In this example, cognitive abilities of 64 students from a middle school were measured. The fictitious data contain nine cognitive test scores. Three of the scores were for reading skills, three others were for math skills, and the remaining three were for writing skills. The covariance matrix for the nine variables was obtained. A confirmatory factor analysis with three factors was conducted. The following is the input data set:
title "Confirmatory Factor Analysis Using the FACTOR Modeling Language"; title2 "Cognitive Data"; data cognitive1(type=cov); _type_='cov'; input _name_ $ reading1 reading2 reading3 math1 math2 math3 writing1 writing2 writing3; datalines; reading1 83.024 . . . . . . . . reading2 50.924 108.243 . . . . . . . reading3 62.205 72.050 99.341 . . . . . . math1 22.522 22.474 25.731 82.214 . . . . . math2 14.157 22.487 18.334 64.423 96.125 . . . . math3 22.252 20.645 23.214 49.287 58.177 88.625 . . . writing1 33.433 42.474 41.731 25.318 14.254 27.370 90.734 . . writing2 24.147 20.487 18.034 22.106 26.105 22.346 53.891 96.543 . writing3 13.340 20.645 23.314 19.387 28.177 38.635 55.347 52.999 98.445 ;
You first fit a confirmatory factor model with uncorrelated factors to the data, as shown in the following statements:
proc calis data=cognitive1 nobs=64 modification; factor Read_Factor ===> reading1reading3 , Math_Factor ===> math1math3 , Write_Factor ===> writing1writing3 ; pvar Read_Factor Math_Factor Write_Factor = 3 * 1.; cov Read_Factor Math_Factor Write_Factor = 3 * 0.; run;
In the PROC CALIS statement, the number of observations is specified with the NOBS= option. With the MODIFICATION in the PROC CALIS statement, LM (Lagrange Multiplier) tests are conducted. The results of LM tests can suggest the inclusion of additional parameters for a better model fit.
The FACTOR modeling language is most handy when you specify confirmatory factor models. You use the FACTOR statement to invoke the FACTOR modeling language. Entries in the FACTOR statement are for specifying factorvariables relationships and are separated by commas. In each entry, you first specify
a latent factor, followed by the right arrow sign ===>
(you can use >
, =>
, ==>
, or ===>
). Then you specify the observed variables that have nonzero loadings on the factor. For example, in the first entry of FACTOR
statement, you specify that latent factor Read_Factor
has nonzero loadings (free parameters) on variables reading1
–reading3
. Optionally, you can specify the parameter list after you specify the factorvariable relationships. For example, you can
name the loading parameters as in the following specification:
factor Read_Factor ===> reading1reading3 = load1load3;
This way, you name the factor loadings with parameter names load1
, load2
, and load3
, respectively. However, in the current example, because the loading parameters are all unconstrained, you can just let PROC
CALIS to generate the parameter names for you. In this example, there are three factors: Read_Factor
, Math_Factor
, and Write_Factor
. These factors have simple cluster structures with the nine observed variables. Each observed variable has only one loading
on exactly one factor.
In the PVAR statement, you can specify the variances of the factors and the error variances of the observed variables. The factor variances in this model are all fixed at 1.0 for identification purposes. You do not need to specify the error variances of the observed variables in the current model because PROC CALIS assumes these are free parameters by default.
In the COV statement, you specify that the covariances among the factors are fixed zeros. There are three covariances among
the three latent factors and therefore you put 3 * 0.
for their fixed values. This means that the factors in the current model are uncorrelated. Note that you must specify uncorrelated
factors explicitly in the COV statement because all latent factors are correlated by default.
In Output 29.20.1, the initial model specification is echoed in matrix form. The observed variables and factors are also displayed.
Output 29.20.1: Uncorrelated Factor Model Specification
Variables in the Model  

Variables  reading1 reading2 reading3 math1 math2 math3 writing1 writing2 writing3 
Factors  Read_Factor Math_Factor Write_Factor 
Number of Variables = 9 Number of Factors = 3 
Initial Factor Loading Matrix  

