STA313 F 2004 Handout 8
Path Model 1 with SAS
/* path1.sas */
options linesize=79 noovp formdlim='_';
title 'STA313f04 Path 1 Example';
data path1;
infile 'path1.dat';
input x1 x2 y1 y2;
proc calis cov; /* Analyze the covariance matrix (Default is corr) */
title2 'Full (unrestricted) Model';
var x1 x2 y1 y2; /* Manifest vars are in the data set */
lineqs /* Simultaneous equations, separated by commas */
y1 = b1 x1 + e1,
y2 = b2 y1 + b3 x2 + e2;
std /* Variances (not standard deviations) */
x1 = sigsqx1, /* Optional starting values in parentheses */
x2 = sigsqx2,
e1 = sigsqe1,
e2 = sigsqe2;
cov /* Covariances */
x1 x2 = sigma12; /* Unmentioned pairs get covariance zero */
bounds 0.0 < sigsqx1,
0.0 < sigsqx2,
0.0 < sigsqe1,
0.0 < sigsqe2;
proc calis cov; /* Analyze the covariance matrix (Default is corr) */
title2 'Reduced (restricted) Model: b3=0';
var x1 x2 y1 y2; /* Manifest vars are in the data set */
lineqs /* Simultaneous equations, separated by commas */
y1 = b1 x1 + e1,
y2 = b2 y1 + e2;
std /* Variances (not standard deviations) */
x1 = sigsqx1, /* Optional starting values in parentheses */
x2 = sigsqx2,
e1 = sigsqe1,
e2 = sigsqe2;
cov /* Covariances */
x1 x2 = sigma12; /* Unmentioned pairs get covariance zero */
bounds 0.0 < sigsqx1,
0.0 < sigsqx2,
0.0 < sigsqe1,
0.0 < sigsqe2;
proc iml;
title2 'Compute G two ways';
print " ";
print "Based on Fit Function";
G1 = 300*(3.3328-0.0227); pval1 = 1-probchi(G1,1);
print "G = " G1 ", df = 1, p = " pval1;
print " ";
print "Based on chi-square";
G2 = 300/299 * (996.5153-6.7874); pval2 = 1-probchi(G2,1);
print "G = " G2 ", df = 1, p = " pval2;
Before looking at the list file, here is a little discussion of how the test statistic G is being computed with proc iml. Notice that except for getting the p-value, these calculations could be done with a hand calculator.
Let us use the term "saturated model" for a model with no constraints on the covariance matrix of the manifest variables. This is the language we have been using in class. Any (identified) model with the same number of parameters as the unique elements of the covariance matrix is also saturated, and yields the same -2 Log Likelihood -- that is, any saturated model has a -2 Log likelihood equal to
n p ( 1 +log(2 pi) ) + n log(|Sigma_hat|) .
The equality of the -2 LL quantities for any saturated model follows from the invariance principle of maximum likelihood estimation, just for the record.
Now suppose you fit a non-saturated model. The difference between the quantity above and -2 LL for the model you fit is a reasonable test for the "goodness of fit" of your model. The null hypothesis is that your model holds, versus the alternative that there are no restrictions at all on the variance-covariance matrix of the manifest variables. The difference between the two -2LL quantities is a G; it's asymptotically chi-square, with degrees of freedom the difference between the number of parameters in your model and the number of parameters (unique elements of the covariance matrix) of the saturated model. This "goodness of fit" chisquare will equal zero (with df=0) only if you are fitting a model that is one-to-one with the saturated model.
If you fit an unrestricted model (but still maybe restricted compared to the saturated model) and you also fit a (more) restricted model, the DIFFERENCE between the 2 goodness of fit chi-square statistics is exactly our test statistic G for testing the null hypothesis that the restricted model is true versus the alternative that the unrestricted model is true. There are two ways to get the goodness of fit chisquare statistic from the SAS output for a model. Of course you need to fit a restriced and an unrestricted model, and subtract to get G.
