STA313 F 2004 Handout 6
Simple Regression as a Structural Equation model
/* simplereg.sas */
options linesize=79 noovp formdlim='_';
title 'Simple regression as a structural equation model with proc calis';
data simple;
infile 'simplereg.dat';
input x y;
proc reg simple;
title2 'For comparison, simple regression with proc reg';
model y = x /noint;
proc calis cov; /* Analyze the covariance matrix (Default is corr) */
title2 'Full (unrestricted) Model';
var x y; /* Manafest vars are in the data set */
lineqs /* Simultaneous equations, separated by commas */
y = b x + e;
std /* Variances (not standard deviations) */
x = sigxx , /* Optional starting values in parentheses */
e = sigee ;
cov /* Covariances */
x e = 0;
bounds 0.0 < sigxx,
0.0 < sigee;
proc calis cov;
title2 'Reduced (restricted) Model';
var x y;
lineqs
y = e; /* Setting b = 0 */
std
x = sigxx ,
e = sigee ;
cov
x e = 0;
bounds 0.0 < sigxx,
0.0 < sigee;
/* Moral: You never have to fit a saturated full model */
Here is simplereg.lst
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Simple regression as a structural equation model with proc calis
1
For comparison, simple regression with proc reg
14:08 Friday, October 15,
2004
Descriptive Statistics
Variables Sum Mean Uncorrected SS
INTERCEP 250 1 250
X -56.27871787 -0.225114871 1014.0011343
Y 90.83210452 0.3633284181 2313.4725451
Variables Variance Std Deviation
INTERCEP 0 0
X 4.0214134856 2.0053462259
Y 9.1585167079 3.0263041334
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Simple regression as a structural equation model with proc calis
2
For comparison, simple regression with proc reg
14:08 Friday, October 15,
2004
Model: MODEL1
NOTE: No intercept in model. R-square is redefined.
Dependent Variable: Y
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Prob>F
Model 1 2.25923 2.25923 0.243 0.6222
Error 249 2311.21332 9.28198
U Total 250 2313.47255
Root MSE 3.04663 R-square 0.0010
Dep Mean 0.36333 Adj R-sq -0.0030
C.V. 838.53457
Parameter Estimates
Parameter Standard T for H0:
Variable DF Estimate Error Parameter=0 Prob > |T|
X 1 -0.047202 0.09567558 -0.493 0.6222
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Simple regression as a structural equation model with proc calis
3
Full (unrestricted) Model
14:08 Friday, October 15,
2004
Covariance Structure Analysis: Pattern and Initial Values
LINEQS Model Statement
-------------------------------
Matrix Rows & Cols Matrix Type
TERM 1-----------------------------------------------------------
1 _SEL_ 2 3 SELECTION
2 _BETA_ 3 3 EQSBETA IMINUSINV
3 _GAMMA_ 3 2 EQSGAMMA
4 _PHI_ 2 2 SYMMETRIC
Number of endogenous variables = 1
Manifest: Y
Number of exogenous variables = 2
Manifest: X
Error: E
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Simple regression as a structural equation model with proc calis
4
Full (unrestricted) Model
14:08 Friday, October 15,
2004
Covariance Structure Analysis: Pattern and Initial Values
Manifest Variable Equations
Initial Estimates
Y = . *X + 1.0000 E
B
Variances of Exogenous Variables
-------------------------------------
Variable Parameter Estimate
-------------------------------------
X SIGXX .
E SIGEE .
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Simple regression as a structural equation model with proc calis
5
Full (unrestricted) Model
14:08 Friday, October 15,
2004
Covariance Structure Analysis: Maximum Likelihood Estimation
250 Observations Model Terms 1
2 Variables Model Matrices 4
3 Informations Parameters 3
VARIABLE Mean Std Dev
X -.2251148715 2.005346226
Y 0.3633284181 3.026304133
Set covariances of exogenous manifest variables:
X
Some initial estimates computed by two-stage LS method.
Vector of Initial Estimates
B 1 -0.02738 Matrix Entry: _GAMMA_[1:1]
SIGXX 2 4.02141 Matrix Entry: _PHI_[1:1]
SIGEE 3 9.15550 Matrix Entry: _PHI_[2:2]
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Simple regression as a structural equation model with proc calis
6
Full (unrestricted) Model
14:08 Friday, October 15,
2004
Covariance Structure Analysis: Maximum Likelihood Estimation
Levenberg-Marquardt Optimization
Scaling Update of More (1978)
Number of Parameter Estimates 3
Number of Functions (Observations) 3
Number of Lower Bounds 2
Number of Upper Bounds 0
Optimization Start: Active Constraints= 0 Criterion= 0.000
Maximum Gradient Element= 0.000 Radius= 1.000
Iter rest nfun act optcrit difcrit maxgrad lambda rho
Optimization Results: Iterations= 0 Function Calls= 2 Jacobian Calls= 1
Active Constraints= 0 Criterion= 0 Maximum Gradient Element= 3.20357E-17
Lambda= 0 Rho= 0 Radius= 1
NOTE: ABSGCONV convergence criterion satisfied.
