Structural%20Equation%20Modeling%20(SEM)

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Structural Equation Modeling (SEM)

• Niina Kotamäki

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SEM

• Covariance structure analysis
• Causal modeling
• Simultaneous equations modeling
• Path analysis
• Confirmatory factor analysis
• Latent variable modeling
• LISREL-modeling
• Highly flexible modeling toolbox
• Extension of the general linear model (GLM)

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SEM

• Quite recent innovation (late 1960s early 1970 ?)
• Extensively applied in social sciences,
  psychology, economy, chemistry and biology
• Applications in ecology and environmental
  sciences are limited
• Even less common in aquatic ecosystems
• tests theoretical hypothesis about causal
  relationships
• tests relationships between observed and
  unobserved variables
• combines regression analysis (path analysis) and
  factor analysis
• researchers use SEM to determine whether a
  certain model is valid

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X1
a
Regression model YaX1bX2e
Y
e
corr
b
X2
DEPENDENT
INDEPENDENT
M

• LIMITATIONS
• Multiple dependent (Y) variables are not
  permitted
• Each independent variable (X) is assumed to be
  measured without error
• controlled experiments ? measurement errors are
  negligible and uncontrolled variation is at
  minimum
• observational studies ? all variables are subject
  to measurement error and uncontrolled variation
• Strong correlation (multicollinearity) may cause
  biased parameter estimates and inflated standard
  errors
• Indirect effects (mediating variables) cannot be
  included
• The error or residual variable is the only
  unobserved variable

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SEM deals with these limitations

• Works with multiple, related equations
  simultaneously
• Allows reciprocal relationships
• Ability to model constructs as latent variables
• Allows the modeller to explicitly capture
  unreliability of measurement in the model
• Indirect effects / mediating variables
• Compares the performance of a model across
  multiple populations

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Simultaneous equation models
ex2
X2
a21
a32
X1
X4
ex4
a41
a31
a43
X3
x2 a21x1ex2
ex3
x3 a31x1a32x2 ex3
x4 a41x1a43x3ex4
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Reciprocal relationship
X1
X3
ex3
X2
X4
ex4
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Latent variables

• also called factors (comparison to factor
  analysis)
• unobserved
• not measured directly, can be expressed in terms
  of one or more directly measurable variables
  (indicators)
• measurement error in indicators
• correlated variables are grouped together and
  separated from other variables with low or no
  correlation

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Latent variable
X1
d1
?
X2
d2
X3
d3
Latent variable (ksi)
Errors (delta)
Indicators
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Measurement error

• Latent variables, measurement error in indicators
  ?allows the structural relations between latent
  variables to be accurately estimated (unbiased).

X1
X4
d1
d4
?2
?1
X2
X5
d2
d5
X3
X6
d3
d6
11
Indirect effect, mediator

• Unmediated model

c
X
Y
Mediated model
c
Complete mediation c0 Partial mediation 0ltcltc
X
Y
a
b
ctotal effect cdirect effect
M
X affects Y through M
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Steps of SEM analysis

• Development of hypothesis / theory
• Construction of path diagram
• Model specification
• Model identification
• Parameter estimation
• Model evaluation
• Model modification

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1. Development of hypothesis

• SEM is a confirmatory technique
• researcher needs to have established theory about
  the relationships
• suited for theory testing, rather than theory
  development

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2. Construction of path diagram
error
coefficients
?
path
error
?
correlation
path
?
error
Endogenous latent variable
Exogenous latent variable
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3. Model Specification

• Creating a hypothesized model that you think
  explains the relationships among multiple
  variables
• Converting the model to multiple equations

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4. Model Identification

• (Just) identified
• a unique estimate for each parameter
• number of equations number of parameters to be
  estimated
• ab5, a-b2
• Under-identified (not identified)
• number of equations lt number of parameters
• infinite number of solutions
• ab7
• model can not be estimated
• Over-identified
• number of equations gt number of parameters
• the model can be wrong

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Just identified model
?2
?1
?2
?1
?3
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Over-identified model (SEM usually)
?1
?1
?2
?2
?3
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5. Parameter estimation

• technique used to calculate parameters
• testing how well a model fits the data
• expected covariance structure is tested against
  the covariance matrix of oberved data H0 SS(?)
• estimating methods e.g. maximum likelihood (ML),
  ordinary least Squares (OLS), etc.

