02a Causal Models with Directly Observed Variables 2002
SOC 681 – Causal Models with
Directly Observed Variables
James G. Anderson, Ph.D.
Purdue University
Types of SEMs
Regression Models
Path Models
Recursive
Nonrecursive
Class Exercise: Example 7
SEMs with Directly Observed Variables
Felson and Bohrnstedt’s study of 209 girls
from 6th through 8th grade
Variables
Academic: Perceived academic ability
Attract: Perceived attractiveness
GPA: Grade point average
Height: Deviation of height from the mean
height
Weight: Weight adjusted for height
Rating: Rating of physical attractiveness
GPA
ACADEMIC
1
e1
HEIGHT
WEIGHT
RATING
ATTRACT
1
e2
GPA
ACADEMIC
1
e1
HEIGHT
WEIGHT
RATING
ATTRACT
1
e2
Assumptions
Relations among variables in the
model are linear, additive and causal.
Curvilinear, multiplicative and
interaction relations are excluded.
Variables not included in the model
but subsumed under the residuals are
assumed to be not correlated with the
model variables.
Assumptions
Variables are measured on an
interval scale.
Variables are measured without
error.
Objectives
Estimate the effect parameters (i.e., path
coefficients). These parameters indicate the
direct effects of a variable hypothesized as a
cause of a variable taken as an effect.
Decompose the correlations between an
exogenous and endogenous or two endogenous
variables into direct and indirect effects.
Determine the goodness of fit of the model to
the data (i.e., how well the model reproduces
the observed covariances/correlations among
the observed variable).
AMOS Input
ASCII
SPSS
Microsoft Excel
Microsoft Access
Microsoft FoxPro
dBase
Lotus
AMOS Output
Path diagram
Structural equations effect
coefficients, standard errors, tscores, R2 values
Goodness of fit statistics
Direct and Indirect Effects
Modification Indices.
Model One
Decomposing the Effects of Variables
on Achievement
Variables
Direct
Indirect
Total
-.03
-
-.03
FatherEd
.17
-
.17
Ethnic
.17
-
.17
IndTrng
.23*
-
.23*
AStress
-.17*
-
-.17*
ActMast
.02
-
.02
SelfCon
.42*
-
.42*
Sex
Model Two
Goodness of Fit: Model 2
Chi-Square = 29.07
df = 15
p < 0.06
Chi-Square/df = 1.8
RMSEA = 0.086
GFI = 0.94
AGFI = 0.85
AIC = 67.82
Chi Square: 2
Best for models with N=75 to N=100
For N>100, chi square is almost always
significant since the magnitude is
affected by the sample size
Chi square is also affected by the size of
correlations in the model: the larger the
correlations, the poorer the fit
Chi Square to df Ratio: 2/df
There are no consistent standards for
what is considered an acceptable model
Some authors suggest a ratio of 2 to 1
In general, a lower chi square to df ratio
indicates a better fitting model
Root Mean Square Error of
Approximation (RMSEA)
Value: [ (2/df-1)/(N-1) ]
If 2 < df for the model, RMSEA is set to
0
Good models have values of < .05;
values of > .10 indicate a poor fit.
GFI and AGFI
(LISREL measures)
Values close to .90 reflect a good fit.
These indices are affected by sample
size and can be large for poorly specified
models.
These are usually not the best measures
to use.
Akaike Information Criterion (AIC)
Value: 2 + k(k-1) - 2(df)
where k= number of variables in the model
A better fit is indicated when AIC is smaller
Not standardized and not interpreted for a
given model.
For two models estimated from the same
data, the model with the smaller AIC is
preferred.
Model Building
Standardized Residuals
ACH – Ethnic = 3.93
Modification Index
ACH – Ethnic = 10.05
Model Three
Goodness of Fit: Model 3
Chi-Square = 16.51
df = 14
p < 0.32
Chi-Square/df = 1.08
RMSEA = 0.037
GFI = 0.96
AGFI = 0.90
AIC = 59.87
Comparing Models
Chi-Square Difference = 12.56
df Difference = 1
p < .0005
AIC Difference = 7.95
Difference in Chi Square
Value: X2diff = X2
DFdiff = DF
model 1
model 1
-X2 model 2
–DFmodel 2
Decomposing the Effects of Variables
on Achievement
Variables
Direct
Indirect
Total
Sex
-
.09
.09
FatherEd
.-
.06
.06
Ethnic
.29
.05
.34
IndTrng
.25
.04
.29
AStress
-.14
-.03
-.17
ActMast
-
.13
.13
SelfCon
.44
-
.44
Class Exercise: Example 7
SEMs with Directly Observed Variables
Attach the data for female subjects
from the Felson and Bohrnstedt
study (SPSS file Fels_fem.sav)
Fit the non-recursive model
Delete the non-significant path
between Attract and Academic and
refit the model
Compare the chi square values and
the AIC values for the two models
Class Exercise: Example 7
SEMs with Directly Observed Variables
Felson and Bohrnstedt’s study of 209 girls
from 6th through 8th grade
Variables
Academic: Perceived academic ability
Attract: Perceived attractiveness
GPA: Grade point average
Height: Deviation of height from the mean
height
Weight: Weight adjusted for height
Rating: Rating of physical attractiveness
GPA
ACADEMIC
1
e1
HEIGHT
WEIGHT
RATING
ATTRACT
1
e2
GPA
ACADEMIC
1
e1
HEIGHT
WEIGHT
RATING
ATTRACT
1
e2
Directly Observed Variables
James G. Anderson, Ph.D.