Read_Factor  Math_Factor  Write_Factor  
reading1 




reading2 




reading3 




math1 




math2 




math3 




writing1 




writing2 




writing3 



Initial Factor Covariance Matrix  

Read_Factor  Math_Factor  Write_Factor  
Read_Factor  1.0000  0  0 
Math_Factor  0  1.0000  0 
Write_Factor  0  0  1.0000 
Initial Error Variances  

Variable  Parameter  Estimate 
reading1  _Add1  . 
reading2  _Add2  . 
reading3  _Add3  . 
math1  _Add4  . 
math2  _Add5  . 
math3  _Add6  . 
writing1  _Add7  . 
writing2  _Add8  . 
writing3  _Add9  . 
NOTE: Parameters with prefix '_Add' are added by PROC CALIS. 
In the table for initial factor loading matrix, the nine loading parameters are shown to have simple cluster relations with the factors. In the table for initial factor covariance matrix, the diagonal matrix shows that the factors are not correlated. The diagonal elements are fixed at ones so that this matrix is also a correlation matrix for the factors. In the table for initial error variances, the nine variance parameters are shown. As described previously, these error variances are generated by PROC CALIS as default parameters.
In Output 29.20.2, initial estimates are generated by the instrumental variable method and the McDonald method.
Output 29.20.2: Optimization of the Uncorrelated Factor Model: Initial Estimates
Initial Estimation Methods  

1  Instrumental Variables Method 
2  McDonald Method 
Optimization Start Parameter Estimates 


N  Parameter  Estimate  Gradient 
1  _Parm1  7.15372  0.00851 
2  _Parm2  7.80225  0.00170 
3  _Parm3  8.70856  0.00602 
4  _Parm4  7.68637  0.00272 
5  _Parm5  8.01765  0.01096 
6  _Parm6  7.05012  0.00932 
7  _Parm7  8.76776  0.0009955 
8  _Parm8  5.96161  0.01335 
9  _Parm9  7.23168  0.01665 
10  _Add1  31.84831  0.00179 
11  _Add2  47.36790  0.0003461 
12  _Add3  23.50199  0.00257 
13  _Add4  23.13374  0.0008384 
14  _Add5  31.84224  0.00280 
15  _Add6  38.92075  0.00167 
16  _Add7  13.86035  0.00579 
17  _Add8  61.00217  0.00115 
18  _Add9  46.14784  0.00300 
Value of Objective Function = 0.9103815918 
These initial estimates turn out to be pretty good, in the sense that only three more iterations are needed to converge to the maximum likelihood estimates and the final function value 0.784 does not change much from the initial function value 0.910, as shown in Output 29.20.3.
Output 29.20.3: Optimization of the Uncorrelated Factor Model: Iteration Summary
Iteration  Restarts  Function Calls 
Active Constraints 
Objective Function 
Objective Function Change 
Max Abs Gradient Element 
Lambda  Ratio Between Actual and Predicted Change 


1  0  4  0  0.78792  0.1225  0.00175  0  0.932  
2  0  6  0  0.78373  0.00419  0.000037  0  1.051  
3  0  8  0  0.78373  5.087E7  3.715E9  0  1.001 
Optimization Results  

Iterations  3  Function Calls  11 
Jacobian Calls  5  Active Constraints  0 
Objective Function  0.783733415  Max Abs Gradient Element  3.7146571E9 
Lambda  0  Actual Over Pred Change  1.0006660673 
Radius  0.0025042942 
Convergence criterion (ABSGCONV=0.00001) satisfied. 
The fit summary is shown in Output 29.20.4.
Output 29.20.4: Fit of the Uncorrelated Factor Model
Fit Summary  