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• Measurement Model
• The part of the model that relates indicators to
  latent factors
• The measurement model is the factor analytic part
  of SEM
• The respective regression coefficient is called
  lambda (?) / loading
• Structural model
• This is the part of the model that includes the
  relationships between the latent variables
• relation between endogenous and exogenous
  construct is called gamma (?) and relation
  between two endogenous constructs is called beta
  (ß)

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Measurement model
?x11
X1
d1
Structural model
?1
?x21
?11
X2
d2
?y11
y1
e1
?21
?1
?y21
X3
d1
?x32
?12
y2
e2
?2
?31
ß21
?x42
?22
X4
d2
?y32
y3
e3
?32
? 2
?y42
?x53
X5
d1
?23
y4
e4
?3
?x63
X6
d2
Endogenous latent variables
Exogenous latent variables
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6. Model evaluation

• Total model
• Chi Square (?2) test
• the theoretically expected values vs. the
  empirical data
• Because we are dealing with a measure of misfit,
  the p-value for ?2 should be larger than .05 to
  decide that the theoretical model fits the data
• fit indices e.g. RMSEA, CFI, NNFI etc.
• Model parts
• t-value for the estimated parameters showing
  whether they are different from 0 (or any other
  value that we want to fix!) t gt 1.96, p lt .05

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7. Model modification

• Simplify the model (i.e., delete non-significant
  parameters or parameters with large standard
  error)
• Expand the model (i.e., include new paths)
• Confirmatory vs. explanatory
• Dont go too far with model modification!

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Advantages of SEM

• use of confirmatory factor analysis to reduce
  measurement error by having multiple indicators
  per latent variable
• graphical modeling interface
• testing models overall rather than coefficients
  individually
• testing models with multiple dependents
• modeling indirect variables
• testing coefficients across multiple
  between-subjects groups
• handling difficult data (time series with
  autocorrelated error, non-normal data, incomplete
  data).

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SEM in ecology, example
Structural model
Physical environment
Water clarity
Phytoplankton dynamics
Nutrients
Herbivore
Example from G.B. Arhonditsis, C.A. Atow, L.J.
Steinberg, M.A. Kenney, R.C. Lathrop, S.j.
McBride, K.H. Reckhow. Exploring ecological
patterns with structural equation modeling and
Bayesian analysis. Ecological Modeling
192 (2006) 385-409
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Chlorophyll a
Biovolume
water clarity
Epilimnion depth
Phytoplankton dynamics
Nutrients
Herbivore
Phosphorus (SRP)
Zooplankton
Daphnia
Nitrogen (DIN)
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e1
e2
Chlorophyll a
Biovolume
?22
?4
?5
Epilimnion depth (physical environment)
water clarity
?1
ß1
Phytoplankton dynamics
f12
?2
ß2
Nutrients
Herbivore
?33
?11
?6
?2
?7
?3
Phosphorus (SRP)
Zooplankton
Daphnia
Nitrogen (DIN)
d2
e4
e5
d3
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?2 22.473 df19 p0.261 gt0.05 OK!
0.67
0.79
Chlorophyll a
Biovolume
0.84
0.89
0.82
Epilimnion depth (physical environment)
water clarity
-0.07
-0.92
Phytoplankton dynamics
0.42
-0.66
-0.84
Nutrients
Herbivore
0.43
0.76
0.96
0.91
0.99
0.84
Phosphorus (SRP)
Zooplankton
Daphnia
Nitrogen (DIN)
0.71
0.83
0.93
0.98
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SEM Software packages

• LISREL
• AMOS
• Function sem in R
• MPlus
• EQS
• Mx
• SEPATH

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References http//www.upa.pdx.edu/IOA/newsom/sem
refs.htm