Example 25.27 Second-Order Confirmatory Factor Analysis

A second-order confirmatory factor-analysis model is applied to a correlation matrix of Thurstone reported by McDonald (1985). The data set is shown in the following DATA step:

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.
;

Using the LINEQS modeling language, you specify the three-term second-order factor-analysis model in the following statements:

proc calis data=Thurst nobs=213 corr nose;
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;                                                      
variance                                                                         
   f4      = 1.,                                                            
   e1-e9   = u1-u9,                                                       
   e10-e12 = 3 * 1.;                                                         
bounds                                                                      
   0. <= u1-u9;                                                            
run;

In the PROC CALIS statement, you specify the data set in the DATA= option and the number of observations in the NOBS= option. With the CORR option, you request the correlations be analyzed. You use the NOSE option to suppress the computation of standard error estimates.

In the LINEQS statement, the first-order loadings for the three factors, f1, f2, and f3, each refer to three variables, X1-X3, X4-X6, and X7-X9, respectively. One second-order factor, f4, reflects the correlations among the three first-order factors, f1, f2, and f3.

In the VARIANCE statement, you fix the variance of f4 to 1.0 for identification. The variances of error terms e1–e9 are free parameters u1–u9. The error variances for the three first-order factors are also fixed at 1.0 for identification purposes.

You also specify the boundary constraints for the error variance parameters u1–u9. You require them to be positive in the estimation.

Output 25.27.1 shows the estimation results.

Alternatively, you can use the COSAN model specification for analyzing the same data set. First, under the second-order factor model, the covariance structures of the observed variables can be derived as

where is the 9 3 first-order loading matrix for the observed variables, is the 3 1 second-order loading matrix for the first-order factors, is the 1 1 covariance matrix for the second-order factor f4, is the 3 3 error covariance matrix of the first-order factors f1–f3 (or the covariance matrix of the error terms e10–12), and is the 9 9 error covariance matrix for the observed variables (or the covariance matrix of the error terms e1–9).

Matrix contains the loading parameters x1–x9 and matrix contains the loading parameters x10–x12. Because there is only one second-order factor f4 in the model, matrix is a scalar, which is a fixed constant 1 in the LINEQS model. Matrix is an identity matrix because all error variances are fixed at 1 and they are not correlated. Matrix is a diagonal matrix that contains the parameters u1–u9. Given this information, you can use the following statements to specify the second-order factor model as a COSAN model:

proc calis data=Thurst nobs=213 corr nose;                        
   cosan
      var = obs1-obs9,
      F1(3) * F2(1) * P(1,IDE) + F1(3) * U2(3,IDE) + U1(9,DIA);            
   matrix F1                                                                  
      [1 , @1] = x1-x3,                                                      
      [4 , @2] = x4-X6,                                               
      [7 , @3] = x7-x9;                                               
   matrix F2                                                                  
      [ ,1]    = x10-x12;                                                    
   matrix U1                                                                  
      [1,1]    = u1-u9;                                                    
   bounds                                                                     
      0. <= u1-u9;
   vnames 
      F1 = [f1 f2 f3],
      F2 = [f4],
      U1 = [e1-e9];
run;  

In the PROC CALIS statement, you specify the observed variables in the VAR= option and the covariance structures for the observed variables. In the terms of the covariance structure formula, you need to specify the expressions only up the central symmetric matrices. The latter parts of these expressions are redundant and can be generated automatically by PROC CALIS, as shown in Output 25.27.2.

Output 25.27.2 shows that the intended covariance structures for the observed variables are being analyzed. The matrix types are shown next. Matrix is a rectangular matrix and matrix is a vector, although they have the default general (GEN) matrix type. Matrices and are fixed identity (IDE) matrices in the model. For these two matrices, you do not need to specify any of their elements by using the MATRIX statement because they are already well-defined with the IDE type. Lastly, matrix is a diagonal (DIA) matrix in the model.

Output 25.27.3 shows the estimates of the first-order factor loading matrix .

Output 25.27.3 Estimation of the Matrix of the Second-Order Factor Model: COSAN Model

Model Matrix F1
(9 x 3 General Rectangular Matrix)
	f1	f2	f3
Obs1	0.5151 [x1]	0	0
Obs2	0.5203 [x2]	0	0
Obs3	0.4874 [x3]	0	0
Obs4	0	0.5211 [x4]	0
Obs5	0	0.4971 [x5]	0
Obs6	0	0.4381 [x6]	0
Obs7	0	0	0.4524 [x7]
Obs8	0	0	0.4173 [x8]
Obs9	0	0	0.4076 [x9]

In the MATRIX statement for , you specify the pattern of the loadings. In the first entry of the MATRIX statement, you specify the loadings in the following elements: [1,1], [2,1], and [3,1]. They are free parameters x1–x3, respectively. Notice that the @ sign is necessary in the first entry because the elements being defined would have been [1,1], [2,2], and [3,3] otherwise. The @ sign fixes the column number to 1. See the MATRIX statement for more details about the notation. Similarly, you define the other clusters of loading in the second and third entries in the MATRIX statement for . This explains the pattern of factor loadings in Output 25.27.3. These loading estimates x1–x9 match those by the LINEQS model specification, as shown in Output 25.27.1.

Output 25.27.3 shows the estimates of the second-order factor loading matrix .

Output 25.27.4 Estimation of the Matrix of the Second-Order Factor Model: COSAN Model

Model Matrix F2
(3 x 1 Column Vector)
	f4
f1	1.4438 [x10]
f2	1.2538 [x11]
f3	1.4066 [x12]

In the MATRIX statement for , you do not specify the row numbers in the [ ,1] specification. PROC CALIS interprets this as stating that all the valid elements in the first column are being specified in the parameter list. In the current example, this means that elements , , and are filled with the free parameters x10, x11, and x12, respectively. Output 25.27.3 shows these specification and the corresponding estimates, which match those of the LINEQS model specification, as shown in Output 25.27.1.

Output 25.27.5 shows the estimates of the error covariance matrix .

Output 25.27.5 Estimation of the Matrix of the Second-Order Factor Model: COSAN Model

Model Matrix U1
(9 x 9 Diagonal Matrix)
	e1	e2	e3	e4	e5	e6	e7	e8	e9
e1	0.1815 [u1]	0	0	0	0	0	0	0	0
e2	0	0.1649 [u2]	0	0	0	0	0	0	0
e3	0	0	0.2671 [u3]	0	0	0	0	0	0
e4	0	0	0	0.3015 [u4]	0	0	0	0	0
e5	0	0	0	0	0.3645 [u5]	0	0	0	0
e6	0	0	0	0	0	0.5064 [u6]	0	0	0
e7	0	0	0	0	0	0	0.3903 [u7]	0	0
e8	0	0	0	0	0	0	0	0.4814 [u8]	0
e9	0	0	0	0	0	0	0	0	0.5051 [u9]

In the MATRIX statement for , you specify the diagonal elements of the matrix by using the starting element at [1,1]. The parameter assignment proceeds to [2,2], [3,3] and so on such that all the trailing parameters u1–u9 are filled. This means that the last element is a free parameter named u9. Output 25.27.5 confirms this intended pattern. Again, all these error variance estimates match those by the LINEQS model specification, as shown in Output 25.27.1.