Modeling Info  Number of Observations  64 
Number of Variables  9  
Number of Moments  45  
Number of Parameters  18  
Number of Active Constraints  0  
Baseline Model Function Value  4.3182  
Baseline Model ChiSquare  272.0467  
Baseline Model ChiSquare DF  36  
Pr > Baseline Model ChiSquare  <.0001  
Absolute Index  Fit Function  0.7837 
ChiSquare  49.3752  
ChiSquare DF  27  
Pr > ChiSquare  0.0054  
ZTest of Wilson & Hilferty  2.5474  
Hoelter Critical N  52  
Root Mean Square Residual (RMR)  19.5739  
Standardized RMR (SRMR)  0.2098  
Goodness of Fit Index (GFI)  0.8555  
Parsimony Index  Adjusted GFI (AGFI)  0.7592 
Parsimonious GFI  0.6416  
RMSEA Estimate  0.1147  
RMSEA Lower 90% Confidence Limit  0.0617  
RMSEA Upper 90% Confidence Limit  0.1646  
Probability of Close Fit  0.0271  
ECVI Estimate  1.4630  
ECVI Lower 90% Confidence Limit  1.2069  
ECVI Upper 90% Confidence Limit  1.8687  
Akaike Information Criterion  85.3752  
Bozdogan CAIC  142.2351  
Schwarz Bayesian Criterion  124.2351  
McDonald Centrality  0.8396  
Incremental Index  Bentler Comparative Fit Index  0.9052 
BentlerBonett NFI  0.8185  
BentlerBonett Nonnormed Index  0.8736  
Bollen Normed Index Rho1  0.7580  
Bollen Nonnormed Index Delta2  0.9087  
James et al. Parsimonious NFI  0.6139 
Using the chisquare model test criterion, the uncorrelated factor model should be rejected at . The RMSEA estimate is 0.1147, which is not indicative of a good fit according to Browne and Cudeck (1993) Other indices might suggest only a marginal good fit. For example, Bentler’s comparative fit index and Bollen nonnormed index delta2 are both above 0.90. However, many other do not attain this 0.90 level. For example, adjusted GFI is only 0.759. It is thus safe to conclude that there could be some improvements on the model fit.
The MODIFICATION option in the PROC CALIS statement has been used to request for computing the LM test indices for model modifications. The results are shown in Output 29.20.5.
Output 29.20.5: Lagrange Multiplier Tests
Rank Order of the 10 Largest LM Stat for Factor Loadings  

Variable  Factor  LM Stat  Pr > ChiSq  Parm Change 
writing1  Read_Factor  9.76596  0.0018  2.95010 
math3  Write_Factor  3.58077  0.0585  1.89703 
math1  Read_Factor  2.15312  0.1423  1.17976 
writing3  Math_Factor  1.87637  0.1707  1.41298 
math3  Read_Factor  1.02954  0.3103  0.95427 
reading2  Write_Factor  0.91230  0.3395  0.99933 
writing2  Math_Factor  0.86221  0.3531  0.95672 
reading1  Write_Factor  0.63403  0.4259  0.73916 
math1  Write_Factor  0.55602  0.4559  0.63906 
reading2  Math_Factor  0.55362  0.4568  0.74628 
Rank Order of the 3 Largest LM Stat for Covariances of Factors  

Var1  Var2  LM Stat  Pr > ChiSq  Parm Change 
Write_Factor  Read_Factor  8.95268  0.0028  0.44165 
Write_Factor  Math_Factor  7.07904  0.0078  0.40132 
Math_Factor  Read_Factor  4.61896  0.0316  0.30411 
Rank Order of the 10 Largest LM Stat for Error Variances and Covariances  

