STA429/1007 F 2004 Handout 15
Unidentified Models with SAS
/* path2a.sas */
options linesize=79 noovp formdlim='_';
title 'STA429f04 Path 2a: Non-identified Simple Regression with Meas Error';
title2 'Just try to fit the model';
data path1;
infile 'path2.dat';
input x1 x2 y;
proc calis cov pshort; /* Analyze the covariance matrix (Default is corr) */
var x1 y; /* Manifest vars are in the data set */
lineqs /* Simultaneous equations, separated by commas */
y = b F + e1,
x1 = F + e2;
std /* Variances (not standard deviations) */
F = sigsqF, /* Optional starting values in parentheses */
e1 = sigsqe1,
e2 = sigsqe2;
bounds 0.0 < sigsqF,
0.0 < sigsqe1,
0.0 < sigsqe2;
/* path2b.sas */
options linesize=79 noovp formdlim='_';
title 'STA429f04 Path 2b: Identified Simple Regression with Meas Error';
title2 'Test H0: b=0';
data path1;
infile 'path2.dat';
input x1 x2 y;
proc calis cov; /* Analyze the covariance matrix (Default is corr) */
title3 'Full model';
var x1 x2 y; /* Manifest vars are in the data set */
lineqs /* Simultaneous equations, separated by commas */
y = b F + e1,
x1 = F + e2,
x2 = F + e3;
std /* Variances (not standard deviations) */
F = sigsqF, /* Optional starting values in parentheses */
e1 = sigsqe1,
e2 = sigsqe,
e3 = sigsqe;
bounds 0.0 < sigsqF,
0.0 < sigsqe,
0.0 < sigsqe1;
proc calis cov; /* Analyze the covariance matrix (Default is corr) */
title3 'Reduced model';
var x1 x2 y; /* Manifest vars are in the data set */
lineqs /* Simultaneous equations, separated by commas */
y = b F + e1,
x1 = F + e2,
x2 = F + e3;
std /* Variances (not standard deviations) */
F = sigsqF, /* Optional starting values in parentheses */
e1 = sigsqe1,
e2 = sigsqe,
e3 = sigsqe;
bounds 0.0 < sigsqF,
0.0 < sigsqe,
0.0 < sigsqe1;
lincon b=0;
_______________________________________________________________________________
STA429f04 Path 2a: Non-identified Simple Regression with Meas Error 1
Just try to fit the model
00:07 Thursday, December 2, 2004
The CALIS Procedure
Covariance Structure Analysis: Pattern and Initial Values
LINEQS Model Statement
Matrix Rows Columns ------Matrix Type-------
Term 1 1 _SEL_ 2 5 SELECTION
2 _BETA_ 5 5 EQSBETA IMINUSINV
3 _GAMMA_ 5 3 EQSGAMMA
4 _PHI_ 3 3 SYMMETRIC
The 2 Endogenous Variables
Manifest x1 y
Latent
The 3 Exogenous Variables
Manifest
Latent F
Error e1 e2
_______________________________________________________________________________
STA429f04 Path 2a: Non-identified Simple Regression with Meas Error 2
Just try to fit the model
00:07 Thursday, December 2, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Levenberg-Marquardt Optimization
Scaling Update of More (1978)
Parameter Estimates 4
Functions (Observations) 3
Lower Bounds 3
Upper Bounds 0
Optimization Start
Active Constraints 0 Objective Function 0.0765191703
Max Abs Gradient Element 12.435221312 Radius 452.22765911
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 4.49913E-6 0.0765 0.1639 0 1.309
2* 0 3 0 1.5469E-13 4.499E-6 0.000031 111E-16 1.002
3 0 4 0 0 1.55E-13 1.28E-10 0 1.000
Optimization Results
Iterations 3 Function Calls 5
Jacobian Calls 4 Active Constraints 0
Objective Function 0 Max Abs Gradient Element 1.280183E-10
Lambda 0 Actual Over Pred Change 0.9998970626
Radius 0.0002251807
GCONV2 convergence criterion satisfied.
