using structural equation modeling to analyze monitoring data jim grace nwrc
TRANSCRIPT
What is structural equation modeling?
A framework for using statistical methods to ask complex questions of data.
Macrohabitat
ηc1
Microhabitat
ηc2
Diversity
ηe3
Litter
ηe2
Herbaceous
ηe1
lake, x1
impound, x2
swale, x3
vhit1, y1
vhit2, y2
herbl, y3
herbc, y4
wlitr, y5
litrd, y6
litrc, y7
rich, y8
0
0
.26
.75
.98
.92.64
.90
.81 -.47
.95
1.13
.44
ns
ns
1.0 1.0
.80
1.00
.59.77
LISREL: A flexible, multiequational framework
y1 = α1 + β1x + ε1i y2 = α2 + β2x + β3y1 + ε2i
y3 = α3 + β4y1 + β5y2 + ε3i
y4 = α4 + β6y1 + β7y3 + ε4i
Can include observed, latent, and composite variables.
x = Λxξ + δy = Λyη + εη = α + Β η + Γξ + ζ
The LISREL Equations
Jöreskög 1973
Σ = {σ11
σ12 σ22
σ13 σ23 σ33
}Implied Covariance Matrix
compareAbsolute Model Fit
x1
y1
y2
Hypothesized Model Observed Covariance Matrix
{1.3.24 .41.01 9.7 12.3}S =
+
Estimation and Evaluation
ParameterEstimates
estimation
LS, ML, and BA
1. It is a “model-oriented” method, not a null-hypothesis-oriented method.
Some Properties of SEM
2. Highly flexible modeling toolbox.
3. Can be applied in either confirmatory (testing) or exploratory (model building) mode.
4. Variety of estimation approaches can be used, including likelihood and Bayesian.
1. Seeks to model uncertainty rather than probabilities.
A Bit about the Bayesian Approach
2. Philosophically well suited for supporting decision making.
3. Popularity partly based on new algorithms that create great flexibility in modeling.
4. It's indeterminant solution procedure, contributes to some uncertainty about results for more complex models(?)
Do the conventional methods meet your needs?
All your greatscientific ideas!
ANOVA result you hope to get!
modified from Starfield and Bleloch (1991)
How do data relate to learning?
Understanding of Processes
univariate descriptive statistics
exploration, methodology and
theory development
realistic predictive models
abstract models
multivariate descriptive statistics
more detailed theoretical models
univariate data modeling
multivariate data modeling
Data
SEM
Larson & Grace (2004) Biol. Ctl. 29:207-214; Larson et al. (2007) Biol. Ctl. 40:1-8.
spurge flea beetlesAphthona nigriscutusAphthona lacertosa
- beetles released since 1989
The Use of Biocontrol Insects On Leafy Spurge
- data collected since 1999
Based on the Available Data, What Have the Beetles Been Doing?
0
5
10
15
20
25
30
35
40
45
1999 2000 2001
DensityFlowering
Spurge isin decline
-100
-80
-60
-40
-20
0
20
40
0 2 4 6
log A. nigriscutis
% C
han
ge i
n S
pu
rge r = -0.21, p = 0.01
Change in spurge densityas a function of A. nigriscutisdensity
-100
-80
-60
-40
-20
0
20
40
0 2 4 6 8
log A. lacertosa
% C
han
ge i
n S
pu
rge r = -0.40, p < 0.001
Change in spurge densityas a function of A. lacertosadensity
How Does Spurge Decline Relate to Beetle Density?
