1. Hierarchical Models

2. Conceptual: What are we talking about? – What makes a statistical model hierarchical? – How does that fit into population analysis? Estimators for hierarchical models – Maximum Likelihood Estimators – Bayesian estimation (MCMC)

3. Hierarchical Models Hierarchical model, multi-level model, random effects model, repeated measures, randomized blocks, sub-sampling Characterized by random processes at one-level being a function of random processes at another level Typically involve a nested/hierarchical structure

4. Simple representation  β  Non-hierarchical Hierarchical Single random process = process generating the data (stat. distribution) True state of Nature (fixed) Y: Data (variable) Additional random process = process generating Z Z is variable (stat. distribution) Y (variable) Z (variable) higher level (fixed) parameter Y is only function of a fixed parameter:  Y is function of , but also conditional on a R.V (Z)

5. Typical Hierarchical Model for Population Analysis We can often divide data generating mechanisms hierarchically Ecological Process: survival, occupancy, abundance Observation Process: detection, mis-classification, sampling error Data: counts, recapture history, # of detections for a patch  ECOLOGICAL PROCESS  OBSERVATION PROCESS Z Y

6. Example of a Random Effects Model Interest in counts (Y) of migrating shorebirds: several sources of variation (at different levels) – Refuges differ in ‘mean abundance’ ψ (process variation) – Ponds within refuges differ in abundance (f[ N|ψ]) (process variation) – Sampling error in counts (Y), due to imperfect detection (  ): f[Y| N  ]  Counts REFUGES PONDS Y Abundance  

7. Statistical model Joint probability distribution:  Data     Data

8. Statistical model Joint probability distribution:  Data     Data Likelihood: In simple cases we can write out the joint distribution and calculate MLE by optimization Bayesian: (1) Integrate to obtain  |Data] (2) Marginalize posterior to obtain  |Data] (likelihood fct) (3) Simulate using Markov Chain Monte Carlo

9. Simple Likelihood Example We go to a site 5 times and record whether a species is present Ecological Process: Occupancy f(z| ψ) ~ Bern(ψ) Observation Process: Probability of detecting the species y times is conditional on whether the species was present f(y|z) ~ Bin(p*z,5) p  DATA (times detected in 5 visits) OCCUPANCY DETECTION Y

10. Statistical model Joint probability distribution: f  y  p  f  p  f  p  DATA OCCUPANCY DETECTION Y  bin(y|1*p,5) + (1-  bin(y|0*p,5)

11. More complicated example Let’s say p varies around a mean value for a set of sites and that ψ varies annually around a mean value The full model is the joint distribution of all the parts: f(y|p, ψ, a,b, c,d) Y ~ bin(z*p,5) p ~ beta(a,b) z ~ bern( ψ ) ψ ~ beta(c,d) MLE methods rapidly become difficult so we typically turn to a Bayesian approach (MCMC estimation)

12. Markov Chain Monte Carlo MCMC is simulation based approximation of posterior distributions using dependent sequences of random variables. Bayesian (need to specify priors) Simulation (sample from the posterior distribution) Prior Knowledge + Insights from Data= Basis of Inference

13. For the sake of estimation, the model can be specified by its components Y ~ bin(z*p,5) p ~ beta(a,b) z ~ bern( ψ ) ψ ~ beta(c,d) And priors a ~ f( θ a ) b ~ f( θ b ) c ~ f( θ c ) d ~ f( θ d )

14. Summary of MCMC Metropolis-Hastings is one approach to MCMC (omnibus and easily described, but not terribly efficient) Best method: Gibbs sampling (many flavors) BUGS (WinBUGS, OpenBUGS, JAGS) automatically selects appropriate algorithm Issues: Starting values Convergence of MC markov chains Autocorrelation MCMC is simulation based evaluation of posterior distributions using dependent sequences of random variables.

15. Why consider MCMC Flexible approach for fitting models typical of ecological systems (hierarchical) Is intuitive Random-effects, spatial correlation, partially observed covariates can all readily be incorporated Calculating derived parameters and their uncertainty When priors are “non-informative” or data is abundant answers similar to MLE methods Naturally incorporate prior knowledge/belief