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:
sigsqF = -283260 + 21760 * b -
170.062427 * sigsqe1 + 1.000000
* sigsqe2
_______________________________________________________________________________
STA429f04 Path 2a: Non-identified Simple Regression with Meas Error 3
Just try to fit the model
00:07 Thursday, December 2, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Fit Function 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) -1.0000
Chi-Square 0.0000
Chi-Square DF -1
Pr > Chi-Square .
Independence Model Chi-Square 0.0013
Independence Model Chi-Square DF 1
RMSEA Estimate 0.0000
RMSEA 90% Lower Confidence Limit .
RMSEA 90% Upper Confidence Limit .
ECVI Estimate 0.0000
ECVI 90% Lower Confidence Limit .
ECVI 90% Upper Confidence Limit .
Probability of Close Fit .
Bentler's Comparative Fit Index .
Normal Theory Reweighted LS Chi-Square 0.0000
Akaike's Information Criterion 2.0000
Bozdogan's (1987) CAIC 6.2983
Schwarz's Bayesian Criterion 5.2983
McDonald's (1989) Centrality 0.9975
Bentler & Bonett's (1980) Non-normed Index .
Bentler & Bonett's (1980) NFI 1.0000
James, Mulaik, & Brett (1982) Parsimonious NFI -1.0000
Z-Test of Wilson & Hilferty (1931) .
Bollen (1986) Normed Index Rho1 .
Bollen (1988) Non-normed Index Delta2 0.0013
Hoelter's (1983) Critical N .
_______________________________________________________________________________
STA429f04 Path 2a: Non-identified Simple Regression with Meas Error 4
Just try to fit the model
00:07 Thursday, December 2, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Manifest Variable Equations with Estimates
x1 = 1.0000 F + 1.0000 e2
y = 13.0408*F + 1.0000 e1
b
Variances of Exogenous Variables
Variable Parameter Estimate
F sigsqF 0.0005993
e1 sigsqe1 3.01230
e2 sigsqe2 2.97844
_______________________________________________________________________________
STA429f04 Path 2a: Non-identified Simple Regression with Meas Error 5
Just try to fit the model
00:07 Thursday, December 2, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Manifest Variable Equations with Standardized Estimates
x1 = 0.0142 F + 0.9999 e2
y = 0.1809*F + 0.9835 e1
b
Squared Multiple Correlations
Error Total
Variable Variance Variance R-Square
1 x1 2.97844 2.97904 0.000201
2 y 3.01230 3.11421 0.0327
/* factor1.sas*/
options linesize=79 noovp formdlim='_';
title 'Three-variables and one factor using LINEQS';
data chain3;
infile 'chain.dat';
input y1 y2 y3;
proc calis cov pshort;
title2 'Unidentified model';
var y1 y2 y3; /* Manifest vars are in the data set */
lineqs
y1 = L1 F + e1, /* L stands for Loading */
y2 = L2 F + e2,
y3 = L3 F + e3;
std /* Variances not standard deviation */
e1 = sige1,
e2 = sige2,
e3 = sige3,
F = 1;
bounds
-1.0 <= L1 <= 1.0, /* Because the loadings are correlations */
-1.0 <= L2 <= 1.0,
-1.0 <= L3 <= 1.0;
/* Now define functions of the parameters, functions that I am going to
constrain in the nlincon statement. First name them and give (initial)
numerical values. */
parameters vy1 vy2 vy3 (1.);
vy1 = sige1 + L1**2;
vy2 = sige2 + L2**2;
vy3 = sige3 + L3**2;
nlincon vy1 = 1, vy2 = 1 , vy3 = 1;
proc calis cov pshort;
title2 'Identified model 1';
var y1 y2 y3; /* Manifest vars are in the data set */
lineqs
y1 = L1 F + e1, /* L stands for Loading */
y2 = L2 F + e2,
y3 = L3 F + e3;
std /* Variances not standard deviation */
e1 = sige1,
e2 = sige2,
e3 = sige3,
F = 1;
bounds
0.0 <= L1 <= 1.0, /* Make L1 Positive */
-1.0 <= L2 <= 1.0,
-1.0 <= L3 <= 1.0;
/* Now define functions of the parametes, functions that I am going to
