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 35.1 through Figure 35.5.
As displayed in Figure 35.1, the prior communality estimates are set to the squared multiple correlations. Figure 35.1 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 35.1 states that three factors are retained, as determined by the PROPORTION criterion.
Figure 35.1: Table of Eigenvalues from PROC FACTOR
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 |
Figure 35.2 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 35.2: 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 35.3 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 35.3: 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 35.4 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 35.4) by the transformation matrix.
Figure 35.4: 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 35.5 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 35.3), 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 35.5: 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 |