# The PRINCOMP Procedure

### Example 91.3 Analyzing Job Ratings of Police Officers

This example uses the PRINCOMP procedure to analyze job performance. Police officers were rated by their supervisors in 14 categories as part of standard police department 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;
datalines;
2 6 8 3 8 8 5 3 8 7 9 8 6 7 7 4 7 5 8 8 7 6 8 5 7 6 6 7 5 6 7 5 7 8 6 3 7 7 5
8 7 5 6 7 8 6 9 7 7 7 9 8 8 9 9 7 9 9 9 9 7 7 9 8 8 7 8 8 8 8 8 9 8 9 7 8 9 9
8 8 8 7 9 9 8 9 9 9 9 8 8 9 8 9 9 7 9 8 8 7 7 9 4 7 9 8 4 6 8 8 8 6 3 5 6 5 2

... more lines ...

7 8 9 9 7 9 9 7 9 9 9 9 8 9 9 8 9 9 8 9 9 8 9 9 7 6 6 5 6 3 9 9 5 6 7 4 8 6
;
```

The `Jobratings` data set contains 14 variables. Each variable contains the job ratings, which use 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 of how each officer performs.

The following statements request a principal component analysis of the `Jobratings` data set, output the scores to the `Scores` data set (OUT= `Scores`), and produce default plots. Note that the variable `Overall Rating` is excluded from the analysis.

```ods graphics on;

