1 General Structural General Structural Equations (LISREL) Equations (LISREL) Week 1 #4 Week 1 #4
Jan 13, 2016
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General Structural General Structural Equations (LISREL)Equations (LISREL)
Week 1 #4Week 1 #4
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Today:
Quick look at more AMOS examples… Extending the work with AMOS:
1. Moving from factor model to causal model (construct equations among latent variables)
2. adding single-indicator exogenous variables (assume no measurement error)
3. adding single-indicator exogenous variables with assumed measurement error
Equality constraints in structural equation models Dummy exogenous variables in structural equation
models SEM equivalents to contrasts Block tests for dummy variables AMOS example
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Also Today:
Model fit (an overview) The SIMPLIS program (part of
LISREL) Moving from Standard Stats
packages into SEM software Conceptualizing SEM models in
Matrix terms (some basics)
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The SIMPLIS interface for LISREL Works in scalar, not matrix, terms Fairly easy to use Sometimes, output is provided in regular
LISREL matrix form (can be a bit confusing) Requires a lower-triangular covariance matrix
(most stats packages produce “square” matrices) OR a special “.dsf” file (both can be created by the PRELIS program which accompanies LISREL).
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Two examples of SIMPLIS programsExample 1SIMPLIS Example for Religion Sexual Morality Data
System file from file f:\Classes\ICPSR2005\Week1Examples\ReligSexMoral-SIMPLIS\ReligSex1.dsf
Latent Variables
Relig Sexmor
Relationships:
V9 V175 V176 = Relig
V147 = 1*Relig
V304 V305 V307 V309 = Sexmor
V308 = 1*Sexmor
End of problem
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SIMPLIS Example for Religion Sexual Morality Data
Covariance Matrix
V9 V147 V175 V176 V304 V305
-------- -------- -------- -------- -------- --------
V9 0.82
V147 1.34 6.50
V175 0.31 0.75 0.48
V176 -1.64 -3.49 -1.09 6.77
V304 0.40 1.06 0.29 -1.52 2.90
V305 0.46 0.98 0.27 -1.45 1.34 3.52
V307 0.79 1.85 0.46 -2.57 1.70 1.69
V308 0.65 1.51 0.37 -1.93 1.59 1.61
V309 1.11 2.39 0.58 -3.12 1.61 1.83
Covariance Matrix
V307 V308 V309
-------- -------- --------
V307 7.26
V308 3.13 4.61
V309 4.02 2.83 7.76
Output
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LISREL Estimates (Maximum Likelihood)
Measurement Equations
V9 = 0.44*Relig, Errorvar.= 0.28 , Rý = 0.66
(0.018) (0.015)
25.20 18.41
V147 = 1.00*Relig, Errorvar.= 3.73 , Rý = 0.43
(0.16)
23.93
V175 = 0.27*Relig, Errorvar.= 0.27 , Rý = 0.44
(0.013) (0.011)
21.54 23.78
V176 = - 1.35*Relig, Errorvar.= 1.74 , Rý = 0.74
(0.052) (0.12)
-25.96 14.68
Output
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V304 = 0.63*Sexmor, Errorvar.= 1.96 , Rý = 0.33
(0.033) (0.082)
19.06 24.02
V305 = 0.66*Sexmor, Errorvar.= 2.49 , Rý = 0.29
(0.036) (0.10)
18.11 24.46
V307 = 1.25*Sexmor, Errorvar.= 3.57 , Rý = 0.51
(0.054) (0.17)
23.14 20.54
V308 = 1.00*Sexmor, Errorvar.= 2.23 , Rý = 0.52
(0.11)
20.33
V309 = 1.26*Sexmor, Errorvar.= 3.96 , Rý = 0.49
(0.055) (0.19)
22.81 21.01
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Covariance Matrix of Independent Variables
Relig Sexmor
-------- --------
Relig 2.77
(0.21)
13.18
Sexmor 1.59 2.38
(0.11) (0.17)
14.25 14.32
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Degrees of Freedom = 26 Minimum Fit Function Chi-Square = 213.09 (P = 0.0)
Normal Theory Weighted Least Squares Chi-Square = 217.29 (P = 0.0)
Estimated Non-centrality Parameter (NCP) = 191.29
90 Percent Confidence Interval for NCP = (147.95 ; 242.10)
