SAS Example 3: Deliberately create numerical problems
Four experiments
1. Try to fit this model, failing the parameter
count rule.
2. Set φ
12
=0 to pass the parameter count rule, but
still not identifiable.
3. Set β
1
=β
2
instead -- still not identifiable.
4. Set β
1
=β
2
, but do not force ω>0.
Page 1 of 18
/* calculus3.sas */
options linesize=79 pagesize=500 noovp formdlim='_' nodate;
title 'Calculus 3: Deliberately cause trouble with identifiability';
title2 'By adding measurement error to response variable';
data math;
infile 'calculus.data' firstobs=2;
input id hscalc hsengl hsgpa test grade;
/* Exclude some output you really don't want to see.
The (persist) option means keep doing it. */
ods exclude
Calis.ModelSpec.LINEQSEqInit (persist)
Calis.ModelSpec.LINEQSVarExogInit (persist)
Calis.ModelSpec.LINEQSCovExogInit (persist)
Calis.ML.SqMultCorr (persist)
Calis.StandardizedResults.LINEQSEqStd (persist)
Calis.StandardizedResults.LINEQSVarExogStd (persist)
Calis.StandardizedResults.LINEQSCovExogStd (persist)
;
proc calis cov ; /* Analyze the covariance matrix (Default is corr) */
title3 'Fails parameter count test';
var grade hsgpa test; /* Declare observed vars */
lineqs /* Simultaneous equations, separated by commas */
Fgrade = beta1 hsgpa + beta2 test + epsilon,
grade = Fgrade + e;
std /* Variances (not standard deviations) */
hsgpa = phi11,
test = phi22,
epsilon = psi,
e = omega;
cov /* Covariances */
hsgpa test = phi12; /* Unmentioned pairs get covariance zero */
bounds 0.0 < phi11,
0.0 < phi22,
0.0 < psi,
0.0 < omega;
Log file says:
WARNING: The estimation problem is not identified: There are more parameters
to estimate ( 7 ) than the total number of mean and covariance
elements ( 6 ).
NOTE: Convergence criterion (ABSGCONV=0.00001) satisfied.
NOTE: The Moore-Penrose inverse is used in computing the covariance matrix for
parameter estimates.
WARNING: Standard errors and t values might not be accurate with the use of
the Moore-Penrose inverse.
WARNING: Critical N is not computable for df= -1.
Page 2 of 18
_______________________________________________________________________________
Calculus 3: Deliberately cause trouble with identifiability 1
By adding measurement error to response variable
Fails parameter count test
The CALIS Procedure
Covariance Structure Analysis: Model and Initial Values
Modeling Information
Data Set WORK.MATH
N Records Read 287
N Records Used 287
N Obs 287
Model Type LINEQS
Analysis Covariances
Variables in the Model
Endogenous Manifest grade
Latent Fgrade
Exogenous Manifest hsgpa test
Latent
Error e epsilon
Number of Endogenous Variables = 2
Number of Exogenous Variables = 4
_______________________________________________________________________________
Calculus 3: Deliberately cause trouble with identifiability 2
By adding measurement error to response variable
Fails parameter count test
The CALIS Procedure
Covariance Structure Analysis: Descriptive Statistics
Simple Statistics
Variable Mean Std Dev
grade 60.98955 17.73355
hsgpa 80.98293 5.97063
test 8.81533 3.56910
_______________________________________________________________________________
Calculus 3: Deliberately cause trouble with identifiability 3
By adding measurement error to response variable
Fails parameter count test
The CALIS Procedure
Covariance Structure Analysis: Optimization
Initial Estimation Methods
1 Observed Moments of Variables
2 McDonald Method
3 Two-Stage Least Squares
Optimization Start
Parameter Estimates
Page 3 of 18
N Parameter Estimate Gradient Lower Bound Upper Bound
1 beta1 1.46805 1.2905E-17 . .
2 beta2 1.34956 1.404E-17 . .
3 phi11 35.64841 1.0173E-18 0 .
4 phi22 12.73851 2.8469E-18 0 .
5 psi 185.72484 2.89877E-7 0 .
6 omega 0.01000 2.89877E-7 0 .
7 phi12 7.24928 -5.581E-18 . .
