Example 50.6: Permutations and Combinations
Occasionally, you may need to generate all possible permutations of
n things, or all possible combinations of n things taken m at a
time.
For example, suppose you are planning an experiment in cognitive
psychology where you want to present four successive stimuli to each
subject. You want to observe each permutation of the four stimuli.
The following statements use PROC PLAN to create a data set containing
all possible permutations of 4 numbers in random order.
title 'All Permutations of 1,2,3,4';
proc plan seed=60359;
factors Subject = 24
Order = 4 ordered;
treatments Stimulus = 4 perm;
output out=Psych;
proc sort data=Psych out=Psych;
by Subject Order;
proc tabulate formchar=' ' noseps;
class Subject Order;
var Stimulus;
table Subject, Order*(Stimulus*f=8.)*sum=' ' / rts=9;
run;
The variable Subject is set at 24 levels because there are 4!=24
total permutations to be listed. If Subject>24, the list repeats.
Output 50.6.1 displays the PROC PLAN output. Note that
the variable Subject is listed in random order.
Output 50.6.1: List of Permutations
|
| All Permutations of 1,2,3,4 |
| Plot Factors |
| Factor |
Select |
Levels |
Order |
| Subject |
24 |
24 |
Random |
| Order |
4 |
4 |
Ordered |
| Treatment Factors |
| Factor |
Select |
Levels |
Order |
| Stimulus |
4 |
4 |
Perm |
|
|
| All Permutations of 1,2,3,4 |
| Subject |
Order |
Stimulus |
| 4 |
1 |
2 |
3 |
4 |
1 |
2 |
3 |
4 |
| 15 |
1 |
2 |
3 |
4 |
1 |
2 |
4 |
3 |
| 24 |
1 |
2 |
3 |
4 |
1 |
3 |
2 |
4 |
| 1 |
1 |
2 |
3 |
4 |
1 |
3 |
4 |
2 |
| 5 |
1 |
2 |
3 |
4 |
1 |
4 |
2 |
3 |
| 17 |
1 |
2 |
3 |
4 |
1 |
4 |
3 |
2 |
| 19 |
1 |
2 |
3 |
4 |
2 |
1 |
3 |
4 |
| 14 |
1 |
2 |
3 |
4 |
2 |
1 |
4 |
3 |
| 6 |
1 |
2 |
3 |
4 |
2 |
3 |
1 |
4 |
| 23 |
1 |
2 |
3 |
4 |
2 |
3 |
4 |
1 |
| 8 |
1 |
2 |
3 |
4 |
2 |
4 |
1 |
3 |
| 2 |
1 |
2 |
3 |
4 |
2 |
4 |
3 |
1 |
| 13 |
1 |
2 |
3 |
4 |
3 |
1 |
2 |
4 |
| 16 |
1 |
2 |
3 |
4 |
3 |
1 |
4 |
2 |
| 12 |
1 |
2 |
3 |
4 |
3 |
2 |
1 |
4 |
| 18 |
1 |
2 |
3 |
4 |
3 |
2 |
4 |
1 |
| 21 |
1 |
2 |
3 |
4 |
3 |
4 |
1 |
2 |
| 9 |
1 |
2 |
3 |
4 |
3 |
4 |
2 |
1 |
| 22 |
1 |
2 |
3 |
4 |
4 |
1 |
2 |
3 |
| 10 |
1 |
2 |
3 |
4 |
4 |
1 |
3 |
2 |
| 7 |
1 |
2 |
3 |
4 |
4 |
2 |
1 |
3 |
| 11 |
1 |
2 |
3 |
4 |
4 |
2 |
3 |
1 |
| 3 |
1 |
2 |
3 |
4 |
4 |
3 |
1 |
2 |
| 20 |
1 |
2 |
3 |
4 |
4 |
3 |
2 |
1 |
|
The output data set Psych contains 96 observations of the 3
variables (Subject, Order, and Stimulus).
Sorting the output data set by Subject and by
Order within Subject
results in all possible permutations of Stimulus in random order.
PROC TABULATE displays these permutations in
Output 50.6.2.
