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