Value of Objective Function = 0
_______________________________________________________________________________
Calculus 3: Deliberately cause trouble with identifiability 4
By adding measurement error to response variable
Fails parameter count test
The CALIS Procedure
Covariance Structure Analysis: Optimization
Levenberg-Marquardt Optimization
Scaling Update of More (1978)
Parameter Estimates 7
Functions (Observations) 6
Lower Bounds 4
Upper Bounds 0
Optimization Start
Active Constraints 0 Objective Function 0
Max Abs Gradient Element 2.8987665E-7 Radius 1
Optimization Results
Iterations 0 Function Calls 4
Jacobian Calls 1 Active Constraints 0
Objective Function 0 Max Abs Gradient Element 2.8987665E-7
Lambda 0 Actual Over Pred Change 0
Radius 1
Convergence criterion (ABSGCONV=0.00001) satisfied.
NOTE: The Moore-Penrose inverse is used in computing the covariance matrix for
parameter estimates.
WARNING: Standard errors and t values might not be accurate with the use of
the Moore-Penrose inverse.
NOTE: Covariance matrix for the estimates is not full rank.
Page 4 of 18
NOTE: The variance of some parameter estimates is zero or some parameter
estimates are linearly related to other parameter estimates as shown in
the following equations:
omega = -6407030 + 34497 * psi
_______________________________________________________________________________
Calculus 3: Deliberately cause trouble with identifiability 5
By adding measurement error to response variable
Fails parameter count test
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Fit Summary
Modeling Info N Observations 287
N Variables 3
N Moments 6
N Parameters 7
N Active Constraints 0
Baseline Model Function Value 0.6496
Baseline Model Chi-Square 185.7968
Baseline Model Chi-Square DF 3
Pr > Baseline Model Chi-Square <.0001
Absolute Index Fit Function 0.0000
Chi-Square 0.0000
Chi-Square DF -1
Pr > Chi-Square .
Z-Test of Wilson & Hilferty .
Hoelter Critical N .
Root Mean Square Residual (RMSR) 0.0041
Standardized RMSR (SRMSR) 0.0000
Goodness of Fit Index (GFI) 1.0000
Parsimony Index Adjusted GFI (AGFI) .
Parsimonious GFI -0.3333
RMSEA Estimate .
Probability of Close Fit .
ECVI Estimate 0.0426
ECVI Lower 90% Confidence Limit .
ECVI Upper 90% Confidence Limit .
Akaike Information Criterion 14.0000
Bozdogan CAIC 46.6164
Schwarz Bayesian Criterion 39.6164
McDonald Centrality 0.9983
Incremental Index Bentler Comparative Fit Index 0.9945
Bentler-Bonett NFI 1.0000
Bentler-Bonett Non-normed Index .
Bollen Normed Index Rho1 .
Bollen Non-normed Index Delta2 0.9946
James et al. Parsimonious NFI -0.3333
WARNING: Indices for models with negative
degrees of freedom may not be interpretable.
_______________________________________________________________________________
Page 5 of 18
Calculus 3: Deliberately cause trouble with identifiability 6
By adding measurement error to response variable
Fails parameter count test
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Linear Equations
Fgrade = 1.4680*hsgpa + 1.3496*test + 1.0000 epsilon
Std Err 0.1435 beta1 0.2401 beta2
t Value 10.2280 5.6206
grade = 1.0000 Fgrade + 1.0000 e
Estimates for Variances of Exogenous Variables
Variable Standard
Type Variable Parameter Estimate Error t Value
Observed hsgpa phi11 35.64841 2.98107 11.95826
test phi22 12.73851 1.06525 11.95826
Disturbance epsilon psi 185.72484 7.76596 23.91523
Error e omega 0.01000 7.76596 0.00129
Covariances Among Exogenous Variables
Standard
Var1 Var2 Parameter Estimate Error t Value
hsgpa test phi12 7.24928 1.33099 5.44653
_______________________________________________________________________________
Earlier output
Linear Equations
grade = 1.4680*hsgpa + 1.3496*test + 1.0000 epsilon
Std Err 0.1435 beta1 0.2401 beta2
t Value 10.2283 5.6207
Estimates for Variances of Exogenous Variables
Variable Standard
Type Variable Parameter Estimate Error t Value
Observed hsgpa phi11 35.64841 2.98107 11.95826
test phi22 12.73851 1.06525 11.95826
Error epsilon psi 185.72484 15.53109 11.95826
Covariances Among Exogenous Variables
Standard
Var1 Var2 Parameter Estimate Error t Value
hsgpa test phi12 7.24928 1.33099 5.44653
Page 6 of 18
proc calis cov ; /* Analyze the covariance matrix (Default is corr) */
title3 'Non-ident but passes parameter count test with phi12=0';
var grade hsgpa test; /* Declare observed vars */
lineqs /* Simultaneous equations, separated by commas */
Fgrade = beta1 hsgpa + beta2 test + epsilon,
grade = Fgrade + e;
std /* Variances (not standard deviations) */
hsgpa = phi11,
test = phi22,
epsilon = psi,
e = omega;
/* No covariance between expl vars */
cov /* Covariances */
hsgpa test = 0;
bounds 0.0 < phi11,
0.0 < phi22,
0.0 < psi,
0.0 < omega;
Log file says
NOTE: Convergence criterion (ABSGCONV=0.00001) satisfied.
