Example 19.4: Linear Relations Among Factor Loadings
The correlation matrix from Kinzer and Kinzer (N=326) is used
by Guttman (1957) as an example that yields an
approximate simplex. McDonald (1980) uses this
data set as an example
of factor analysis where he supposes that the loadings of
the second factor are a linear function of the loadings on the
first factor, for example
This example is also discussed in Browne (1982).
The matrix specification of the model is

C = F_{1}F'_{1}
with
This example is recomputed by PROC CALIS to illustrate a simple
application of the COSAN model statement combined
with program statements. This example also serves to illustrate the
identification problem.
data Kinzer(TYPE=CORR);
Title "Data Matrix of Kinzer & Kinzer, see GUTTMAN (1957)";
_TYPE_ = 'CORR'; INPUT _NAME_ $ Obs1Obs6;
datalines;
Obs1 1.00 . . . . .
Obs2 .51 1.00 . . . .
Obs3 .46 .51 1.00 . . .
Obs4 .46 .47 .54 1.00 . .
Obs5 .40 .39 .49 .57 1.00 .
Obs6 .33 .39 .47 .45 .56 1.00
;
In a first test run of PROC CALIS, the same model is used as reported
in McDonald (1980). Using the LevenbergMarquardt optimization
algorithm, this example specifies maximum likelihood estimation in the
following code:
proc calis data=Kinzer method=max outram=ram nobs=326;
Title2 "Linearly Related Factor Analysis, (Mcdonald,1980)";
Title3 "Identification Problem";
Cosan F(8,Gen) * I(8,Ide);
Matrix F
[ ,1]= X1X6,
[ ,2]= X7X12,
[1,3]= X13X18;
Parms Alfa = .5 Beta = .5;
X7 = Alfa + Beta * X1;
X8 = Alfa + Beta * X2;
X9 = Alfa + Beta * X3;
X10 = Alfa + Beta * X4;
X11 = Alfa + Beta * X5;
X12 = Alfa + Beta * X6;
Bounds X13X18 >= 0.;
Vnames F Fact1 Fact2 Uvar1Uvar6;
run;
The pattern of the initial values is displayed in vector
and in matrix form. You should always read this output very
carefully, particularly when you use your own programming
statements to constrain the matrix elements. The vector form
shows the mapping of the model parameters to indices of the
vector X that is optimized. The matrix form indicates
parameter elements that are constrained by program statements
by indices of X in angle brackets ( < > ). An asterisk trailing the
iteration number in the displayed optimization history of the
LevenbergMarquardt algorithm indicates that the optimization
process encountered a singular Hessian matrix.
When this happens, especially in the last iterations,
the model may not be properly identified.
The computed value of 10.337 for 7 degrees of
freedom and the computed unique loadings agree with those reported by
McDonald (1980), but the maximum likelihood
estimates for the common factor loadings differ to some degree.
The common factor loadings can be subjected to transformations
that do not increase the value of the optimization criterion
because the problem is not identified.
An estimation problem that is not fully identified can lead to
different solutions caused only by different initial values,
different optimization techniques, or computers with different
machine precision or floatingpoint arithmetic.
To overcome the identification problem in the first model,
restart PROC CALIS with a simple modification to the model in which
the former parameter X1 is fixed to 0. This
leads to 8 instead of 7 degrees of freedom. The following code
produces results that are partially displayed in Output 19.4.1.
data ram2(TYPE=RAM); set ram;
if _type_ = 'ESTIM' then
if _name_ = 'X1' then do;
_name_ = ' '; _estim_ = 0.;
end;
run;
proc calis data=Kinzer method=max inram=ram2 nobs=326;
Title2 "Linearly Related Factor Analysis, (Mcdonald,1980)";
Title3 "Identified Model";
Parms Alfa = .5 Beta = .5;
X7 = Alfa;
X8 = Alfa + Beta * X2;
X9 = Alfa + Beta * X3;
X10 = Alfa + Beta * X4;
X11 = Alfa + Beta * X5;
X12 = Alfa + Beta * X6;
Bounds X13X18 >= 0.;
run;
Output 19.4.1: Linearly Related Factor Analysis: Identification Problem
Data Matrix of Kinzer & Kinzer, see GUTTMAN (1957) 
Linearly Related Factor Analysis, (Mcdonald,1980) 
Identified Model 
The CALIS Procedure 
Covariance Structure Analysis: Pattern and Initial Values 
COSAN Model Statement 

