Structural Equation Modeling (SEM)

Niina Kotamäki

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)

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

X1

Y

X2

a

b

εRegression model:

Y=aX1+bX2+ε

LIMITATIONSMultiple 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 minimumobservational 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 includedThe error or residual variable is the only unobserved variable

corr

DEPENDENT INDEPENDENT

M

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

1. Development of hypothesis / theory

2. Construction of path diagram

3. Model specification

4. Model identification

5. Parameter estimation

6. Model evaluation

7. Model modification

Steps of SEM analysis

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

2. Construction of path diagram

ηξ

η

correlationpath

coefficients

erro

r

error

error

path

Endogenous latent variable

Exogenous latent variable

Creating a hypothesized model that you think explains the relationships among multiple variables

Converting the model to multiple equations

3. Model Specification

4. Model Identification

(Just) identified• a unique estimate for each parameter• number of equations = number of parameters to be estimated• a+b=5, a-b=2

Under-identified (not identified)• number of equations < number of parameters• infinite number of solutions • a+b=7 • model can not be estimated

Over-identified • number of equations > number of parameters• the model can be wrong

ξ1

ξ2

ξ3

η2

η1

Just identified model

ξ1

ξ2

ξ3

η1

η2

Over-identified model (SEM usually)

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: Σ=Σ(θ)

estimating methods: e.g. maximum likelihood (ML), ordinary least Squares (OLS), etc.

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

(γ) and relation between two endogenous constructs is called beta (β)

ξ1

X1

X2

δ1

δ2

λx11

λx21

ξ2

X3

X4

δ1

δ2

λx32

λx42

ξ3

X5

X6

δ1

δ2

λx53

λx63

η1

η 2

y1

y2

y3

y4

ε1

ε2

ε3

ε4

λy11

λy21

λy32

λy42

Measurement model

Structural model

β21

γ11

γ12

γ22

γ23

ϕ21

ϕ32

ϕ31

Endogenous latent variables

Exogenous latent variables

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 > 1.96, p < .05

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• Don’t go too far with model modification!

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).

Advantages of SEM

SEM in ecology, example

Phytoplankton dynamics

Nutrients Herbivore

Physical environment Water clarity

Structural model

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

Phytoplankton dynamics

Nutrients Herbivore

Phosphorus (SRP)

Chlorophyll aBiovolume

ZooplanktonDaphniaNitrogen (DIN)

Epilimnion depthwater clarity

Phytoplankton dynamics

Nutrients Herbivore

Phosphorus (SRP)

Chlorophyll aBiovolume

ZooplanktonDaphniaNitrogen (DIN)

Epilimnion depth (physical environment)

water clarity

ε1 ε2

ε4ε5

β2

β1

φ12

ψ33

ψ22

ψ11

δ2 δ3

γ1

γ2

λ2 λ3λ6 λ7

λ4 λ5

Phytoplankton dynamics

Nutrients Herbivore

Phosphorus (SRP)

Chlorophyll aBiovolume

ZooplanktonDaphniaNitrogen (DIN)

Epilimnion depth (physical environment)

water clarity

0.67 0.79

0.830.93

-0.66

0.82

-0.92

0.91

0.89

0.990.96

0.84

0.42

0.43

0.84

0.76

0.71 0.98

-0.07

-0.84

2 =22.473; df=19

p=0.261 >0.05 OK!

SEM Software packages

LISREL AMOS Function sem in R MPlus EQS Mx SEPATH

References: http://www.upa.pdx.edu/IOA/newsom/semrefs.htm