Statistical Models for Stream Ecology Data: Random Effects Graphical Models

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Statistical Models for Stream Ecology
DataRandom Effects Graphical Models

• Devin S. Johnson
• Jennifer A. Hoeting
• STARMAP
• Department of Statistics
• Colorado State University

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The work reported here was developed under the
STAR Research Assistance Agreement CR-829095
awarded by the U.S. Environmental Protection
Agency (EPA) to Colorado State University. This
presentation has not been formally reviewed by
EPA.  The views expressed here are solely those
of presenter and the STARMAP, the Program he
represents. EPA does not endorse any products or
commercial services mentioned in this
presentation.
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Motivating Problem

• Various stream sites in the Mid-Atlantic region
  of the United States were visited in Summer 1994.
• For each site, each observed fish species was
  cross categorized according to several traits
• Environmental variables are also measured at each
  site (e.g. precipitation, chloride
  concentration,)
• Relative proportions are more informative.
• How can we determine if collected environmental
  variables affect species richness compositions
  (which ones)?

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Outline

• Introduction
• Compositional data
• Probability models
• Brief introduction to chain graphs
• A graphical model for compositional data
• Modeling individual probabilities
• Markov properties of random effects graphical
  models
• Analysis of fish species richness compositional
  data
• Conclusions and Future Research

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Discrete Compositions and Probability Models

• Compositional data are multivariate observations
• Z (Z1,,ZD) subject to the constraints that
  SiZi 1 and Zi ? 0.
• Compositional data are usually modeled with the
  Logistic-Normal distribution (Aitchison 1986).
• Scale and location parameters provide a large
  amount of flexibility compared to the Dirichlet
  model
• LN model defined for positive compositions only
• Problem With discrete counts one has a
  non-trivial probability of observing 0
  individuals in a particular category

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Existing Compositional Data Models

• Billhiemer and Guttorp (2001) proposed using a
  multinomial state-space model for a single
  composition,
• where Yij is the number of individuals belonging
  to category j 1,,D at site i 1,,S.
• Limitations
• Models proportions of a single categorical
  variable.
• Abstract interpretation of included covariate
  effects

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Existing Graphical Models

• Graph model theory (see Lauritzen 1996) has been
  used for many years to
• model cell probabilities for high dimensional
  contingency tables
• determine dependence relationships among
  categorical and continuous variables
• Limitation
• Graphical models are designed for a single sample
  (or site in the case of the Oregon stream data).
  Compositional data may arise at many sites

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New Improvements for Compositional Data Models

• The Billhiemer and Guttorp model can be
  generalized by the application of graphical model
  theory.
• Generalized models can be applied to
  cross-classified compositions
• Simple interpretation of covariate effects as a
  variable in a Markov random field
• Conversely, graphical model theory can be
  expanded to include models for multiple site
  sampling schemes

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Chain Graphs
b
a
c
d
e

• Mathematical graphs are used to illustrate
  complex dependence relationships in a
  multivariate distribution.
• A random vector is represented as a set of
  vertices, V .
• Pairs of vertices are connected by directed edges
  if a causal relationship is assumed, undirected
  if the relationship is mutual

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Probability Model for Individuals (Unobserved
Composition)

• Response variables
• Set F of discrete categorical variables
• Notation y is a specific cell
• Explanatory variables
• Set G ? D of categorical (D) and/or continuous
  (G) variables
• Notation x refers to a specific explanatory
  observation
• Random effects
• Allows flexibility when sampling many sites
• Unobserved covariates
• Notation ef, f ? F, refers to a random effect.

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Probability Model and Extended Chain Graph, Ge

• Joint distribution
• f (y, x, e) f (yx, e) ? f (x) ? f (e)
• Graph illustrating possible dependence
  relationships for the full model, Ge.

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Random Effects Discrete Regression Model(REDR)

• Sampling of individuals occurs at many different
  random sites, i 1,,S, where covariates are
  measured only once per site
• Hierarchical model for individual probabilities

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Random Effects Discrete Regression Model(REDR)

• Response parameters constraints
• The function aF(x,e) is a normalizing constant
  w.r.t. y(x,e), and therefore, is not a function
  of y.
• The parameters bfcd(y, xD), wfg dm(y, xD), and
  ef (y) are interaction effects that depend on y
  and xD through the levels of the variables in f
  and d only.
• Interaction parameters (and random effects) are
  set to zero for identifiability of the model if
  the cells y or xD are indexed by the first level
  of any variable in f or d.

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Random Effects Discrete Regression Model(REDR)

• Model for explanatory variables (CG
  distribution)
• Again, interactions depend on xD through the
  levels of the variables in the set d only, and
  identifiability constraints are imposed.

