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methods@manchester
What is Structural Equation Modelling?
Nick ShryaneInstitute for Social ChangeUniversity of Manchester
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Topics
•
Where SEM fits in the families of statistical models
•
Causality–
SEM is useful for representing causal models, but can’t demonstrate causality on its own
•
Useful applications: –
measurement error, missing data, mediation models, group differences
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History
•
Structural Equation Modelling–
Was cobbled together out of
•
Regression •
Path Analysis (Wright, 1921)
•
Confirmatory Factor Analysis (Jöreskog, 1969)
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Families of Statistical Models
General Latent Variable Modelling (e.g. Mplus, GLLAMM)
Structural Equation Modelling
Path Analysis
Factor Analysis Latent Class Analysis
Mixed (multilevel) Models
Regression
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History
•
Structural Equation Modelling–
Used to be known as:
•
Covariance Structure Modelling•
Covariance Structure Analysis
•
Causal Modelling
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Causality
X Y
We observe a correlation between two variables. Why?
X Y
1. X causes Y ?
X Y
2. Y causes X ?
X Y
3. Reciprocal causation ?
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Causality
X Y
We observe a correlation between two variables. Why?
X Y
Z
4. A third, unmeasured variable ? Whichever is ‘true’, there will be a whole causal chain of mechanisms between the supposed cause and effect variables.
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CausalityCausal statement:Eating sugary foods causes tooth decay
Eating sugar
Tooth decay
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CausalityIf the causal statement is true, then there must be a causal mechanism…
Eating sugar
Tooth fairy
unhappy
Tooth decay
?
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Causality
Eating
Sugars Bacteria
Plaque
Acid Tooth decay
Gum disease
If the causal statement is true, then there must be a causal mechanism…
?
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Regression Models
y = x1
b1
+ x2
b2
+ e
x1b1 y
ε
x2 b2 Regression is not well suited to describing these causal sequences
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Regression Models
y = x1
b1
+ x2
b2
+ x3
b3
+ e
x1b1 y
ε
x2 b2
x3 b3
Adding new explanatory variables makes the model more comprehensive and complicated…
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Regression Models
y = x1
b1
+ x2
b2
+ x3
b3
+ x4
b4
+ e
x1b1 y
ε
x2 b2
x3 b3
x4 b4
…but doesn’t easily admit causal sequences or unmeasured causes
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Path Models
z = xb1
+ e1
y = zb2
+ e2
x
b1
z
ε1
y
ε2
b2
(z predicted by x)(y predicted by z)
Path models allow us to fit chains of conditional relationships (here a mediation hypotheses, i.e. that z
mediates the
relationship between x and y)
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Path Models
•
Regression analysis -
does x affect/predict y?•
Path/mediation analysis –
how/why does x
affect/predict y? Via the action of some intervening variable Z?
•
Brushing your teeth (x) reduces tooth decay (y) by removing bacteria (z)–
Measuring and testing mediators can help in evaluating a causal hypothesis
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Factor ModelsVariables may be correlated due to the action of unobserved influences.Sometimes these are confounding variables, but many constructs of interest are not directly observed (or even observable)
Unobserved Construct
Observed MeasuresSocial Capital
Bowling club membershipLocal newspaper reading
Ethnic prejudice
Housing segregationEthnic intermarriage
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Factor Models
y1 y2 y3 y4
y1 1.0
y2 0.6 1.0
y3 0.7 0.6 1.0
y4 0.5 0.6 0.8 1.0
y1 y3 y4y2
Correlations may not be due to causal relations among the observed variables at all, but due to these unmeasured, latent influences -
factors
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Factor Models
F
y1 y3 y4y2
e1 e2 e3 e4
y1
= F + e1
y2
= F + e2
y3
= F + e3
y4
= F + e4
The observed correlations may due to each observed measure sharing an unobserved component (F)
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Factor Models
Hypothesised Factor Model Observed data
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Factor Models
F
y1 y3 y4y2
λ1 λ2 λ3 λ4
e1 e2 e3 e4
y1
= Fλ1
+ e1
y2
= Fλ2
+ e2
y3
= Fλ3
+ e3
y4
= Fλ4
+ e4
The full factor model allows the strength of relationship between F and the observed ‘indicators’
y
to
vary –
different loadings (λ).
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Structural Equation ModelsThe full Structural Equation model is a combination of some or all of these elements:–
Regression model
–
Path model–
Factor model
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Structural Equation Models
x1 F1
y1 y3y2
e1 e2 e3
d1
F2
y4 y6y5
e4 e5 e6
d2
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Structural Equation Models
x1 F1
y1 y3y2
e1 e2 e3
d1
F2
y4 y6y5
e4 e5 e6
d2
Structural part
(Regression / Path Analysis)
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Structural Equation Models
x1 F1
y1 y3y2
e1 e2 e3
d1
F2
y4 y6y5
e4 e5 e6
d2
Measurement part
(Factor Analysis)
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Model Fit
•
Often the goal of the analysis is to assess the plausibility of the model as a whole
•
Some aspects of plausibility are nothing to do with statistics
•
If I claim X →Y, does that make sense? Does eating turkey cause Christmas…?
