Applied Statistical Modelling for Ecologists

1. Introduction

1.1 Statistical models and why you need them

1.2 Why linear statistical models?

1.3 Why go beyond the linear model?

1.4 Random effects and why you need them

1.5 Why do you need both Bayesian and non-Bayesian statistical inference?

1.6 More reasons for why you should really understand maximum likelihood

1.7 The data simulation/model fitting dualism

1.8 The 'experimental approach' to statistics

1.9 The first principle of modeling: Start simple!

1.10 Overview of the ASM book and additional resources

1.11 How to use the ASM book for self-study and in courses

1.12 Summary and outlook


2. Introduction to statistical inference

2.1 Probability as the basis for statistical inference

2.2 Random variables and their description by probability distributions

2.2.1 Discrete and continuous random variables

2.2.2 Description of random variables by probability distributions

2.2.3 Location and spread of a rand. variable, and what "modeling a parameter" means

2.2.4 Short summary on random variables and probability distributions

2.3 Statistical models and their usages

2.4 The likelihood function

2.5 Classical inference by max. likelihood and its application to a single-parameter model

2.5.1 Introduction to maximum likelihood

2.5.2 Confidence intervals by inversion of the likelihood ratio test

2.5.3 Stand. errors and conf. intervals based on approximate normality of the MLEs

2.5.4 Obtaining standard errors and confidence intervals using the bootstrap

2.5.5 A short summary on maximum likelihood estimation

2.6 Maximum likelihood estimation in a two-parameter model

2.7 Bayesian inference using posterior distributions

2.7.1 Probability as a general measure of uncertainty, or of imperfect knowledge

2.7.2 The posterior distribution as a fundamental target of inference

2.7.3 Prior distributions

2.8 Bayesian computation by Markov chain Monte Carlo (MCMC)

2.8.1 Monte Carlo integration

2.8.2 MCMC for the Swiss bee-eaters

2.8.3 A practical in MCMC for the bee-eaters with function demoMCMC() (*)

2.9 So should you now be a Bayesian or a frequentist?

2.10 Summary and outlook


3. Linear regression models and their extensions to generalized linear, hierarchical and integrated models

3.1 Introduction

3.2 Statistical distributions for the random variables in our model

3.2.1 Normal distribution

3.2.2 Uniform distribution

3.2.3 Binomial distribution: The “coin-flip distribution”

3.2.4 Poisson distribution

3.2.5 Summary remarks on statistical distributions

3.3 Link functions to model parameters on a transformed scale

3.4 Linear modeling of covariate effects

3.4.1 The "model of the mean"

3.4.2 Comparison of two groups

3.4.3 Simple linear regression: modeling the effect of one continuous covariate

3.4.4 Modeling the effects of one categorical covariate, or factor

3.4.5 Modeling the effects of two factors

3.4.6 Modeling the effects of a factor and a continuous covariate

3.5 Brief overview of linear, generalized linear, (generalized) linear mixed, hierarchical and integrated models

3.6 Summary and outlook


4. Introduction to general-purpose model-fitting engines and the model of the mean

4.1 Introduction

4.2 Data generation

4.3 Analysis using a canned function in R

4.4 JAGS

4.4.1 Introduction to JAGS

4.4.2 Fit the model with JAGS

4.5 NIMBLE

4.5.1 Introduction to NIMBLE

4.5.2 Fit the model with NIMBLE

4.6 Stan

4.6.1 Introduction to Stan

4.6.2 Fit the model with Stan

4.7 Maximum likelihood in R

4.7.1 Introduction to maximum likelihood in R

4.7.2 Fit the model using maximum likelilhood in R

4.8 Maximum likelihood using Template Model Builder (TMB)

4.8.1 Introduction to TMB

4.8.2 Fit the model with TMB

4.9 Comparison of engines and concluding remarks


5. Simple linear regression with Normal errors

5.1 Introduction

5.2 Data generation

5.3 Likelihood analysis using canned functions in R

5.4 Bayesian analysis with JAGS

5.4.1 Fitting the model

5.4.3 Forming predictions

5.4.4 Interpretation of confidence vs. credible intervals

5.5 Bayesian analysis with Stan

5.6 Do-it-yourself MLEs

5.7 Likelihood analysis with TMB

5.8 Comparison of the engines

5.9 Summary and outlook


6. Comparison of two groups

6.1 Introduction

6.2 Comparing two groups with equal variances

6.2.1 Data generation

6.2.2 Likelihood analysis with canned functions in R

6.2.3 Bayesian analysis with JAGS

6.2.4 Bayesian analysis with Stan

6.2.5 Do-it-yourself MLEs

6.2.6 Likelihood analysis with TMB

6.2.7 Comparison of the engines