Error of 
Error of 
LM Stat  Pr > ChiSq  Parm Change 
writing1  math2  5.45986  0.0195  13.16822 
writing1  math1  5.05573  0.0245  12.32431 
writing3  math3  3.93014  0.0474  13.59149 
writing3  math1  2.83209  0.0924  9.86342 
writing2  reading1  2.56677  0.1091  10.15901 
writing2  math2  1.94879  0.1627  8.40273 
writing2  reading3  1.75181  0.1856  7.82777 
writing3  reading1  1.57978  0.2088  7.97915 
writing1  reading2  1.34894  0.2455  7.77158 
writing2  math3  1.11704  0.2906  7.23762 
Three different tables for ranking the LM test results are shown. In the first table, the new loading parameters that would
improve the model fit the most are shown first. For example, in the first row a new factor loading of writing1
on the Read_Factor
is suggested to improve the model fit the most. The 'LM Stat' value is 9.77. This is an approximation of the chisquare drop
if this parameter was included in the model. The 'Pr > ChiSq' value of 0.0018 indicates a significant improvement of model
fit at . Nine more new loading parameters are suggested in the table, with less and less statistical significance in the change of
model fit chisquare. Note that these approximate chisquares are oneatatime chisquare changes. That means that the overall
chisquare drop is not a simple sum of individual chisquare changes when you include two or more new parameters in the modified
model.
The other two tables in Output 29.20.5 shows the new parameters in factor covariances, error variances, or error covariances that would result in a better model fit. The table for the new parameters of the factor covariance matrix indicates that adding each of the covariances among factors might lead to a statistically significant improvement in model fit. The largest 'LM Stat' value in this table is 8.95, which is smaller than that of the largest 'LM Stat' for the factor loading parameters. Despite this, it is more reasonable to add the covariance parameters among factors first to determine whether that improves the model fit.
To fit the corresponding confirmatory factor model with correlated factors, you can remove the fixed zeros from the COV statement in the preceding specification, as shown in the following statements:
proc calis data=cognitive1 nobs=64 modification; factor Read_Factor ===> reading1reading3 , Math_Factor ===> math1math3 , Write_Factor ===> writing1writing3 ; pvar Read_Factor Math_Factor Write_Factor = 3 * 1.; cov Read_Factor Math_Factor Write_Factor /* = 3 * 0. */; run;
In the COV statement, you comment out the fixed zeros so that the covariances among the latent factors are now free parameters. An alternative way is to delete the entire COV statement so that the covariances among factors are free parameters by the FACTOR model default.
The fit summary of the correlated factor model is shown in Output 29.20.6.
Output 29.20.6: Fit of the Correlated Factor Model
Fit Summary  

Modeling Info  Number of Observations  64 
Number of Variables  9  
Number of Moments  45  
Number of Parameters  21  
Number of Active Constraints  0  
Baseline Model Function Value  4.3182  
Baseline Model ChiSquare  272.0467  
Baseline Model ChiSquare DF  36  
Pr > Baseline Model ChiSquare  <.0001  
Absolute Index  Fit Function  0.4677 
ChiSquare  29.4667  
ChiSquare DF  24  
Pr > ChiSquare  0.2031  
ZTest of Wilson & Hilferty  0.8320  
Hoelter Critical N  78  
Root Mean Square Residual (RMR)  5.7038  
Standardized RMR (SRMR)  0.0607  
Goodness of Fit Index (GFI)  0.9109  
Parsimony Index  Adjusted GFI (AGFI)  0.8330 
Parsimonious GFI  0.6073  
RMSEA Estimate  0.0601  
RMSEA Lower 90% Confidence Limit  0.0000  
RMSEA Upper 90% Confidence Limit  0.1244  
Probability of Close Fit  0.3814  
ECVI Estimate  1.2602  
ECVI Lower 90% Confidence Limit  1.2453  
ECVI Upper 90% Confidence Limit  1.5637  
Akaike Information Criterion  71.4667  
Bozdogan CAIC  137.8032  
Schwarz Bayesian Criterion  116.8032  
McDonald Centrality  0.9582  
Incremental Index  Bentler Comparative Fit Index  0.9768 
BentlerBonett NFI  0.8917  
BentlerBonett Nonnormed Index  0.9653  
Bollen Normed Index Rho1  0.8375  
Bollen Nonnormed Index Delta2  0.9780  
James et al. Parsimonious NFI  0.5945 
The model fit chisquare value is 29.47, which is about 20 less than the model with uncorrelated factors. The pvalue is 0.20, indicating a satisfactory model fit. The RMSEA value is 0.06, which is close to 0.05, a value recommended as an indication of good model fit by Browne and Cudeck (1993) More fit indices that do not attain the 0.9 level with the uncorrelated factor model now have values close to or above 0.9. These include the goodnessoffit index (GFI), McDonald centrality, BentlerBonnet NFI, and BentlerBonnet nonnormed index. By all counts, the correlated factor model is a much better fit than the uncorrelated factor model.
In Output 29.20.7, the estimation results for factor loadings are shown. All these loadings are statistically significant, indicating nonchance relationships with the factors.
Output 29.20.7: Estimation of the Factor Loading Matrix
Factor Loading Matrix: Estimate/StdErr/tvalue  