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Simple regression as a structural equation model with proc calis
7
Full (unrestricted) Model
14:08 Friday, October 15,
2004
Covariance Structure Analysis: Maximum Likelihood Estimation
Fit criterion . . . . . . . . . . . . . . . . . . 0.0000
Goodness of Fit Index (GFI) . . . . . . . . . . . 1.0000
GFI Adjusted for Degrees of Freedom (AGFI). . . . .
Root Mean Square Residual (RMR) . . . . . . . . . 0.0000
Parsimonious GFI (Mulaik, 1989) . . . . . . . . . 0.0000
Chi-square = 0.0000 df = 0 Prob>chi**2 = 0.0001
Null Model Chi-square: df = 1 0.0820
RMSEA Estimate . . . . . . . . . . . 0.0000 90%C.I.[., .]
Probability of Close Fit . . . . . . . . . . . . .
ECVI Estimate . . . . . . . . . . . . 0.0244 90%C.I.[., .]
Bentler's Comparative Fit Index . . . . . . . . . .
Normal Theory Reweighted LS Chi-square . . . . . 0.0000
Akaike's Information Criterion. . . . . . . . . . 0.0000
Bozdogan's (1987) CAIC. . . . . . . . . . . . . . 0.0000
Schwarz's Bayesian Criterion. . . . . . . . . . . 0.0000
McDonald's (1989) Centrality. . . . . . . . . . . 1.0000
Bentler & Bonett's (1980) Non-normed Index. . . . .
Bentler & Bonett's (1980) NFI . . . . . . . . . . 1.0000
James, Mulaik, & Brett (1982) Parsimonious NFI. . 0.0000
Z-Test of Wilson & Hilferty (1931). . . . . . . . .
Bollen (1986) Normed Index Rho1 . . . . . . . . . .
Bollen (1988) Non-normed Index Delta2 . . . . . . 1.0000
Hoelter's (1983) Critical N . . . . . . . . . . . .
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Simple regression as a structural equation model with proc calis
8
Full (unrestricted) Model
14:08 Friday, October 15,
2004
Covariance Structure Analysis: Maximum Likelihood Estimation
Manifest Variable Equations
Y = - 0.0274*X + 1.0000 E
Std Err 0.0956 B
t Value -0.2863
Variances of Exogenous Variables
---------------------------------------------------------------------
Standard
Variable Parameter Estimate Error t Value
---------------------------------------------------------------------
X SIGXX 4.021413 0.360408 11.158
E SIGEE 9.155502 0.820536 11.158
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Simple regression as a structural equation model with proc calis
9
Full (unrestricted) Model
14:08 Friday, October 15,
2004
Covariance Structure Analysis: Maximum Likelihood Estimation
Equations with Standardized Coefficients
Y = - 0.0181*X + 0.9998 E
B
Squared Multiple Correlations
----------------------------------------------------------
Error Total
Variable Variance Variance R-squared
----------------------------------------------------------
1 Y 9.155502 9.158517 0.000329
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Simple regression as a structural equation model with proc calis
10
Reduced (restricted) Model
14:08 Friday, October 15,
2004
Covariance Structure Analysis: Pattern and Initial Values
LINEQS Model Statement
-------------------------------
Matrix Rows & Cols Matrix Type
TERM 1-----------------------------------------------------------
1 _SEL_ 2 3 SELECTION
2 _BETA_ 3 3 EQSBETA IMINUSINV
3 _GAMMA_ 3 2 EQSGAMMA
4 _PHI_ 2 2 SYMMETRIC
Number of endogenous variables = 1
Manifest: Y
Number of exogenous variables = 2
Manifest: X
Error: E
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Simple regression as a structural equation model with proc calis
11
Reduced (restricted) Model
14:08 Friday, October 15,
2004
Covariance Structure Analysis: Pattern and Initial Values
Manifest Variable Equations
Initial Estimates
Y = 1.0000 E
Variances of Exogenous Variables
-------------------------------------
Variable Parameter Estimate
-------------------------------------
X SIGXX .
E SIGEE .