NOTE: The Moore-Penrose inverse is used in computing the covariance matrix for
parameter estimates.
WARNING: Standard errors and t values might not be accurate with the use of
the Moore-Penrose inverse.
WARNING: Critical N is not computable for df= 0.
List file says
Optimization Start
Active Constraints 0 Objective Function 0.1229884791
Max Abs Gradient Element 2.8987665E-7 Radius 1
Optimization Results
Iterations 0 Function Calls 4
Jacobian Calls 1 Active Constraints 0
Objective Function 0.1229884791 Max Abs Gradient Element 2.8987665E-7
Lambda 0 Actual Over Pred Change 0
Radius 1
Convergence criterion (ABSGCONV=0.00001) satisfied.
NOTE: The Moore-Penrose inverse is used in computing the covariance matrix for
parameter estimates.
WARNING: Standard errors and t values might not be accurate with the use of
the Moore-Penrose inverse.
NOTE: Covariance matrix for the estimates is not full rank.
Page 7 of 18
NOTE: The variance of some parameter estimates is zero or some parameter
estimates are linearly related to other parameter estimates as shown in
the following equations:
omega = -6407030 + 34497 * psi
_______________________________________________________________________________
Calculus 3: Deliberately cause trouble with identifiability 11
By adding measurement error to response variable
Non-ident but passes parameter count test with phi12=0
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Fit Summary
Modeling Info N Observations 287
N Variables 3
N Moments 6
N Parameters 6
N Active Constraints 0
Baseline Model Function Value 0.6496
Baseline Model Chi-Square 185.7968
Baseline Model Chi-Square DF 3
Pr > Baseline Model Chi-Square <.0001
Absolute Index Fit Function 0.1230
Chi-Square 35.1747
Chi-Square DF 0
Pr > Chi-Square .
Z-Test of Wilson & Hilferty .
Hoelter Critical N .
Root Mean Square Residual (RMSR) 13.4541
Standardized RMSR (SRMSR) 0.1637
Goodness of Fit Index (GFI) 0.9284
Parsimony Index Adjusted GFI (AGFI) .
Parsimonious GFI 0.0000
RMSEA Estimate .
Probability of Close Fit .
ECVI Estimate 0.0426
ECVI Lower 90% Confidence Limit .
ECVI Upper 90% Confidence Limit .
Akaike Information Criterion 47.1747
Bozdogan CAIC 75.1316
Schwarz Bayesian Criterion 69.1316
McDonald Centrality 0.9406
Incremental Index Bentler Comparative Fit Index 0.8076
Bentler-Bonett NFI 0.8107
Bentler-Bonett Non-normed Index .
Bollen Normed Index Rho1 .