Matrix 
Rows 
Columns 
Matrix Type 
Term 1 
1 
F 
6 
8 
GENERAL 


2 
I 
8 
8 
IDENTITY 

Data Matrix of Kinzer & Kinzer, see GUTTMAN (1957) 
Linearly Related Factor Analysis, (Mcdonald,1980) 
Identified Model 
The CALIS Procedure 
Covariance Structure Analysis: Maximum Likelihood Estimation 
Parameter Estimates 
13 
Functions (Observations) 
21 
Lower Bounds 
6 
Upper Bounds 
0 
Optimization Start 
Active Constraints 
0 
Objective Function 
0.3234289189 
Max Abs Gradient Element 
2.2633860283 
Radius 
5.8468569273 

Data Matrix of Kinzer & Kinzer, see GUTTMAN (1957) 
Linearly Related Factor Analysis, (Mcdonald,1980) 
Identified Model 
The CALIS Procedure 
Covariance Structure Analysis: Maximum Likelihood Estimation 
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 

0.07994 
0.2435 
0.3984 
0 
0.557 
2 

0 
3 
0 

0.03334 
0.0466 
0.0672 
0 
1.202 
3 

0 
4 
0 

0.03185 
0.00150 
0.00439 
0 
1.058 
4 

0 
5 
0 

0.03181 
0.000034 
0.00236 
0 
0.811 
5 

0 
6 
0 

0.03181 
3.982E6 
0.000775 
0 
0.591 
6 

0 
7 
0 

0.03181 
9.275E7 
0.000490 
0 
0.543 
7 

0 
8 
0 

0.03181 
2.402E7 
0.000206 
0 
0.526 
8 

0 
9 
0 

0.03181 
6.336E8 
0.000129 
0 
0.514 
9 

0 
10 
0 

0.03181 
1.687E8 
0.000054 
0 
0.505 
10 

0 
11 
0 

0.03181 
4.521E9 
0.000034 
0 
0.498 
11 

0 
12 
0 

0.03181 
1.217E9 
0.000014 
0 
0.493 
12 

0 
13 
0 

0.03181 
3.29E10 
8.971E6 
0 
0.489 
Optimization Results 
Iterations 
12 
Function Calls 
14 
Jacobian Calls 
13 
Active Constraints 
0 
Objective Function 
0.0318073951 
Max Abs Gradient Element 
8.9711916E6 
Lambda 
0 
Actual Over Pred Change 
0.4888109559 
Radius 
0.0002016088 


ABSGCONV convergence criterion satisfied. 

Data Matrix of Kinzer & Kinzer, see GUTTMAN (1957) 
Linearly Related Factor Analysis, (Mcdonald,1980) 
Identified Model 
The CALIS Procedure 
Covariance Structure Analysis: Maximum Likelihood Estimation 
Fit Function 
0.0318 
Goodness of Fit Index (GFI) 
0.9897 
GFI Adjusted for Degrees of Freedom (AGFI) 
0.9730 
Root Mean Square Residual (RMR) 
0.0409 
Parsimonious GFI (Mulaik, 1989) 
0.5278 
ChiSquare 
10.3374 
ChiSquare DF 
8 
Pr > ChiSquare 
0.2421 
Independence Model ChiSquare 
682.87 
Independence Model ChiSquare DF 
15 
RMSEA Estimate 
0.0300 
RMSEA 90% Lower Confidence Limit 
. 
RMSEA 90% Upper Confidence Limit 
0.0756 
ECVI Estimate 
0.1136 
ECVI 90% Lower Confidence Limit 
. 
ECVI 90% Upper Confidence Limit 
0.1525 
Probability of Close Fit 
0.7137 
Bentler's Comparative Fit Index 
0.9965 
Normal Theory Reweighted LS ChiSquare 
10.1441 
Akaike's Information Criterion 
5.6626 
Bozdogan's (1987) CAIC 
43.9578 
Schwarz's Bayesian Criterion 
35.9578 
McDonald's (1989) Centrality 
0.9964 
Bentler & Bonett's (1980) Nonnormed Index 
0.9934 
Bentler & Bonett's (1980) NFI 
0.9849 
James, Mulaik, & Brett (1982) Parsimonious NFI 
0.5253 
ZTest of Wilson & Hilferty (1931) 
0.7019 
Bollen (1986) Normed Index Rho1 
0.9716 
Bollen (1988) Nonnormed Index Delta2 
0.9965 
Hoelter's (1983) Critical N 
489 