constrain in the nlincon statement. First name them and give (initial)
numerical values. */
parameters vy1 vy2 vy3 (1.);
vy1 = sige1 + L1**2;
vy2 = sige2 + L2**2;
vy3 = sige3 + L3**2;
nlincon vy1 = 1, vy2 = 1 , vy3 = 1;
proc calis cov pshort;
title2 'Identified model 2';
var y1 y2 y3; /* Manifest vars are in the data set */
lineqs
y1 = L1 F + e1, /* L stands for Loading */
y2 = L2 F + e2,
y3 = L3 F + e3;
std /* Variances not standard deviation */
e1 = sige1,
e2 = sige2,
e3 = sige3,
F = 1;
bounds
-1.0 <= L1 <= 1.0,
-1.0 <= L2 <= 1.0,
0.0 <= L3 <= 1.0; /* Make L3 Positive */
/* Now define functions of the parametes, functions that I am going to
constrain in the nlincon statement. First name them and give (initial)
numerical values. */
parameters vy1 vy2 vy3 (1.);
vy1 = sige1 + L1**2;
vy2 = sige2 + L2**2;
vy3 = sige3 + L3**2;
nlincon vy1 = 1, vy2 = 1 , vy3 = 1;
_______________________________________________________________________________
Three-variables and one factor using LINEQS 1
Unidentified model
22:55 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 7 SELECTION
2 _BETA_ 7 7 EQSBETA IMINUSINV
3 _GAMMA_ 7 4 EQSGAMMA
4 _PHI_ 4 4 SYMMETRIC
The 3 Endogenous Variables
Manifest y1 y2 y3
Latent
The 4 Exogenous Variables
Manifest
Latent F
Error e1 e2 e3
_______________________________________________________________________________
Three-variables and one factor using LINEQS 2
Unidentified model
22:55 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-variables and one factor using LINEQS 3
Unidentified model
22:55 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Dual Quasi-Newton Optimization
Modified VMCWD Algorithm of Powell (1978, 1982)
Dual Broyden - Fletcher - Goldfarb - Shanno Update (DBFGS)
Lagrange Multiplier Update of Powell(1982)
Parameter Estimates 6
Functions (Observations) 6
Lower Bounds 3
Upper Bounds 3
Nonlinear Constraints 3
Nonlinear Equality Constraints 3
Optimization Start
Objective Function 0.9685074192 Maximum Constraint 0.0001388008
Violation
Maximum Gradient of the 0.7932665457
Lagran Func
Maximum
Gradient
Element
Maximum Predicted of the
Function Objective Constraint Function Step Lagrange
Iter Restarts Calls Function Violation Reduction Size Function
1 0 15 0.47335 0.6293 0.4388 1.000 0.974
2 0 16 0.59895 0.1015 0.1114 1.000 0.259
3 0 17 0.56294 0.0149 0.0730 1.000 0.216
4 0 19 0.54488 0.0107 0.0319 0.479 0.0374
5 0 20 0.56087 0.000218 0.000237 1.000 0.0170
6 0 21 0.56071 0.000036 0.00126 1.000 0.0154
7* 0 22 0.55802 0.00121 0.00192 1.000 0.0318
8 0 24 0.55762 0.00114 0.00285 0.306 0.0361
9* 0 25 0.54663 0.00744 0.0212 1.000 0.0597
10 0 26 0.55637 0.000432 0.00193 1.000 0.0260
11 0 27 0.55373 0.00212 0.00224 1.000 0.0108
12* 0 28 0.54473 0.00911 0.0184 1.000 0.0359
13 0 29 0.55374 0.000046 0.00104 1.000 0.0167
14* 0 30 0.54887 0.00482 0.00810 1.000 0.0153
15* 0 31 0.55153 0.00134 0.00237 1.000 0.0252
16 0 32 0.55271 0.000019 0.000038 1.000 0.00244
17 0 33 0.55272 4.118E-7 3.859E-7 1.000 0.00012
Optimization Results
Iterations 17 Function Calls 35
Gradient Calls 20 Active Constraints 4
Objective Function 0.5527247315 Maximum Constraint 4.1183614E-7
Violation
_______________________________________________________________________________
Three-variables and one factor using LINEQS 4
Unidentified model
22:55 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Optimization Results
Maximum Projected 0.0001202793 Value Lagrange Function 0.5527249264
Gradient
Maximum Gradient of the 0.0001179179 Slope of Search Direction -3.859116E-7
Lagran Func
FCONV2 convergence criterion satisfied.