Bollen Non-normed Index Delta2 0.8107
James et al. Parsimonious NFI 0.0000
Page 8 of 18
_______________________________________________________________________________
Calculus 3: Deliberately cause trouble with identifiability 12
By adding measurement error to response variable
Non-ident but passes parameter count test with phi12=0
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Linear Equations
Fgrade = 1.4680*hsgpa + 1.3496*test + 1.0000 epsilon
Std Err 0.1350 beta1 0.2258 beta2
t Value 10.8767 5.9771
grade = 1.0000 Fgrade + 1.0000 e
Estimates for Variances of Exogenous Variables
Variable Standard
Type Variable Parameter Estimate Error t Value
Observed hsgpa phi11 35.64841 2.98107 11.95826
test phi22 12.73851 1.06525 11.95826
Disturbance epsilon psi 185.72484 7.76596 23.91523
Error e omega 0.01000 7.76596 0.00129
Covariances Among Exogenous Variables
Standard
Var1 Var2 Estimate Error t Value
hsgpa test 0
Page 9 of 18
proc calis cov ; /* Analyze the covariance matrix (Default is corr) */
title3 'beta1 = beta2 = beta but phi12 ne 0';
var grade hsgpa test; /* Declare observed vars */
lineqs /* Simultaneous equations, separated by commas */
Fgrade = beta hsgpa + beta test + epsilon,
grade = Fgrade + e;
std /* Variances (not standard deviations) */
hsgpa = phi11,
test = phi22,
epsilon = psi,
e = omega;
/* No covariance between expl vars */
cov /* Covariances */
hsgpa test = phi12;
bounds 0.0 < phi11,
0.0 < phi22,
0.0 < psi,
0.0 < omega;
proc calis cov ; /* Analyze the covariance matrix (Default is corr) */
title3 'beta1 = beta2 = beta, phi12 ne 0, no bound on omega';
var grade hsgpa test; /* Declare observed vars */
lineqs /* Simultaneous equations, separated by commas */
Fgrade = beta hsgpa + beta test + epsilon,
grade = Fgrade + e;
std /* Variances (not standard deviations) */
hsgpa = phi11,
test = phi22,
epsilon = psi,
e = omega;
cov /* Covariances */
hsgpa test = phi12;
bounds 0.0 < phi11,
0.0 < phi22,
0.0 < psi;
Log file says
WARNING: There are 1 active boundary or linear inequality constraints at the
solution. The standard errors and chi-square test statistic assume
that the solution is located in the interior of the parameter space;
hence, they 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. The number of parameters used to calculate fit indices is
decreased by the number of active constraints. To turn off the
adjustment, use the NOADJDF option.
WARNING: The estimated variance of error variable e is zero or very close to
zero.
WARNING: Although all predicted variances for the observed and latent
variables are positive, the corresponding predicted covariance matrix
is not positive definite. It has one zero eigenvalue.
Page 10 of 18
Optimization Start
Active Constraints 0 Objective Function 0.0005028148
Max Abs Gradient Element 0.0052378258 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 4 1 0.0004825 0.000020 4.088E-7 111E-16 1.000
Optimization Results
Iterations 1 Function Calls 7
Jacobian Calls 3 Active Constraints 1
Objective Function 0.0004825473 Max Abs Gradient Element 4.08842E-7
Lambda 1.110223E-14 Actual Over Pred Change 0.9999215546
Radius 2
Convergence criterion (ABSGCONV=0.00001) satisfied.
Earlier results with beta1=beta2
Optimization Start
Active Constraints 0 Objective Function 0.0005028133
Max Abs Gradient Element 0.0052381077 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 4 0 0.0004825 0.000020 1.091E-7 0 1.000
Optimization Results
Iterations 1 Function Calls 7
Jacobian Calls 3 Active Constraints 0
Objective Function 0.0004825446 Max Abs Gradient Element 1.0907832E-7
Lambda 0 Actual Over Pred Change 1
Radius 0.0127338107
Page 11 of 18
Covariance Structure Analysis: Maximum Likelihood Estimation
Linear Equations
Fgrade = 1.4304*hsgpa + 1.4304*test + 1.0000 epsilon
Std Err 0.1016 beta 0.1016 beta
t Value 14.0720 14.0720
grade = 1.0000 Fgrade + 1.0000 e
Estimates for Variances of Exogenous Variables
Variable Standard
Type Variable Parameter Estimate Error t Value
Observed hsgpa phi11 35.64841 2.98107 11.95826
test phi22 12.73851 1.06525 11.95826
Disturbance epsilon psi 185.82860 15.53977 11.95826
Error e omega 0 0 .
Covariances Among Exogenous Variables
Standard
Var1 Var2 Parameter Estimate Error t Value
hsgpa test phi12 7.24928 1.33099 5.44653
Earlier results with beta1=beta2
Reduced model with beta1 = beta2 = beta
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Linear Equations
grade = 1.4304*hsgpa + 1.4304*test + 1.0000 epsilon
Std Err 0.1016 beta 0.1016 beta
t Value 14.0724 14.0724
Estimates for Variances of Exogenous Variables
Variable Standard
Type Variable Parameter Estimate Error t Value
Observed hsgpa phi11 35.64841 2.98107 11.95826
test phi22 12.73851 1.06525 11.95826
Error epsilon psi 185.81825 15.53890 11.95826
Covariances Among Exogenous Variables
Standard
Var1 Var2 Parameter Estimate Error t Value
hsgpa test phi12 7.24928 1.33099 5.44653
Page 12 of 18
With omega unconstrained, log file says
NOTE: The Moore-Penrose inverse is used in computing the covariance matrix for
parameter estimates.
WARNING: Standard errors and t values might not be accurate with the use of
the Moore-Penrose inverse.
