The following example uses the PRINCOMP procedure to analyze job performance. Police officers were rated by their supervisors in 14 categories as part of standard police departmental administrative procedure.
The following statements create the Jobratings data set:
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 . . . 99899899899899 76656399567486 ;
The data set Jobratings contains 14 variables . Each variable contains the job ratings using a scale measurement from 1 to 10 (1=fail to comply , 10=exceptional). The last variable Overall Rating contains a score as an overall index on how each officer performs .
The following statement requests a principal component analysis on the Jobratings data set and outputs the scores to the Scores data set (OUT= Scores ). Note that variable Overall Rating is excluded from the analysis.
proc princomp data=Jobratings(drop='Overall Rating'n) out=scores; run;
Figure 58.1 to Figure 58.3 display the PROC PRINCOMP output, beginning with simple statistics followed by the correlation matrix. The PROC PRINCOMP statement requests by default principal components computed from the correlation matrix, so the total variance is equal to the number of variables, 13. In this example, it would also be reasonable to use the COV option, which would cause variables with a high variance (such as Dependability )tohavemoreinfluence on the results than variables with a low variance (such as Learning Ability ). If you used the COV option, scores would be computed from centered rather than standardized variables.
The PRINCOMP Procedure Observations 103 Variables 13 Simple Statistics Judgment Communication Problem Learning Under Observational Skills Solving Ability Pressure Skills Mean 6.650485437 6.631067961 6.990291262 6.737864078 6.932038835 StD 1.764068036 1.590352602 1.339411238 1.731830976 1.761584269 Simple Statistics Willingness to Confront Interest Interpersonal Desire for Problems in People Sensitivity Self-Improvement Appearance Mean 7.291262136 6.708737864 6.621359223 6.572815534 7.000000000 StD 1.525155524 1.892353385 1.760773587 1.729796212 1.798692335 Simple Statistics Physical Dependability Ability Integrity Mean 6.825242718 7.203883495 7.213592233 StD 1.917040123 1.555251845 1.845240223
The PRINCOMP Procedure Correlation Matrix Judgment Communication Problem Learning Under Skills Solving Ability Pressure Communication Skills 1.0000 0.6280 0.5546 0.5538 Problem Solving 0.6280 1.0000 0.5690 0.6195 Learning Ability 0.5546 0.5690 1.0000 0.4892 Judgment Under Pressure 0.5538 0.6195 0.4892 1.0000 Observational Skills 0.5381 0.4284 0.6230 0.3733 Willingness to Confront Problems 0.5265 0.5015 0.5245 0.4004 Interest in People 0.4391 0.3972 0.2735 0.6226 Interpersonal Sensitivity 0.5030 0.4398 0.1855 0.6134 Desire for Self-Improvement 0.5642 0.4090 0.5737 0.4826 Appearance 0.4913 0.3873 0.3988 0.2266 Dependability 0.5471 0.4546 0.5110 0.5471 Physical Ability 0.2192 0.3201 0.2269 0.3476 Integrity 0.5081 0.3846 0.3142 0.5883 Correlation Matrix Willingness Interest Observational to Confront in Skills Problems People Communication Skills 0.5381 0.5265 0.4391 Problem Solving 0.4284 0.5015 0.3972 Learning Ability 0.6230 0.5245 0.2735 Judgment Under Pressure 0.3733 0.4004 0.6226 Observational Skills 1.0000 0.7300 0.2616 Willingness to Confront Problems 0.7300 1.0000 0.2233 Interest in People 0.2616 0.2233 1.0000 Interpersonal Sensitivity 0.1655 0.1291 0.8051 Desire for Self-Improvement 0.5985 0.5307 0.4857 Appearance 0.4177 0.4825 0.2679 Dependability 0.5626 0.4870 0.6074 Physical Ability 0.4274 0.4872 0.3768 Integrity 0.3906 0.3260 0.7452
Correlation Matrix Interpersonal Desire for Sensitivity Self-Improvement Appearance Communication Skills 0.5030 0.5642 0.4913 Problem Solving 0.4398 0.4090 0.3873 Learning Ability 0.1855 0.5737 0.3988 Judgment Under Pressure 0.6134 0.4826 0.2266 Observational Skills 0.1655 0.5985 0.4177 Willingness to Confront Problems 0.1291 0.5307 0.4825 Interest in People 0.8051 0.4857 0.2679 Interpersonal Sensitivity 1.0000 0.3713 0.2600 Desire for Self-Improvement 0.3713 1.0000 0.4474 Appearance 0.2600 0.4474 1.0000 Dependability 0.5408 0.5981 0.5089 Physical Ability 0.2182 0.3752 0.3820 Integrity 0.6920 0.5664 0.4135 Correlation Matrix Physical Dependability Ability Integrity Communication Skills 0.5471 0.2192 0.5081 Problem Solving 0.4546 0.3201 0.3846 Learning Ability 0.5110 0.2269 0.3142 Judgment Under Pressure 0.5471 0.3476 0.5883 Observational Skills 0.5626 0.4274 0.3906 Willingness to Confront Problems 0.4870 0.4872 0.3260 Interest in People 0.6074 0.3768 0.7452 Interpersonal Sensitivity 0.5408 0.2182 0.6920 Desire for Self-Improvement 0.5981 0.3752 0.5664 Appearance 0.5089 0.3820 0.4135 Dependability 1.0000 0.4461 0.6536 Physical Ability 0.4461 1.0000 0.3810 Integrity 0.6536 0.3810 1.0000