WARNING: The point x is feasible only at the LCEPSILON= 1E-6 range.
WARNING: There are 4 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-variables and one factor using LINEQS 5
Unidentified model
22:55 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Fit Function 0.5527
Goodness of Fit Index (GFI) 0.7842
GFI Adjusted for Degrees of Freedom (AGFI) 0.6763
Root Mean Square Residual (RMR) 0.6535
Parsimonious GFI (Mulaik, 1989) 1.0456
Chi-Square 109.9922
Chi-Square DF 4
Pr > Chi-Square <.0001
Independence Model Chi-Square 65.979
Independence Model Chi-Square DF 3
RMSEA Estimate 0.3649
RMSEA 90% Lower Confidence Limit 0.3079
RMSEA 90% Upper Confidence Limit 0.4253
ECVI Estimate 0.5732
ECVI 90% Lower Confidence Limit 0.4184
ECVI 90% Upper Confidence Limit 0.7662
Probability of Close Fit 0.0000
Bentler's Comparative Fit Index -0.6830
Normal Theory Reweighted LS Chi-Square 186.5728
Akaike's Information Criterion 101.9922
Bozdogan's (1987) CAIC 84.7990
Schwarz's Bayesian Criterion 88.7990
McDonald's (1989) Centrality 0.7672
Bentler & Bonett's (1980) Non-normed Index -0.2622
Bentler & Bonett's (1980) NFI -0.6671
James, Mulaik, & Brett (1982) Parsimonious NFI -0.8895
Z-Test of Wilson & Hilferty (1931) 8.7988
Bollen (1986) Normed Index Rho1 -0.2503
Bollen (1988) Non-normed Index Delta2 -0.7101
Hoelter's (1983) Critical N 19
WARNING: The central parameter matrix _PHI_ has probably 1 zero eigenvalue(s).
_______________________________________________________________________________
Three-variables and one factor using LINEQS 6
Unidentified model
22:55 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Manifest Variable Equations with Estimates
y1 = 0.1006*F + 1.0000 e1
L1
y2 = -0.3740*F + 1.0000 e2
L2
y3 = -1.0000*F + 1.0000 e3
L3
Variances of Exogenous Variables
Variable Parameter Estimate
F 1.00000
e1 sige1 0.98989
e2 sige2 0.86015
e3 sige3 -4.215E-11
_______________________________________________________________________________
Three-variables and one factor using LINEQS 7
Unidentified model
22:55 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Manifest Variable Equations with Standardized Estimates
y1 = 0.1006*F + 0.9949 e1
L1
y2 = -0.3740*F + 0.9274 e2
L2
y3 = -1.0000*F + 1.0000 e3
L3
Squared Multiple Correlations
Error Total
Variable Variance Variance R-Square
1 y1 0.98989 1.00000 0.0101
2 y2 0.86015 1.00000 0.1398
3 y3 -4.215E-11 1.00000 1.0000
_______________________________________________________________________________
Three-variables and one factor using LINEQS 8
Identified model 1
22:55 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 7 SELECTION
