Getting Started | SAS/STAT 9.1 Users Guide Volume 2 only

The following example demonstrates how you can use the FACTOR procedure to perform common factor analysis and use a transformation to rotate the extracted factors.

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   35652335143113   89888879576867   76557899446397   97889998898989   76766677598888   77667676779677   63839932588856   25738811284915   88879966797988   87979877959679   87989975878798   99889988898888   78876765687677   88889888899899   88889988878988   67646577384776   78778788799997   76888866768667   67678665746776   33424476664855   65656765785766   54566676565866   56655566656775   88889988868887   89899999898799   98889999899899   57554776468878   53687777797887   68666716475767   78778889798997   67364767565846   77678865886767   68698955669998   55546866663886   68888999998989   97787888798999   76677899799997   44754687877787   77876678798888   76668778799797   57653634361543   76777745653656   76766665656676   88888888878789   88977888869778   58894888747886   58674565473676   76777767777777   77788878789798   98989987999868   66729911474713   98889976999988   88786856667748   77868887897889   99999986999999   46688587616886   66755778486776   87777788889797   65666656545976   73574488887687   74755556586596   76677778789797   87878746777667   86776955874877   77888767778678   65778787778997   58786887787987   65787766676778   86777875468777   67788877757777   77778967855867   67887876767777   24786585535866   46532343542533   35566766676784   11231214211211   76886588536887   57784788688589   56667766465666   66787778778898   77687998877997   76668888546676   66477987589998   86788976884597   77868765785477   99988888987888   65948933886457   99999877988898   96636736876587   98676887798968   87878877898979   88897888888788   99997899799799   99899899899899   76656399567486   ;

The following statements invoke the FACTOR procedure:

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

The DATA= optionin 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 the following figures.

As displayed in Figure 27.1, the prior communality estimates are set to the squared multiple correlations. Figure 27.1 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 27.1 states that three factors are retained, as determined by the PROPORTION criterion.

  The FACTOR Procedure   Initial Factor Method: Principal Factors   Prior Communality Estimates: SMC   Judgment   Communication         Problem        Learning           Under      Observational   Skills         Solving         Ability        Pressure             Skills   0.62981394      0.58657431      0.61009871      0.63766021         0.67187583   Willingness   to Confront        Interest      Interpersonal            Desire for   Problems       in People        Sensitivity      Self-Improvement   0.64779805      0.75641519         0.75584891            0.57460176   Physical   Appearance      Dependability         Ability       Integrity   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 27.1: Table of Eigenvalues from PROC FACTOR

Figure 27.2 displays the initial factor pattern matrix. The factor pattern matrix represents standardized regression coefficients for predicting the variables 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.

  Initial Factor Method: Principal Factors   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

Figure 27.2: Factor Pattern Matrix from PROC FACTOR

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 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 27.3 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.

  Initial Factor Method: Principal Factors   Variance Explained by Each Factor   Factor1         Factor2         Factor3   6.1776055       1.4622860       0.5604525   Final Communality Estimates: Total = 8.200344   Judgment   Communication         Problem        Learning           Under      Observational   Skills         Solving         Ability        Pressure             Skills   0.64036292      0.59791844      0.61924167      0.63972863         0.69237485   Willingness   to Confront        Interest      Interpersonal            Desire for   Problems       in People        Sensitivity      Self-Improvement   0.67538695      0.79833968         0.79951357            0.56879171   Physical   Appearance      Dependability         Ability       Integrity   0.39403630         0.66056907      0.39504805      0.71903222

Figure 27.3: Variance Explained and Final Communality Estimates

Figure 27.4 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 27.2) by the transformation matrix.

  Rotation Method: Varimax   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

Figure 27.4: Transformation Matrix and Rotated Factor Pattern

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 using the coverage displays. See the section Confidence Intervals and the Salience of Factor Loadings on page 1327 for more details.

Figure 27.5 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 27.3), 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 may simplify the interpretations of the factors and redistribute the variance explained by the factors.

  Rotation Method: Varimax   Variance Explained by Each Factor   Factor1         Factor2         Factor3   3.1024330       2.7684489       2.3294622   Final Communality Estimates: Total = 8.200344   Judgment   Communication         Problem        Learning           Under      Observational   Skills         Solving         Ability        Pressure             Skills   0.64036292      0.59791844      0.61924167      0.63972863         0.69237485   Willingness   to Confront        Interest      Interpersonal            Desire for   Problems       in People        Sensitivity      Self-Improvement   0.67538695      0.79833968         0.79951357            0.56879171   Physical   Appearance      Dependability         Ability       Integrity   0.39403630         0.66056907      0.39504805      0.71903222

Figure 27.5: Variance Explained and Final Communality Estimates after Rotation