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Graphical Models for Discrete Compositions

• Sampling many individuals at a site results in
  cell counts,
• C(y)i individuals in cell y at site i.
• Conditional count likelihood
• C(y)iy xi, Ni multinomial(Ni fRE(yxi,
  ei)y ),
• or
• C(y)i xi indep. Poisson(k (xi) ? fRE(yxi,
  ei) )
• Joint covariate count likelihood
• multinomial(Ni fRE(yxi, ei)y ) ? CG(l, t, ?Ø)

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Markov Properties of Chain Graph Models

• Let P denote a probability measure on the product
  space
• X ?a?V X a
• Markov (Global) property
• The probability measure P is Markovian with
  respect to a chain graph G if for any triple (A,
  B, S) of disjoint sets in V, such that S
  separates A from B in Gan(A?B?S)m, we have
• A ? B S.
• There are two weaker Markov properties, pairwise
  and local Markov properties.

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Markov Properties of the REDR Model

• Proposition 1. A REDR model is Ge Markovian if
  and only if the following six constraints are
  satisfied for a given extended graph Ge.
• Response model
• bfcd(y, xD) 0 unless f ? c ? d is complete for
• c ? d ? Ø.
• wfgdm(y , xD) 0 for m 1,,M, unless f ? g ?
  d is complete, where g ? G and d ? D.
• ef (y) -bf ØØ (y) with probability 1 if f is
  not complete.

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Markov Properties of the REDR Model

• Proposition 1. A REDR model is Ge Markovian if
  and only if the following six constraints are
  satisfied for a given extended graph Ge.
• Covariate model
• ld(xd) 0 unless d is complete .
• tdg(xd) 0 unless g ? c is complete, where g
  ? G and d ? D.
• ?mg. 0 unless m, g is complete, where g, m ? G
  and ?mg is the (m, g) element of ?Ø.

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Markov Properties of the REDR Model

• Sketch of proof.
• Lauritzen and Wermuth (1989) prove conditions
  concerning the l, t, and ?Ø parameters for the CG
  distribution.
• If the b and w parameters are 0 for the specified
  sets then the density factorizes according to
  Frydenburgs (1990) theorem.
• A modified version of the proof of the
  Hammersley-Clifford Theorem shows that if f (yx,
  e) separates into complete factors, then, the
  corresponding b and w vectors for non-complete
  sets must be 0.

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Preservative REDR Models

• Preservative REDR models are defined by the
  following conditions
• All connected components aq, q 1,,Q, of F in
  Ge are complete, where Q is the total number of
  connected components.
• Any d ? G?D that is a parent of f ? aq is also a
  parent of every other f ? aq, q 1,,Q.

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Markov Properties of the REDR Model

• Proposition 2. If P is a preservative REDR model,
  and P is Ge Markovian, then the marginal
  distribution, PF?G ?D, of the covariates and
  response variables is G (Ge)F?G ?D Markovian.

Sketch of Proof. The integrated REDR density
follows Frydenbergs (1990) factorization
criterion. The factor functions, however, do not
exist in closed form.
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Parameter Estimation

• A Gibbs sampling approach is used for parameter
  estimation
• Hierarchical centering
• Produces Gibbs samplers which converge to the
  posterior distributions faster
• Most parameters have standard full conditionals
  if given conditional conjugate distributions.
• Independent priors imply that covariate and
  response models can be analyzed with separate
  MCMC procedures.

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Fish Species Richness in the Mid-Atlantic
Highlands

• 91 stream sites in the Mid Atlantic region of the
  United States were visited in an EPA EMAP study
• Response composition
• Observed fish species were cross-categorized
  according to 2 discrete variables

• Habit
• Column species
• Benthic species

• Pollution tolerance
• Intolerant
• Intermediate
• Tolerant

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Stream Covariates

• Environmental covariates
• Values were measured at each site for the
  following covariates
• Mean watershed precipitation (m)
• Minimum watershed elevation (m)
• Turbidity (ln NTU)
• Chloride concentration (ln meq/L)
• Sulfate concentration (ln meq/L)
• Watershed area (ln km2)

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Fish Species Richness Model

• Composition Graphical Model
• and
• Prior distributions

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Model Selection

• Three different models are considered
• Independent response
• (i.e. bfg(yi) ef (yi) 0 for f H, T )
• Depended response w/ independent errors
• Dependent response w/ correlated errors
• (equivalent to Billheimer Guttorp model)

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Fish Species Functional Groups
Posterior suggested chain graph for independence
model (lowest DIC model)

• Edge exclusion determined from 95 HPD intervals
  for b parameters and off-diagonal elements of ?Ø.

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Comments and Conclusions

• Using Discrete Response model with random
  effects, the Billheimer-Guttorp model can be
  generalized
• Relationships evaluated though a graphical model
• Multi-way compositions can be analyzed with
  specified dependence structure between cells
• MVN random effects imply that the cell
  probabilities have a constrained LN distribution
• DR models also extend the capabilities of
  graphical models
• Data can be analyzed from many multiple sites
• Over dispersion in cell counts can be added

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Future Work

• Model determination under a Bayesian framework
• Models involve regression coefficients as well as
  many random effects
• Initial investigation suggests selection based on
  exclusion/inclusion of parameters not edges
  produces models with higher posterior mass
• Accounting for spatial correlation