The PRINCOMP Procedure Eigenvalues of the Correlation Matrix Eigenvalue Difference Proportion Cumulative 1 6.54740242 4.77468744 0.5036 0.5036 2 1.77271499 0.76747933 0.1364 0.6400 3 1.00523565 0.26209665 0.0773 0.7173 4 0.74313901 0.06479499 0.0572 0.7745 5 0.67834402 0.22696368 0.0522 0.8267 6 0.45138034 0.06922167 0.0347 0.8614 7 0.38215866 0.08432613 0.0294 0.8908 8 0.29783254 0.02340663 0.0229 0.9137 9 0.27442591 0.01208809 0.0211 0.9348 10 0.26233782 0.01778332 0.0202 0.9550 11 0.24455450 0.04677622 0.0188 0.9738 12 0.19777828 0.05508241 0.0152 0.9890 13 0.14269586 0.0110 1.0000
Figure 58.5 and Figure 58.6 display the eigenvectors. From the eigenvectors matrix, you can represent the first principal component Prin1 as a linear combination of the original variables
The PRINCOMP Procedure Eigenvectors Prin1 Prin2 Prin3 Prin4 Communication Skills 0.303548 0.052039 .329181 .227039 Problem Solving 0.278034 0.057046 .400112 0.300476 Learning Ability 0.266521 0.288152 .354591 .020735 Judgment Under Pressure 0.294376 .199458 .255164 0.397306 Observational Skills 0.276641 0.366979 0.065959 0.035711 Willingness to Confront Problems 0.267580 0.392989 0.098723 0.184409 Interest in People 0.278060 .432916 0.118113 0.046047 Interpersonal Sensitivity 0.253814 .495662 .064547 .060000 Desire for Self-Improvement 0.299833 0.099077 0.061097 .211279 Appearance 0.237358 0.190065 0.248353 .544587 Dependability 0.319480 .049742 0.169476 .156070 Physical Ability 0.213868 0.097499 0.614959 0.514519 Integrity 0.298246 .301812 0.190222 .169062 Eigenvectors Prin5 Prin6 Prin7 Prin8 Communication Skills 0.181087 .416563 0.143543 0.333846 Problem Solving 0.453604 0.096750 0.048904 0.199259 Learning Ability .219329 0.578388 .114808 0.064088 Judgment Under Pressure .030188 0.102087 0.068204 .591822 Observational Skills .325257 .301254 .297894 0.163484 Willingness to Confront Problems 0.038278 .458585 .044796 .365684 Interest in People .111279 0.030870 .011105 0.154829 Interpersonal Sensitivity 0.107807 .170305 .088194 0.192725 Desire for Self-Improvement .427477 0.105369 0.689011 0.087453 Appearance 0.568044 0.221643 0.049267 .257497 Dependability .130575 0.202301 .594850 0.081242 Physical Ability 0.203995 0.173168 0.169247 0.302536 Integrity .130757 .100039 0.029456 .317545
Eigenvectors Prin9 Prin10 Prin11 Prin12 Communication Skills .430955 0.375983 0.028370 .252778 Problem Solving 0.256098 .372914 .434417 0.069863 Learning Ability 0.224706 0.287031 0.210540 .284355 Judgment Under Pressure .358618 0.178270 0.118318 0.306490 Observational Skills 0.258377 0.223793 .079692 0.565290 Willingness to Confront Problems 0.129976 .330710 0.275249 .386151 Interest in People 0.321200 .081470 0.393841 .210915 Interpersonal Sensitivity 0.137468 .074821 0.285447 0.276824 Desire for Self-Improvement .121474 .363854 .052085 0.151436 Appearance 0.087395 0.061890 0.168369 0.236655 Dependability .495598 .377561 .164909 .090904 Physical Ability .149625 0.258321 .006202 .055828 Integrity 0.271060 0.297010 .612497 .276273 Eigenvectors Prin13 Communication Skills .122809 Problem Solving .116642 Learning Ability 0.248555 Judgment Under Pressure .126636 Observational Skills .168555 Willingness to Confront Problems 0.177688 Interest in People .610215 Interpersonal Sensitivity 0.643410 Desire for Self-Improvement 0.053834 Appearance .113705 Dependability .018094 Physical Ability 0.133430 Integrity 0.114965
Prin1 =0 . 303548 — ( Communication Skills )
+0 . 278034 — ( Problem Solving )
+0 . 266521 — ( Learning Ability )
.
.
.
+0 . 298246 — ( Integrity )
and, similarly, the second principal component Prin2 is
Prin2 =0 . 052039 — ( Communication Skills )
+0 . 057046 — ( Problem Solving )
+0 . 288152 — ( Learning Ability )
.
.
.
ˆ’ . 301812 — ( Integrity )
where the variables are standardized.
The first component reflects overall performance since the first eigenvector shows approximately equal loadings on all variables. The second eigenvector has high positive loadings on the variables Observational Skills and Willingness to Confront Problems but even higher negative loadings on the variables Interest in People and Interpersonal Sensitivity. This component seems to reflect the ability to take action, but it also reflects a lack of interpersonal skills. The third eigenvector has a very high positive loading on the variable Physical Ability and high negative loadings on the variables Problem Solving and Learning Ability. This component seems to reflect physical strength, but also shows poor learning and problem-solving skills.
In short, the three components represent:
First Component: overall performance
Second Component: smart, tough, and introverted
Third Component: superior strength and average intellect