Data Matrix of Kinzer & Kinzer, see GUTTMAN (1957) 
Linearly Related Factor Analysis, (Mcdonald,1980) 
Identified Model 
The CALIS Procedure 
Covariance Structure Analysis: Maximum Likelihood Estimation 
Estimated Parameter Matrix F[6:8] Standard Errors and t Values General Matrix 

Fact1 
Fact2 
Uvar1 
Uvar2 
Uvar3 
Uvar4 
Uvar5 
Uvar6 
Obs1 
0
0
0

0.7151
0.0405
17.6382
<X7> 
0.7283
0.0408
17.8276
[X13] 
0
0
0

0
0
0

0
0
0

0
0
0

0
0
0

Obs2 
0.0543
0.1042
0.5215
[X2] 
0.7294
0.0438
16.6655
<X8> 
0
0
0

0.6707
0.0472
14.2059
[X14] 
0
0
0

0
0
0

0
0
0

0
0
0

Obs3 
0.1710
0.0845
2.0249
[X3] 
0.6703
0.0396
16.9077
<X9> 
0
0
0

0
0
0

0.6983
0.0324
21.5473
[X15] 
0
0
0

0
0
0

0
0
0

Obs4 
0.2922
0.0829
3.5224
[X4] 
0.6385
0.0462
13.8352
<X10> 
0
0
0

0
0
0

0
0
0

0.6876
0.0319
21.5791
[X16] 
0
0
0

0
0
0

Obs5 
0.5987
0.1003
5.9665
[X5] 
0.5582
0.0730
7.6504
<X11> 
0
0
0

0
0
0

0
0
0

0
0
0

0.5579
0.0798
6.9938
[X17] 
0
0
0

Obs6 
0.4278
0.0913
4.6844
[X6] 
0.6029
0.0586
10.2928
<X12> 
0
0
0

0
0
0

0
0
0

0
0
0

0
0
0

0.7336
0.0400
18.3580
[X18] 

Data Matrix of Kinzer & Kinzer, see GUTTMAN (1957) 
Linearly Related Factor Analysis, (Mcdonald,1980) 
Identified Model 
The CALIS Procedure 
Covariance Structure Analysis: Maximum Likelihood Estimation 
Additional PARMS and Dependent Parameters 
The Number of Dependent Parameters is 6 
Parameter 
Estimate 
Standard Error 
t Value 
Alfa 
0.71511 
0.04054 
17.64 
Beta 
0.26217 
0.12966 
2.02 
X7 
0.71511 
0.04054 
17.64 
X8 
0.72936 
0.04376 
16.67 
X9 
0.67027 
0.03964 
16.91 
X10 
0.63851 
0.04615 
13.84 
X11 
0.55815 
0.07296 
7.65 
X12 
0.60295 
0.05858 
10.29 

The lambda value of the iteration history indicates that
Newton steps can always be
performed. Because no singular Hessian matrices (which
can slow down the convergence rate considerably) are computed,
this example needs just 12 iterations compared to the 17 needed
in the previous example. Note that the number of iterations may
be machinedependent.
The value of the fit funciton, the residuals, and the
value agree with the values obtained in fitting
the first model. This indicates that this second model is better
identified than the first one. It is
fully identified, as indicated by the fact that the Hessian
matrix is nonsingular.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.