Minimum Fit Function Value = 0.15
Population Discrepancy Function Value (F0) = 0.13
90 Percent Confidence Interval for F0 = (0.10 ; 0.17)
Root Mean Square Error of Approximation (RMSEA) = 0.071
90 Percent Confidence Interval for RMSEA = (0.063 ; 0.080)
P-Value for Test of Close Fit (RMSEA < 0.05) = 0.00
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Normed Fit Index (NFI) = 0.97
Non-Normed Fit Index (NNFI) = 0.97
Parsimony Normed Fit Index (PNFI) = 0.70
Comparative Fit Index (CFI) = 0.98
Incremental Fit Index (IFI) = 0.98
Relative Fit Index (RFI) = 0.96
Critical N (CN) = 312.87
Root Mean Square Residual (RMR) = 0.15
Standardized RMR = 0.035
Goodness of Fit Index (GFI) = 0.97
Adjusted Goodness of Fit Index (AGFI) = 0.94
Parsimony Goodness of Fit Index (PGFI) = 0.56
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Another SIMPLIS Example(Same 2 latent variables with single-indicator exogenous variables added)SIMPLIS Example for Religion Sexual Morality DataObserved variables: V9 V147 V175 V176 V304 V305 V307 V308 V309 V310 V355 V356 SEX OCC1 OCC2 OCC3 OCC4 OCC5Covariance matrix from file e:\ICPSR2005\RSM1.COVSample size = 1457Latent Variables: Relig SexmorRelationships: V9 V175 V176 = Relig V147 = 1*Relig V304 V305 V307 V309 = Sexmor V308 = 1*SexmorEquations: Relig = V355 V356 SEX Sexmor = V355 V356 SEXLet the error covariance of Relig and Sexmor be freeLet the error covariance of V175 and V176 be freeOptions MI ND=3 SCEnd of problem
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Output
Covariance Matrix (portion)
V307 V308 V309 V355 V356 SEX
-------- -------- -------- -------- -------- -------- V307 7.264 V308 3.132 4.606 V309 4.023 2.832 7.758 V355 -7.317 -5.385 -4.860 305.580 V356 1.447 0.656 1.455 -8.744 4.869 SEX -0.101 0.123 0.019 -0.107 0.090 0.250
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Error Covariance for V176 and V175 = -0.210
(0.0327)
-6.417
Structural Equations
Relig = - 0.0139*V355 + 0.0801*V356 + 0.443*SEX, Errorvar.= 2.735 , R2 = 0.0575
(0.00281) (0.0222) (0.0958) (0.204)
-4.962 3.607 4.626 13.425
Sexmor = - 0.0148*V355 + 0.155*V356 + 0.0413*SEX, Errorvar.= 2.089 , R2 = 0.0973
(0.00255) (0.0204) (0.0860) (0.149)
-5.795 7.587 0.480 13.995
Error Covariance for Sexmor and Relig = 1.454
(0.104)
13.954
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Standardized
• In Simplis: OPTIONS SC
Completely Standardized Solution
LAMBDA-Y
Relig Sexmor
-------- --------
V9 0.848 - -
V147 0.668 - -
V175 0.598 - -
V176 -0.821 - -
V304 - - 0.560
V305 - - 0.540
V307 - - 0.721
V308 - - 0.709
V309 - - 0.706
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Standardized
GAMMA
V355 V356 SEX
-------- -------- --------
Relig -0.143 0.104 0.130
Sexmor -0.170 0.225 0.014
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Moving from Stat Package System files to SEM Software
SPSS
SYSTEM
FILE
AMOS (reads
Directly from SPSS
system files)
SPSS SYSTEM FILE
A ‘DSF’ file created by PRELIS
LISREL reads DSF files
Use PRELIS
A raw covariancematrix (lower triangle) created by PRELIS
SAS, Stata, etc. SYSTEM FILE
LISREL reads lower triangular matrices
AMOS
LISREL
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Fit of a model
How far apart are Σ and S? Test of significance for H0: Σ=S
chi-square test Note: “Independence chi-square” is a different test! It
tests H0: S=0 Test is a simple function of N:
Χ2 = F*(N-1) “Perfect fit” (non-significant chi-square) much
easier to obtain in small samples
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Fit of a model
Search for “fit indices” that are not a function of N Desirable properties of fit indices:
Not a direct, linear function of N Not affected by N (expect wider sampling distribution with
