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 The CALIS Procedure

## 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 FunctionCalls ActiveConstraints ObjectiveFunction ObjectiveFunctionChange Max AbsGradientElement Lambda RatioBetweenActualandPredictedChange 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 8.209 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 0.059 X5 t Value 8.4213 Obs6 = 0.4381 * F2 + 1.0000 E6 Std Err 0.056 X6 t Value 7.8283 Obs7 = 0.4524 * F3 + 1.0000 E7 Std Err 0.066 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 5.932 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 StandardError 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.

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