# Presentation: 458582

## Hierarchical Models And The Analysis Of Survey Information

- John R. Sauer,William A. Link,
- and J Andrew Royle*USGS Patuxent Wildlife Research Center,*
*and U.S. Fish and Wildlife Service**11510 American Holly Drive** Laurel, Maryland 20708, USA*

## Often in Situation in Which We Need to Summarize Information: 

- Over Space:
- Mapping abundances
- Over Time:
- Estimation of trends
- Over Species
- Species group characteristics

## Problem Is Hierarchical

- Want to summarize collections of estimates
- Trends among regions
- Trends among species
- All we have are estimates
- Traditional approaches
- Route regression
- Robust, but cumbersome and sometimes controversial
- Weighting
- Better approach:
- Hierarchical models

## Hierarchical Models

- Data and parameters both random variables
- Data are observed
- Distributions described conditionally on realized values of parameters that are also random variables
- Need to include additional level(s) of distributional assumptions
-  Analysis of hierarchical models
- Bayesian methods
- Model-based
- Probability statements about s

##  Bayesian Probability Distributions Describing Data and Parameters.  

*f*(Ys|s)  ~  Sampling distribution of the data given the unknown parameter.
- (s |)      ~  Prior distribution of the parameters;  hyperparameter.
*f*(s| Ys)  ~  Posterior distribution
- Bayesian inference about s
- Based on posterior distribution.
- “Posterior mean"

## Historically, Bayes Analyses Were Difficult

- Technical complexity
- Could only do simple models
- Heavy mathematics
- Philosophical complexity
- Modeling associated with prior (assumptions about distribution of parameters) influence results
- Controversial

## Markov Chain Monte Carlo (MCMC)

- New procedure for fitting hierarchical models
- Define models
- Write as “full conditionals” (distribution with all other parameters fixed)
- Do an interative sampling from the full conditionals (Gibbs sampling)
- Converges on posterior distributions

## Estimation of Population Change

- Notation:
- λ is expected value of Y
-  Stratum-specific intercept, slope, and year effects (S,β, and γ),
- Observer/route effects (ω),
- and overdispersion effects (ε)

- Observer/route effects, Year Effects, and Overdispersion effects are treated as mean zero normal random variables.

## Annual Indices and Trend

- Z : proportion of routes on which the species was encountered

- Trend is ratio of year effects:

## Composite Regional Estimates

- Regional Indexes are sums of regional totals..the indexes multiplied by the regional areas

- The composite trend is calculated as

## Example Application:Northern Bobwhite in Appalachians

- Divided Appalachians into 3 groups of states, estimated trend and indexes separately for each, then estimated composite
- Route regression results:
- -5.77, p < 0.001, n = 195
- MCMC results:
- -5.86, (-6.69,-5.02)

## Slide 12

## Benefits of Hierarchical Models

- Appropriate conceptual framework for dealing with collections of estimates
- Accommodate error in summaries of estimates
- Allow for estimation of derived attributes
- Population stability metric
- Ranking
- Can be implemented directly in simulations

## Comments

- Many other applications for hierarchical models
- Spatial analyses
- (Relatively) easy to implement in programs such as BUGS
- Avoid ad-hoc procedures of past

## The Santa Rosa Bird Data

- These data are similar to the BBS
- Spatial structure
- Points
- Transects
- Island
- Bad Features
- Missing data
- Confounding

## Distance Data Are Available

- Can use program DISTANCE
- Estimate observer-specific detection rates
- Effective Radii (EDR)
- Easier to adjust areas of counting in analysis
- 3 observers:
- TJC:  (EDR: 16.258, area= 0.0830 km^2)
- GSA:  (EDR: 27.8, area= 0.2428 km^2)
- LCD:  (EDR: 16.60, area= 0.0866 km^2)

## Set up Hierarchical Model

- Points analogous to BBS Routes
- Transects similar to strata
- Island result aggregate of transects
- Adjust for differing areas of counting
- Unbalanced data greatly complicated analysis
- Can’t implement full model
- Fitting issues

## Model Structure (BUGS Code)

- for( k in 1 : ncounts ) {
- log(lambda[k]) <- pts[point[k]] + noise[k] +Yeareffect[year[k]]  + logareas[year[k]]
- noise[k] ~ dnorm(0.0, taunoise),  count[k] ~ dpois(lambda[k]) }
- taunoise ~ dgamma(0.001,0.001)
- for( i in 1 : npoints ) { pts[i] ~ dnorm( 0.0,taupts)       }
- taupts ~ dgamma(0.001,0.001)
- for( y in 1 : nyears ) {yeareffect[y] ~ dnorm( 0.0, 1.0E-3)
- logareas[y] <-log(area[y])}
- for( t in 1 : nyears ) { n[t] <- exp(yeareffect[t])   }
- # nyears for this example, due to lost year!
- 	  B <- pow(n[nyears]/n[1],1/(nyears))

## Hierarchical Model Plotted vs Area-Adjusted Means

## Estimated Trend

- Hierarchical model -15.54 %/year (-23.65,-5.95)
- “Naïve” trend:  0.29 %/year

## Diagnostics include:

- Autocorrelations of the MCMC “Replicates.” and

- Plots of Estimates During Process

- Modeling requires some expertise!