6.3 Comparing two groups with unequal variances

6.3.1 Data generation

6.3.2 Frequentist analysis using canned functions in R

6.3.3 Bayesian analysis with JAGS

6.3.4 Bayesian analysis with Stan

6.3.5 Do-it-yourself MLEs

6.3.6 Likelihood analysis with TMB

6.3.7 Comparison of the engines

6.4 Summary and outlook


7. Comparisons among multiple groups

7.1 Introduction: A first encounter with fixed and random effects

7.2 Fixed effects of categorical covariates

7.2.1 Data generation

7.2.2 Likelihood analysis using canned functions in R

7.2.3 Bayesian analysis with JAGS

7.2.4 Bayesian analysis with Stan

7.2.5 Do-it-yourself MLEs

7.2.6 Likelihood analysis with TMB

7.2.7 Comparison of the engines

7.3 Random effects of categorical covariates

7.3.1 Data generation

7.3.2 Likelihood analysis using canned functions in R

7.3.3 Bayesian analysis with JAGS ... and a quick fixed-random comparison

7.3.4 Bayesian analysis with Stan

7.3.5 Do-it-yourself MLEs

7.3.6 Likelihood analysis with TMB

7.3.7 Comparison of the engines

7.4 Summary and outlook


8. Comparisons in two classifications or with two categorical covariates

8.1 Introduction: Main and interaction effects

8.2 Data generation

8.3 Likelihood analysis using canned functions in R

8.4 An aside: Using simulation to assess bias and precision of an estimator ...
and to see what a standard error is

8.5 Bayesian analysis with JAGS

8.5.1 Main-effects model

8.5.2 Interaction-effects model

8.5.3 Forming predictions

8.6 Bayesian analysis with Stan

8.6.1 Main-effects model

8.6.2 Interaction-effects model

8.7 Do-it-yourself MLEs

8.7.1 Main-effects model

8.7.2 Interaction-effects model

8.8 Likelihood analysis with TMB

8.8.1 Main-effects model

8.8.2 Interaction-effects model

8.9 Comparison of the engines

8.10 Summary and outlook


9. General linear model with continuous and categorical explanatory variables

9.1 Introduction

9.2 Data generation

9.3 Likelihood analysis with canned functions in R

9.4 Bayesian analysis with JAGS

9.5 Bayesian analysis with Stan

9.6 Do-it-yourself MLEs

9.7 Likelihood analysis with TMB

9.8 Comparison of the engines

9.9 Summary


10. Linear mixed-effects model

10.1 Introduction: Mixed effects

10.2 Data generation

10.3 Analysis under a random-intercepts model

10.3.1 ML and REML estimates using canned functions in R

10.3.2 Bayesian analysis with JAGS

10.3.3 Bayesian analysis with Stan

10.3.4 Do-it-yourself MLEs

10.3.5 Likelihood analysis with TMB

10.3.6 Comparison of the engines

10.4 Analysis under a random-coefficients model without correlation between
intercept and slope

10.4.1 REML and ML estimates using canned functions in R

10.4.2 Bayesian analysis with JAGS

10.4.3 Bayesian analysis with Stan

10.4.4 Do-it-yourself MLEs

10.4.5 Likelihood analysis with TMB

10.4.6 Comparison of the engines

10.5 The random-coefficients model with correlation between intercept and slope

10.5.1 Introduction

10.5.2 Data generation

10.5.3 REML and ML estimates using canned functions in R

10.5.4. Bayesian analysis with JAGS

10.5.5 Bayesian analysis with Stan

10.5.6 Do-it-yourself MLEs

10.5.7 Likelihood analysis with TMB

10.5.8 Comparison of the the engines

10.6 Summary and outlook


11. Introduction to the Generalized linear model (GLM): Comparing two groups
in a Poisson regression

11.1 Introduction

11.2 An important but often forgotten issue with count data

11.3 Data generation

11.4 Likelihood analysis with canned functions in R

11.5 Bayesian analysis with JAGS

11.6 Bayesian analysis with Stan

11.7 Do-it-yourself MLEs

11.8 Likelihood analysis with TMB

11.9 Comparison of the engines

11.10 Summary and outlook


12. Overdispersion, zero-inflation and offsets in a GLM

12.1 Introduction

12.2 Overdispersion

12.2.1 Data generation

12.2.2 Likelihood analysis with canned functions in R

12.2.3 Bayesian analysis with JAGS

12.2.4 Bayesian analysis with Stan

12.2.5 Do-it-yourself MLEs

12.2.6 Likelihood analysis with TMB

12.2.7 Comparison of the engines

12.3 Zero-inflation

12.3.1 Data generation

12.3.2 Likelihood analysis with canned functions in R

12.3.3 Bayesian analysis with JAGS

12.3.4 Bayesian analysis with Stan

12.3.5 Do-it-yourself MLEs

12.3.6 Likelihood analysis with TMB

12.3.7 Comparison of the engines

12.4 Offsets

12.4.1 Data generation

12.4.2 Likelihood analysis with canned functions in R

12.4.3 Bayesian analysis with JAGS

12.4.4 Bayesian analysis with Stan

12.4.5 Do-it-yourself MLEs

12.4.6 Likelihood analysis with TMB

12.4.7 Comparison of the engines

