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The FACTOR Procedure

Example 33.3 Maximum Likelihood Factor Analysis

This example uses maximum likelihood factor analyses for one, two, and three factors. It is already apparent from the principal factor analysis that the best number of common factors is almost certainly two. The one- and three-factor ML solutions reinforce this conclusion and illustrate some of the numerical problems that can occur. The following statements produce Output 33.3.1 through Output 33.3.3:

   title3 'Maximum Likelihood Factor Analysis with One Factor';
   proc factor data=SocioEconomics method=ml heywood n=1;
   run;

   title3 'Maximum Likelihood Factor Analysis with Two Factors';
   proc factor data=SocioEconomics method=ml heywood n=2;
   run;

   title3 'Maximum Likelihood Factor Analysis with Three Factors';
   proc factor data=SocioEconomics method=ml heywood n=3;
   run;

Output 33.3.1 displays the results of the analysis with one factor.

Output 33.3.1 Maximum Likelihood Factor Analysis

Five Socioeconomic Variables

See Page 14 of Harman: Modern Factor Analysis, 3rd Ed

Maximum Likelihood Factor Analysis with One Factor

The FACTOR Procedure

Initial Factor Method: Maximum Likelihood

Prior Communality Estimates: SMC
Population	School	Employment	Services	HouseValue
0.96859160	0.82228514	0.96918082	0.78572440	0.84701921

Preliminary Eigenvalues: Total = 76.1165859 Average = 15.2233172
	Eigenvalue	Difference	Proportion	Cumulative
1	63.7010086	50.6462895	0.8369	0.8369
2	13.0547191	12.7270798	0.1715	1.0084
3	0.3276393	0.6749199	0.0043	1.0127
4	-0.3472805	0.2722202	-0.0046	1.0081
5	-0.6195007		-0.0081	1.0000

1 factor will be retained by the NFACTOR criterion.

Iteration	Criterion	Ridge	Change	Communalities
1	6.5429218	0.0000	0.1033	0.93828	0.72227	1.00000	0.71940	0.74371
2	3.1232699	0.0000	0.7288	0.94566	0.02380	1.00000	0.26493	0.01487

Convergence criterion satisfied.

Significance Tests Based on 12 Observations
Test	DF	Chi-Square	Pr > ChiSq
H0: No common factors	10	54.2517	<.0001
HA: At least one common factor
H0: 1 Factor is sufficient	5	24.4656	0.0002
HA: More factors are needed

Chi-Square without Bartlett's Correction	34.355969
Akaike's Information Criterion	24.355969
Schwarz's Bayesian Criterion	21.931436
Tucker and Lewis's Reliability Coefficient	0.120231

Squared Canonical Correlations
Factor1
1.0000000

Eigenvalues of the Weighted Reduced Correlation Matrix: Total = 0 Average = 0
	Eigenvalue	Difference
1	Infty	Infty
2	1.92716032	2.15547340
3	-.22831308	0.56464322
4	-.79295630	0.11293464
5	-.90589094

Factor Pattern
	Factor1
Population	0.97245
School	0.15428
Employment	1.00000
Services	0.51472
HouseValue	0.12193

Variance Explained by Each Factor
Factor	Weighted	Unweighted
Factor1	17.8010629	2.24926004

Final Communality Estimates and Variable Weights
Total Communality: Weighted = 17.801063 Unweighted = 2.249260
Variable	Communality	Weight
Population	0.94565561	18.4011648
School	0.02380349	1.0243839
Employment	1.00000000	Infty
Services	0.26493499	1.3604239
HouseValue	0.01486595	1.0150903

The solution on the second iteration is so close to the optimum that PROC FACTOR cannot find a better solution; hence you receive this message:

   Convergence criterion satisfied.

When this message appears, you should try rerunning PROC FACTOR with different prior communality estimates to make sure that the solution is correct. In this case, other prior estimates lead to the same solution or possibly to worse local optima, as indicated by the information criteria or the chi-square values.

The variable Employment has a communality of 1.0 and, therefore, an infinite weight that is displayed next to the final communality estimate as a missing/infinite value. The first eigenvalue is also infinite. Infinite values are ignored in computing the total of the eigenvalues and the total final communality.

Output 33.3.2 displays the results of the analysis with two factors. The analysis converges without incident. This time, however, the Population variable is a Heywood case.

The results of the three-factor analysis are shown in Output 33.3.3.

In the results, a warning message is displayed:

   WARNING:  Too many factors for a unique solution.

The number of parameters in the model exceeds the number of elements in the correlation matrix from which they can be estimated, so an infinite number of different perfect solutions can be obtained. The criterion approaches zero at an improper optimum, as indicated by this message:

   Converged, but not to a proper optimum.

The degrees of freedom for the chi-square test are , so a probability level cannot be computed for three factors. Note also that the variable Employment is a Heywood case again.

The probability levels for the chi-square test are 0.0001 for the hypothesis of no common factors, 0.0002 for one common factor, and 0.1382 for two common factors. Therefore, the two-factor model seems to be an adequate representation. Akaike’s information criterion and Schwarz’s Bayesian criterion attain their minimum values at two common factors, so there is little doubt that two factors are appropriate for these data.

Copyright © 2009 by SAS Institute Inc., Cary, NC, USA. All rights reserved.