The first way is based on the "Fit Function" of the SAS output, which equals 0.0227 for the Full (unrestricted) model in the path1 example, and 3.3328 for the reduced (restricted) model. Multiply it by n, and you get that goodness of fit chisquare, directly. Multiply the difference by n, and you get the test statistic we are seeking. Thus, what we want is G = 300*(3.3328-0.0227) = 993.03. That's G1 in the proc iml above.
The second way to get G is from the SAS "Chi-square" statistic; Chi-square is equal to 6.7874 for the Full (unrestricted) model, and 996.5153 for the Reduced (restricted) model in the SAS output. This is almost the right number. It's what we want, but multiplied by (n-1)/n. Don't ask me why they do this, but of course for very large samples, (n-1)/n has no effect, and the G test is based on large-sample theory. We will multiply by n/(n-1) to get the traditional likelihood ratio test. Thus, G = 300/299 * (996.5153-6.7874) = 993.038. That's G2 in the proc iml; it's equal to G1 except for rounding error.
For comparison, when we did this example with R (in Handout 7) we got G = 993.038.
Now here is path1.lst.
_______________________________________________________________________________
STA313f04 Path 1 Example 1
Full (unrestricted) Model
10:17 Friday, November 5, 2004
The CALIS Procedure
Covariance Structure Analysis: Pattern and Initial Values
LINEQS Model Statement
Matrix Rows Columns ------Matrix Type-------
Term 1 1 _SEL_ 4 6 SELECTION
2 _BETA_ 6 6 EQSBETA IMINUSINV
3 _GAMMA_ 6 4 EQSGAMMA
4 _PHI_ 4 4 SYMMETRIC
The 2 Endogenous Variables
Manifest y1 y2
Latent
The 4 Exogenous Variables
Manifest x1 x2
Latent
Error e1 e2
_______________________________________________________________________________
STA313f04 Path 1 Example 2
Full (unrestricted) Model
10:17 Friday, November 5, 2004
The CALIS Procedure
Covariance Structure Analysis: Pattern and Initial Values
Manifest Variable Equations with Initial Estimates
y1 = .*x1 + 1.0000 e1
b1
y2 = .*y1 + .*x2 + 1.0000 e2
b2 b3
Variances of Exogenous Variables
Variable Parameter Estimate
x1 sigsqx1 .
x2 sigsqx2 .
e1 sigsqe1 .
e2 sigsqe2 .
Covariances Among Exogenous Variables
Var1 Var2 Parameter Estimate
x1 x2 sigma12 .
_______________________________________________________________________________
STA313f04 Path 1 Example 3
Full (unrestricted) Model
10:17 Friday, November 5, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Observations 300 Model Terms 1
Variables 4 Model Matrices 4
Informations 10 Parameters 8
Variable Mean Std Dev
x1 0.16588 2.27566
x2 -0.20103 3.08880
y1 0.11331 2.68760
y2 -0.41910 10.88369
Set Covariances of Exogenous Manifest Variables
x1 x2
NOTE: Some initial estimates computed by two-stage LS method.