•
Plausibility is often assessed by the ability of the model to reproduce or ‘account for’
the
observed variances and covariances.
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Model Fit
•
Many indices have been suggested to assess global model fit to the observed data, e.g. –
Comparative Fit Index (CFI; Bentler, 1990)
–
Root Mean Square Error of Approximation (RMSEA; Brown & Cudeck, 1993; Steiger, 1990)
•
These suggest whether the model is statistically plausible, not whether it is ‘true’
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Causality again•
Ultimately, the SEM is a hypothesis, which is either supported or not by the data–
Poor model fit
•
Some aspect of the model (structural/measurement) is not a plausible description of the ‘data-generating mechanism’
–
Good model fit•
The model is a plausible, candidate explanation
•
But so might be lots of equivalent or alternative models:–
MacCallum, R. C., Wegener, D. T., Uchino, B. N. et al. (1993). The problem of equivalent models in applications of covariance structure analysis. Psychological Bulletin, 114(1), 185-199.
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Equivalent Models•
These two models would give identical fit
F1
y1 y3y2
e1 e2 e3
F2
y4 y6y5
e4 e5 e6
d2
F1
y1 y3y2
e1 e2 e3
F2
y4 y6y5
e4 e5 e6
d1
•
A well fitting model does not, on it’s own, mean that the causal hypothesis is true
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Applications: Covariate Measurement Error
•
Standard regression analysis assumes covariates/ predictors are measured without error
•
Effects are biased (weaker); more error, greater bias
•
With multiple indicators you can build a factor model to ‘measure’
your construct, excluding
error
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Applications: Covariate Measurement Error
F1
z1 z3z2
e1 e2 e3
y4
e4
xCovariate (x) assumed error free (e.g. sex)
Factor represents shared z variance
Unique/error z variance tucked away here
(This model suitable for multicolinearity
too)
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Applications: Correlated outcomes with missing data
•
Three highly correlated observed outcomes (y) but with lots of missing data in each
F1
y1 y3y2
e1 e2 e3
x
Individuals with just one non-missing y measure can still be included
(if we make certain assumptions about the missing data)
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Applications: Multiple Group Model
Ff
yf1 yf
3yf2
ef1 ef
2 ef3
3-item scale
Hypothesis: Item 3 is sex-
biased
Can test for Differential Item Functioning (DIF)
Fm
ym1 ym
3ym2
em1 em
2 em3
Female Male
λf1 λf
2 λf3 λm
1 λm2 λm
3
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Applications: Multiple Group Model
Ff
yf1 yf
3yf2
ef1 ef
2 ef3
Model 1:All item loadings constrained to be equal across groups
λf1
= λm1
λf2
= λm2
λf3
= λm3
Fm
ym1 ym
3ym2
em1 em
2 em3
Female Male
λf1 λf
2 λf3 λm
1 λm2 λm
3
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Applications: Multiple Group Model
Ff
yf1 yf
3yf2
ef1 ef
2 ef3
Fm
ym1 ym
3ym2
em1 em
2 em3
Female Male
λf1 λf
2 λf3 λm1 λm
2 λm3
Model 2:Item 3 loading can be different across groups
λf1
= λm1
λf2
= λm2
λf3
≠
λm3
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Applications: Multiple Group Model
Ff
yf1 yf
3yf2
ef1 ef
2 ef3
Does Model 2 fit the data significantly better than Model 1? If so, Item 3 is sex-biased (its ‘meaning’
is
different between the groups)
Fm
ym1 ym
3ym2
em1 em
2 em3
Female Male
λf1 λf
2 λf3 λm
1 λm2 λm
3
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Application: Mediation Model
Test of partial/full mediation
x
b1
z
ε1
y
ε2
b2
Model 1: Fit full modelModel 2: Fit model with path b3
fixed to zero
Does Model 2 fit significantly worse than Model 1? (Likelihood ratio test.)
b3
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Features / Limitations
•
Need large-ish
sample size, depending upon the strength of relationships –
(E.g. simple factor model with strong loadings, N>200; complicated multilevel SEM with weak loadings, N>5000)
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Features / Limitations
•
You need to think carefully about your theory–
Each arrow is a hypothesis
–
The absence of an arrow is a hypothesis–
It’s easy to specify models, harder to interpret them
–
Where to stop? Tempting to always try and add a bit more
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Features / Limitations•
Good model fit statistics DO NOT IMPLY THAT YOUR CAUSAL SEM MODEL IS TRUE!!!! Just that it may not be false.–
Can you defend the assumptions underlying your model (E.g. Cross-sectional survey data? Why should one item predict another and not vice-versa?)
–
Have you considered equivalent models? (Different but statistically indistinguishable ones.)
–
Have you compared your model with alternative but similar model variations?
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SEM software
•
AMOS–
Nice graphical interface (GUI); works with SPSS; available on campus network
•
Mplus–
Unparalleled range of models; expensive (but student version available); syntax, not GUI
•
R–
At least two libraries: OpenMX, sem; R is free!
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Further SEM
•
ISC / CCSR 1-day course–
“Introduction to SEM using Mplus”
–
March 22nd
2010. Book here: http://www.ccsr.ac.uk/courses/sem/
•
Introductory Text–
Kline (2004). Principles and Practice of SEM