2 _BETA_ 7 7 EQSBETA IMINUSINV
3 _GAMMA_ 7 4 EQSGAMMA
4 _PHI_ 4 4 SYMMETRIC
The 3 Endogenous Variables
Manifest y1 y2 y3
Latent
The 4 Exogenous Variables
Manifest
Latent F
Error e1 e2 e3
_______________________________________________________________________________
Three-variables and one factor using LINEQS 9
Identified model 1
22:55 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-variables and one factor using LINEQS 10
Identified model 1
22:55 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Dual Quasi-Newton Optimization
Modified VMCWD Algorithm of Powell (1978, 1982)
Dual Broyden - Fletcher - Goldfarb - Shanno Update (DBFGS)
Lagrange Multiplier Update of Powell(1982)
Parameter Estimates 6
Functions (Observations) 6
Lower Bounds 3
Upper Bounds 3
Nonlinear Constraints 3
Nonlinear Equality Constraints 3
Optimization Start
Objective Function 0.9685074192 Maximum Constraint 0.0001388008
Violation
Maximum Gradient of the 0.7932665457
Lagran Func
Maximum
Gradient
Element
Maximum Predicted of the
Function Objective Constraint Function Step Lagrange
Iter Restarts Calls Function Violation Reduction Size Function
1 0 15 0.47335 0.6293 0.4388 1.000 0.974
2 0 16 0.59895 0.1015 0.1114 1.000 0.259
3 0 17 0.56294 0.0149 0.0730 1.000 0.216
4 0 19 0.54488 0.0107 0.0319 0.479 0.0374
5 0 20 0.56087 0.000218 0.000237 1.000 0.0170
6 0 21 0.56071 0.000036 0.00126 1.000 0.0154
7* 0 22 0.55802 0.00121 0.00192 1.000 0.0318
8 0 24 0.55762 0.00114 0.00285 0.306 0.0361
9* 0 25 0.54663 0.00744 0.0212 1.000 0.0597
10 0 26 0.55637 0.000432 0.00193 1.000 0.0260
11 0 27 0.55373 0.00212 0.00224 1.000 0.0108
12* 0 28 0.54473 0.00911 0.0184 1.000 0.0359
13 0 29 0.55374 0.000046 0.00104 1.000 0.0167
14* 0 30 0.54887 0.00482 0.00810 1.000 0.0153
15* 0 31 0.55153 0.00134 0.00237 1.000 0.0252
16 0 32 0.55271 0.000019 0.000038 1.000 0.00244
17 0 33 0.55272 4.118E-7 3.859E-7 1.000 0.00012
Optimization Results
Iterations 17 Function Calls 35
Gradient Calls 20 Active Constraints 4
Objective Function 0.5527247315 Maximum Constraint 4.1183614E-7
Violation
_______________________________________________________________________________
Three-variables and one factor using LINEQS 11
Identified model 1
22:55 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Optimization Results
Maximum Projected 0.0001202793 Value Lagrange Function 0.5527249264
Gradient
Maximum Gradient of the 0.0001179179 Slope of Search Direction -3.859116E-7
Lagran Func
FCONV2 convergence criterion satisfied.
WARNING: The point x is feasible only at the LCEPSILON= 1E-6 range.
