Example 19.3: Second-Order Confirmatory Factor Analysis
A second-order confirmatory factor analysis model is applied to
a correlation matrix of Thurstone reported by
McDonald (1985).
Using the LINEQS statement, the three-term
second-order factor analysis model is specified in equations notation.
The first-order loadings for the three factors, F1, F2, and F3,
each refer to three variables, X1-X3, X4-X6, and X7-X9.
One second-order factor, F4, reflects the correlations among
the three first-order factors. The second-order
factor correlation matrix P is defined as a 1 ×1 identity matrix.
Choosing the second-order uniqueness matrix U2 as a diagonal matrix
with parameters U21-U23 gives an unidentified model. To compute
identified maximum likelihood estimates, the matrix
U2 is defined as a 3 ×3 identity matrix. The following code
generates results that are partially displayed in Output 19.3.1.
data Thurst(TYPE=CORR);
Title "Example of THURSTONE resp. McDONALD (1985, p.57, p.105)";
_TYPE_ = 'CORR'; Input _NAME_ $ Obs1-Obs9;
Label Obs1='Sentences' Obs2='Vocabulary' Obs3='Sentence Completion'
Obs4='First Letters' Obs5='Four-letter Words' Obs6='Suffices'
Obs7='Letter series' Obs8='Pedigrees' Obs9='Letter Grouping';
datalines;
Obs1 1. . . . . . . . .
Obs2 .828 1. . . . . . . .
Obs3 .776 .779 1. . . . . . .
Obs4 .439 .493 .460 1. . . . . .
Obs5 .432 .464 .425 .674 1. . . . .
Obs6 .447 .489 .443 .590 .541 1. . . .
Obs7 .447 .432 .401 .381 .402 .288 1. . .
Obs8 .541 .537 .534 .350 .367 .320 .555 1. .
Obs9 .380 .358 .359 .424 .446 .325 .598 .452 1.
;
proc calis data=Thurst method=max edf=212 pestim se;
Title2 "Identified Second Order Confirmatory Factor Analysis";
Title3 "C = F1 * F2 * P * F2' * F1' + F1 * U2 * F1' + U1, With P=U2=Ide";
Lineqs
Obs1 = X1 F1 + E1,
Obs2 = X2 F1 + E2,
Obs3 = X3 F1 + E3,
Obs4 = X4 F2 + E4,
Obs5 = X5 F2 + E5,
Obs6 = X6 F2 + E6,
Obs7 = X7 F3 + E7,
Obs8 = X8 F3 + E8,
Obs9 = X9 F3 + E9,
F1 = X10 F4 + E10,
F2 = X11 F4 + E11,
F3 = X12 F4 + E12;
Std
F4 = 1. ,
E1-E9 = U11-U19 ,
E10-E12 = 3 * 1.;
Bounds
0. <= U11-U19;
run;
Output 19.3.1: Second-Order Confirmatory Factor Analysis
Example of THURSTONE resp. McDONALD (1985, p.57, p.105) |
Identified Second Order Confirmatory Factor Analysis |
C = F1 * F2 * P * F2' * F1' + F1 * U2 * F1' + U1, With P=U2=Ide |
The CALIS Procedure |
Covariance Structure Analysis: Maximum Likelihood Estimation |
Parameter Estimates |
21 |
Functions (Observations) |
45 |
Lower Bounds |
9 |
Upper Bounds |
0 |
Optimization Start |
Active Constraints |
0 |
Objective Function |
0.7151823452 |
Max Abs Gradient Element |
0.4067179803 |
Radius |
2.2578762496 |
Iteration |
|
Restarts |
Function Calls |
Active Constraints |
|
Objective Function |
Objective Function Change |
Max Abs Gradient Element |
Lambda |
Ratio Between Actual and Predicted Change |
1 |
|
0 |
2 |
0 |
|
0.23113 |
0.4840 |
0.1299 |
0 |
1.363 |
2 |
|
0 |
3 |
0 |
|
0.18322 |