_______________________________________________________________________________
Simple regression as a structural equation model with proc calis
12
Reduced (restricted) Model
14:08 Friday, October 15,
2004
Covariance Structure Analysis: Maximum Likelihood Estimation
250 Observations Model Terms 1
2 Variables Model Matrices 4
3 Informations Parameters 2
VARIABLE Mean Std Dev
X -.2251148715 2.005346226
Y 0.3633284181 3.026304133
Set covariances of exogenous manifest variables:
X
Vector of Initial Estimates
SIGXX 1 4.02141 Matrix Entry: _PHI_[1:1]
SIGEE 2 9.15852 Matrix Entry: _PHI_[2:2]
Predetermined Elements of the Predicted Moment Matrix
X Y
X . 0
Y 0 .
WARNING: The predicted moment matrix has 1 constant elements whose values
differ from those of the observed moment matrix.
The sum of squared differences is 0.012122346 .
NOTE: Only 1 elements of the moment matrix are used in the model
specification.
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Simple regression as a structural equation model with proc calis
13
Reduced (restricted) Model
14:08 Friday, October 15,
2004
Covariance Structure Analysis: Maximum Likelihood Estimation
Levenberg-Marquardt Optimization
Scaling Update of More (1978)
Number of Parameter Estimates 2
Number of Functions (Observations) 3
Number of Lower Bounds 2
Number of Upper Bounds 0
Optimization Start: Active Constraints= 0 Criterion= 0.000
Maximum Gradient Element= 0.000 Radius= 1.000
Iter rest nfun act optcrit difcrit maxgrad lambda rho
Optimization Results: Iterations= 0 Function Calls= 2 Jacobian Calls= 1
Active Constraints= 0 Criterion= 0.0003291958
Maximum Gradient Element= 2.76078E-17 Lambda= 0 Rho= 0 Radius= 1
NOTE: ABSGCONV convergence criterion satisfied.
_______________________________________________________________________________
Simple regression as a structural equation model with proc calis
14
Reduced (restricted) Model
14:08 Friday, October 15,
2004
Covariance Structure Analysis: Maximum Likelihood Estimation
Fit criterion . . . . . . . . . . . . . . . . . . 0.0003
Goodness of Fit Index (GFI) . . . . . . . . . . . 0.9997
GFI Adjusted for Degrees of Freedom (AGFI). . . . 0.9990
Root Mean Square Residual (RMR) . . . . . . . . . 0.0636
Parsimonious GFI (Mulaik, 1989) . . . . . . . . . 0.9997
Chi-square = 0.0820 df = 1 Prob>chi**2 = 0.7746
Null Model Chi-square: df = 1 0.0820
RMSEA Estimate . . . . . . . . . 0.0000 90%C.I.[., 0.1115]
Probability of Close Fit . . . . . . . . . . . . 0.8335
ECVI Estimate . . . . . . . . . . 0.0166 90%C.I.[., 0.0329]
Bentler's Comparative Fit Index . . . . . . . . . .
Normal Theory Reweighted LS Chi-square . . . . . 0.0820
Akaike's Information Criterion. . . . . . . . . . -1.9180
Bozdogan's (1987) CAIC. . . . . . . . . . . . . . -6.4395
Schwarz's Bayesian Criterion. . . . . . . . . . . -5.4395
McDonald's (1989) Centrality. . . . . . . . . . . 1.0018
Bentler & Bonett's (1980) Non-normed Index. . . . -0.0000
Bentler & Bonett's (1980) NFI . . . . . . . . . . 0.0000
James, Mulaik, & Brett (1982) Parsimonious NFI. . 0.0000
Z-Test of Wilson & Hilferty (1931). . . . . . . . -0.7284
Bollen (1986) Normed Index Rho1 . . . . . . . . . 0.0000
Bollen (1988) Non-normed Index Delta2 . . . . . . -0.0000
Hoelter's (1983) Critical N . . . . . . . . . . . 11671
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Simple regression as a structural equation model with proc calis
15
Reduced (restricted) Model
14:08 Friday, October 15,
2004
Covariance Structure Analysis: Maximum Likelihood Estimation
Manifest Variable Equations
Y = 1.0000 E
Variances of Exogenous Variables
---------------------------------------------------------------------
Standard
Variable Parameter Estimate Error t Value
---------------------------------------------------------------------
X SIGXX 4.021413 0.360408 11.158
E SIGEE 9.158517 0.820806 11.158
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Simple regression as a structural equation model with proc calis
16
Reduced (restricted) Model
14:08 Friday, October 15,
2004
Covariance Structure Analysis: Maximum Likelihood Estimation
Equations with Standardized Coefficients
Y = 1.0000 E
Squared Multiple Correlations
----------------------------------------------------------
Error Total
Variable Variance Variance R-squared
----------------------------------------------------------
1 Y 9.158517 9.158517 0