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Bayesian Structural Equation Modeling: A New

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Bayesian Structural Equation Modeling: A New
Bayesian Structural Equation Modeling: A New
Methodological Tool for Unraveling Ecological
Patterns
George B.
B Arhonditsis
University of Toronto
C i A.
Craig
A St
Stow
University of South Carolina
Kenneth H. Reckhow
Duke University
What is missing from the common
ecological
g
ppractice is a statistical
method that…
i) can translate fairly complicated ecological phenomena
and express them as a function of several conceptual
environmental factors
Physical
environment
Water
clarity
Phytoplankton
Nutrients
Herbivory
Epilimnetic phytoplankton dynamics
ii) link the conceptual factors of interest with routinely
measured variables by explicitly acknowledging that
none of those reflects perfectly the underlying property
Bi l
Biovolume
Measurement
error
Chl
Chlorophyll
h ll a
Epilimnion
d h
depth
Water
clarity
Structural
error
Phytoplankton
SRP
Nutrients
Herbivory
DIN
O
Observed
variable
i
Latent variable
Daphnia
Zooplankton
iii) test both direct and indirect paths of this ecological
structure and identify the importance of their role
Bi l
Biovolume
Chl
Chlorophyll
h ll a
Epilimnion
d h
depth
Water
clarity
Phytoplankton
SRP
Bottom-up
Top-down
Nutrients
Herbivory
Nutrient
recycling
DIN
Daphnia
Zooplankton
Advantages of Structural Equation
Modeling
• In contrast with regression analysis:
analysis:
i) the predictor variables are NOT assumed to be free of
measurement error or uncontrolled variation,
ii) hypotheses are formulated in a way that allows for the
inclusion of unobserved, “latent” variables and NOT only
directly observed variables,
iii) provides a flexible tool for testing both direct and indirect
paths of ecological structures and identify the importance of
their role
• Principal component analysis also has the ability to reduce a set
of correlated variables to higherhigher-order components but has a
limited flexibility to specify the model structure prior to the
analysis and does not account for measurement error
“…Thinking only in terms of directly observable variables confines
our horizons and limits our assessment of complex systems….
systems….”
Malaeb et al. (2000, pg 95)
Application
pp
off Structural Equation
q
Modeling
Lake Washington (mesotrophic environment)
Lake Mendota (eutrophic environment)
Lake Washington
0.07
C
Cyanobacteria
b
i
Epilimnion
d h
depth
0.20
-
+
SRP
0.56
Chl
Chlorophyll
h ll a
0.32
-
+
-
Phytoplankton
+
+
+
Water
clarity
Nutrients
Herbivory
+
0.21
+
DIN
0.51
Daphnia
+
0.79
Zooplankton
Lake Mendota
0.67
Bi l
Biovolume
Epilimnion
d h
depth
0.71
-
+
SRP
0.79
Chl
Chlorophyll
h ll a
0.84
+
+
-
Phytoplankton
-
-
+
Water
clarity
Nutrients
Herbivory
+
0.98
+
DIN
0.93
Daphnia
+
0.83
Zooplankton
Bayes’
y Theorem
p(θ
p(
p
θ|D) ∝ p
p(θ
p(
θ) L(D|θ
L(D||θ))
Posterior belief on θ after
observing the new data
P i beliefs
Prior
b li f
on θ
likelihood function:
function:
used to update the
prior beliefs on θ
to account for the new data
Sequential updating
• Repeated use of the Bayes’ Theorem
• Current
C
t posterior
t i becomes
b
prior
i when
h new d
data
t are available
il bl
• Realistic ecological structures subject to sequential updating with routinely
monitored environmental variables
• Predictions that account for the uncertainty in both model parameters and
predictions
Watershed Model
Nutrient
Loading
Bayesian SEM
Follow--up studies
Follow
A fframework
k that
h tests the
h compatibility
ibili off different
diff
pre
pre-conceptualizations of the phytoplankton community structure with
the observed ecological
g
p
patterns.
Determination of the optimal phytoplankton community
aggregation level
F t
Future
perspectives
Flexible modeling tool for biodiversity studies
Hypothesis 1 for the phytoplankton community
composition
S li it
Salinity
Light
Attenuation
Temperature
Diatoms
Physical
environment
Flow rates
DIN
Dino
flagellates
Functional
Groupp
A
Chlorophytes
Nitrogen
Cryptophytes
NOx
PO4
Cyano
bacteria
Hypothesis 2 for the phytoplankton community
composition
S li it
Salinity
Light
Attenuation
Temperature
Dino
flagellates
Diatoms
Physical
environment
Flow rates
DIN
Functional
Group
B
Functional
Groupp
C
Chlorophytes
Nitrogen
Cryptophytes
NOx
PO4
Cyano
bacteria
References
1) Arhonditsis, G.B., Stow, C.A., Steinberg, L.J., Kenney, M.A.,
Lathrop, R.C., McBride, S.J., Reckhow, K.H. 2006.
Exploring ecological patterns with structural equation modeling
and Bayesian analysis.
Ecological Modelling, 192 (3(3-4): 385
385--409
2) Arhonditsis, G.B., Stow, C.A., Paerl, H.W., ValdesValdes-Weaver,
LM S
L.M.,
Steinberg,
i b
L
L.J.,
J R
Reckhow,
kh
K
K.H.
H
Delineation of the role of nutrient dynamics and hydrologic
forcing on phytoplankton patterns along a freshwaterfreshwater-marine
continuum.
Submitted Manuscript.
Manuscript
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