_______________________________________________________________________________
STA313f04 Path 1 Example 4
Full (unrestricted) Model
10:17 Friday, November 5, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Vector of Initial Estimates
Parameter Estimate Type
1 b2 2.09453 Matrix Entry: _BETA_[2:1]
2 b1 1.00190 Matrix Entry: _GAMMA_[1:1]
3 b3 2.98282 Matrix Entry: _GAMMA_[2:2]
4 sigsqx1 5.17862 Matrix Entry: _PHI_[1:1]
5 sigma12 0.44648 Matrix Entry: _PHI_[2:1]
6 sigsqx2 9.54070 Matrix Entry: _PHI_[2:2]
7 sigsqe1 2.02490 Matrix Entry: _PHI_[3:3]
8 sigsqe2 3.23198 Matrix Entry: _PHI_[4:4]
_______________________________________________________________________________
STA313f04 Path 1 Example 5
Full (unrestricted) Model
10:17 Friday, November 5, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Levenberg-Marquardt Optimization
Scaling Update of More (1978)
Parameter Estimates 8
Functions (Observations) 10
Lower Bounds 4
Upper Bounds 0
Optimization Start
Active Constraints 0 Objective Function 0.0274155346
Max Abs Gradient Element 0.2050710365 Radius 1
Actual
Max Abs Over
Rest Func Act Objective Obj Fun Gradient Pred
Iter arts Calls Con Function Change Element Lambda Change
1 0 2 0 0.02272 0.00469 0.0127 0 0.994
2 0 3 0 0.02270 0.000025 0.000596 0 0.998
3 0 4 0 0.02270 3.992E-8 0.000037 0 0.983
4 0 5 0 0.02270 1.18E-10 1.741E-6 0 0.978
Optimization Results
Iterations 4 Function Calls 6
Jacobian Calls 5 Active Constraints 0
Objective Function 0.0227001914 Max Abs Gradient Element 1.7411686E-6
Lambda 0 Actual Over Pred Change 0.9775696387
Radius 0.000047076
ABSGCONV convergence criterion satisfied.
_______________________________________________________________________________
STA313f04 Path 1 Example 6
Full (unrestricted) Model
10:17 Friday, November 5, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Fit Function 0.0227
Goodness of Fit Index (GFI) 0.9889
GFI Adjusted for Degrees of Freedom (AGFI) 0.9446
Root Mean Square Residual (RMR) 1.7981
Parsimonious GFI (Mulaik, 1989) 0.3296
Chi-Square 6.7874
Chi-Square DF 2
Pr > Chi-Square 0.0336
Independence Model Chi-Square 1466.5
Independence Model Chi-Square DF 6
RMSEA Estimate 0.0895
RMSEA 90% Lower Confidence Limit 0.0215
RMSEA 90% Upper Confidence Limit 0.1675
ECVI Estimate 0.0771
ECVI 90% Lower Confidence Limit 0.0620
ECVI 90% Upper Confidence Limit 0.1176
Probability of Close Fit 0.1372
Bentler's Comparative Fit Index 0.9967
Normal Theory Reweighted LS Chi-Square 6.7003
Akaike's Information Criterion 2.7874
Bozdogan's (1987) CAIC -6.6202
Schwarz's Bayesian Criterion -4.6202
McDonald's (1989) Centrality 0.9921
Bentler & Bonett's (1980) Non-normed Index 0.9902
Bentler & Bonett's (1980) NFI 0.9954
James, Mulaik, & Brett (1982) Parsimonious NFI 0.3318
Z-Test of Wilson & Hilferty (1931) 1.8416
Bollen (1986) Normed Index Rho1 0.9861
Bollen (1988) Non-normed Index Delta2 0.9967
Hoelter's (1983) Critical N 265
_______________________________________________________________________________
STA313f04 Path 1 Example 7
Full (unrestricted) Model
10:17 Friday, November 5, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Manifest Variable Equations with Estimates
y1 = 1.0019*x1 + 1.0000 e1
Std Err 0.0362 b1
t Value 27.7053
y2 = 2.0486*y1 + 2.9828*x2 + 1.0000 e2
Std Err 0.0386 b2 0.0336 b3
t Value 53.0060 88.6977
Variances of Exogenous Variables
Standard
Variable Parameter Estimate Error t Value
x1 sigsqx1 5.17862 0.42354 12.23
x2 sigsqx2 9.54070 0.78030 12.23
e1 sigsqe1 2.02490 0.16561 12.23
e2 sigsqe2 3.21678 0.26309 12.23
Covariances Among Exogenous Variables
Standard
Var1 Var2 Parameter Estimate Error t Value
x1 x2 sigma12 0.44648 0.40732 1.10
