Getting Started: FACTOR Procedure

The following example demonstrates how you can use the FACTOR procedure to perform common factor analysis and factor rotation.

In this example, 103 police officers were rated by their supervisors on 14 scales (variables). You conduct a common factor analysis on these variables to see what latent factors are operating behind these ratings. The overall rating variable is excluded from the factor analysis.

The following DATA step creates the SAS data set `jobratings`:

```options validvarname=any;
data jobratings;
input ('Communication Skills'n
'Problem Solving'n
'Learning Ability'n
'Judgment Under Pressure'n
'Observational Skills'n
'Willingness to Confront Problems'n
'Interest in People'n
'Interpersonal Sensitivity'n
'Desire for Self-Improvement'n
'Appearance'n
'Dependability'n
'Physical Ability'n
'Integrity'n
'Overall Rating'n) (1.);
datalines;
26838853879867
74758876857667
56757863775875
67869777988997
99997798878888
89897899888799
89999889899798
87794798468886

... more lines ...

99899899899899
76656399567486
;
```

The following statements invoke the FACTOR procedure:

```proc factor data=jobratings(drop='Overall Rating'n) priors=smc
rotate=varimax;
run;
```

The DATA= option in PROC FACTOR specifies the SAS data set `jobratings` as the input data set. The DROP= option drops the `Overall Rating` variable from the analysis. To conduct a common factor analysis, you need to set the prior communality estimate to less than one for each variable. Otherwise, the factor solution would simply be a recast of the principal components solution, in which factors are linear combinations of observed variables. However, in the common factor model you always assume that observed variables are functions of underlying factors. In this example, the PRIORS= option specifies that the squared multiple correlations (SMC) of each variable with all the other variables are used as the prior communality estimates. Note that squared multiple correlations are usually less than one. By default, the principal factor extraction is used if the METHOD= option is not specified. To facilitate interpretations, the ROTATE= option specifies the VARIMAX orthogonal factor rotation to be used.

The output from the factor analysis is displayed in Figure 37.1 through Figure 37.5.

As displayed in Figure 37.1, the prior communality estimates are set to the squared multiple correlations. Figure 37.1 also displays the table of eigenvalues (the variances of the principal factors) of the reduced correlation matrix. Each row of the table pertains to a single eigenvalue. Following the column of eigenvalues are three measures of each eigenvalue’s relative size and importance. The first of these displays the difference between the eigenvalue and its successor. The last two columns display the individual and cumulative proportions that the corresponding factor contributes to the total variation. The last line displayed in Figure 37.1 states that three factors are retained, as determined by the PROPORTION criterion.

Figure 37.1: Table of Eigenvalues from PROC FACTOR

The FACTOR Procedure
Initial Factor Method: Principal Factors

Prior Communality Estimates: SMC
Communication Skills Problem Solving Learning Ability Judgment Under
Pressure
Observational Skills Willingness to
Confront Problems
Interest in People Interpersonal Sensitivity Desire for Self-Improvement Appearance Dependability Physical Ability Integrity
0.62981394 0.58657431 0.61009871 0.63766021 0.67187583 0.64779805 0.75641519 0.75584891 0.57460176 0.45505304 0.63449045 0.42245324 0.68195454

Eigenvalues of the Reduced Correlation Matrix: Total = 8.06463816 Average = 0.62035678
Eigenvalue Difference Proportion Cumulative
1 6.17760549 4.71531946 0.7660 0.7660
2 1.46228602 0.90183348 0.1813 0.9473
3 0.56045254 0.28093933 0.0695 1.0168
4 0.27951322 0.04766016 0.0347 1.0515
5 0.23185305 0.16113428 0.0287 1.0802
6 0.07071877 0.07489624 0.0088 1.0890
7 -.00417747 0.03387533 -0.0005 1.0885
8 -.03805279 0.04776534 -0.0047 1.0838
9 -.08581814 0.02438060 -0.0106 1.0731
10 -.11019874 0.01452741 -0.0137 1.0595
11 -.12472615 0.02356465 -0.0155 1.0440
12 -.14829080 0.05823605 -0.0184 1.0256
13 -.20652684   -0.0256 1.0000

 3 factors will be retained by the PROPORTION criterion.

Figure 37.2 displays the initial factor pattern matrix. The factor pattern matrix represents standardized regression coefficients for predicting the variables by using the extracted factors. Because the initial factors are uncorrelated, the pattern matrix is also equal to the correlations between variables and the common factors.

Figure 37.2: Factor Pattern Matrix from PROC FACTOR

Factor Pattern
Factor1 Factor2 Factor3
Communication Skills 0.75441 0.07707 -0.25551
Problem Solving 0.68590 0.08026 -0.34788
Learning Ability 0.65904 0.34808 -0.25249
Judgment Under Pressure 0.73391 -0.21405 -0.23513
Observational Skills 0.69039 0.45292 0.10298
Willingness to Confront Problems 0.66458 0.47460 0.09210
Interest in People 0.70770 -0.53427 0.10979
Interpersonal Sensitivity 0.64668 -0.61284 -0.07582
Desire for Self-Improvement 0.73820 0.12506 0.09062
Appearance 0.57188 0.20052 0.16367
Dependability 0.79475 -0.04516 0.16400
Physical Ability 0.51285 0.10251 0.34860
Integrity 0.74906 -0.35091 0.18656