smaller Ns.. this might imply that some types of fit indices yield “better” values for the same model in larger samples
Easily interpretable metric (e.g., 0 1) Consistent across estimation methods Not affected by metric of variables (e.g., same results
whether variables standardized or not)
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Fit of a model Desirable properties of fit indices (more):
Do not reward data dredging (vs. construction of parsimonious models) So-called “parsimony” measures include a penalty function for
adding parameters to a model
Commonly-used fit measures: Joreskog & Sorbom’s GFI (affected by N though) Bentler’s Normed Fit Index (and NNFI) Incremental, Comparative fit indices Root Mean Square Error of Approximation (RMSEA)
(for this index, low values are good)
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Improving the fit of a model: diagnostics Residuals:
Matrix of differences between sigma and S Would need to standardize before we could
determine where a model should be improved A residual is not necessarily connected to one
single parameter: A high residual might imply any one of 3 or 4 parameters
could/should be added to the model
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Improving the fit of a model: diagnostics Modification indices
Based on 2nd order derivative matrix Estimate the improvement in model fit if a
particular parameter is added Metric: chi-square (difference) Any value greater than 3.84 is “significant” at
p<.05 BUT criteria other than straight significance can/have been employed Reason: otherwise, sensitive to N; in large samples will
never get parsimonious model, etc.
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Modification Indices
• In AMOS, click “modification indices” under output options
• In SIMPLIS, Options MI
Modification Indices and Expected Change (SIMPLIS model discussed ealrier)
The Modification Indices Suggest to Add the
Path to from Decrease in Chi-Square New Estimate
V9 Sexmor 8.4 -0.06
V176 Sexmor 11.6 -0.18
V307 Relig 14.2 -0.21
V309 Relig 34.0 0.33
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Important note on modification indices It is not always the case that the parameter
with the highest MI should be added to a model
Some MIs will not make substantive sense (e.g., in a causal model, an MI suggesting a path from respondent’s social status to parent’s social status).
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Improving the fit of a model: diagnostics Estimated parameter change values
Estimated value of a parameter that is currently fixed (if this parameter is “freed” [included in the model]).
Standardized values can be helpful in determining whether adding a parameter is substantively important
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Equality Constraints in Structural Equation Models We can set “equality
constraints” on any two (or more) parameters in a model
E.g.: b1=b2 E.g.: VAR(e1) =
VAR(e2)
VAR-E1
E1
1
1
VAR-E2
E2
b1
1
VAR-E3
E3
b2
1
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Equality Constraints in Structural Equation Models We can set “equality
constraints” on any two (or more) parameters in a model
In AMOS we do this by giving parameters names, and then using the same name in the locations where we want to impose equality constraints
VAR-E1
E1
1
1
VAR-E2
E2
b1
1
VAR-E3
E3
b1
1
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Equality Constraints in Structural Equation Models We can set “equality
constraints” on any two (or more) parameters in a model
In SIMPLIS, we do this by adding statements:
VAR-E1
E1
1
1
VAR-E2
E2
b1
1
VAR-E3
E3
b1
1
Let the path from Relig to V176 be equal to the path from Relig to V167.