12.5 Summary and outlook


13. Poisson regression with both continuous and categorical explanatory variables

13.1 Introduction

13.2 Data generation

13.3 Likelihood analysis with canned functions in R

13.4 Bayesian analysis with JAGS

13.4.1 Fitting the model

13.4.2 Forming predictions

13.5 Bayesian analysis with Stan

13.6 Do-it-yourself MLEs

13.7 Likelihood analysis with TMB

13.8 Comparison of the engines

13.9 Summary and outlook


14. Poisson mixed-effects model or Poisson GLMM

14.1. Introduction

14.2 Data generation

14.3 Likelihood analysis with canned functions in R

14.4 Bayesian analysis with JAGS

14.5 Bayesian analysis with Stan

14.6 Do-it-yourself MLEs

14.7 Likelihood analysis with TMB

14.8 Comparison of the engines

14.9 Summary and outlook


15. Comparing two groups in a Binomial regression

15.1 Introduction

15.2 Data generation

15.3 Likelihood analysis with canned functions in R

15.4 Bayesian analysis with JAGS

15.5 Bayesian analysis with Stan

15.6 Do-it-yourself MLEs

15.7 Likelihood analysis with TMB

15.8 Comparison of the engines

15.9 Summary and outlook


16. Binomial GLM with both continuous and categorical explanatory variables

16.1 Introduction

16.2 Data generation

16.3 Likelihood analysis with canned functions in R

16.4 Bayesian analysis with JAGS

16.5 Bayesian analysis with Stan

16.6 Do-it-yourself MLEs

16.7 Likelihood analysis with TMB

16.8 Comparison of the engines

16.9 Summary and outlook


17. Binomial mixed-effects model or Binomial GLMM

17.1 Introduction

17.2 Data generation

17.3 Likelihood analysis with canned functions in R

17.4 Bayesian analysis with JAGS

17.5 Bayesian analysis with Stan

17.6 Do-it-yourself MLEs

17.7 Likelihood analysis with TMB

17.8 Comparison of the engines

17.9 Summary and outlook


18. Model building, model checking and model selection

18.1 Introduction

18.2 Why do we build a statistical model ?

18.3 Tickly rattlesnakes

18.4 How do we build a statistical model ?

18.4.1 Model expansion and iterating on a single model

18.4.2 Causal, path and structural equation models

18.4.3 Performance optimization in a predictive model

18.5 Model checking and goodness-of-fit testing

18.5.1 Traditional residual diagnostics

18.5.2 Bootstrapped and posterior predictive distributions

18.5.3 Cross-validation for goodness-of-fit assessments

18.5.4 Identifying and correcting for overdispersion

18.5.5 Measures of the absolute fit of a model

18.5.6 Parameter identifiability and robustness to assumptions

18.6 Model selection

18.6.1 When to do model selection ... and when not

18.6.1 Measuring the predictive performance of a model

18.6.2 Fully external model validation using "out-of-sample" data

18.6.3 Approximating external validation for "in-sample" data with cross-validation

18.6.4 Approximating external validation for "in-sample" data with information criteria: AIC, DIC, WAIC, and LOO-IC

18.7 Model averaging

18.8 Regularization: penalization, shrinkage, ridge and lasso

18.9 Summary and outlook


19. General hierarchical models: Site-occupancy species distribution model (SDM)

19.1 Introduction

19.2 Data generation

19.3 Likelihood analysis with canned functions in the R package unmarked

19.4 Bayesian analysis with JAGS

19.5 Bayesian analysis with Stan

19.6 Bayesian analysis with canned functions in the R package ubms

19.7 Do-it-yourself MLEs

19.8 Likelihood analysis with TMB

19.9 Comparison of the engines

19.10 Summary and outlook


20. Integrated models

20.1 Introduction

20.2 Data generation: Simulating three abundance data sets with different observation/aggregation models

20.3 Fitting models to the three individual data sets first

20.3.1 Fitting a standard Poisson GLM in R and JAGS to data set 1

20.3.2 Fitting the zero-truncated Poisson GLM in R and JAGS to data set 2

20.3.3 Fitting the cloglog Bernoulli GLM in R and JAGS to data set 3

20.4 Fitting the integrated model to all three data sets simultaneously

20.4.1 Fitting the integrated model with a Do-it-yourself likelihood function

20.4.2 Fitting the integrated model with JAGS

20.4.3 Fitting the integrated model with NIMBLE

20.4.4 Fitting the integrated model with Stan

20.4.5 Fitting the integrated model with TMB

20.4.6 Comparison of the parameter estimates for the integrated model

20.5 What do we gain by analysing the joint likelihood in our analysis?

20.6 Summary and outlook


21. Conclusion

21.1 Looking back

21.2 Looking ahead

21.2.1 Study design

21.2.2 Do-it-yourself MCMC

21.2.2 Machine learning

21.2.4 Hierarchical models