STA429/1007 F 2004 Handout 14
Structural Equation Models with proc calis
In the classical structural equation models, where everything is normal with expected value zero (accomplished by centering all variables by subtracting off the mean), there is a "saturated model" -- one that imposes no constraints on the variance-covariance matrix of the manifest variables, other than the obscure constraints implied by multivariate normal distribution. The saturated model is most easily estimated by just estimating all the population varianes and covariances by the corresponding sample quantities. For any data set, there are infinitely many potential saturated models. They all have the same number of parameters as the number of variances and covariances of manifest variables, and their parameters are one-to-one functions of the variances and covariances. Each member of this class of saturated models has the same -2 log likelihood. In other words, they are all equivalent, and they all fit the data equally well. Rather than minimizing -2 Likelihood (equivalent to maximizing the likelihood), proc calis does something else equivalent; it minimizes
Once this quantity is minimized over the set of parameters, the parameter values at which the minimum occurs are the Maximum Likelihood Estimates.
In the output, the minimum value is called either the "objective function" or the "fit function," depending on where you look. The number called objective function is preferable, because it is displayed to more decimal places of accuracy. Multiply it by n, and you get a likelihood ratio test for goodness of fit, comparing the specified model to the saturated model.
So the likelihood ratio test for goodness of fit is
and the degrees of freedom for the test are
where there are p manifest variables and k parameters in the model.
To compare a full and a reduced model, you can use the difference of the chisquare values for goodness of fit, with degrees of freedom equal to the difference in degrees of freedom. The result is a classical likelihood ratio test. There is another way to get a chisquare test for goodness of fit. In the massive table of fit information, the value labelled "Chi-square" is
(n - 1)
Chi-square = -------- * G
n
I have no idea why they multiply by (n-1)/n. Of course, since these are large-sample likelihood ratio tests, the effect of this multiplier is negligible, and the result is still a valid chi-square test -- a bit more conservative than the standard one. It's convenient, too, because it comes with a p-value. And as you might expect, the difference between Chi-square values for a full and a reduced model is also a valid chisquare test, with df equal to the diference in degrees of freedom.
There's one more useful test that is given by default. The "Independence Model Chi-Square" compares the saturated model to one that has zero covariances (no relationship at all) among all the variables. It is a simultaneous test for all the covariances (or equivalently, all the correlations) among manifest variables. If it's not significant, then there's not much point in fitting a structural equation model.
/* chain3.sas*/
options linesize=79 noovp formdlim='_';
title 'Three-var chain model';
data chain3;
infile 'chain.dat';
input y1 y2 y3;
proc calis cov pshort simple;
title2 'Full (unrestricted) Model: y1 -> y2 -> y3';
var y1 y2 y3; /* Manifest vars in the data set */
lineqs
y2 = b1 y1 + e1,
y3 = b2 y2 + e2;
std /* Variances not standard deviation */
y1 = sigy1,
e1 = sigee1,
e2 = sigee2;
/* cov statement is unnecessary */
cov /* Covariances */
y1 e1 = 0,
y1 e2 = 0,
e1 e2 = 0;
bounds
0.0 < sigy1,
0.0 < sigee1,
0.0 < sigee2;
proc calis cov pshort;
title2 'Full (unrestricted) Model: y3 -> y2 -> y1';
var y1 y2 y3; /* Manifest vars are in the data set */
lineqs
y2 = b1 y1 + e1,
y3 = b2 y2 + e2;
std /* Variances not standard deviation */
y1 = sigy1,
e1 = sigee1,
e2 = sigee2;
/* cov statement is unnecessary */
cov /* Covariances */
y1 e1 = 0,
y1 e2 = 0,
e1 e2 = 0;
bounds
0.0 < sigy1,
0.0 < sigee1,
0.0 < sigee2;
proc calis cov pshort;
title2 'Reduced Model: y1 -> y2 y3';
var y1 y2 y3; /* Manifest vars are in the data set */
lineqs
y2 = b1 y1 + e1,
y3 = b2 y2 + e2;
std /* Variances not standard deviation */
y1 = sigy1,
e1 = sigee1,
e2 = sigee2;
/* cov statement is unnecessary */
cov /* Covariances */
y1 e1 = 0,
y1 e2 = 0,
e1 e2 = 0;
bounds
0.0 < sigy1,
0.0 < sigee1,
0.0 < sigee2;
lincon b2=0; /* Linear constraints separated by commas.
