# The PRINCOMP Procedure

The data in this example are rankings of 35 college basketball teams. The rankings were made before the start of the 1985–86 season by 10 news services.

The purpose of the principal component analysis is to compute a single variable that best summarizes all 10 of the preseason rankings.

Note that the various news services rank different numbers of teams, varying from 20 through 30 (there is a missing rank in one of the variables, `WashPost`). And, of course, not all services rank the same teams, so there are missing values in these data. Each of the 35 teams is ranked by at least one news service.

The PRINCOMP procedure omits observations with missing values. To obtain principal component scores for all of the teams, it is necessary to replace the missing values. Since it is the best teams that are ranked, it is not appropriate to replace missing values with the mean of the nonmissing values. Instead, an ad hoc method is used that replaces missing values with the mean of the unassigned ranks. For example, if 20 teams are ranked by a news service, then ranks 21 through 35 are unassigned. The mean of ranks 21 through 35 is 28, so missing values for that variable are replaced by the value 28. To prevent the method of missing-value replacement from having an undue effect on the analysis, each observation is weighted according to the number of nonmissing values it has. See Example 74.2 in Chapter 74: The PRINQUAL Procedure, for an alternative analysis of these data.

Since the first principal component accounts for 78% of the variance, there is substantial agreement among the rankings. The eigenvector shows that all the news services are about equally weighted; this is also suggested by the nearly horizontal line of the pattern profile plot in Output 73.2.3. So a simple average would work almost as well as the first principal component. The following statements produce Output 73.2.1.

```/*-----------------------------------------------------------*/
/*                                                           */
/* Pre-season 1985 College Basketball Rankings               */
/* (rankings of 35 teams by 10 news services)                */
/*                                                           */
/* Note: (a) news services rank varying numbers of teams;    */
/*       (b) not all teams are ranked by all news services;  */
/*       (c) each team is ranked by at least one service;    */
/*       (d) rank 20 is missing for UPI.                     */
/*                                                           */
/*-----------------------------------------------------------*/

data HoopsRanks;
input School \$13. CSN DurSun DurHer WashPost USAToday
Sport InSports UPI AP SI;
label CSN      = 'Community Sports News (Chapel Hill, NC)'
DurSun   = 'Durham Sun'
DurHer   = 'Durham Morning Herald'
WashPost = 'Washington Post'
USAToday = 'USA Today'
Sport    = 'Sport Magazine'
InSports = 'Inside Sports'
UPI      = 'United Press International'
AP       = 'Associated Press'
SI       = 'Sports Illustrated'
;
format CSN--SI 5.1;
datalines;
Louisville     1  8  1  9  8  9  6 10  9  9
Georgia Tech   2  2  4  3  1  1  1  2  1  1
Kansas         3  4  5  1  5 11  8  4  5  7
Michigan       4  5  9  4  2  5  3  1  3  2
Duke           5  6  7  5  4 10  4  5  6  5
UNC            6  1  2  2  3  4  2  3  2  3
Syracuse       7 10  6 11  6  6  5  6  4 10
Notre Dame     8 14 15 13 11 20 18 13 12  .
Kentucky       9 15 16 14 14 19 11 12 11 13
LSU           10  9 13  . 13 15 16  9 14  8
DePaul        11  . 21 15 20  . 19  .  . 19
Georgetown    12  7  8  6  9  2  9  8  8  4
Navy          13 20 23 10 18 13 15  . 20  .
Illinois      14  3  3  7  7  3 10  7  7  6
Iowa          15 16  .  . 23  .  . 14  . 20
Arkansas      16  .  .  . 25  .  .  .  . 16
Memphis State 17  . 11  . 16  8 20  . 15 12
Washington    18  .  .  .  .  .  . 17  .  .
UAB           19 13 10  . 12 17  . 16 16 15
UNLV          20 18 18 19 22  . 14 18 18  .
NC State      21 17 14 16 15  . 12 15 17 18
Maryland      22  .  .  . 19  .  .  . 19 14
Pittsburgh    23  .  .  .  .  .  .  .  .  .
Oklahoma      24 19 17 17 17 12 17  . 13 17
Indiana       25 12 20 18 21  .  .  .  .  .
Virginia      26  . 22  .  . 18  .  .  .  .
Old Dominion  27  .  .  .  .  .  .  .  .  .
Auburn        28 11 12  8 10  7  7 11 10 11
St. Johns     29  .  .  .  . 14  .  .  .  .
UCLA          30  .  .  .  .  .  . 19  .  .
St. Joseph's   .  . 19  .  .  .  .  .  .  .
Tennessee      .  . 24  .  . 16  .  .  .  .
Montana        .  .  . 20  .  .  .  .  .  .
Houston        .  .  .  . 24  .  .  .  .  .
Virginia Tech  .  .  .  .  .  . 13  .  .  .
;
```
```/* PROC MEANS is used to output a data set containing the      */
/* maximum value of each of the newspaper and magazine         */
/* rankings.  The output data set, maxrank, is then used       */
/* to set the missing values to the next highest rank plus     */
/* thirty-six, divided by two (that is, the mean of the        */
/* missing ranks).  This ad hoc method of replacing missing    */
/* values is based more on intuition than on rigorous          */
/* statistical theory.  Observations are weighted by the       */
/* number of nonmissing values.                                */
/*                                                             */

title 'Pre-Season 1985 College Basketball Rankings';
proc means data=HoopsRanks;
output out=MaxRank
max=CSNMax DurSunMax DurHerMax
WashPostMax USATodayMax SportMax
InSportsMax UPIMax APMax SIMax;
run;
```