WARNING: The estimated variance of error variable e is negative.
WARNING: Although all predicted variances for the observed and latent
variables are positive, the corresponding predicted covariance matrix
is not positive definite. It has one negative eigenvalue.
WARNING: Critical N is not computable for df= 0.
List file is very similar, except ...
Estimates for Variances of Exogenous Variables
Variable Standard
Type Variable Parameter Estimate Error t Value
Observed hsgpa phi11 35.64841 2.98107 11.95826
test phi22 12.73851 1.06525 11.95826
Disturbance epsilon psi 185.82860 7.76945 23.91785
Error e omega -0.01035 7.76945 -0.00133
Page 13 of 18
Stay in the parameter space with different starting values
From calculus2 with grade = beta hsgpa + beta test + epsilon, had
Linear Equations
grade = 1.4304*hsgpa + 1.4304*test + 1.0000 epsilon
Std Err 0.1016 beta 0.1016 beta
t Value 14.0724 14.0724
Estimates for Variances of Exogenous Variables
Variable Standard
Type Variable Parameter Estimate Error t Value
Observed hsgpa phi11 35.64841 2.98107 11.95826
test phi22 12.73851 1.06525 11.95826
Error epsilon psi 185.81825 15.53890 11.95826
Covariances Among Exogenous Variables
Standard
Var1 Var2 Parameter Estimate Error t Value
hsgpa test phi12 7.24928 1.33099 5.44653
/* calculus3b.sas */
options linesize=79 pagesize=500 noovp formdlim='_' ;
title 'Calculus 3b: Start at another point on the maximum surface';
title2 'Deeper in parameter space';
data math;
infile 'calculus.data' firstobs=2;
input id hscalc hsengl hsgpa test grade;
/* Exclude less output this time */
ods exclude
Calis.ML.SqMultCorr (persist)
Calis.StandardizedResults.LINEQSEqStd (persist)
Calis.StandardizedResults.LINEQSVarExogStd (persist)
Calis.StandardizedResults.LINEQSCovExogStd (persist)
;
/* Specify starting values to be final answer from calculus2, except
split the variance of 185.81825 between psi and omega. */
proc calis cov ; /* Analyze the covariance matrix (Default is corr) */
title3 'beta1 = beta2 = beta, no bound on variances';
var grade hsgpa test; /* Declare observed vars */
lineqs /* Simultaneous equations, separated by commas */
Fgrade = beta (1.4304) hsgpa + beta (1.4304) test + epsilon,
grade = Fgrade + e;
std /* Variances (not standard deviations) */
hsgpa = phi11 (35.64841),
test = phi22 (12.73851),
epsilon = psi (100),
e = omega (85.81825);
cov /* Covariances */
hsgpa test = phi12 (7.24928);
Page 14 of 18
_______________________________________________________________________________
Calculus 3b: Start at another point on the maximum surface 1
Deeper in parameter space
beta1 = beta2 = beta, no bound on variances
The CALIS Procedure
Covariance Structure Analysis: Model and Initial Values
Modeling Information
Data Set WORK.MATH
N Records Read 287
N Records Used 287
N Obs 287
Model Type LINEQS
Analysis Covariances
Variables in the Model
Endogenous Manifest grade
Latent Fgrade
Exogenous Manifest hsgpa test
Latent
Error e epsilon
Number of Endogenous Variables = 2
Number of Exogenous Variables = 4
Initial Estimates for Linear Equations
Fgrade = 1.4304*hsgpa + 1.4304*test + 1.0000 epsilon
beta beta
grade = 1.0000 Fgrade + 1.0000 e
Initial Estimates for Variances of Exogenous Variables
Variable
Type Variable Parameter Estimate
Observed hsgpa phi11 35.64841
test phi22 12.73851
Disturbance epsilon psi 100.00000
Error e omega 85.81825
Initial Estimates for Covariances Among Exogenous Variables
Var1 Var2 Parameter Estimate
hsgpa test phi12 7.24928
_______________________________________________________________________________
Page 15 of 18
Calculus 3b: Start at another point on the maximum surface 2
Deeper in parameter space
beta1 = beta2 = beta, no bound on variances
The CALIS Procedure
Covariance Structure Analysis: Descriptive Statistics
Simple Statistics
Variable Mean Std Dev
grade 60.98955 17.73355
hsgpa 80.98293 5.97063
test 8.81533 3.56910
_______________________________________________________________________________
Calculus 3b: Start at another point on the maximum surface 3