The independence model would be
lincon b1=0 b2=0;
*/
_______________________________________________________________________________
Three-var chain model 1
Full (unrestricted) Model: y1 -> y2 -> y3
20:39 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Pattern and Initial Values
LINEQS Model Statement
Matrix Rows Columns ------Matrix Type-------
Term 1 1 _SEL_ 3 5 SELECTION
2 _BETA_ 5 5 EQSBETA IMINUSINV
3 _GAMMA_ 5 3 EQSGAMMA
4 _PHI_ 3 3 SYMMETRIC
The 2 Endogenous Variables
Manifest y2 y3
Latent
The 3 Exogenous Variables
Manifest y1
Latent
Error e1 e2
_______________________________________________________________________________
Three-var chain model 2
Full (unrestricted) Model: y1 -> y2 -> y3
20:39 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Observations 200 Model Terms 1
Variables 3 Model Matrices 4
Informations 6 Parameters 5
Variable Mean Std Dev
y1 0.07600 1.07182
y2 -0.09045 1.47679
y3 0.03195 1.35460
_______________________________________________________________________________
Three-var chain model 3
Full (unrestricted) Model: y1 -> y2 -> y3
20:39 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Levenberg-Marquardt Optimization
Scaling Update of More (1978)
Parameter Estimates 5
Functions (Observations) 6
Lower Bounds 3
Upper Bounds 0
Optimization Start
Active Constraints 0 Objective Function 2.2667947436
Max Abs Gradient Element 0.7827946588 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 1.42926 0.8375 0.1975 0.338 1.000
2 0 3 0 0.09884 1.3304 0.1695 0 3.263
3 0 4 0 0.01771 0.0811 2.93E-16 0 1.320
Optimization Results
Iterations 3 Function Calls 5
Jacobian Calls 4 Active Constraints 0
Objective Function 0.0177065358 Max Abs Gradient Element 2.925504E-16
Lambda 0 Actual Over Pred Change 1.3197130985
Radius 0.7012876847
ABSGCONV convergence criterion satisfied.
_______________________________________________________________________________
Three-var chain model 4
Full (unrestricted) Model: y1 -> y2 -> y3
20:39 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Fit Function 0.0177
Goodness of Fit Index (GFI) 0.9884
GFI Adjusted for Degrees of Freedom (AGFI) 0.9306
Root Mean Square Residual (RMR) 0.0671
Parsimonious GFI (Mulaik, 1989) 0.3295
Chi-Square 3.5236
Chi-Square DF 1
Pr > Chi-Square 0.0605
Independence Model Chi-Square 65.979
Independence Model Chi-Square DF 3
RMSEA Estimate 0.1126
RMSEA 90% Lower Confidence Limit .
RMSEA 90% Upper Confidence Limit 0.2497
ECVI Estimate 0.0690
ECVI 90% Lower Confidence Limit .
ECVI 90% Upper Confidence Limit 0.1193
Probability of Close Fit 0.1255
Bentler's Comparative Fit Index 0.9599
Normal Theory Reweighted LS Chi-Square 3.4926
Akaike's Information Criterion 1.5236
Bozdogan's (1987) CAIC -2.7747
Schwarz's Bayesian Criterion -1.7747
McDonald's (1989) Centrality 0.9937
Bentler & Bonett's (1980) Non-normed Index 0.8798
Bentler & Bonett's (1980) NFI 0.9466
James, Mulaik, & Brett (1982) Parsimonious NFI 0.3155
Z-Test of Wilson & Hilferty (1931) 1.5781
Bollen (1986) Normed Index Rho1 0.8398
Bollen (1988) Non-normed Index Delta2 0.9612
Hoelter's (1983) Critical N 218
_______________________________________________________________________________
Three-var chain model 5
Full (unrestricted) Model: y1 -> y2 -> y3
20:39 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Manifest Variable Equations with Estimates
y2 = -0.0626*y1 + 1.0000 e1
b1
y3 = 0.4747*y2 + 1.0000 e2
b2
Variances of Exogenous Variables
Variable Parameter Estimate
y1 sigy1 1.14880
e1 sigee1 2.17641
e2 sigee2 1.34344
_______________________________________________________________________________
Three-var chain model 6
Full (unrestricted) Model: y1 -> y2 -> y3
20:39 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Manifest Variable Equations with Standardized Estimates
y2 = -0.0454*y1 + 0.9990 e1
b1
y3 = 0.5175*y2 + 0.8557 e2
b2
Squared Multiple Correlations
Error Total
Variable Variance Variance R-Square
1 y2 2.17641 2.18091 0.00207
2 y3 1.34344 1.83494 0.2679
_______________________________________________________________________________
Three-var chain model 7
Full (unrestricted) Model: y3 -> y2 -> y1