Output 50.6.2: Randomized Permutations
|
| All Permutations of 1,2,3,4 |
| |
Order |
| 1 |
2 |
3 |
4 |
| Stimulus |
Stimulus |
Stimulus |
Stimulus |
| Subject |
1 |
3 |
4 |
2 |
| 1 |
| 2 |
2 |
4 |
3 |
1 |
| 3 |
4 |
3 |
1 |
2 |
| 4 |
1 |
2 |
3 |
4 |
| 5 |
1 |
4 |
2 |
3 |
| 6 |
2 |
3 |
1 |
4 |
| 7 |
4 |
2 |
1 |
3 |
| 8 |
2 |
4 |
1 |
3 |
| 9 |
3 |
4 |
2 |
1 |
| 10 |
4 |
1 |
3 |
2 |
| 11 |
4 |
2 |
3 |
1 |
| 12 |
3 |
2 |
1 |
4 |
| 13 |
3 |
1 |
2 |
4 |
| 14 |
2 |
1 |
4 |
3 |
| 15 |
1 |
2 |
4 |
3 |
| 16 |
3 |
1 |
4 |
2 |
| 17 |
1 |
4 |
3 |
2 |
| 18 |
3 |
2 |
4 |
1 |
| 19 |
2 |
1 |
3 |
4 |
| 20 |
4 |
3 |
2 |
1 |
| 21 |
3 |
4 |
1 |
2 |
| 22 |
4 |
1 |
2 |
3 |
| 23 |
2 |
3 |
4 |
1 |
| 24 |
1 |
3 |
2 |
4 |
|
As another example, suppose you have six alternative treatments, any
four of which can occur together in a block (in no particular order).
The following statements use PROC PLAN to create a data set containing
all possible combinations of six numbers taken four at a time. In this
case, you use ODS to create the data set.
title 'All Combinations of (6 Choose 4) Integers';
ods output Plan=Combinations;
proc plan;
factors Block=15 ordered
Treat= 4 of 6 comb;
run;
proc print data=Combinations noobs;
run;
The variable Block has 15 levels since there are a total of
6!/(4!2!)=15 combinations of four integers chosen from six integers.
The data set formed by ODS from the displayed plan has one row for each
block, with the four values of Treat corresponding to four different
variables, as shown in Output 50.6.3.
Output 50.6.3: List of Combinations
|
| All Combinations of (6 Choose 4) Integers |
| Factor |
Select |
Levels |
Order |
| Block |
15 |
15 |
Ordered |
| Treat |
4 |
6 |
Comb |
| Block |
Treat |
| 1 |
1 |
2 |
3 |
4 |
| 2 |
1 |
2 |
3 |
5 |
| 3 |
1 |
2 |
3 |
6 |
| 4 |
1 |
2 |
4 |
5 |
| 5 |
1 |
2 |
4 |
6 |
| 6 |
1 |
2 |
5 |
6 |
| 7 |
1 |
3 |
4 |
5 |
| 8 |
1 |
3 |
4 |
6 |
| 9 |
1 |
3 |
5 |
6 |
| 10 |
1 |
4 |
5 |
6 |
| 11 |
2 |
3 |
4 |
5 |
| 12 |
2 |
3 |
4 |
6 |
| 13 |
2 |
3 |
5 |
6 |
| 14 |
2 |
4 |
5 |
6 |
| 15 |
3 |
4 |
5 |
6 |
|
Output 50.6.4: Combinations Data Set Created by ODS
|
| All Combinations of (6 Choose 4) Integers |
| Block |
Treat1 |
Treat2 |
Treat3 |
Treat4 |
| 1 |
1 |
2 |
3 |
4 |
| 2 |
1 |
2 |
3 |
5 |
| 3 |
1 |
2 |
3 |
6 |
| 4 |
1 |
2 |
4 |
5 |
| 5 |
1 |
2 |
4 |
6 |
| 6 |
1 |
2 |
5 |
6 |
| 7 |
1 |
3 |
4 |
5 |
| 8 |
1 |
3 |
4 |
6 |
| 9 |
1 |
3 |
5 |
6 |
| 10 |
1 |
4 |
5 |
6 |
| 11 |
2 |
3 |
4 |
5 |
| 12 |
2 |
3 |
4 |
6 |
| 13 |
2 |
3 |
5 |
6 |
| 14 |
2 |
4 |
5 |
6 |
| 15 |
3 |
4 |
5 |
6 |
|
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.