proc princomp data=Jobratings(drop='Overall Rating'n);
run;
```

Output 91.3.1 and Output 91.3.2 display the PROC PRINCOMP output, beginning with simple statistics and then the correlation matrix. By default, PROC PRINCOMP computes principal components 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 that have a high variance (such as `Dependability`) to influence the results more than variables that have a low variance (such as `Learning Ability`). If you used the COV option, scores would be computed from centered rather than standardized variables.

Output 91.3.1: Simple Statistics and Correlation Matrix from Using PROC PRINCOMP

The PRINCOMP Procedure

 Observations 37 13

Simple Statistics
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
Mean 6.567567568 6.675675676 6.891891892 6.378378378 7.081081081 6.756756757 6.675675676 6.540540541 7.027027027 7.135135135 7.027027027 7.162162162 7.081081081
StD 1.878837414 1.748873511 1.696135866 2.252792728 1.816259563 2.126622327 1.871631108 2.218540494 1.707605316 1.436859271 1.499749729 1.343988953 1.460182226

Correlation Matrix
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
Communication Skills 1.0000 0.7254 0.3685 0.6107 0.4338 0.5708 0.4646 0.2975 0.0211 -.0086 -.2619 -.1145 -.2096
Problem Solving 0.7254 1.0000 0.6715 0.6877 0.6207 0.6504 0.3828 0.3113 0.1890 0.1064 -.0389 -.0361 -.1852
Learning Ability 0.3685 0.6715 1.0000 0.5126 0.7603 0.3545 0.1024 0.3112 0.3079 0.1885 0.0121 0.0932 -.1085
Judgment Under Pressure 0.6107 0.6877 0.5126 1.0000 0.5761 0.6227 0.5635 0.4915 0.1489 0.1382 -.1347 -.1217 -.1025
Observational Skills 0.4338 0.6207 0.7603 0.5761 1.0000 0.4655 0.2449 0.4921 0.4113 0.0915 -.1640 0.0741 -.0549
Willingness to Confront Problems 0.5708 0.6504 0.3545 0.6227 0.4655 1.0000 0.4751 0.2170 0.1931 0.1111 0.2286 0.1114 -.1813
Interest in People 0.4646 0.3828 0.1024 0.5635 0.2449 0.4751 1.0000 0.5652 0.3765 0.2750 0.1220 0.0215 0.1115
Interpersonal Sensitivity 0.2975 0.3113 0.3112 0.4915 0.4921 0.2170 0.5652 1.0000 0.5460 0.4121 0.0790 0.1747 0.1747
Desire for Self-Improvement 0.0211 0.1890 0.3079 0.1489 0.4113 0.1931 0.3765 0.5460 1.0000 0.5645 0.2166 0.3248 0.3667
Appearance -.0086 0.1064 0.1885 0.1382 0.0915 0.1111 0.2750 0.4121 0.5645 1.0000 0.5525 0.3479 0.4183
Dependability -.2619 -.0389 0.0121 -.1347 -.1640 0.2286 0.1220 0.0790 0.2166 0.5525 1.0000 0.5628 0.3415
Physical Ability -.1145 -.0361 0.0932 -.1217 0.0741 0.1114 0.0215 0.1747 0.3248 0.3479 0.5628 1.0000 0.5027
Integrity -.2096 -.1852 -.1085 -.1025 -.0549 -.1813 0.1115 0.1747 0.3667 0.4183 0.3415 0.5027 1.0000

Output 91.3.2 displays the eigenvalues. The first principal component accounts for about 50% of the total variance, the second principal component accounts for about 13.6%, and the third principal component accounts for about 7.7%. Note that the eigenvalues sum to the total variance. The eigenvalues indicate that three to five components provide a good summary of the data: three components account for about 71.7% of the total variance, and five components account for about 82.7%. Subsequent components account for less than 5% each.

Output 91.3.2: Eigenvalues and Eigenvectors from Using PROC PRINCOMP

Eigenvalues of the Correlation Matrix
Eigenvalue Difference Proportion Cumulative
1 4.69468687 1.81899683 0.3611 0.3611
2 2.87569003 1.67100277 0.2212 0.5823
3 1.20468727 0.03118935 0.0927 0.6750
4 1.17349791 0.45846322 0.0903 0.7653
5 0.71503470 0.15713583 0.0550 0.8203
6 0.55789887 0.09269082 0.0429 0.8632
7 0.46520805 0.04118763 0.0358 0.8990
8 0.42402041 0.13454552 0.0326 0.9316
9 0.28947489 0.06869311 0.0223 0.9539
10 0.22078178 0.03221769 0.0170 0.9708
11 0.18856410 0.06620108 0.0145 0.9853
12 0.12236302 0.05427092 0.0094 0.9948
13 0.06809210   0.0052 1.0000

Eigenvectors
Prin1 Prin2 Prin3 Prin4 Prin5 Prin6 Prin7 Prin8 Prin9 Prin10 Prin11 Prin12 Prin13
Communication Skills 0.323548 -.236730 0.206727 0.092655 0.293138 0.260352 -.215988 -.550645 -.050648 0.107002 0.262509 0.341232 0.291574
Problem Solving 0.383857 -.160898 -.091224 0.212751 0.025258 0.252518 -.140816 -.104392 0.283104 0.221940 -.548010 -.492803 -.073999
Learning Ability 0.322899 -.050464 -.553565 0.056656 -.138393 0.168405 0.150062 0.055518 0.391053 -.223399 0.132338 0.442471 -.307096
Judgment Under Pressure 0.379958 -.142821 0.155157 -.025467 0.043612 0.175269 0.361045 0.391055 -.315796 -.392714 -.286021 0.111225 0.382730
Observational Skills 0.359246 -.067434 -.424397 -.148191 0.093417 -.221005 0.022944 0.177808 -.141401 0.225326 0.502509 -.416669 0.278776
Willingness to Confront Problems 0.333754 -.064285 0.183338 0.459764 -.024447 -.304704 -.247094 0.259896 -.387665 0.158552 0.047611 0.168464 -.459746
Interest in People 0.296160 0.082187 0.575827 -.140226 0.023973 -.159653 -.015476 0.131682 0.540942 -.277206 0.299254 -.197252 -.112818
Interpersonal Sensitivity 0.302693 0.180810 0.119231 -.432281 -.047507 -.238610 0.501550 -.303435 -.097727 0.393688 -.196906 0.137833 -.222427
Desire for Self-Improvement 0.225795 0.344251 -.123236 -.333516 -.174557 -.266896 -.621875 0.020842 0.018350 -.105222 -.293349 0.219662 0.263644
Appearance 0.158341 0.425329 0.052469 -.022665 -.441729 0.494677 -.051864 -.204081 -.350816 -.186793 0.226289 -.256802 -.177399
Dependability 0.025597 0.427337 0.079019 0.520679 -.289013 -.044047 0.221520 0.079762 0.250326 0.336689 0.049300 0.146711 0.445532
Physical Ability 0.052980 0.418985 -.185687 0.312555 0.486621 -.299641 0.145579 -.340453 -.072184 -.432711 -.090520 -.154868 -.034075
Integrity -.006172 0.435225 0.015874 -.147905 0.578186 0.421421 -.087126 0.396179 0.030130 0.284351 0.021483 0.113790 -.129601

PROC PRINCOMP produces the scree plot as shown in Output 91.3.3 by default when ODS Graphics is enabled. You can obtain more plots by specifying the PLOTS= option in the PROC PRINCOMP statement.

The scree plot on the left shows that the eigenvalue of the first component is approximately 6.5 and the eigenvalue of the second component is largely decreased to under 2.0. The variance explained plot on the right shows that the first four principal components account for nearly 80% of the total variance.

Output 91.3.3: Scree Plot from Using PROC PRINCOMP

The first component reflects overall performance, because 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 it also shows poor learning and problem-solving skills.

In short, the three components represent the following:

First component:

overall performance

Second component:

smartness, toughness, and introversion

Third component:

superior strength and average intellect

PROC PRINCOMP also produces other plots besides the scree plot, that help interpret the results. The following statements request plots from the PRINCOMP procedure:

```proc princomp data=Jobratings(drop='Overall Rating'n)
n=5 plots(ncomp=3)=all;
run;
```

The N=5 option sets the number of principal components to five. The option PLOTS(NCOMP=3)=ALL produces all plots but limits to three the number of components that are displayed in the component pattern plots and the component score plots.

Output 91.3.4 shows a matrix plot of component scores for the first five principal components. The histogram of each component is displayed in the diagonal element of the matrix. The histograms indicate that the first principal component is skewed to the left and the second principal component is slightly skewed to the right.

Output 91.3.4: Matrix Plot of Component Scores

The pairwise component pattern plots are shown in Output 91.3.5 through Output 91.3.7. The pattern plots show the following:

• All variables positively and evenly correlate with the first principal component (Output 91.3.5 and Output 91.3.6).

• The variables `Observational Skills` and `Willingness to Confront Problems` correlate highly with the second component, and the variables `Interest in People` and `Interpersonal Sensitivity` correlate highly but negatively with the second component (Output 91.3.5).

• The variable `Physical Ability` correlates highly with the third component, and the variables `Problem Solving` and `Learning Ability` correlate highly but negatively with the third component (Output 91.3.6).

• The variables `Observational Skills`, `Willingness to Confront Problems`, `Interest in People`, and `Interpersonal Sensitivity` correlate highly (either positively or negatively) with the second component, but all these variables have very low correlations with the third component; the variables `Physical Ability` and `Problem Solving` correlate highly (either positively or negatively) with the third component, but both variables have very low correlations with the second component (Output 91.3.7).

Output 91.3.5: Pattern Plot of Component 2 by Component 1

Output 91.3.6: Pattern Plot of Component 3 by Component 1

Output 91.3.7: Pattern Plot of Component 3 by Component 2

Output 91.3.8 shows a component pattern profile. As is shown in the pattern plots, the nearly horizontal profile from the first component indicates that the first component is mostly correlated evenly across all variables.

Output 91.3.8: Component Pattern Profile Plot from Using PROC PRINCOMP

Output 91.3.9 through Output 91.3.11 display the pairwise component score plots. Observation numbers are used as the plotting symbol.

Output 91.3.9 shows a scatter plot of the first and second components. Observations 4 and 31 seem like outliers on the first component. Observations 22 and 30 can be potential outliers on the second component.

Output 91.3.9: Component 2 versus Component 1

Output 91.3.10 shows a scatter plot of the first and third components. Observations 4 and 31 seem like outliers on the first component.

Output 91.3.10: Component 3 versus Component 1

Output 91.3.11 shows a scatter plot of the second and third components. Observations 22 and 30 can be potential outliers on the second component.

Output 91.3.11: Component 3 versus Component 2

Output 91.3.12 shows a scatter plot of the second and third components, displaying the first component in color. Color interpolation ranges from red (minimum) to blue (middle) to green (maximum).

Output 91.3.12: Component 3 versus Component 2, Painted by Component 1