scale and shape 176 6.5.3 Visualizing the gamma distribution 176 6.5.4 Different link functions 178 6.5.5 Running the Gamma GLM using the glm function 179 6.5.6 Scale confusion 179 6.5.7 Identity link and inverse link function 182 6.6 FITTING A GAMMA GLM USING JAGS 183 6.6.1 Specifying the data for JAGS 183 6.6.2 JAGS modelling code 185 6.6.3 Priors 185 6.6.4 Likelihood function 185 6.6.5 Initial values and parameters to save 186 6.6.6 Running JAGS from R 186 6.6.7 JAGS results presented within R 187 6.6.8 Model interpretation 190 6.6.9 Model validation 193 6.7 ADDING MORE COVARIATES TO THE GAMMA GLM IN JAGS 195 6.8 GAMMA GLMM 195 6.8.1 R code for a gamma GLMM in JAGS 196 6.8.2 Results from JAGS for the gamma GLMM 198 6.9 TRUNCATED GAUSSIAN LINEAR REGRESSION 199 6.9.1 Zero trick to fit any statistical distribution in JAGS 199 6.9.2 Multiple linear regression in JAGS with the zero trick 200 6.9.3 Tobit model in JAGS 202 6.9.4 Tobit model with random effects in JAGS 205 6.10 DISCUSSION 205 6.11 WHAT TO PRESENT IN A PAPER 206 7 BINOMIAL, BETA-BINOMIAL, AND BETA GLMM APPLIED TO CHEETAH DATA 207 7.1 STEREOTYPIC BEHAVIOURS IN CAPTIVE CHEETAHS 207 7.2 IMPORTING THE DATA 209 7.3 DATA EXPLORATION 209 7.3.1 Outliers 209 7.3.2 Collinearity 210 7.4 BINOMIAL GLMM USING A FREQUENTIST APPROACH 212 7.4.1 Standardizing covariates 212 7.4.2 Binomial GLMM with random intercept zoo 213 7.4.3 Executing the GLMM using the glmer function 213 7.4.4 Overdispersion 215 7.4.5 Binomial GLMM with observation level random intercept 216 7.4.6 Visualization of results 220 7.5 BINOMIAL GLMM WITH RANDOM INTERCEPT ZOO IN JAGS 223 7.5.1 Data for JAGS 223 7.5.2 JAGS modelling code for a binomial GLMM 224 7.5.3 Results for the binomial GLMM 226 7.5.4 Overdispersion 226 7.6 BETA-BINOMIAL GLMM IN JAGS 228 7.6.1 The Beta distribution 228 7.6.2 From beta to beta-binomial distribution 229 7.6.3 JAGS code for beta-binomial GLMM 230 7.6.4 Beta-binomial GLMM results 231 7.6.5 Model validation of the beta-binomial GLMM 232 7.7 USING A BETA GLMM FOR PROPORTIONS 234 7.8 COMPARING ESTIMATED PARAMETERS FROM ALL MODELS 237 7.9 MODEL SELECTION FROM A FREQUENTIST POINT OF VIEW 239 7.10 MODEL SELECTION FROM A BAYESIAN POINT OF VIEW 241 7.10.1 Using the DIC, AIC and BIC 242 7.10.2 Inclusion probabilities 246 7.11 WHAT TO PRESENT IN A PAPER 246 REFERENCES 247 INDEX 251 BOOKS BY HIGHLAND STATISTICS 255 UPCOMING BOOKS IN 2013 AND 2014 256