Hierarchical Models with Ecological Data

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Hierarchical Models with Ecological Data

• Edward Boone
• Keying Ye
• Eric P. Smith
• Department of Statistics, Virginia Tech

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Outline

• An ecological data set
• Hierarchical modeling with missing data and
  spatial correlation
• Analysis of the data
• Hierarchical model or not, does it make a
  difference?

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The Data

• Water Quality data from the Ohio EPA.
• Response IBI Index of biotic integrity.
• Predictors
• QHEI Quality of Habitat Environment Index
• DO Dissolved Oxygen.
• Others such as NH3, Talk Alkalinity, PH, AL,
  Hardness, PB, MN, TSS and more,

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The Problem

• Understanding the relationship between biology
  and environmental conditions.
• Which variables are important?
• Is the relationship similar across the state?
• Some of the attributes common to ecological data
• (1) Measures at different scales. (i.e. site,
  river basin, state)
• (2) Spatial correlation.
• (3) Missing data.

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Which Predictors are Important?

• We performed forward, backward, stepwise and
  highest posterior probability selections in the
  regression, over the entire data set to determine
  which models could be candidates for the
  Hierarchical Models.
• After data reduction only 190 of 2087
  observations were used. So results may not be
  reliable.

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The Hierarchical Model

• Site Level
• Yi N (Xibi ,S(q) )
• Basin Level
• bi N( Tg , G(f) )
• Hyperparameters
• g N( a, A )

Spatial Continuous
Spatial Lattice
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• where we have
• S(q) is a continuous (geostatistical) spatial
  covariance model.
• G(f) is a lattice spatial model.
• This allows for modeling of spatial correlation
  among the site level variables and the basin
  level variables.

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Missing Data

• Missing data is a common feature to ecological
  data sets. Data augmentation is used to deal
  with the problem.
• Our goal is to incorporate all uncertainties into
  the model. So our imputed values vary during
  simulation. This feature is not present in many
  analyses.

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Data Augmentation Algorithm

• Partition the response vector and data matrix
  into the following form
• where Z is the vector of missing values.

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For the Normal case

• Prior for Z
• ZN(mZ,SZ)
• The full conditional for Z
• ZothersN(m,F)
• where
• and

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Estimation

• To estimate our model we will use the Gibbs
  Sampler.
• Suppose we have a parameter vector q
  (q1,q2,...,qk) we wish to determine the posterior
  distribution of. We can sample the posterior
  distribution by the following method
• q1 q2,q3,...,qk
• q2 q1,q3,...,qk
• qi q1,...,qi-1,qi1,...,qk
• qk q1,q2,...,qk-1

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Analysis of the data (no spatial)

• Base Model
• QHEI and DO are relatively mound shaped, thus
  normal priors with m 50 and s 100 for QHEI and
  m 5 and s 10 for DO are used (flat priors.)

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Model estimation using Gibbs sampling

• Full Conditionals

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Model convergence checking

• Posterior distribution is simulated. The samples
  had autocorrelation at the first lag so we used a
  thinning interval of 2.

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• Hierarchical Model Results of IBI vs. QHEI and DO
  with no spatial correlation using Data
  Augmentation. (Significance was determined via a
  95 probability interval.)

Significance of QHEI
Significance of DO
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Data Analysis (with spatial correlation)

• Conditional Autoregressive Model
• This is a model for Lattice data. It is similar
  to the Time Series model MA. The main assumption
  is pair wise dependence in the data.
• where aii0 and ajiaij.

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• This can be translated into a covariance matrix.
• S(I-A)-1D
• where D is a diagonal matrix and A is a matrix
  of the aij.
• The model we use here is the one parameter model,
• S(I-rA)-1D
• with constraints on r to ensure that S is
  positive definite, and aij1 if basin i is a
  neighbor of basin j, zero otherwise. Our model
  will have three r parameters, one for each
  predictor variable at level one. This is a CAR
  model.

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Estimation using Gibbs sampling with CAR modeling

• Full Conditionals

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Model convergence checking

• Posterior distribution is simulated. The samples
  had autocorrelation at the first lag so we used a
  thinning interval of 2.

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• Hierarchical Model Results of IBI vs. QHEI and DO
  with no spatial correlation using Data
  Augmentation. (Significance was determined via a
  95 probability interval.)

Significance of QHEI
Significance of DO
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(1) No spatial (2) with CAR model
QHEI variable
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(1) No spatial (2) with CAR model
DO variable
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Possible Interpretations

• Quality of Habitat Environment Index
• (1) The non-CAR model showed overall
  significance of QHEI, while the CAR model did
  not
• (2) Since QHEI was significant in three basins
  under either model, we should not deem QHEI as
  unimportant.
• (3) Any policy decisions should be made on a
  river basin level than a statewide level.

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Possible Interpretations (cont.)

• Dissolved Oxygen
• (1) DO is significant in both models so DO
  should be treated as an important predictor
• (2) There is significant variation in the mean
  parameter for DO. So the effect of DO differs
  across basins
• (3) Any policy decisions should be made on a
  river basin level than a statewide level.

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Can this be done without a hierarchical structure?

• Separable covariograms
• Suppose a covariogram function C(?) of h can be
  written as C(h)C1(h1)C2(h2) where h(h1, h2).
  Then C is called a separable covariogram.
• The advantage of this is that two spatial
  analyses maybe done separately.

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• Consider the problem we are interested in as
  follows.
• A Hierarchical Stochastic Process
• Level I Y(i,s)s? Dib(i)
• where EY(i,s)b(i)Xi(s)b(i) and
• VarY(i,s)b(i)Si
• Level II b(i)i? D2g
• where Eb(i)gTg and Varb(i)gW.
• For hyperparameters gp(g).
• Note we have two types of spatial problems here
  one with continuous (s) and the other is lattice
  (i).

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• Matrix notation

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• Rewriting the model
• and
• Variance-covariance matrix for Y(s)

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• On the individual observation base, we have the
  following covariance structures
• where si(jl) is the j,l element of the matrix
  Si, and
• for i? k.
• Both spatial components are additively separable.
  

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• Comments
• (1) Y(i,s) is not second-order stationary for
  any i, unless W0 (which we do not have the
  lattice spatial component) so many standard
  spatial methods do not work in this case
• (2) Y(s) is not second-order stationary
• (3) The spatial components of s and i (the
  lattice) are NOT separable so we cannot model
  the covariance structure of the both separately.

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• What can we do?
• Hierarchical modeling with spatial correlation
  of the Y(i,s) and lattice spatial component of i
  (such as CAR or SAR models) is definitely one of
  the natural answers.