_______________________________________________________________________________
STA313f04 Path 1 Example 8
Full (unrestricted) Model
10:17 Friday, November 5, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Manifest Variable Equations with Standardized Estimates
y1 = 0.8483*x1 + 0.5295 e1
b1
y2 = 0.4947*y1 + 0.8278*x2 + 0.1611 e2
b2 b3
Squared Multiple Correlations
Error Total
Variable Variance Variance R-Square
1 y1 2.02490 7.22317 0.7197
2 y2 3.21678 123.88541 0.9740
Correlations Among Exogenous Variables
Var1 Var2 Parameter Estimate
x1 x2 sigma12 0.06352
_______________________________________________________________________________
STA313f04 Path 1 Example 9
Reduced (restricted) Model: b3=0
10:17 Friday, November 5, 2004
The CALIS Procedure
Covariance Structure Analysis: Pattern and Initial Values
LINEQS Model Statement
Matrix Rows Columns ------Matrix Type-------
Term 1 1 _SEL_ 4 6 SELECTION
2 _BETA_ 6 6 EQSBETA IMINUSINV
3 _GAMMA_ 6 4 EQSGAMMA
4 _PHI_ 4 4 SYMMETRIC
The 2 Endogenous Variables
Manifest y1 y2
Latent
The 4 Exogenous Variables
Manifest x1 x2
Latent
Error e1 e2
_______________________________________________________________________________
STA313f04 Path 1 Example 10
Reduced (restricted) Model: b3=0
10:17 Friday, November 5, 2004
The CALIS Procedure
Covariance Structure Analysis: Pattern and Initial Values
Manifest Variable Equations with Initial Estimates
y1 = .*x1 + 1.0000 e1
b1
y2 = .*y1 + 1.0000 e2
b2
Variances of Exogenous Variables
Variable Parameter Estimate
x1 sigsqx1 .
x2 sigsqx2 .
e1 sigsqe1 .
e2 sigsqe2 .
Covariances Among Exogenous Variables
Var1 Var2 Parameter Estimate
x1 x2 sigma12 .
_______________________________________________________________________________
STA313f04 Path 1 Example 11
Reduced (restricted) Model: b3=0
10:17 Friday, November 5, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Observations 300 Model Terms 1
Variables 4 Model Matrices 4
Informations 10 Parameters 7
Variable Mean Std Dev
x1 0.16588 2.27566
x2 -0.20103 3.08880
y1 0.11331 2.68760
y2 -0.41910 10.88369
Set Covariances of Exogenous Manifest Variables
x1 x2
NOTE: Some initial estimates computed by two-stage LS method.
_______________________________________________________________________________
STA313f04 Path 1 Example 12
Reduced (restricted) Model: b3=0
10:17 Friday, November 5, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Vector of Initial Estimates
Parameter Estimate Type
1 b2 2.09622 Matrix Entry: _BETA_[2:1]
2 b1 1.00190 Matrix Entry: _GAMMA_[1:1]
3 sigsqx1 5.17862 Matrix Entry: _PHI_[1:1]
4 sigma12 0.44648 Matrix Entry: _PHI_[2:1]
5 sigsqx2 9.54070 Matrix Entry: _PHI_[2:2]
6 sigsqe1 2.02490 Matrix Entry: _PHI_[3:3]
7 sigsqe2 88.11855 Matrix Entry: _PHI_[4:4]
_______________________________________________________________________________
STA313f04 Path 1 Example 13
Reduced (restricted) Model: b3=0
10:17 Friday, November 5, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Levenberg-Marquardt Optimization
Scaling Update of More (1978)
Parameter Estimates 7
Functions (Observations) 10
Lower Bounds 4
Upper Bounds 0
Optimization Start
Active Constraints 0 Objective Function 3.3330031833
Max Abs Gradient Element 0.0075986595 Radius 1
Actual
Max Abs Over
Rest Func Act Objective Obj Fun Gradient Pred
Iter arts Calls Con Function Change Element Lambda Change
1 0 2 0 3.33283 0.000176 1.998E-6 0 1.000
Optimization Results
Iterations 1 Function Calls 3
Jacobian Calls 2 Active Constraints 0
Objective Function 3.3328270857 Max Abs Gradient Element 1.9984173E-6
Lambda 0 Actual Over Pred Change 1
Radius 0.0375337328
ABSGCONV convergence criterion satisfied.