0.0479 |
0.0721 |
0 |
1.078 |
3 |
|
0 |
4 |
0 |
|
0.18051 |
0.00271 |
0.0200 |
0 |
1.006 |
4 |
|
0 |
5 |
0 |
|
0.18022 |
0.000289 |
0.00834 |
0 |
1.093 |
5 |
|
0 |
6 |
0 |
|
0.18018 |
0.000041 |
0.00251 |
0 |
1.201 |
6 |
|
0 |
7 |
0 |
|
0.18017 |
6.523E-6 |
0.00114 |
0 |
1.289 |
7 |
|
0 |
8 |
0 |
|
0.18017 |
1.085E-6 |
0.000388 |
0 |
1.347 |
8 |
|
0 |
9 |
0 |
|
0.18017 |
1.853E-7 |
0.000173 |
0 |
1.380 |
9 |
|
0 |
10 |
0 |
|
0.18017 |
3.208E-8 |
0.000063 |
0 |
1.399 |
10 |
|
0 |
11 |
0 |
|
0.18017 |
5.593E-9 |
0.000028 |
0 |
1.408 |
11 |
|
0 |
12 |
0 |
|
0.18017 |
9.79E-10 |
0.000011 |
0 |
1.414 |
Optimization Results |
Iterations |
11 |
Function Calls |
13 |
Jacobian Calls |
12 |
Active Constraints |
0 |
Objective Function |
0.1801712147 |
Max Abs Gradient Element |
0.0000105805 |
Lambda |
0 |
Actual Over Pred Change |
1.4135857595 |
Radius |
0.0002026368 |
|
|
GCONV convergence criterion satisfied. |
|
Example of THURSTONE resp. McDONALD (1985, p.57, p.105) |
Identified Second Order Confirmatory Factor Analysis |
C = F1 * F2 * P * F2' * F1' + F1 * U2 * F1' + U1, With P=U2=Ide |
The CALIS Procedure |
Covariance Structure Analysis: Maximum Likelihood Estimation |
Fit Function |
0.1802 |
Goodness of Fit Index (GFI) |
0.9596 |
GFI Adjusted for Degrees of Freedom (AGFI) |
0.9242 |
Root Mean Square Residual (RMR) |
0.0436 |
Parsimonious GFI (Mulaik, 1989) |
0.6397 |
Chi-Square |
38.1963 |
Chi-Square DF |
24 |
Pr > Chi-Square |
0.0331 |
Independence Model Chi-Square |
1101.9 |
Independence Model Chi-Square DF |
36 |
RMSEA Estimate |
0.0528 |
RMSEA 90% Lower Confidence Limit |
0.0153 |
RMSEA 90% Upper Confidence Limit |
0.0831 |
ECVI Estimate |
0.3881 |
ECVI 90% Lower Confidence Limit |
. |
ECVI 90% Upper Confidence Limit |
0.4888 |
Probability of Close Fit |
0.4088 |
Bentler's Comparative Fit Index |
0.9867 |
Normal Theory Reweighted LS Chi-Square |
40.1947 |
Akaike's Information Criterion |
-9.8037 |
Bozdogan's (1987) CAIC |
-114.4747 |
Schwarz's Bayesian Criterion |
-90.4747 |
McDonald's (1989) Centrality |
0.9672 |
Bentler & Bonett's (1980) Non-normed Index |
0.9800 |
Bentler & Bonett's (1980) NFI |
0.9653 |
James, Mulaik, & Brett (1982) Parsimonious NFI |
0.6436 |
Z-Test of Wilson & Hilferty (1931) |
1.8373 |
Bollen (1986) Normed Index Rho1 |
0.9480 |
Bollen (1988) Non-normed Index Delta2 |
0.9868 |
Hoelter's (1983) Critical N |
204 |
|
Example of THURSTONE resp. McDONALD (1985, p.57, p.105) |
Identified Second Order Confirmatory Factor Analysis |
C = F1 * F2 * P * F2' * F1' + F1 * U2 * F1' + U1, With P=U2=Ide |
The CALIS Procedure |
Covariance Structure Analysis: Maximum Likelihood Estimation |
Obs1 |
= |
0.5151 |
* |
F1 |
+ |
1.0000 |
|
E1 |
Std Err |
|
0.0629 |
|
X1 |
|
|
|
|
t Value |
|
8.1868 |
|
|
|
|
|
|
Obs2 |
= |
0.5203 |
* |
F1 |
+ |
1.0000 |
|
E2 |
Std Err |
|
0.0634 |
|
X2 |
|
|
|
|
t Value |
|
8.2090 |
|
|
|
|
|
|
Obs3 |
= |
0.4874 |
* |
F1 |
+ |