Multivariate View: Hypothesized Model
A. lacertosa2001
Number ofStems2000
A. nigriscutis2000
A. lacertosa2000
A. nigriscutis2001
Change inStems
2000-2001
Change inStems
2000-2001
A. nigriscutis2001
Number ofStems2000
A. nigriscutis2000
A. lacertosa2001
A. lacertosa2000
R2 = 0.54
R2 = 0.61
-.20
.66
.57
.31-.14
-.55
.23
.51
.17
-.24R2 = 0.42
.08
Results for 2000 - 2001note: rawcorrelationwas r = -.21
note: rawcorrelationwas r = -.40
ns
Example #2: Coastal Prairie Vegetation and Soil
Properties
Summary of CommunityCharacteristics using Ordination
Axis 1
Axi
s 2
SEM model results
Mg
Mn
N
Ca
Zn
K
P
pH
C
elevELEV
MINRL
HYDR
-.67
.71
-.33 AXIS2
AXIS1
a2
a1
.57
-.30
-.28
-.23
R2 = .63
R2 = .09
The Problem:A variety of theoriesabout diversitylead to a similar setof bivariate expectations
Stress/Disturbance/Productivity
Sp
ecie
s R
ich
ne
ss
Stress/Disturbance/Productivity
Sp
ecie
s R
ich
ne
ss
Stress/Disturbance Effects
Stress/Disturbance/Productivity
Sp
ec
ies
Ric
hn
ess species pool 1
species pool 2
species pool 3
Stress/Disturbance Effects
Stress/Disturbance/Productivity
Sp
ec
ies
Ric
hn
ess species pool 1
species pool 2
species pool 3
Stress/Disturbance/Productivity
Sp
ec
ies
Ric
hn
ess species pool 1
species pool 2
species pool 3
Dynamic Equilibrium Effects
Rate of Displacement
Fre
qu
en
cy
of
Re
du
cti
on
Dynamic Equilibrium Effects
Rate of Displacement
Fre
qu
en
cy
of
Re
du
cti
on
Resource Variance Effects
Resource Supply Rate
Sp
eci
es
Ric
hn
es
s variance 1
variance 2
variance 3
Resource Variance Effects
Resource Supply Rate
Sp
eci
es
Ric
hn
es
s variance 1
variance 2
variance 3
Species Pool Effects
Productivity
Sp
ec
ies
Ric
hn
es
s
species pool
local richness
Species Pool Effects
Productivity
Sp
ec
ies
Ric
hn
es
s
species pool
local richness
Example #3:EvaluatingTheories ofDiversity
Suspected Underlying Processes
com
peti
tive
exc
lusi
oncom
petitive inhibition
sampling effect
AbioticConditions
stress
filtering
Hetero-geneity
facilitationof coexistence
expansionof niche space
Disturbance
damage
mortality
recruitment extinctionSpeciesRichness
SpeciesLost
Local SpeciesPool
Biomassproduction biomass loss Biomass
RemovedNet
Photosyn.
nich
ecom
plem
entarity
National Center for Ecological Analysis and Synthesis Project
Finnish meadows
Kansas prairie
Louisiana prairie
Minn. prairie
Texas grasslands
Louisiana marsh2
Indian tropicalsavanna
Louisiana marsh1
Wisconsin prairie
Miss. prairie
Utah grassland
Africa grassland
AbioticConditions
stress
filtering
Disturbance
damage
mortality
recruitment extinctionSpeciesRichness
SpeciesLost
Local SpeciesPool
Biomassproduction biomass loss Biomass
RemovedNet
Photosyn.
Interpretations
nic
he
com
ple
men
tari
tycom
petitive exclu
sion
Collaborative Applications of Multivariate Modeling
• USGS - Numerous units and individuals
• Univ. California - Davis
• Univ. Northern Arizona
• Univ. North Carolina
• Univ. Alabama
• Univ. Minnesota
• Nat. Ctr. Ecol. Analysis
• Univ. New Mexico
• Purdue Univ.
• Univ. Texas - Arlington
• Michigan State Univ.
• Univ. Groenegen (The Netherlands)
• Syracuse Univ.
• Rice Univ.
• Univ. Houston
• LSU
• US Forest Service
• Colorado State Univ.
• Univ. California - Irving
• Oregon State Univ.
• Yale Univ.
• Univ. Wisc. - Eau Claire
• Univ. Connecticut
• Univ. Newcastle - UK
• Univ. Montpellier - France