20:39 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Pattern and Initial Values
LINEQS Model Statement
Matrix Rows Columns ------Matrix Type-------
Term 1 1 _SEL_ 3 5 SELECTION
2 _BETA_ 5 5 EQSBETA IMINUSINV
3 _GAMMA_ 5 3 EQSGAMMA
4 _PHI_ 3 3 SYMMETRIC
The 2 Endogenous Variables
Manifest y2 y3
Latent
The 3 Exogenous Variables
Manifest y1
Latent
Error e1 e2
_______________________________________________________________________________
Three-var chain model 8
Full (unrestricted) Model: y3 -> y2 -> y1
20:39 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Levenberg-Marquardt Optimization
Scaling Update of More (1978)
Parameter Estimates 5
Functions (Observations) 6
Lower Bounds 3
Upper Bounds 0
Optimization Start
Active Constraints 0 Objective Function 2.2667947436
Max Abs Gradient Element 0.7827946588 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 1.42926 0.8375 0.1975 0.338 1.000
2 0 3 0 0.09884 1.3304 0.1695 0 3.263
3 0 4 0 0.01771 0.0811 2.93E-16 0 1.320
Optimization Results
Iterations 3 Function Calls 5
Jacobian Calls 4 Active Constraints 0
Objective Function 0.0177065358 Max Abs Gradient Element 2.925504E-16
Lambda 0 Actual Over Pred Change 1.3197130985
Radius 0.7012876847
ABSGCONV convergence criterion satisfied.
_______________________________________________________________________________
Three-var chain model 9
Full (unrestricted) Model: y3 -> y2 -> y1
20:39 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Fit Function 0.0177
Goodness of Fit Index (GFI) 0.9884
GFI Adjusted for Degrees of Freedom (AGFI) 0.9306
Root Mean Square Residual (RMR) 0.0671
Parsimonious GFI (Mulaik, 1989) 0.3295
Chi-Square 3.5236
Chi-Square DF 1
Pr > Chi-Square 0.0605
Independence Model Chi-Square 65.979
Independence Model Chi-Square DF 3
RMSEA Estimate 0.1126
RMSEA 90% Lower Confidence Limit .
RMSEA 90% Upper Confidence Limit 0.2497
ECVI Estimate 0.0690
ECVI 90% Lower Confidence Limit .
ECVI 90% Upper Confidence Limit 0.1193
Probability of Close Fit 0.1255
Bentler's Comparative Fit Index 0.9599
Normal Theory Reweighted LS Chi-Square 3.4926
Akaike's Information Criterion 1.5236
Bozdogan's (1987) CAIC -2.7747
Schwarz's Bayesian Criterion -1.7747
McDonald's (1989) Centrality 0.9937
Bentler & Bonett's (1980) Non-normed Index 0.8798
Bentler & Bonett's (1980) NFI 0.9466
James, Mulaik, & Brett (1982) Parsimonious NFI 0.3155
Z-Test of Wilson & Hilferty (1931) 1.5781
Bollen (1986) Normed Index Rho1 0.8398
Bollen (1988) Non-normed Index Delta2 0.9612
Hoelter's (1983) Critical N 218
_______________________________________________________________________________
Three-var chain model 10
Full (unrestricted) Model: y3 -> y2 -> y1
20:39 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Manifest Variable Equations with Estimates
y2 = -0.0626*y1 + 1.0000 e1
b1
y3 = 0.4747*y2 + 1.0000 e2
b2
Variances of Exogenous Variables
Variable Parameter Estimate
y1 sigy1 1.14880
e1 sigee1 2.17641
e2 sigee2 1.34344
_______________________________________________________________________________
Three-var chain model 11
Full (unrestricted) Model: y3 -> y2 -> y1
20:39 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Manifest Variable Equations with Standardized Estimates
y2 = -0.0454*y1 + 0.9990 e1
b1
y3 = 0.5175*y2 + 0.8557 e2
b2
Squared Multiple Correlations
Error Total
Variable Variance Variance R-Square
1 y2 2.17641 2.18091 0.00207
2 y3 1.34344 1.83494 0.2679
_______________________________________________________________________________
Three-var chain model 12
Reduced Model: y1 -> y2 y3
20:39 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Pattern and Initial Values
LINEQS Model Statement
Matrix Rows Columns ------Matrix Type-------
Term 1 1 _SEL_ 3 5 SELECTION
2 _BETA_ 5 5 EQSBETA IMINUSINV
3 _GAMMA_ 5 3 EQSGAMMA
4 _PHI_ 3 3 SYMMETRIC
The 2 Endogenous Variables
Manifest y2 y3
Latent
The 3 Exogenous Variables
Manifest y1
Latent
Error e1 e2
_______________________________________________________________________________
Three-var chain model 13
Reduced Model: y1 -> y2 y3
20:39 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
NOTE: Initial point was changed to be feasible for boundary and linear
constraints.
