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Fun With Structural Equation Modelling
in Psychological Research
Jeremy Miles
IBS, Derby University
• Structural Equation Modelling
• Analysis of Moment Structures
• Covariance Structure Analysis
• Analysis of Linear Structural Relationships (LISREL)
• Covariance Structure Models
• Path Analysis
Normal Statistics
• Modelling process– What is the best model to describe a set of data– Mean, sd, median, correlation, factor structure,
t-value
Data Model
SEM
• Modelling process– Could this model have led to the data that I
have?
Model Data
• Theory driven process– Theory is specified as a model
• Alternative theories can be tested– Specified as models
Data
Theory A Theory B
Ooohh, SEM Is Hard• It was. Now its not
• Jöreskog and Sörbom developed LISREL– Matrices:xy
– Variables: X Y
– Intercepts:
The Joy of Path Diagrams
Variable
Causal Arrow
Correlational Arrow
Doing “Normal” Statistics
x y
Correlation
Doing “Normal” Statistics
x y
T-Test
Doing “Normal” Statistics
x1
y
One way ANOVA(Dummy coding)
x2
x3
Doing “Normal” Statistics
x1
y
Two- way ANOVA(Dummy coding)
x2
x1 * x2
Doing “Normal” Statistics
x
y
Regression
x
x
Doing “Normal” Statistics
MANOVA
x1
x2
y1
y2
y3
Doing “Normal” Statistics
ANCOVA
x y
z
etc . . .
Identification
• Often thought of as being a very sticky issue
• Is a fairly sticky issue
• The extent to which we are able to estimate everything we want to estimate
X = 4
Unknown: x
x = 4y = 7
Unknown: x, y
x + y= 4x - y = 1
Unknown: x, y
x + y = 4
Unknown: x, y
Things We Know
Things We Want to Know
=
x=4x + y = 4, x - y = 2
Just identified
Can never be wrong
“Normal” statistics are just identified
Things We Know Things We Want
to Know <
x + y = 7
Not identified
Can never be solved
Things We Know Things We Want
to Know >
x + y = 4, x - y = 2, 2x - y = 3
over-identified
Can be wrong
SEM models are over-identified
Identification• We have information
– (Correlations, means, variances)
• “Normal” statistics– Use all of the information to estimate the
parameters of the model– Just identified
• All parameters estimated
• Model cannot be wrong
Over-identification
• SEM– Over-identified– The model can be wrong
• If a model is a theory– Enables the testing of theories
Parameter Identificationx - 2 = y
x + 2 = y
• Should be identified according to our previous rules– it’s not though
• There is model identification– there is not parameter identification
Sampling Variation and 2
• Equations and numbers– Easy to determine if its correct
• Sample data may vary from the model– Even if the model is correct in the population
• Use the 2 test to measure difference between the data and the model– Some difference is OK– Too much difference is not OK
Simple Over-identification
x y
Estimate 1 parameter-just-identified
x y Estimate 0 parameters-over-identified
Example 1
• Rab = 0.3, N = 100
• Estimate = 0.3, SE = 0.105, C.R. = 2.859• The correlation is significantly different from 0
a b
• Model
• Tests the hypothesis that the correlation in the population is equal to zero– It will never be zero, because of sampling
variation– The 2 tells us if the variation is significantly
different from zero
a b
Example 2• Test the model
• Force the value to be zero– Input parameters = 1– Parameters estimated = 0
• The model is now over-identified and can therefore be wrong
a b
• The program gives a 2 statistic
• The significance of difference between the data and the model– Distributed with df = known parameters - input
parameters
• 2 = 9.337, df = 1 - 0 = 1, p = 0.002
• So what? A correlation of 0.3 is significant?
Hardly a Revelation
• No. We have tested a correlation for significance. Something which is much more easily done in other ways
• But– We have introduced a very flexible technique– Can be used in a range of other ways
Testing Other Than Zero• Estimated parameters usually tested against zero
– Reasonable?
• Model testing allows us to test against other values
• 2 = 2.3, n.s.• Example 3
a b
0.15
Example 4: Comparing correlations
• 4 variables– mothers' sensitivity– mothers' parental bonding– fathers' sensitivity– fathers' parental bonding
• Does the correlation differ between mothers and fathers?