Output 73.2.1: Summary Statistics for Basketball Rankings Using PROC MEANS

The MEANS Procedure

Variable Label N Mean Std Dev Minimum Maximum
CSN
DurSun
DurHer
WashPost
USAToday
Sport
InSports
UPI
AP
SI
Community Sports News (Chapel Hill, NC)
Durham Sun
Durham Morning Herald
Washington Post
USA Today
Sport Magazine
Inside Sports
United Press International
Associated Press
Sports Illustrated
 30 20 24 19 25 20 20 19 20 20
 15.5 10.5 12.5 10.4211 13 10.5 10.5 10 10.5 10.5
 8.80341 5.91608 7.07107 6.06736 7.3598 5.91608 5.91608 5.62731 5.91608 5.91608
 1 1 1 1 1 1 1 1 1 1
 30 20 24 20 25 20 20 19 20 20

The following statements produce Output 73.2.2 and Output 73.2.3:

```data Basketball;
set HoopsRanks;
if _n_=1 then set MaxRank;
array Services{10} CSN--SI;
array MaxRanks{10} CSNMax--SIMax;
keep School CSN--SI Weight;
Weight=0;
do i=1 to 10;
if Services{i}=. then Services{i}=(MaxRanks{i}+36)/2;
else Weight=Weight+1;
end;
run;
```
```ods graphics on;

plots=patternprofile;
var CSN--SI;
weight Weight;
run;
```

Output 73.2.2: Principal Components Analysis of Basketball Rankings Using PROC PRINCOMP

The PRINCOMP Procedure

 Observations 35 10

Simple Statistics
CSN DurSun DurHer WashPost USAToday Sport InSports UPI AP SI
Mean 13.33640553 13.06451613 12.88018433 13.83410138 12.55760369 13.83870968 13.24423963 13.59216590 12.83410138 13.52534562
StD 22.08036285 21.66394183 21.38091837 23.47841791 20.48207965 23.37756267 22.20231526 23.25602811 21.40782406 22.93219584

Correlation Matrix
CSN DurSun DurHer WashPost USAToday Sport InSports UPI AP SI
CSN Community Sports News (Chapel Hill, NC) 1.0000 0.6505 0.6415 0.6121 0.7456 0.4806 0.6558 0.7007 0.6779 0.6135
DurSun Durham Sun 0.6505 1.0000 0.8341 0.7667 0.8860 0.6940 0.7702 0.9015 0.8437 0.7518
DurHer Durham Morning Herald 0.6415 0.8341 1.0000 0.7035 0.8877 0.7788 0.7900 0.7676 0.8788 0.7761
WashPost Washington Post 0.6121 0.7667 0.7035 1.0000 0.7984 0.6598 0.8717 0.6953 0.7809 0.5952
USAToday USA Today 0.7456 0.8860 0.8877 0.7984 1.0000 0.7716 0.8475 0.8539 0.9479 0.8426
Sport Sport Magazine 0.4806 0.6940 0.7788 0.6598 0.7716 1.0000 0.7176 0.6220 0.8217 0.7701
InSports Inside Sports 0.6558 0.7702 0.7900 0.8717 0.8475 0.7176 1.0000 0.7920 0.8830 0.7332
UPI United Press International 0.7007 0.9015 0.7676 0.6953 0.8539 0.6220 0.7920 1.0000 0.8436 0.7738
AP Associated Press 0.6779 0.8437 0.8788 0.7809 0.9479 0.8217 0.8830 0.8436 1.0000 0.8212
SI Sports Illustrated 0.6135 0.7518 0.7761 0.5952 0.8426 0.7701 0.7332 0.7738 0.8212 1.0000

Eigenvalues of the Correlation Matrix
Eigenvalue Difference Proportion Cumulative
1 7.88601647   0.7886 0.7886

Eigenvectors
Prin1
CSN Community Sports News (Chapel Hill, NC) 0.270205
DurSun Durham Sun 0.326048
DurHer Durham Morning Herald 0.324392
WashPost Washington Post 0.300449
USAToday USA Today 0.345200
Sport Sport Magazine 0.293881
InSports Inside Sports 0.324088
UPI United Press International 0.319902
AP Associated Press 0.342151
SI Sports Illustrated 0.308570

Output 73.2.3: Pattern Profile Plot

The following statements produce Output 73.2.4:

```proc sort data=PCBasketball;
by Prin1;
run;
```
```proc print;
var School Prin1;
title 'Pre-Season 1985 College Basketball Rankings';
title2 'College Teams as Ordered by PROC PRINCOMP';
run;
```

Output 73.2.4: Basketball Rankings Using PROC PRINCOMP

 Pre-Season 1985 College Basketball Rankings College Teams as Ordered by PROC PRINCOMP

Obs School Prin1
1 Georgia Tech -0.58068
2 UNC -0.53317
3 Michigan -0.47874
4 Kansas -0.40285
5 Duke -0.38464
6 Illinois -0.33586
7 Syracuse -0.31578
8 Louisville -0.31489
9 Georgetown -0.29735
10 Auburn -0.09785
11 Kentucky 0.00843
12 LSU 0.00872
13 Notre Dame 0.09407
14 NC State 0.19404
15 UAB 0.19771
16 Oklahoma 0.23864
17 Memphis State 0.25319
18 Navy 0.28921
19 UNLV 0.35103
20 DePaul 0.43770
21 Iowa 0.50213
22 Indiana 0.51713
23 Maryland 0.55910
24 Arkansas 0.62977
25 Virginia 0.67586
26 Washington 0.67756
27 Tennessee 0.70822
28 St. Johns 0.71425
29 Virginia Tech 0.71638
30 St. Joseph's 0.73492
31 UCLA 0.73965
32 Pittsburgh 0.75078
33 Houston 0.75534
34 Montana 0.75790
35 Old Dominion 0.76821