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 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, each service does not 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 by 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 53.2 in Chapter 53, "The PRINQUAL Procedure," for an alternative analysis of these data.

Since the first principal component accounts for 78 percent of the variance, there is substantial agreement among the rankings. The eigenvector shows that all the news services are about equally weighted, so a simple average would work almost as well as the first principal component. The following statements produce Output 52.2.1 through Output 52.2.3:

```   /*----------------------------------------------------------*/
/*                                                          */
/* Preseason 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.                    */
/*                                                          */
/*----------------------------------------------------------*/
title1 'Preseason 1985 College Basketball Rankings';
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.                             */

proc means data=HoopsRanks;
output out=MaxRank
max=CSNMax DurSunMax DurHerMax
WashPostMax USATodayMax SportMax
InSportsMax UPIMax APMax SIMax;
run;

/* The following method of filling in missing values is a   */
/* reasonable method for this specific example.  It would   */
/* be inappropriate to use this method for other data sets. */
/* sets.  In addition, any method of filling in missing     */
/* values can result in incorrect statistics. The choice    */
/* of whether to fill in missing values, and what method    */
/* to use to do so, is the responsibility of the person     */
/* performing the analysis.                                 */

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;

/* Use the PRINCOMP procedure to transform the observed */
/* ranks. Use n=1 because the data should be related to */
/* a single underlying variable. Sort the data and      */
/* display the resulting component.                     */                */

standard;
var CSN--SI;
weight Weight;
run;

by Prin1;
run;

proc print;
var School Prin1;
title2 'College Teams as Ordered by PROC PRINCOMP';
run;
```

Output 52.2.1: 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 15.5000000 10.5000000 12.5000000 10.4210526 13.0000000 10.5000000 10.5000000 10.0000000 10.5000000 10.5000000 8.8034084 5.9160798 7.0710678 6.0673607 7.3598007 5.9160798 5.9160798 5.6273143 5.9160798 5.9160798 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 30.0000000 20.0000000 24.0000000 20.0000000 25.0000000 20.0000000 20.0000000 19.0000000 20.0000000 20.0000000

Output 52.2.2: Principal Components Analysis of Basketball Rankings Using PROC PRINCOMP

 The PRINCOMP Procedure

 Observations 35 Variables 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

 The PRINCOMP Procedure

 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 52.2.3: 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

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