Purdue University
Types of SEMs
Regression Models
Path Models
Recursive
Nonrecursive
Class Exercise: Example 7
SEMs with Directly Observed Variables
Felson and Bohrnstedt’s study of 209 girls
from 6th through 8th grade
Variables
Academic: Perceived academic ability
Attract: Perceived attractiveness
GPA: Grade point average
Height: Deviation of height from the mean
height
Weight: Weight adjusted for height
Rating: Rating of physical attractiveness
GPA
ACADEMIC
1
e1
HEIGHT
WEIGHT
RATING
ATTRACT
1
e2
GPA
ACADEMIC
1
e1
HEIGHT
WEIGHT
RATING
ATTRACT
1
e2
Assumptions
Relations among variables in the
model are linear, additive and causal.
Curvilinear, multiplicative and
interaction relations are excluded.
Variables not included in the model
but subsumed under the residuals are
assumed to be not correlated with the
model variables.
Assumptions
Variables are measured on an
interval scale.
Variables are measured without
error.
Objectives
Estimate the effect parameters (i.e., path
coefficients). These parameters indicate the
direct effects of a variable hypothesized as a
cause of a variable taken as an effect.
Decompose the correlations between an
exogenous and endogenous or two endogenous
variables into direct and indirect effects.
Determine the goodness of fit of the model to
the data (i.e., how well the model reproduces
the observed covariances/correlations among
the observed variable).
AMOS Input
ASCII
SPSS
Microsoft Excel
Microsoft Access
Microsoft FoxPro
dBase
Lotus
AMOS Output
Path diagram
Structural equations effect
coefficients, standard errors, tscores, R2 values
Goodness of fit statistics
Direct and Indirect Effects
Modification Indices.
Model One
Decomposing the Effects of Variables
on Achievement
Variables
Direct
Indirect
Total
-.03
-
-.03
FatherEd
.17
-
.17
Ethnic
.17
-
.17
IndTrng
.23*
-
.23*
AStress
-.17*
-
-.17*
ActMast
.02
-
.02
SelfCon
.42*
-
.42*
Sex
Model Two
Goodness of Fit: Model 2
Chi-Square = 29.07
df = 15
p < 0.06
Chi-Square/df = 1.8
RMSEA = 0.086
GFI = 0.94
AGFI = 0.85
AIC = 67.82
Chi Square: 2
Best for models with N=75 to N=100
For N>100, chi square is almost always
significant since the magnitude is
affected by the sample size
Chi square is also affected by the size of
correlations in the model: the larger the
correlations, the poorer the fit
Chi Square to df Ratio: 2/df
There are no consistent standards for
what is considered an acceptable model
Some authors suggest a ratio of 2 to 1
In general, a lower chi square to df ratio
indicates a better fitting model
Root Mean Square Error of
Approximation (RMSEA)
Value: [ (2/df-1)/(N-1) ]
If 2 < df for the model, RMSEA is set to
0
Good models have values of < .05;
values of > .10 indicate a poor fit.
GFI and AGFI
(LISREL measures)
Values close to .90 reflect a good fit.
These indices are affected by sample
size and can be large for poorly specified
models.
These are usually not the best measures
to use.
Akaike Information Criterion (AIC)
Value: 2 + k(k-1) - 2(df)
where k= number of variables in the model
A better fit is indicated when AIC is smaller
Not standardized and not interpreted for a
given model.
For two models estimated from the same
data, the model with the smaller AIC is
preferred.
Model Building
Standardized Residuals
ACH – Ethnic = 3.93
Modification Index
ACH – Ethnic = 10.05
Model Three
Goodness of Fit: Model 3
Chi-Square = 16.51
df = 14
p < 0.32
Chi-Square/df = 1.08
RMSEA = 0.037
GFI = 0.96
AGFI = 0.90
AIC = 59.87
Comparing Models
Chi-Square Difference = 12.56
df Difference = 1
p < .0005
AIC Difference = 7.95
Difference in Chi Square
Value: X2diff = X2
DFdiff = DF
model 1
model 1
-X2 model 2
–DFmodel 2
Decomposing the Effects of Variables
on Achievement
Variables
Direct
Indirect
Total
Sex
-
.09
.09
FatherEd
.-
.06
.06
Ethnic
.29
.05
.34
IndTrng
.25
.04
.29
AStress
-.14
-.03
-.17
ActMast
-
.13
.13
SelfCon
.44
-
.44
Class Exercise: Example 7
SEMs with Directly Observed Variables
Attach the data for female subjects
from the Felson and Bohrnstedt
study (SPSS file Fels_fem.sav)
Fit the non-recursive model
Delete the non-significant path
between Attract and Academic and
refit the model
Compare the chi square values and
the AIC values for the two models
Class Exercise: Example 7
SEMs with Directly Observed Variables
Felson and Bohrnstedt’s study of 209 girls
from 6th through 8th grade
Variables
Academic: Perceived academic ability
Attract: Perceived attractiveness
GPA: Grade point average
Height: Deviation of height from the mean
height
Weight: Weight adjusted for height
Rating: Rating of physical attractiveness
GPA
ACADEMIC
1
e1
HEIGHT
WEIGHT
RATING
ATTRACT
1
e2
GPA
ACADEMIC
1
e1
HEIGHT
WEIGHT
RATING
ATTRACT
1
e2