Deeper in parameter space
beta1 = beta2 = beta, no bound on variances
The CALIS Procedure
Covariance Structure Analysis: Optimization
Initial Estimation Methods
1 User Specifications
2 Observed Moments of Variables
Optimization Start
Parameter Estimates
N Parameter Estimate Gradient
1 beta 1.43040 7.73673E-6
2 phi11 35.64841 8.2795E-19
3 phi22 12.73851 2.9964E-18
4 psi 100.00000 1.0903E-7
5 omega 85.81825 1.0903E-7
6 phi12 7.24928 -5.421E-18
Value of Objective Function = 0.0004825446
_______________________________________________________________________________
Calculus 3b: Start at another point on the maximum surface 4
Deeper in parameter space
beta1 = beta2 = beta, no bound on variances
The CALIS Procedure
Covariance Structure Analysis: Optimization
Levenberg-Marquardt Optimization
Scaling Update of More (1978)
Parameter Estimates 6
Functions (Observations) 6
Optimization Start
Active Constraints 0 Objective Function 0.0004825446
Max Abs Gradient Element 7.7367309E-6 Radius 1
Page 16 of 18
Optimization Results
Iterations 0 Function Calls 4
Jacobian Calls 1 Active Constraints 0
Objective Function 0.0004825446 Max Abs Gradient Element 7.7367309E-6
Lambda 0 Actual Over Pred Change 0
Radius 1
Convergence criterion (ABSGCONV=0.00001) satisfied.
NOTE: The Moore-Penrose inverse is used in computing the covariance matrix for
parameter estimates.
WARNING: Standard errors and t values might not be accurate with the use of
the Moore-Penrose inverse.
NOTE: Covariance matrix for the estimates is not full rank.
NOTE: The variance of some parameter estimates is zero or some parameter
estimates are linearly related to other parameter estimates as shown in
the following equations:
omega = -3452756 + 34528 * psi
_______________________________________________________________________________
Calculus 3b: Start at another point on the maximum surface 5
Deeper in parameter space
beta1 = beta2 = beta, no bound on variances
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Fit Summary
Modeling Info N Observations 287
N Variables 3
N Moments 6
N Parameters 6
N Active Constraints 0
Baseline Model Function Value 0.6496
Baseline Model Chi-Square 185.7968
Baseline Model Chi-Square DF 3
Pr > Baseline Model Chi-Square <.0001
Absolute Index Fit Function 0.0005
Chi-Square 0.1380
Chi-Square DF 0
Pr > Chi-Square .
Z-Test of Wilson & Hilferty .
Hoelter Critical N .
Root Mean Square Residual (RMSR) 0.4367
Standardized RMSR (SRMSR) 0.0057
Goodness of Fit Index (GFI) 0.9997
Parsimony Index Adjusted GFI (AGFI) .
Parsimonious GFI 0.0000
RMSEA Estimate .
Probability of Close Fit .
Page 17 of 18
_______________________________________________________________________________
Calculus 3b: Start at another point on the maximum surface 6
Deeper in parameter space
beta1 = beta2 = beta, no bound on variances
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Linear Equations
Fgrade = 1.4304*hsgpa + 1.4304*test + 1.0000 epsilon
Std Err 0.1016 beta 0.1016 beta
t Value 14.0725 14.0725
grade = 1.0000 Fgrade + 1.0000 e
Estimates for Variances of Exogenous Variables
Variable Standard
Type Variable Parameter Estimate Error t Value
Observed hsgpa phi11 35.64841 2.98107 11.95826
test phi22 12.73851 1.06525 11.95826
Disturbance epsilon psi 100.00000 7.76945 12.87092
Error e omega 85.81825 7.76945 11.04560
Covariances Among Exogenous Variables
Standard
Var1 Var2 Parameter Estimate Error t Value
hsgpa test phi12 7.24928 1.33099 5.44653
Again, compare
Linear Equations
grade = 1.4304*hsgpa + 1.4304*test + 1.0000 epsilon
Std Err 0.1016 beta 0.1016 beta
t Value 14.0724 14.0724
Estimates for Variances of Exogenous Variables
Variable Standard
Type Variable Parameter Estimate Error t Value
Observed hsgpa phi11 35.64841 2.98107 11.95826
test phi22 12.73851 1.06525 11.95826
Error epsilon psi 185.81825 15.53890 11.95826
Covariances Among Exogenous Variables
Standard
Var1 Var2 Parameter Estimate Error t Value
hsgpa test phi12 7.24928 1.33099 5.44653
Page 18 of 18