_______________________________________________________________________________
STA313f04 Path 1 Example 14
Reduced (restricted) Model: b3=0
10:17 Friday, November 5, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Fit Function 3.3328
Goodness of Fit Index (GFI) 0.6694
GFI Adjusted for Degrees of Freedom (AGFI) -0.1018
Root Mean Square Residual (RMR) 8.7264
Parsimonious GFI (Mulaik, 1989) 0.3347
Chi-Square 996.5153
Chi-Square DF 3
Pr > Chi-Square <.0001
Independence Model Chi-Square 1466.5
Independence Model Chi-Square DF 6
RMSEA Estimate 1.0524
RMSEA 90% Lower Confidence Limit 0.9980
RMSEA 90% Upper Confidence Limit 1.1079
ECVI Estimate 3.3804
ECVI 90% Lower Confidence Limit 3.0430
ECVI 90% Upper Confidence Limit 3.7431
Probability of Close Fit 0.0000
Bentler's Comparative Fit Index 0.3197
Normal Theory Reweighted LS Chi-Square 295.2477
Akaike's Information Criterion 990.5153
Bozdogan's (1987) CAIC 976.4040
Schwarz's Bayesian Criterion 979.4040
McDonald's (1989) Centrality 0.1909
Bentler & Bonett's (1980) Non-normed Index -0.3605
Bentler & Bonett's (1980) NFI 0.3205
James, Mulaik, & Brett (1982) Parsimonious NFI 0.1602
Z-Test of Wilson & Hilferty (1931) 22.0440
Bollen (1986) Normed Index Rho1 -0.3590
Bollen (1988) Non-normed Index Delta2 0.3211
Hoelter's (1983) Critical N 4
_______________________________________________________________________________
STA313f04 Path 1 Example 15
Reduced (restricted) Model: b3=0
10:17 Friday, November 5, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Manifest Variable Equations with Estimates
y1 = 1.0019*x1 + 1.0000 e1
Std Err 0.0362 b1
t Value 27.7053
y2 = 2.0499*y1 + 1.0000 e2
Std Err 0.2020 b2
t Value 10.1483
Variances of Exogenous Variables
Standard
Variable Parameter Estimate Error t Value
x1 sigsqx1 5.17862 0.42354 12.23
x2 sigsqx2 9.54070 0.78030 12.23
e1 sigsqe1 2.02490 0.16561 12.23
e2 sigsqe2 88.11855 7.20687 12.23
Covariances Among Exogenous Variables
Standard
Var1 Var2 Parameter Estimate Error t Value
x1 x2 sigma12 0.44648 0.40732 1.10
_______________________________________________________________________________
STA313f04 Path 1 Example 16
Reduced (restricted) Model: b3=0
10:17 Friday, November 5, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Manifest Variable Equations with Standardized Estimates
y1 = 0.8483*x1 + 0.5295 e1
b1
y2 = 0.5062*y1 + 0.8624 e2
b2
Squared Multiple Correlations
Error Total
Variable Variance Variance R-Square
1 y1 2.02490 7.22317 0.7197
2 y2 88.11855 118.47013 0.2562
Correlations Among Exogenous Variables
Var1 Var2 Parameter Estimate
x1 x2 sigma12 0.06352
_______________________________________________________________________________
STA313f04 Path 1 Example 17
Compute G two ways
10:17 Friday, November 5, 2004
Based on Fit Function
G1 PVAL1
G = 993.03 , df = 1, p = 0
Based on chi-square
G2 PVAL2
G = 993.03803 , df = 1, p = 0