WARNING: There are 4 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-variables and one factor using LINEQS 12
Identified model 1
22:55 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Fit Function 0.5527
Goodness of Fit Index (GFI) 0.7842
GFI Adjusted for Degrees of Freedom (AGFI) 0.6763
Root Mean Square Residual (RMR) 0.6535
Parsimonious GFI (Mulaik, 1989) 1.0456
Chi-Square 109.9922
Chi-Square DF 4
Pr > Chi-Square <.0001
Independence Model Chi-Square 65.979
Independence Model Chi-Square DF 3
RMSEA Estimate 0.3649
RMSEA 90% Lower Confidence Limit 0.3079
RMSEA 90% Upper Confidence Limit 0.4253
ECVI Estimate 0.5732
ECVI 90% Lower Confidence Limit 0.4184
ECVI 90% Upper Confidence Limit 0.7662
Probability of Close Fit 0.0000
Bentler's Comparative Fit Index -0.6830
Normal Theory Reweighted LS Chi-Square 186.5728
Akaike's Information Criterion 101.9922
Bozdogan's (1987) CAIC 84.7990
Schwarz's Bayesian Criterion 88.7990
McDonald's (1989) Centrality 0.7672
Bentler & Bonett's (1980) Non-normed Index -0.2622
Bentler & Bonett's (1980) NFI -0.6671
James, Mulaik, & Brett (1982) Parsimonious NFI -0.8895
Z-Test of Wilson & Hilferty (1931) 8.7988
Bollen (1986) Normed Index Rho1 -0.2503
Bollen (1988) Non-normed Index Delta2 -0.7101
Hoelter's (1983) Critical N 19
WARNING: The central parameter matrix _PHI_ has probably 1 zero eigenvalue(s).
_______________________________________________________________________________
Three-variables and one factor using LINEQS 13
Identified model 1
22:55 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Manifest Variable Equations with Estimates
y1 = 0.1006*F + 1.0000 e1
L1
y2 = -0.3740*F + 1.0000 e2
L2
y3 = -1.0000*F + 1.0000 e3
L3
Variances of Exogenous Variables
Variable Parameter Estimate
F 1.00000
e1 sige1 0.98989
e2 sige2 0.86015
e3 sige3 -4.215E-11
_______________________________________________________________________________
Three-variables and one factor using LINEQS 14
Identified model 1
22:55 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Manifest Variable Equations with Standardized Estimates
y1 = 0.1006*F + 0.9949 e1
L1
y2 = -0.3740*F + 0.9274 e2
L2
y3 = -1.0000*F + 1.0000 e3
L3
Squared Multiple Correlations
Error Total
Variable Variance Variance R-Square
1 y1 0.98989 1.00000 0.0101
2 y2 0.86015 1.00000 0.1398
3 y3 -4.215E-11 1.00000 1.0000
_______________________________________________________________________________
Three-variables and one factor using LINEQS 15
Identified model 2
22:55 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 7 SELECTION
2 _BETA_ 7 7 EQSBETA IMINUSINV
3 _GAMMA_ 7 4 EQSGAMMA
4 _PHI_ 4 4 SYMMETRIC
The 3 Endogenous Variables
Manifest y1 y2 y3
Latent
The 4 Exogenous Variables
Manifest
Latent F
Error e1 e2 e3
_______________________________________________________________________________
Three-variables and one factor using LINEQS 16
Identified model 2
22:55 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-variables and one factor using LINEQS 17
Identified model 2
22:55 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Dual Quasi-Newton Optimization
Modified VMCWD Algorithm of Powell (1978, 1982)
Dual Broyden - Fletcher - Goldfarb - Shanno Update (DBFGS)
Lagrange Multiplier Update of Powell(1982)
Parameter Estimates 6
Functions (Observations) 6