Read_Factor  Math_Factor  Write_Factor  
reading1 




reading2 




reading3 




math1 




math2 




math3 




writing1 




writing2 




writing3 



In Output 29.20.8, the factor covariance matrix is shown. Because the diagonal elements are all ones, the offdiagonal elements are correlations among factors. The correlations range from 0.30–0.5. These factors are moderately correlated.
Output 29.20.8: Estimation of the Correlations of Factors
Factor Covariance Matrix: Estimate/StdErr/tvalue  

Read_Factor  Math_Factor  Write_Factor  
Read_Factor 




Math_Factor 




Write_Factor 



In Output 29.20.9, the error variances for variables are shown.
Output 29.20.9: Estimation of the Error Variances
Error Variances  

Variable  Parameter  Estimate  Standard Error 
t Value 
reading1  _Add1  37.24939  8.33997  4.46637 
reading2  _Add2  46.49695  10.69869  4.34604 
reading3  _Add3  15.90447  9.26097  1.71737 
math1  _Add4  25.22889  7.72269  3.26685 
math2  _Add5  24.89032  8.98327  2.77074 
math3  _Add6  42.12110  9.20362  4.57658 
writing1  _Add7  27.24965  10.36489  2.62903 
writing2  _Add8  49.28881  11.39812  4.32429 
writing3  _Add9  48.10684  11.48868  4.18733 
All t values except the one for reading3
are greater than 2, a value close to a critical t value at . This means that the error variance for reading3
could have been zero in the population, or it could have been nonzero but the current sample just has this insignificant
value by chance (that is, a Type 2 error). Further research is needed to confirm either way.
In addition to the parameter estimation results, PROC CALIS also outputs supplementary results that could be useful for interpretations. In Output 29.20.10, the squared multiple correlations and the factor scores regression coefficients are shown.
Output 29.20.10: Supplementary Estimation Results
Squared Multiple Correlations  

Variable  Error Variance  Total Variance  RSquare 
reading1  37.24939  83.02400  0.5513 
reading2  46.49695  108.24300  0.5704 
reading3  15.90447  99.34100  0.8399 
math1  25.22889  82.21400  0.6931 
math2  24.89032  96.12500  0.7411 
math3  42.12110  88.62500  0.5247 
writing1  27.24965  90.73400  0.6997 
writing2  49.28881  96.54300  0.4895 
writing3  48.10684  98.44500  0.5113 
Factor Scores Regression Coefficients  

Read_Factor  Math_Factor  Write_Factor  
reading1  0.0200  0.000681  0.001985 
reading2  0.0186  0.000633  0.001847 
reading3  0.0633  0.002152  0.006275 
math1  0.001121  0.0403  0.002808 
math2  0.001271  0.0457  0.003183 
math3  0.000607  0.0218  0.001520 
writing1  0.003195  0.002744  0.0513 
writing2  0.001524  0.001309  0.0245 
writing3  0.001611  0.001384  0.0259 
The percentages of variance for the observed variables that can be explained by the factors are shown in the 'RSquare' column of the table for squared multiple correlations (Rsquares). These Rsquares can be interpreted meaningfully because there is no reciprocal relationships among variables or correlated errors in the model. All estimates of Rsquares are bounded between 0 and 1.
In the table for factor scores regression coefficients, entries are coefficients for the variables you can use to create the
factor scores. The larger the coefficient, the more influence of the corresponding variable for creating the factor scores.
It makes intuitive sense to see the cluster pattern of these coefficients—the reading measures are more important to create
the latent variable scores of Read_Factor
and so on.