### Example 37.3 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 37.3.1 through Output 37.3.3:

```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 37.3.1 displays the results of the analysis with one factor.

Output 37.3.1: Maximum Likelihood Factor Analysis

 Maximum Likelihood Factor Analysis with One Factor

The FACTOR Procedure

Input Data Type Raw Data 12 12 12

 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 34.356 24.356 21.9314 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 37.3.2 displays the results of the analysis with two factors. The analysis converges without incident. This time, however, the `Population` variable is a Heywood case.

Output 37.3.2: Maximum Likelihood Factor Analysis: Two Factors

Input Data Type Raw Data 12 12 12

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

 2 factors will be retained by the NFACTOR criterion.

Iteration Criterion Ridge Change Communalities
1 0.3431221 0.0000 0.0471 1.00000 0.80672 0.95058 0.79348 0.89412
2 0.3072178 0.0000 0.0307 1.00000 0.80821 0.96023 0.81048 0.92480
3 0.3067860 0.0000 0.0063 1.00000 0.81149 0.95948 0.81677 0.92023
4 0.3067373 0.0000 0.0022 1.00000 0.80985 0.95963 0.81498 0.92241
5 0.3067321 0.0000 0.0007 1.00000 0.81019 0.95955 0.81569 0.92187

 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: 2 Factors are sufficient 1 2.1982 0.1382
HA: More factors are needed

 Chi-Square without Bartlett's Correction 3.37405 1.37405 0.889146 0.72922

Squared Canonical Correlations
Factor1 Factor2
1.0000000 0.9518891

Eigenvalues of the Weighted Reduced Correlation Matrix: Total = 19.7853157 Average = 4.94632893
Eigenvalue Difference Proportion Cumulative
1 Infty Infty
2 19.7853143 19.2421292 1.0000 1.0000
3 0.5431851 0.5829564 0.0275 1.0275
4 -0.0397713 0.4636411 -0.0020 1.0254
5 -0.5034124   -0.0254 1.0000

Factor Pattern
Factor1 Factor2
Population 1.00000 0.00000
School 0.00975 0.90003
Employment 0.97245 0.11797
Services 0.43887 0.78930
HouseValue 0.02241 0.95989

Variance Explained by Each Factor
Factor Weighted Unweighted
Factor1 24.4329707 2.13886057
Factor2 19.7853143 2.36835294

Final Communality Estimates and Variable
Weights
Total Communality: Weighted = 44.218285
Unweighted = 4.507214
Variable Communality Weight
Population 1.00000000 Infty
School 0.81014489 5.2682940
Employment 0.95957142 24.7246669
Services 0.81560348 5.4256462
HouseValue 0.92189372 12.7996793

The results of the three-factor analysis are shown in Output 37.3.3.

Output 37.3.3: Maximum Likelihood Factor Analysis: Three Factors

Input Data Type Raw Data 12 12 12

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

 3 factors will be retained by the NFACTOR criterion.

 Warning: Too many factors for a unique solution.

Iteration Criterion Ridge Change Communalities
1 0.1798029 0.0313 0.0501 0.96081 0.84184 1.00000 0.80175 0.89716
2 0.0016405 0.0313 0.0678 0.98081 0.88713 1.00000 0.79559 0.96500
3 0.0000041 0.0313 0.0094 0.98195 0.88603 1.00000 0.80498 0.96751
4 0.0000000 0.0313 0.0006 0.98202 0.88585 1.00000 0.80561 0.96735

 ERROR: Converged, but not to a proper optimum.

 Try a different 'PRIORS' statement.

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: 3 Factors are sufficient -2 0.0000 .
HA: More factors are needed

 Chi-Square without Bartlett's Correction 3e-07 4 4.96981 0

Squared Canonical Correlations
Factor1 Factor2 Factor3
1.0000000 0.9751895 0.6894465

Eigenvalues of the Weighted Reduced Correlation Matrix: Total = 41.5254193 Average = 10.3813548
Eigenvalue Difference Proportion Cumulative
1 Infty Infty
2 39.3054826 37.0854258 0.9465 0.9465
3 2.2200568 2.2199693 0.0535 1.0000
4 0.0000875 0.0002949 0.0000 1.0000
5 -0.0002075   -0.0000 1.0000

Factor Pattern
Factor1 Factor2 Factor3
Population 0.97245 -0.11233 -0.15409
School 0.15428 0.89108 0.26083
Employment 1.00000 0.00000 0.00000
Services 0.51472 0.72416 -0.12766
HouseValue 0.12193 0.97227 -0.08473

Variance Explained by Each Factor
Factor Weighted Unweighted
Factor1 54.6115241 2.24926004
Factor2 39.3054826 2.27634375
Factor3 2.2200568 0.11525433

Final Communality Estimates and Variable
Weights
Total Communality: Weighted = 96.137063
Unweighted = 4.640858
Variable Communality Weight
Population 0.98201660 55.6066901
School 0.88585165 8.7607194
Employment 1.00000000 Infty
Services 0.80564301 5.1444261
HouseValue 0.96734687 30.6251078

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 –2, 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.