M S
M PB
F PB
F S
0.5 0.3
0.1
0.1
0.20.2
• Example 4a– analyse with all parameters free– 0 df, model is correct
• Example 4b– fix FS-FPB and MS-MPB to be equal. – See if that model can account for the data
M S
M PB
F PB
F S
dave dave
2 = 1.82, df = 1p = 0.177
dave = 0.41 (s.e. 0.08)
Latent Variables
• The true power of SEM comes from latent variable modelling
• Variables in psychology are rarely (never?) measured directly– the effects of the variable are measured– Intelligence, self-esteem, depression– Reaction time, diagnostic skill
Measuring a Latent Variable
• Latent variables are drawn as ellipses– hypothesised causal relationship
with measured variables
• Measured variable has two causes– latent variable– “other stuff”
• random error
Latent Measured
x = t + e
• Reliability is:• the square root of proportion of variance in x that is
accounted • the correlation between x and e
MeasuredTrue Score
Error
Identification and Latent Variables
• 1 measured variable– not (even close to) identified
• 4 measured variables– 6 known, 4 estimated
• model is identified
• Need four measured variables to identify the model
• Need to identify the variance of the latent variable– fix to 1
Why oh why oh why?
• Why bother with all these tricky latent variables?
• 2 reasons– unidimensional scale construction– attenuation correction
Unidimensionality
• Correlation matrix
• 2 = 3.65, df = 2, p = 0.16
1.00 0.68 1.00 0.73 0.63 1.00 0.68 0.63 0.69 1.00
Attenuation Correction
• Why bother?– Gets accurate measure of correlation between
true scores
• Why bother– theories in psychology are ordinal– attenuation can only cause relationships to
lower
The Multivariate Case• Much more complex and unpredictable
x1 y1
x2 y2
a c
d
e
b
Some More Models
• Multiple Trait Multiple Method Models (MTMM)
• Temporal Stability
• Multiple Indicator Multiple Cause (MIMIC)
MTMM• Multiple Trait
– more than one measure
• Multiple Method– using more than one technique
• Variance in measured score comes from true score, random error variance, and systematic error variance, associated with the shared methods
What?• Example 6 (From Wothke, 1996)
– Three traits• Getting along with others (G)• Dedication (D)• Apply learning (L)
• Three methods• Peer nomination (PN)• Peer Checklist (PC)• Supervisor ratings (SC)
Matrix 1 .524 1 .241 .403 1 .071 .102 -.018 1 .022 .096 .018 .435 1 .076 .102 .100 .342 .347 1 .136 .132 .061 .243 .203 .100 1-.028 .168 .135 .093 .209 .042 .461 1-.054 .162 .252 .053 .108 .108 .294 .280 1 g.pn d.pn l.pn g.pc d.pc l.pc g.sc d.sc l.sc
Analysis
g.pn l.pn d.pc
pn
g.pn l.pc d.pc
pc
g.sc l.sc d.sc
sc
g l d
Temporal Stability• Usually
– sum the items– correlate them
• BUT– items may not be unidimensional – relationship will be attenuated due to
measurement error– relationship will be inflated, due to correlated
error
L1
X3.1 X4.1 X5.1X2.1X1.1
L2
X3.2 X4.2 X5.2X2.2X1.2
•Corrects for attenuation•But - correlated errors may be a problem
• Added correlated errors
• Example 7b
L1
X3.1 X4.1 X5.1X2.1X1.1
L2
X3.2 X4.2 X5.2X2.2X1.2
MIMIC Model• “Conventional wisdom” in psychological
measurement is that a latent variable is the cause of the measured variables
• Assumption is made (implicitly) in many types of measurement– Bollen and Lennox (1989)– not necessarily the case
Value of a Car• Causes
– type, size, age, rustiness– no reason they should, or should not, be
correlated
• Effects– assessment of value by people who know
Level of Depression• Questionnaire items
– causes or effects?• been feeling unhappy and depressed?• been having restless and disturbed
nights?• found everything getting 'on top' of you?
• MIMIC
Example 8: MIMIC
L1
c1
c2
c3
y4y1
LY1
LY2
y2 y3
y5 y6 y7 y8
Concluding remarks
• Given a taster– some may be too simple?
• Much more to say– no time to say it
• See further reading (Books and WWW)
Further Info
• SEMNET - email list– [email protected] (messages)– listserv@ bama.ua.edu (leave)– http://www.gsu.edu/~mkteer/semfaq.html
• the semnet FAQ
Books
• See web pagehttp://ibs.derby.ac.uk/~jeremym/fun/fun/index.htm
References
• See web pagehttp://ibs.derby.ac.uk/~jeremym/fun/fun/index.htm