The pattern matrix suggests that `Factor1` represents general ability. All loadings for `Factor1` in the Factor Pattern are at least 0.5. `Factor2` consists of high positive loadings on certain task-related skills (`Willingness to Confront Problems`, `Observational Skills`, and `Learning Ability`) and high negative loadings on some interpersonal skills (`Interpersonal Sensitivity`, `Interest in People`, and `Integrity`). This factor measures individuals’ relative strength in these skills. Theoretically, individuals with high positive scores on this factor would exhibit better task-related skills than interpersonal skills. Individuals with high negative scores would exhibit better interpersonal skills than task-related skills. Individuals with scores near zero would have those skills balanced. `Factor3` does not have a cluster of very high or very low factor loadings. Therefore, interpreting this factor is difficult.

Figure 37.3 displays the proportion of variance explained by each factor and the final communality estimates, including the total communality. The final communality estimates are the proportion of variance of the variables accounted for by the common factors. When the factors are orthogonal, the final communalities are calculated by taking the sum of squares of each row of the factor pattern matrix.

Figure 37.3: Variance Explained and Final Communality Estimates

Variance Explained by Each Factor
Factor1 Factor2 Factor3
6.1776055 1.4622860 0.5604525

Final Communality Estimates: Total = 8.200344
Communication Skills Problem Solving Learning Ability Judgment Under
Pressure
Observational Skills Willingness to
Confront Problems
Interest in People Interpersonal Sensitivity Desire for Self-Improvement Appearance Dependability Physical Ability Integrity
0.64036292 0.59791844 0.61924167 0.63972863 0.69237485 0.67538695 0.79833968 0.79951357 0.56879171 0.39403630 0.66056907 0.39504805 0.71903222

Figure 37.4 displays the results of the VARIMAX rotation of the three extracted factors and the corresponding orthogonal transformation matrix. The rotated factor pattern matrix is calculated by postmultiplying the original factor pattern matrix (Figure 37.4) by the transformation matrix.

Figure 37.4: Transformation Matrix and Rotated Factor Pattern

Orthogonal Transformation Matrix
1 2 3
1 0.59125 0.59249 0.54715
2 -0.80080 0.51170 0.31125
3 0.09557 0.62219 -0.77701

Rotated Factor Pattern
Factor1 Factor2 Factor3
Communication Skills 0.35991 0.32744 0.63530
Problem Solving 0.30802 0.23102 0.67058
Learning Ability 0.08679 0.41149 0.66512
Judgment Under Pressure 0.58287 0.17901 0.51764
Observational Skills 0.05533 0.70488 0.43870
Willingness to Confront Problems 0.02168 0.69391 0.43978
Interest in People 0.85677 0.21422 0.13562
Interpersonal Sensitivity 0.86587 0.02239 0.22200
Desire for Self-Improvement 0.34498 0.55775 0.37242
Appearance 0.19319 0.54327 0.24814
Dependability 0.52174 0.54981 0.29337
Physical Ability 0.25445 0.57321 0.04165
Integrity 0.74172 0.38033 0.15567

The rotated factor pattern matrix is somewhat simpler to interpret. If a magnitude of at least 0.5 is required to indicate a salient variable-factor relationship, `Factor1` now represents interpersonal skills (`Interpersonal Sensitivity`, `Interest in People`, `Integrity`, `Judgment Under Pressure`, and `Dependability`). `Factor2` measures physical skills and job enthusiasm (`Observational Skills`, `Willingness to Confront Problems`, `Physical Ability`, `Desire for Self-Improvement`, `Dependability`, and `Appearance`). `Factor3` measures cognitive skills (`Communication Skills`, `Problem Solving`, `Learning Ability`, and `Judgment Under Pressure`).

However, using 0.5 for determining a salient variable-factor relationship does not take sampling variability into account. If the underlying assumptions for the maximum likelihood estimation are approximately satisfied, you can output standard error estimates and the confidence intervals with METHOD=ML. You can then determine the salience of the variable-factor relationship by using the coverage displays. See the section Confidence Intervals and the Salience of Factor Loadings for more details.

Figure 37.5 displays the variance explained by each factor and the final communality estimates after the orthogonal rotation. Even though the variances explained by the rotated factors are different from that of the unrotated factor (compare with Figure 37.3), the cumulative variance explained by the common factors remains the same. Note also that the final communalities for variables, as well as the total communality, remain unchanged after rotation. Although rotating a factor solution will not increase or decrease the statistical quality of the factor model, it can simplify the interpretations of the factors and redistribute the variance explained by the factors.

Figure 37.5: Variance Explained and Final Communality Estimates after Rotation

Variance Explained by Each Factor
Factor1 Factor2 Factor3
3.1024330 2.7684489 2.3294622

Final Communality Estimates: Total = 8.200344
Communication Skills Problem Solving Learning Ability Judgment Under
Pressure
Observational Skills Willingness to
Confront Problems
Interest in People Interpersonal Sensitivity Desire for Self-Improvement Appearance Dependability Physical Ability Integrity
0.64036292 0.59791844 0.61924167 0.63972863 0.69237485 0.67538695 0.79833968 0.79951357 0.56879171 0.39403630 0.66056907 0.39504805 0.71903222