1.0000 |
|
E3 |
Std Err |
|
0.0608 |
|
X3 |
|
|
|
|
t Value |
|
8.0151 |
|
|
|
|
|
|
Obs4 |
= |
0.5211 |
* |
F2 |
+ |
1.0000 |
|
E4 |
Std Err |
|
0.0611 |
|
X4 |
|
|
|
|
t Value |
|
8.5342 |
|
|
|
|
|
|
Obs5 |
= |
0.4971 |
* |
F2 |
+ |
1.0000 |
|
E5 |
Std Err |
|
0.0590 |
|
X5 |
|
|
|
|
t Value |
|
8.4213 |
|
|
|
|
|
|
Obs6 |
= |
0.4381 |
* |
F2 |
+ |
1.0000 |
|
E6 |
Std Err |
|
0.0560 |
|
X6 |
|
|
|
|
t Value |
|
7.8283 |
|
|
|
|
|
|
Obs7 |
= |
0.4524 |
* |
F3 |
+ |
1.0000 |
|
E7 |
Std Err |
|
0.0660 |
|
X7 |
|
|
|
|
t Value |
|
6.8584 |
|
|
|
|
|
|
Obs8 |
= |
0.4173 |
* |
F3 |
+ |
1.0000 |
|
E8 |
Std Err |
|
0.0622 |
|
X8 |
|
|
|
|
t Value |
|
6.7135 |
|
|
|
|
|
|
Obs9 |
= |
0.4076 |
* |
F3 |
+ |
1.0000 |
|
E9 |
Std Err |
|
0.0613 |
|
X9 |
|
|
|
|
t Value |
|
6.6484 |
|
|
|
|
|
|
Example of THURSTONE resp. McDONALD (1985, p.57, p.105) |
Identified Second Order Confirmatory Factor Analysis |
C = F1 * F2 * P * F2' * F1' + F1 * U2 * F1' + U1, With P=U2=Ide |
The CALIS Procedure |
Covariance Structure Analysis: Maximum Likelihood Estimation |
F1 |
= |
1.4438 |
* |
F4 |
+ |
1.0000 |
|
E10 |
Std Err |
|
0.2565 |
|
X10 |
|
|
|
|
t Value |
|
5.6282 |
|
|
|
|
|
|
F2 |
= |
1.2538 |
* |
F4 |
+ |
1.0000 |
|
E11 |
Std Err |
|
0.2114 |
|
X11 |
|
|
|
|
t Value |
|
5.9320 |
|
|
|
|
|
|
F3 |
= |
1.4065 |
* |
F4 |
+ |
1.0000 |
|
E12 |
Std Err |
|
0.2689 |
|
X12 |
|
|
|
|
t Value |
|
5.2307 |
|
|
|
|
|
|
|
Example of THURSTONE resp. McDONALD (1985, p.57, p.105) |
Identified Second Order Confirmatory Factor Analysis |
C = F1 * F2 * P * F2' * F1' + F1 * U2 * F1' + U1, With P=U2=Ide |
The CALIS Procedure |
Covariance Structure Analysis: Maximum Likelihood Estimation |
Variances of Exogenous Variables |
Variable |
Parameter |
Estimate |
Standard Error |
t Value |
F4 |
|
1.00000 |
|
|
E1 |
U11 |
0.18150 |
0.02848 |
6.37 |
E2 |
U12 |
0.16493 |
0.02777 |
5.94 |
E3 |
U13 |
0.26713 |
0.03336 |
8.01 |
E4 |
U14 |
0.30150 |
0.05102 |
5.91 |
E5 |
U15 |
0.36450 |
0.05264 |
6.93 |
E6 |
U16 |
0.50642 |
0.05963 |
8.49 |
E7 |
U17 |
0.39032 |
0.05934 |
6.58 |
E8 |
U18 |
0.48138 |
0.06225 |
7.73 |
E9 |
U19 |
0.50509 |
0.06333 |
7.98 |
E10 |
|
1.00000 |
|
|
E11 |
|
1.00000 |
|
|
E12 |
|
1.00000 |
|
|
|
Example of THURSTONE resp. McDONALD (1985, p.57, p.105) |
Identified Second Order Confirmatory Factor Analysis |
C = F1 * F2 * P * F2' * F1' + F1 * U2 * F1' + U1, With P=U2=Ide |
The CALIS Procedure |
Covariance Structure Analysis: Maximum Likelihood Estimation |
Obs1 |
= |
0.9047 |
* |
F1 |
+ |
0.4260 |
|
E1 |
|
|
|
|
X1 |
|
|
|
|
Obs2 |
= |
0.9138 |
* |
F1 |
+ |
0.4061 |
|
E2 |
|
|
|
|
X2 |
|
|
|
|
Obs3 |
= |
0.8561 |
* |
F1 |
+ |
0.5168 |
|
E3 |
|
|
|
|
X3 |
|
|
|
|
Obs4 |
= |
0.8358 |
* |
F2 |
+ |
0.5491 |
|
E4 |
|
|
|
|
X4 |
|
|
|
|
Obs5 |
= |
0.7972 |
* |
F2 |
+ |
0.6037 |
|
E5 |
|
|
|
|
X5 |
|
|
|
|
Obs6 |
= |
0.7026 |
* |
F2 |
+ |
0.7116 |
|
E6 |
|
|
|
|
X6 |
|
|
|
|
Obs7 |
= |
0.7808 |
* |
