1 Course overview

The course will cover several advanced statistical modeling methods using the programming language R, including maximum-likelihood, non-linear, Bayesian, and multi-level (hierarchical) methods as well as techniques for using data simulation to test models. The R function lmer, an accessible yet complex tool for advanced modeling, will be covered in detail. To establish a base for understanding multi-level models, some review of standard regression will be included, plus a session on fitting non-linear models with maximum likelihood.

During the first half of each session, I will explain methods and present examples of their use; in the second half, students will work on assignments using the same methods. Datasets will be provided, but students are encouraged to bring their own data as well. A course web site will provide sample code, data, and a list of key R functions. Students should be familiar with R: manipulating dataframes, graphing, and linear regression.

1.1 To apply

• To join, contact Alexandra Sapoznikow, Oficina de Vinculación Tecnológica, Centro Nacional Patagónico -Conicet

1.2 Schedule

• When: Five Sessions, 9:00-18:00, 9-13 Oct 2012
• Where: Salon Península, Cenpat, Puerto Madryn

2 Software required

• R base package
• R contributed packages lme4, arm, coda, mvtnorm, date, available at http://cran.r-project.org/
• RStudio, or other programming editor such as Geany or Notepad++ (NOT Wordpad NOT Notepad)
• CTFSRPackage from http://ctfs.arnarb.harvard.edu/Public/CTFSRPackage

3 Course web site

• http://ctfs.arnarb.harvard.edu/Public/Workshops/Cenpat
  • http://ctfs.arnarb.harvard.edu/Public/Workshops/Cenpat/outline.html, and outline.pdf
  • http://ctfs.arnarb.harvard.edu/Public/Workshops/Cenpat/assignments.html, and assignments.pdf
  • sample R datasets http://ctfs.arnarb.harvard.edu/Public/Workshops/data
  • R scripts http://ctfs.arnarb.harvard.edu/Public/Workshops/Cenpat/source
  • history or R commands I enter during the course http://ctfs.arnarb.harvard.edu/Public/Workshops/Cenpat/history
• Each will be updated regularly throughout the course

4 Books and other background material

• Bates’ online book http://lme4.r-forge.r-project.org/
• Random vs. fixed effects http://andrewgelman.com/2005/01/why_i_dont_use/
• Gelman: http://www.amazon.com/Analysis-Regression-Multilevel-Hierarchical-Models/dp/052168689X
• Bolker: http://www.amazon.com/Ecological-Models-Data-Benjamin-Bolker/dp/0691125228/ref=sr_1_2?s=books&ie=UTF8&qid=1348667709&sr=1-2&keywords=Bolker
• Kruschke: http://www.amazon.com/Doing-Bayesian-Data-Analysis-Tutorial/dp/0123814855/ref=pd_sim_b_2
• Carlin: http://www.amazon.com/Bayesian-Methods-Analysis-Edition-Statistical/dp/1584886978/ref=pd_sim_b_4
• Albert: http://www.amazon.com/Bayesian-Computation-R-Use/dp/0387922970/ref=pd_sim_b_5
• Robert: http://www.amazon.com/Introducing-Monte-Carlo-Methods-Use/dp/1441915753/ref=pd_sim_b_2

