General Structural Equations (LISREL)

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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


(0.16)

23.93


(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


(0.036) (0.10)

18.11 24.46


(0.054) (0.17)

23.14 20.54


(0.11)

20.33


(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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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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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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11

1

1

SEX 1/0

Educ

Age

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• 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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• 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

General Structural Equations (LISREL)

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