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Equality Constraints in Structural Equation Models The b1=b2 constraint
may not make sense if the metric of the 2 latent variables is not the same (makes most sense if variances are the same – would work if the variables were standardized]
VAR-E1
E1
1
1
VAR-E2
E2
b1
1
VAR-E3
E3
b1
1
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Equality Constraints in Structural Equation Models In this model, we could
test b1=b2, b2=b3, b1=b3 or b1=b2=b3 by setting the parameter names to be the same
Equality constraints only make sense if variances of the 3 exogenous manifest variables are the same, though
1
111
b1
b2
b3
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Equality Constraints in Structural Equation Models
1
111
b1
b2
b3
Formal tests: Model 1 b1, b2 estimated
separately Model 2 b1=b2 (i.e., labels
“b1” in each of 2 locations) Model 2 has 1 more
degree of freedom than model 1
A df=1 test for the equality constraint is obtained by subtracting the model 1 chi-square from the model 2 chi-square
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Dummy Variables in Structural Equation Models
• Dummy variables can be included in structural equation models if they are completely exogenous
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1
1
SEX 1/0
Sex: 0/1 variable
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Dummy Variables in Structural Equation Models
• Dummy variables can be included in structural equation models if they are completely exogenous
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1
1
SEX 1/0
Educ
Age
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Dummy Variables in Structural Equation Models
• Dummy variables cannot be included in structural equation models as indicators of latent constructs
Pol Partic
VOTED
e1
1
1
Trust
e2
1
Pol. Cor
e3
1
VOTED = 0/1 voted/did not vote last election
TRUST = 5 pt. trust in government item
POL COR = 5 pt. agree/disagree politicians corrupt
This model is NOT appropriate
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Dummy Variables in Structural Equation Models
• Dummy variables can be included in structural equation models if they are completely exogenous
• For categorical independent variables with more than 2 categories, sets of dummy variables can be included (just like in regression models)
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Dummy Variables in Structural Equation Models•For categorical independent variables
with more than 2 categories, sets of dummy variables can be included (just like in regression models)
• Design matrix as with Regression (could use effects or indicator coding; example below uses indicator coding):
D1 D2 D3
Catholic 1 0 0
Protestant 0 1 0
Jewish 0 0 1
Atheist 0 0 0
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DUMMY VARIABLES
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1
1
D1
D2
D3
(add curved arrow D1 D2 )
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DUMMY VARIABLES
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1
1
D1
D2
D3
b1
b2
b3
Test H0 for entire religion variable: estimate model with parameters b1, b2 and b3 all set to 0
(add curved arrow D1 D2 )
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DUMMY VARIABLES
Test H0 for entire religion variable: estimate model with parameters b1, b2 and b3 all set to 0
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DUMMY VARIABLES
• Test religion variable : b1=b2=b3=0
Model 1 (3 separate parameters) vs. Model
2 (all parameters = 0) df=3 test
•Test Protestant (category 1) vs. Atheist (reference group):
• Model 1 (3 separate parameters)
• Model 2 (fix b1=0) df=1
•OR: look at t-test for b1 parameter
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DUMMY VARIABLES
•Test Protestant (category 1) vs. Catholic (category2):
• Model 1 (3 separate parameters)
• Model 2 (fix b1=b2) df=1
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LV Structural Equation Models in Matrix terms
Thus far, our work has involved “scalar” equations.
• one equation at a time
•Specify a model (e.g, with software) by writing these equations out, one line per equation
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Matrix formWe can represent the previous 2 equations in
matrix form:
Matrix Form
(single, double subscript)
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There are other matrices in this model
Variance-covariance matrix of error terms (e’s)
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(other matrices, continued)
Variance covariance matrix of exogenous (manifest) variables
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Two scalar equations re-written
scalar
Matrix
Contents of matrices
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More generic form (combines all exogenous variables into single matrix)
More generic:
Where E1 Ξ X1, E2 Ξ X2 and E3 Ξ X3
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More generic form:
All exogenous variables part of a single variance-covariance matrix