5 Contents and approximate scheduling (daily progress will depend on experience of the students

• Modeling with standard regression and maximum likelihood [day 1]
  1. Linear regression with lm (review)
     • Gaussian error
     • Residuals and statistics (coef, summary)
     • Data treemass: log(agb) vs. log(dbh)
     • Centering x in linear regression!
       Use xCenter = x - mean(x)
  2. Numerical estimation with optim
     • maximize likelihood vs. minimize sum of squares
     • alternate methods in optim (Nelder-Mead etc.)
     • comparing models with AIC
     • Non-linear models
  3. Survival models with maximum likelihood [day 2 morning]
     • binomial error instead of Gaussian error
     • logistic function to describe data
• Data simulation [day 2 afternoon]
  1. Two purposes of simulation
     • Understand connection from Process –> Data
     • Test whether models work
  2. R’s probability distribution functions
     • density and random draws (eg, dnorm and rnorm)
     • important distributions: normal, binomial, poisson, negative binomial
  3. Regression with error
  4. Multi-level regression
  5. Extra: Survival
• Multi-level models (mixed effect, hierarchical, random vs. fixed effects) [day 2-3]
  1. Why multi-level modeling?
  2. Limitation: linear (or transformed linear) with normal error
  3. Multi-level vs. standard regression
     Bates Chap 4, Section 4.4; Gelman & Hill pp. 251-259
  4. Regression with one group using lmer
     • output of display
     • graphs using the coefficients
     • variable intercept, slope, or both
  5. Regression with two groups or two predictors x using lmer
     • output of display
     • models with or without covariance
     • group level predictor (see Gelman&Hill p. 265)
     • graphs using the coefficients
  6. Random or fixed?
     • Traditional
       • Random: nuisance effects, unrepeatable (batch, plot)
       • Fixed: permanent group, repeatable (sex
       • Gray area: year? site?
     • Recent issues favoring multi-level approach
       (ie, Gelman, who replaces ’random’ with ’grouping’)
       • Is group-level variation an explicit research topic?
       • Can different groups be thought of as similar?
       • Can information on one group support other groups?
       • Are some groups rare and thus needing support?
       • Are there enough groups? (too few -> little evidence on group-level variation)
• Bayesian methods [day 4-5]
  1. Bayes rule and the posterior distribution
  2. Metropolis, the Gibbs sampler (MCMC)
     1. Another method for fitting parameters
     2. Automatically provides fully accurate confidence
     3. Much more flexible modeling options (ie, non-linear with many parameters)
     4. Any error distribution
     5. Latent states or latent data
  3. Hierarchical modeling
  4. Limitations: long run time, complicated program
  5. Keys to your own program
     1. Getting the correct likelihood functions, and this can be difficult in complex models
     2. Preparing data structures to save all the data and likelihood
     3. Looping through all the parameters and hyperparameters
     4. Returning results
  6. Details
     1. Parameter correlation, autocorrelation and poor convergence
     2. Diagnostics (see coda package)
     3. Fitting the covariance
     4. Special cases where Metropolis not needed

6 Key R functions

• Data extraction
  1. subset
  2. apply
  3. tapply
  4. cut
  5. dim
  6. str
  7. names
  8. ifelse [R base package]
  9. IfElse [CTFSRPackage version]
• Graphics
  1. hist
  2. plot
  3. points
  4. line
  5. curve
  6. abline
  7. box
  8. axis
  9. X11
  10. dev.set
• Modeling
  1. summary
     • mean
     • median
     • sd
     • var
     • cor
     • CI [CTFSRPackage]
  2. model
     • lm
     • glm
     • lmer [lme4 package]
     • coef
     • summary
     • fixef [arm package]
     • ranef [arm package]
     • display [arm package]
     • dotplot [lattice package]
     • xyplot [lattice package]
• Likelihood
  1. optimize
  2. optim
  3. metrop1step [in CTFSRPackage]
• Error functions and probability distributions
  1. dnorm is the standard
  2. dbinom is the standard for survival or occurrence (or similar)
  3. dlnorm
     • for abundances, whether integer or not (but usually not used in favor of log-transformation
     • good match for tree growth rates
     • but cannot handle zeroes
  4. dgamma is similar to log=normal
  5. dpois including zeroes (but does not handle much ecological data well)
     • for integer abundances
     • handles zeroes
     • however, close to Gaussian so not appropriate for much ecological data
  6. dnbinom
     • for integer abundances that are highly skewed
     • very common in ecology
     • R: prob=dnbinom(count,size=k,mu=mu)
     • size is ’clumping parameter’; mu is mean