Lower Bounds 3
Upper Bounds 3
Nonlinear Constraints 3
Nonlinear Equality Constraints 3
Optimization Start
Objective Function 1.5265896183 Maximum Constraint 0.0000825999
Violation
Maximum Gradient of the 2.5393371004
Lagran Func
Maximum
Gradient
Element
Maximum Predicted of the
Function Objective Constraint Function Step Lagrange
Iter Restarts Calls Function Violation Reduction Size Function
1 0 18 0.29691 0.5188 1.0001 0.589 1.456
2 0 19 0.58904 0.0472 0.0692 1.000 0.579
3 0 20 0.57891 0.00570 0.0749 1.000 0.391
4 0 21 0.53572 0.0204 0.0377 1.000 0.161
5 0 22 0.55329 0.00168 0.00360 1.000 0.0143
6 0 23 0.55507 9.889E-6 0.000030 1.000 0.00784
7' 0 24 0.55506 2.856E-6 0.000498 1.000 0.00725
8* 0 25 0.55406 0.000736 0.00103 1.000 0.0185
9 0 27 0.55380 0.000777 0.00309 0.207 0.0213
10 0 29 0.55343 0.000916 0.00337 0.100 0.0227
11 0 31 0.55312 0.000999 0.00309 0.100 0.0216
12 0 33 0.55288 0.00104 0.00253 0.100 0.0193
13 0 35 0.55267 0.00107 0.00199 0.111 0.0162
14 0 37 0.55248 0.00108 0.00152 0.127 0.0123
15 0 39 0.55233 0.00107 0.00152 0.150 0.00791
16* 0 40 0.54817 0.00500 0.00937 1.000 0.0387
17 0 41 0.55276 0.000080 0.000149 1.000 0.0216
18 0 42 0.55268 0.000080 0.000089 1.000 0.00785
19 0 43 0.55272 1.923E-6 4.318E-6 1.000 0.00050
20 0 44 0.55272 1.375E-8 8.01E-9 1.000 0.00004
_______________________________________________________________________________
Three-variables and one factor using LINEQS 18
Identified model 2
22:55 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Optimization Results
Iterations 20 Function Calls 46
Gradient Calls 23 Active Constraints 4
Objective Function 0.5527249205 Maximum Constraint 1.3749371E-8
Violation
Maximum Projected 0.0000425824 Value Lagrange Function 0.5527249249
Gradient
Maximum Gradient of the 0.000041746 Slope of Search Direction -8.009844E-9
Lagran Func
FCONV2 convergence criterion satisfied.
WARNING: The point x is feasible only at the LCEPSILON= 1E-7 range.
WARNING: There are 4 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-variables and one factor using LINEQS 19
Identified model 2
22:55 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Fit Function 0.5527
Goodness of Fit Index (GFI) 0.7842
GFI Adjusted for Degrees of Freedom (AGFI) 0.6763
Root Mean Square Residual (RMR) 0.6535
Parsimonious GFI (Mulaik, 1989) 1.0456
Chi-Square 109.9923
Chi-Square DF 4
Pr > Chi-Square <.0001
Independence Model Chi-Square 65.979
Independence Model Chi-Square DF 3
RMSEA Estimate 0.3649
RMSEA 90% Lower Confidence Limit 0.3079
RMSEA 90% Upper Confidence Limit 0.4253
ECVI Estimate 0.5732
ECVI 90% Lower Confidence Limit 0.4184
ECVI 90% Upper Confidence Limit 0.7662
Probability of Close Fit 0.0000
Bentler's Comparative Fit Index -0.6830
Normal Theory Reweighted LS Chi-Square 186.5722
Akaike's Information Criterion 101.9923
Bozdogan's (1987) CAIC 84.7990
Schwarz's Bayesian Criterion 88.7990
McDonald's (1989) Centrality 0.7672
Bentler & Bonett's (1980) Non-normed Index -0.2622
Bentler & Bonett's (1980) NFI -0.6671
James, Mulaik, & Brett (1982) Parsimonious NFI -0.8895
Z-Test of Wilson & Hilferty (1931) 8.7988
Bollen (1986) Normed Index Rho1 -0.2503
Bollen (1988) Non-normed Index Delta2 -0.7101
Hoelter's (1983) Critical N 19
WARNING: The central parameter matrix _PHI_ has probably 1 zero eigenvalue(s).