F3 |
+ |
0.6248 |
|
E7 |
|
|
|
|
X7 |
|
|
|
|
Obs8 |
= |
0.7202 |
* |
F3 |
+ |
0.6938 |
|
E8 |
|
|
|
|
X8 |
|
|
|
|
Obs9 |
= |
0.7035 |
* |
F3 |
+ |
0.7107 |
|
E9 |
|
|
|
|
X9 |
|
|
|
|
Example of THURSTONE resp. McDONALD (1985, p.57, p.105) |
Identified Second Order Confirmatory Factor Analysis |
C = F1 * F2 * P * F2' * F1' + F1 * U2 * F1' + U1, With P=U2=Ide |
The CALIS Procedure |
Covariance Structure Analysis: Maximum Likelihood Estimation |
F1 |
= |
0.8221 |
* |
F4 |
+ |
0.5694 |
|
E10 |
|
|
|
|
X10 |
|
|
|
|
F2 |
= |
0.7818 |
* |
F4 |
+ |
0.6235 |
|
E11 |
|
|
|
|
X11 |
|
|
|
|
F3 |
= |
0.8150 |
* |
F4 |
+ |
0.5794 |
|
E12 |
|
|
|
|
X12 |
|
|
|
|
|
Example of THURSTONE resp. McDONALD (1985, p.57, p.105) |
Identified Second Order Confirmatory Factor Analysis |
C = F1 * F2 * P * F2' * F1' + F1 * U2 * F1' + U1, With P=U2=Ide |
The CALIS Procedure |
Covariance Structure Analysis: Maximum Likelihood Estimation |
Squared Multiple Correlations |
|
Variable |
Error Variance |
Total Variance |
R-Square |
1 |
Obs1 |
0.18150 |
1.00000 |
0.8185 |
2 |
Obs2 |
0.16493 |
1.00000 |
0.8351 |
3 |
Obs3 |
0.26713 |
1.00000 |
0.7329 |
4 |
Obs4 |
0.30150 |
1.00000 |
0.6985 |
5 |
Obs5 |
0.36450 |
1.00000 |
0.6355 |
6 |
Obs6 |
0.50642 |
1.00000 |
0.4936 |
7 |
Obs7 |
0.39032 |
1.00000 |
0.6097 |
8 |
Obs8 |
0.48138 |
1.00000 |
0.5186 |
9 |
Obs9 |
0.50509 |
1.00000 |
0.4949 |
10 |
F1 |
1.00000 |
3.08452 |
0.6758 |
11 |
F2 |
1.00000 |
2.57213 |
0.6112 |
12 |
F3 |
1.00000 |
2.97832 |
0.6642 |
|
To compute McDonald's unidentified model, you would have to change
the STD and BOUNDS statements to include
three more parameters:
Std
F4 = 1. ,
E1-E9 = U11-U19 ,
E10-E12 = U21-U23 ;
Bounds
0. <= U11-U19,
0. <= U21-U23;
The unidentified model is indicated in the
output by an analysis of the linear dependencies in the approximate
Hessian matrix (not shown).
Because the information matrix is singular,
standard errors are computed based on a Moore-Penrose inverse.
The results computed by PROC CALIS differ from those reported
by McDonald (1985). In the case of an unidentified model, the
parameter estimates are not unique.
To specify the identified model using the COSAN model statement,
you can use the following statements:
Title2 "Identified Second Order Confirmatory Factor Analysis Using COSAN";
Title3 "C = F1*F2*P*F2'*F1' + F1*U2*F1' + U1, With P=U2=Ide";
proc calis data=Thurst method=max edf=212 pestim se;
Cosan F1(3) * F2(1) * P(1,Ide) + F1(3) * U2(3,Ide) + U1(9,Dia);
Matrix F1
[ ,1] = X1-X3,
[ ,2] = 3 * 0. X4-X6,
[ ,3] = 6 * 0. X7-X9;
Matrix F2
[ ,1] = X10-X12;
Matrix U1
[1,1] = U11-U19;
Bounds
0. <= U11-U19;
run;
Because PROC CALIS cannot compute initial estimates for a model
specified by the general COSAN statement, this analysis may require
more iterations than one using the LINEQS statement, depending on
the precision of the processor.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.