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 The PLAN Procedure

## 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

 The PLAN Procedure

 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

 The PLAN Procedure

 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

 The PLAN Procedure

 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

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