Hierarchical Bayesian Models Hierarchical Regression and Spatial models
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Hierarchical Bayesian Models
Hierarchical Regression and Spatial models
What is a hierarchical model? Hierarchical models are used when the data are structured in
groups. e.g. demographically, temporally, spatially Different (but related) parameters are used for each group These group level parameters are can be interpreted to describe
group level differences for predictors we DIDN’T measure Types of groups
Clustered data: when modeling individual plant growth and comparing among populations, some observations are made on individuals (e.g. stem length) while some are made on populations (e.g. temperature, soil) E.g., fit a matrix demographic model and use population random effects on
probability of seedling survival Repeated measures data: Time effects - e.g. in a mark recapture model
survival might vary by year Spatial effects – in a species distribution model, latent, spatially
autocorrelated patterns – cells are more like their neighbors
How are hierarchical parameters modeled?
Consider the three alternatives for modeling groups Pool all groups together
Ignore latent differences Ignore autocorrelation (observations not independent)
Model each group separately Small sample sizes problematic Many more parameters to estimate Ignores latent similarity
Model each group hierarchically Groups are similar but different Intermediate number of parameters
Comparison of pooling alternatives
Proportion of juvenile aloes in 35 populations spanning the species’ range
Multiple samples from each population
Combined estimates across samples from different years to estimate the variance in each population
Hierarchical regression Regressions have intercept and slope terms Hierarchical regressions have different intercepts or
slopes (or both) for each group Group level parameters do not vary independently, but
are constrained by a distribution – often a normal with a variance that we model
Hierarchical regression
30 young rats, weights measured weekly for five weeks
Dependent variable (Yij) is weight for rat “i” at week “j”
Hierarchical: weights (observations) within rats (groups)
Data:
from Gelfand et al (1990)"
Hierarchical regression
Hierarchical regression
Pop line (average growth)
Individual Growth Lines
Rat “i” has its own expected growth line:
E(Yij) = b0i + b1iXj
There is also an overall, average population growth line:
E(Yij) = β0 + β1Xj
Wei
ght
Study Day (centered)
NONheirarchical options
Hierarchical regression
A compromise between pooling and individual estimates:"
Pop line (average growth)
Bayes-Shrunk Individual Growth Lines
Hierarchical regression W
eigh
t
Study Day (centered)
A compromise: Each rat has its own line, but information is borrowed across rats to tell us about individual rat growth"
Summary: Visualizing a hierarchical model
ß|α
µ11|ß µ21|ß µ31|ß
y11 y12 y13 y21 y22 y23 y31 y32 y33
Requires a hyperprior distribution p(α)
Priors for each sub-population through p(ß| α)
Data
Summary: Visualizing a hierarchical model
Simple, non-hierarchical model
Hierarchical model
Why use hierarchical models? Account for sources of uncertainty/variability
Account for individual level and group level variation when estimating group level coefficients (classical models require averaging over individual level variation)
Borrow strength across groups (minimize effects of small sample sizes in some groups)
Random effects absorb variation that’s NOT related to the fixed effects so you potentially get lower bias for fixed effect estimates
Random effects pick up unmeasured variation It’s probably never worse than nonhierarchical models because
the worst case is that you find among group variation to be irrelevant
Exercise: Group means Data structure
J groups, with means θj , each with nj observations yij = ith measurement from the jth group variance of each group is σ2
j (assumed known)
Model structure Estimate θj for each of the J groups Estimate μ, the group mean Estimate τ, the variance of the θj
Formalizing the group means model Data structure
J groups, with means θj , each with nj observations yij = ith measurement from the jth group variance of each group is σ2
j (assumed known) εij = error εij ~ N(0,σ2)
Model structure Estimate θj for each of the J groups Estimate μ, the group mean Estimate τ, the variance of the θj
All groups the same yij = µ + εij
Each group unique yij = θj + εij
Compromise: Groups similar but different yij = θj + εij, with: θj ~ N(µ,τ2)
See Ch. 5 of Gelman et al. 2005. Bayesian Data Analyis for details of this model
Exercise: Hierarchical Normal Model Model from Gelman 2003 Estimate the proportion of juveniles and dead individuals
of Aloe dichotoma across their range for a matrix demographic model
35 populations, each with a subset of ~100 individuals measured 6 years apart
Instructions in Exercise Day 3.R
Exercise: Hierarchical Normal Model
Outline Review Continuation of exercises from last time Hierarchical model Spatial models
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Spatial Models Point-level modelling refers to modelling of spatial data
collected at locations referenced by coordinates (e.g., lat-long, Easting-Northing).
Fundamental concept: Data from a spatial process {Y (s) : s ∈ D}, where D represents some landscape
Example: Y(s) is a pollutant level at site s Conceptually: Pollutant level exists at all possible sites Practically: Data will be a partial realization of a spatial
process – observed at {s1, . . . , sn} Statistical objectives: Inference about the process Y
(s); predict at new locations.
Spatial Models
Slide from Andrew Finley and Sudipto Banerjee
Spatial Models
Slide from Andrew Finley and Sudipto Banerjee
Spatial Gaussian process
Spatial Models
Slide from Andrew Finley and Sudipto Banerjee
Φ is the distance decay of the spatial process
Φ small
Φ medium
Φ large
Spatial Models
Slide from Andrew Finley and Sudipto Banerjee
Spatial Models
Slide from Andrew Finley and Sudipto Banerjee
Spatial Models
Slide from Andrew Finley and Sudipto Banerjee
To summarize, the parameters we want are: β = regression coefficients τ2 = residual variance σ2 = spatial variance ϕ= decay of spatial dependence
Together, σ2 and ϕ define the spatial weights, w Usually we want to look at the surface of spatial weights
to see if there are any latent patterns
Spatial Models
Slide from Andrew Finley and Sudipto Banerjee
Spatial Models
Slide from Andrew Finley and Sudipto Banerjee
Spatial weights w/elevation
Spatial weights w/o elevation
Spatial Models
Slide from Andrew Finley and Sudipto Banerjee
Spatial random effects can: Account for spatial autocorrelation (samples aren’t
independent) Ensure that your regression coefficients aren’t contorted
to accommodate patterns that aren’t actually related to the predictor
Identify spatial patterns in residuals that are related to omitted predictors
Exercise: Spatial Models
Slide from Andrew Finley and Sudipto Banerjee
The exercise is given in the second section of today’s “Exercises Day 3.R” file
This exercise is formulated as a demonstration because some of this stuff is tricky to code
Uses the R package spBayes to do all the manual labor so you just specify Starting values Priors Jump distributions
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Conclusions
Model 1 Prior: Bayesian methods are extremely complicated Data: Go to a Bayesian workshop Posterior: Bayesian methods aren’t too bad†
Model 2 Prior: Posterior from step 1 Data: Read Bayesian Books Posterior: Use Bayesian Methods†
† Metropolis-Hastings sampling recommended
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