_______________________________________________________________________________
Three-variables and one factor using LINEQS 20
Identified model 2
22:55 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Manifest Variable Equations with Estimates
y1 = -0.1006*F + 1.0000 e1
L1
y2 = 0.3740*F + 1.0000 e2
L2
y3 = 1.0000*F + 1.0000 e3
L3
Variances of Exogenous Variables
Variable Parameter Estimate
F 1.00000
e1 sige1 0.98988
e2 sige2 0.86015
e3 sige3 -4.215E-11
_______________________________________________________________________________
Three-variables and one factor using LINEQS 21
Identified model 2
22:55 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Manifest Variable Equations with Standardized Estimates
y1 = -0.1006*F + 0.9949 e1
L1
y2 = 0.3740*F + 0.9274 e2
L2
y3 = 1.0000*F + 1.0000 e3
L3
Squared Multiple Correlations
Error Total
Variable Variance Variance R-Square
1 y1 0.98988 1.00000 0.0101
2 y2 0.86015 1.00000 0.1399
3 y3 -4.215E-11 1.00000 1.0000
_______________________________________________________________________________
Three-variables and one factor using LINEQS 22
Try it with Factor
22:55 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Pattern and Initial Values
FACTOR Model Statement
Matrix Rows Columns ------Matrix Type-------
Term 1 1 _F_ 3 1 GENERAL
2 _P_ 1 1 SYMMETRIC
Term 2 3 _U_ 3 3 SYMMETRIC
_______________________________________________________________________________
Three-variables and one factor using LINEQS 23
Try it with Factor
22:55 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Levenberg-Marquardt Optimization
Scaling Update of More (1978)
Parameter Estimates 6
Functions (Observations) 6
Optimization Start
Active Constraints 0 Objective Function 0.0496168138
Max Abs Gradient Element 0.2356045213 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 0.03439 0.0152 0.2565 0 0.316
2 0 3 0 0.00263 0.0318 0.1734 0 1.104
3 0 4 0 4.97854E-7 0.00263 0.00268 0 1.050
4 0 5 0 0 4.979E-7 2.071E-8 0 1.001
Optimization Results
Iterations 4 Function Calls 6
Jacobian Calls 5 Active Constraints 0
Objective Function 0 Max Abs Gradient Element 2.0706604E-8
Lambda 0 Actual Over Pred Change 1.0006655111
Radius 0.0023771506
GCONV2 convergence criterion satisfied.
_______________________________________________________________________________
Three-variables and one factor using LINEQS 24
Try it with Factor
22:55 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Fit Function 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
Chi-Square DF 0
Pr > Chi-Square <.0001
Independence Model Chi-Square 65.979
Independence Model Chi-Square DF 3
RMSEA Estimate 0.0000
RMSEA 90% Lower Confidence Limit .
RMSEA 90% Upper Confidence Limit .
ECVI Estimate 0.0615
ECVI 90% Lower Confidence Limit .
ECVI 90% Upper Confidence Limit .
Probability of Close Fit .
Bentler's Comparative Fit Index 1.0000
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 .
WARNING: The central parameter matrix _U_ has probably 1 negative
eigenvalue(s).
Estimated Parameter Matrix _P_[1:1]
Identity Matrix
Constant Model Matrix
_______________________________________________________________________________
Three-variables and one factor using LINEQS 25
Try it with Factor
22:55 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Estimated Parameter Matrix _F_[3:1]
Lower Triangular Matrix
FACT1
y1 -0.1096
[_F1]
y2 0.4147
[_F2]
y3 1.2479
[_F3]
_______________________________________________________________________________
Three-variables and one factor using LINEQS 26
Try it with Factor
22:55 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Estimated Parameter Matrix _U_[3:3]
Diagonal Matrix
UVAR1 UVAR2 UVAR3
y1 0.9880 0 0
[_U1]
y2 0 0.8280 0
[_U2]
y3 0 0 -0.5574
[_U3]
_______________________________________________________________________________
Three-variables and one factor using LINEQS 27
Try it with Factor
22:55 Wednesday, December 1, 2004
The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Standardized Factor Loadings
FACT1
y1 -0.1096
y2 0.4147
y3 1.2479
Squared Multiple Correlations
Error Total
Variable Variance Variance R-Square
1 y1 0.98799 1.00000 0.0120
2 y2 0.82801 1.00000 0.1720
3 y3 -0.55736 1.00000 1.5574
Factor Score Regression Coefficients
FACT1
y1 0.0705
y2 -0.3182
y3 1.4222