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 The FACTOR Procedure

## Example 26.3: 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 26.3.1:

```   proc factor data=SocioEconomics method=ml heywood n=1;
title3 'Maximum-Likelihood Factor Analysis with One Factor';
run;
proc factor data=SocioEconomics method=ml heywood n=2;
title3 'Maximum-Likelihood Factor Analysis with Two Factors';
run;
proc factor data=SocioEconomics method=ml heywood n=3;
title3 'Maximum-Likelihood Factor Analysis with Three Factors';
run;
```

Output 26.3.1: Maximum-Likelihood Factor Analysis

 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.

 Maximum-Likelihood Factor Analysis with One Factor

 The FACTOR Procedure Initial Factor Method: Maximum Likelihood

 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 Akaike's Information Criterion 24.356 Schwarz's Bayesian Criterion 21.9314 Tucker and Lewis's Reliability Coefficient 0.120231

 Squared CanonicalCorrelations Factor1 1.0000000

 Eigenvalues of the Weighted ReducedCorrelation 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

 Maximum-Likelihood Factor Analysis with One Factor

 The FACTOR Procedure Initial Factor Method: Maximum Likelihood

 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 VariableWeights 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

Output 26.3.1 displays the results of the analysis with one factor. 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 26.3.2: Maximum-Likelihood Factor Analysis: Two Factors

 Maximum-Likelihood Factor Analysis with Two Factors

 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

 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.

 Maximum-Likelihood Factor Analysis with Two Factors

 The FACTOR Procedure Initial Factor Method: Maximum Likelihood

 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 Akaike's Information Criterion 1.37405 Schwarz's Bayesian Criterion 0.889146 Tucker and Lewis's Reliability Coefficient 0.72922

 Squared Canonical Correlations Factor1 Factor2 1.0000000 0.9518891

 Eigenvalues of the Weighted Reduced CorrelationMatrix: 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

 Maximum-Likelihood Factor Analysis with Two Factors

 The FACTOR Procedure Initial Factor Method: Maximum Likelihood

 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 VariableWeights 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

Output 26.3.2 displays the results of the analysis using two factors. The analysis converges without incident. This time, however, the Population variable is a Heywood case.

Output 26.3.3: Maximum-Likelihood Factor Analysis: Three Factors

 Maximum-Likelihood Factor Analysis with Three Factors

 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

 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.

 Maximum-Likelihood Factor Analysis with Three Factors

 The FACTOR Procedure Initial Factor Method: Maximum Likelihood

 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 Akaike's Information Criterion 4 Schwarz's Bayesian Criterion 4.96981 Tucker and Lewis's Reliability Coefficient 0

 Squared Canonical Correlations Factor1 Factor2 Factor3 1.0000000 0.9751895 0.6894465

 Eigenvalues of the Weighted Reduced CorrelationMatrix: 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

 Maximum-Likelihood Factor Analysis with Three Factors

 The FACTOR Procedure Initial Factor Method: Maximum Likelihood

 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 VariableWeights 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

The three-factor analysis displayed in Output 26.3.3 generates this message:

```   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.

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