_______________________________________________________________________________
Three-var chain model 14
Reduced Model: y1 -> y2 y3
20:39 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Levenberg-Marquardt Optimization
Scaling Update of More (1978)
Parameter Estimates 5
Functions (Observations) 6
Lower Bounds 4
Upper Bounds 1
Optimization Start
Active Constraints 1 Objective Function 1.4108855332
Max Abs Gradient Element 0.0672113951 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 1 0.32948 1.0814 2.94E-16 0 2.952
Optimization Results
Iterations 1 Function Calls 3
Jacobian Calls 2 Active Constraints 1
Objective Function 0.3294830486 Max Abs Gradient Element 2.944582E-16
Lambda 0 Actual Over Pred Change 2.95231001
Radius 1.7118184209
ABSGCONV convergence criterion satisfied.
WARNING: There are 1 active constraints at the solution. The standard errors
and Chi-Square test statistic assume the solution is located in the
interior of the parameter space and hence do not apply if it is
likely that some different set of inequality constraints could be
active.
NOTE: The degrees of freedom are increased by the number of active constraints
(see Dijkstra, 1992). The number of parameters in calculating fit
indices is decreased by the number of active constraints. To turn off
the adjustment, use the NOADJDF option.
_______________________________________________________________________________
Three-var chain model 15
Reduced Model: y1 -> y2 y3
20:39 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Fit Function 0.3295
Goodness of Fit Index (GFI) 0.8424
GFI Adjusted for Degrees of Freedom (AGFI) 0.5271
Root Mean Square Residual (RMR) 0.4304
Parsimonious GFI (Mulaik, 1989) 0.5616
Chi-Square 65.5671
Chi-Square DF 2
Pr > Chi-Square <.0001
Independence Model Chi-Square 65.979
Independence Model Chi-Square DF 3
RMSEA Estimate 0.3996
RMSEA 90% Lower Confidence Limit 0.3199
RMSEA 90% Upper Confidence Limit 0.4855
ECVI Estimate 0.3705
ECVI 90% Lower Confidence Limit 0.2548
ECVI 90% Upper Confidence Limit 0.5242
Probability of Close Fit 0.0000
Bentler's Comparative Fit Index -0.0093
Normal Theory Reweighted LS Chi-Square 55.8602
Akaike's Information Criterion 61.5671
Bozdogan's (1987) CAIC 52.9705
Schwarz's Bayesian Criterion 54.9705
McDonald's (1989) Centrality 0.8531
Bentler & Bonett's (1980) Non-normed Index -0.5140
Bentler & Bonett's (1980) NFI 0.0062
James, Mulaik, & Brett (1982) Parsimonious NFI 0.0042
Z-Test of Wilson & Hilferty (1931) 6.9349
Bollen (1986) Normed Index Rho1 -0.4906
Bollen (1988) Non-normed Index Delta2 0.0064
Hoelter's (1983) Critical N 20
_______________________________________________________________________________
Three-var chain model 16
Reduced Model: y1 -> y2 y3
20:39 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Manifest Variable Equations with Estimates
y2 = -0.0626*y1 + 1.0000 e1
b1
y3 = 0*y2 + 1.0000 e2
b2
Variances of Exogenous Variables
Variable Parameter Estimate
y1 sigy1 1.14880
e1 sigee1 2.17641
e2 sigee2 1.83494
_______________________________________________________________________________
Three-var chain model 17
Reduced Model: y1 -> y2 y3
20:39 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Manifest Variable Equations with Standardized Estimates
y2 = -0.0454*y1 + 0.9990 e1
b1
y3 = 0*y2 + 1.0000 e2
b2
Squared Multiple Correlations
Error Total
Variable Variance Variance R-Square
1 y2 2.17641 2.18091 0.00207
2 y3 1.83494 1.83494 0