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The %MktBSize Macro

Introduction

The %MktBSize autocall macro suggests sizes for balanced incomplete block designs (BIBDs). The sizes that it reports meet necessary but not sufficient conditions for the existence of a BIBD, so a BIBD might not exist for every reported size.

Note: The version of the %MktBSize macro that is documented here requires SAS 940m1 or any subsequent release and the macro is included in the 940m3 release. An earlier version of the %MktBSize macro that is compatible with SAS 9.01 or any subsequent release is also available. The documentation for the earlier version of the macro is published at http://support.sas.com/rnd/app/macros/MktBSize_901/MktBSize_901.htm.

BIBD Parameters

The parameters of a BIBD are as follows:

  • $b$ specifies the number of blocks. In a partial-profile design, this is the number of profiles. In a MaxDiff (best-worst) design, this is the number of sets.

  • $v$ specifies the number of varieties or treatments. In a partial-profile or MaxDiff design, this is the total number of attributes or messages.

  • $k$ specifies the block size, which is the number of treatments in each block. In a partial-profile or MaxDiff design, this is the number of attributes or messages that are shown at one time.

If $r = b k / v$ and $l = r (k - 1) / (v - 1)$ are integers, and $k = v$ and $b \ge v$, then a complete block design might be possible. This is a necessary but not sufficient condition for the existence of a complete block design. If $r = b k / v$ and $l= r (k - 1) / (v - 1)$ are integers, and $k < v$ and $b \ge v$, then a balanced incomplete block design might be possible. This is a necessary but not sufficient condition for the existence of a BIBD.

%MktBSize Macro Syntax

%MktBSize(K=do-list, V=do-list <, optional arguments>)

Required Arguments

K=do-list
SETSIZE=do-list

specifies the block size, or the number of treatments in each block. In a partial-profile or MaxDiff design, this is the number of attributes or messages that are shown at one time in each set. Specify either an integer or a list of integers in the SAS do-list syntax.

V=do-list
T=do-list
NATTRS=do-list

specifies the number of varieties or treatments. In a partial-profile or MaxDiff design, this is the total number of attributes or messages. Specify either an integer or a list of integers in the SAS do-list syntax. When you specify this argument, the output uses the word "Attribute" rather than "Treatment" and the word "Set" rather than "Block."

Optional Arguments

B=do-list
NSETS=do-list

specifies the number of blocks. In a partial-profile design, this is the number of profiles. In a MaxDiff design, this is the number of sets. Specify either an integer or a list of integers in the SAS do-list syntax. By default, B=2 to 500.

MAXREPS=n

specifies the maximum number of replications. By default, MAXREPS=1. By default, this argument prevents the %MktBSize macro from reporting designs of size 2b, 3b, and so on, after it has found a size that has b blocks.

OPTIONS=NOCHECK | UBD

specifies binary arguments. You can specify one or both of the following values:

NOCHECK

turns off certain checks that are performed on b, v, and k by default. This lets you make some creative expressions that otherwise are not permitted.

UBD

lifts the balance restriction on the design. Results are reported when $r = b k / v$ is an integer but $l = r (k - 1) / (v - 1)$ might or might not be an integer. Use this argument when you want to see sizes for which every treatment can occur equally often but the pairwise frequencies can be unequal. The listing can contain both sizes for which a BIBD might be possible ($\lambda $, the expected pairwise frequency, is an integer) and sizes for which a BIBD is not possible ($\lambda $ is not an integer). For example, you might use this argument when the block design is being used to make a partial-profile design.

By default, neither of these arguments is specified.

ORDER=VKB | VBK | BVK | BKV | KBV | KVB
ORDER=TKB | TBK | BTK | BKT | KBT | KTB

specifies the order of the loops; by default, ORDER=VKB. If you specify expressions in the V=, B=, or K= argument, you might need an ordering other than the default. For example, if you specify something like v = 2 to 0.5 * b, then you must specify ORDER=BKV or any other ordering that defines b before v. Alternatively, you can specify this argument to change the default ordering of the results. A "T" in position $i$ is equivalent to a "V" in position $i$.

OUT=SAS-data-set

specifies the output data set to contain the list of potential design sizes. By default, OUT=BIBD.

Help Option

You can specify either of the following to display the option names and simple examples of the macro syntax: 


%mktbsize(help)
%mktbsize(?)

%MktBSize Macro Notes

This macro specifies options nonotes throughout most of its execution. If you want to see all the notes, submit the following statement before running the macro: 


%let mktopts = notes;

To see the macro version, submit the following statement before running the macro: 


%let mktopts = version;

Examples

The following invocation of the %MktBSize macro requests a list of designs that have 12 treatments, between 4 and 8 treatments per block, and between 12 and 30 blocks: 


%mktbsize(v=12, k=4 to 8, b=12 to 30)

The results are shown in Figure 1.

Figure 1: %MktBSize Output


v
Number of
Treatments
k
Block
Size
b
Number
of Blocks
r
Treatment
Frequency
Lambda
Pairwise
Frequencies
n
Total
Sample
Size
12 6 22 11 5 132 



 
You can use this information to create a BIBD by using the %MktBIBD macro as follows: 

%mktbibd(v=12, k=6, b=22, seed=104)

There is no guarantee that the %MktBIBD macro will find a BIBD for every specification, but in this case it does, and it finds the design shown in Figure 2. 
  
Figure 2: BIBD with V=12, K=6, B=22





Balanced Incomplete Block Design






x1
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x5
x6

10
6
8
11
12
7

12
9
3
8
1
6

6
3
7
1
5
12

1
3
2
4
8
9

3
10
12
5
2
4

8
10
2
7
9
5

6
8
5
10
9
3

11
7
6
2
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9

7
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1
12
4
10

9
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12
6
5
11

8
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11
10
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1

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7
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10

2
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1
10
12

12
7
8
3
2
11

7
11
4
5
1
3

10
6
4
3
12
2

5
12
9
8
11
4

5
2
6
7
4
8

4
1
10
9
6
7

2
11
1
4
8
6

1
5
10
11
6
2

9
2
5
12
7
1










 
The design has $b = 22$ blocks (rows), $k = 6$ treatments per block (columns), and $v = 12$ treatments (the entries are the integers 1 to 12). Each of the $v = 12$ treatments occurs $r = 11$ times, and each treatment occurs in a block with every other treatment $\lambda = 5$ times. 
The following invocation of the %MktBSize macro creates a list of more than 50 designs: 

%mktbsize(v=5 to 20, k=3 to v - 1, b=v to 30)

Figure 3 shows a partial listing of the results. 
  
Figure 3: %MktBSize Output









v 
 Number of 
 Treatments		
k 
 Block 
 Size		
b 
 Number 
 of Blocks		
r 
 Treatment 
 Frequency		
Lambda 
 Pairwise 
 Frequencies		
n 
 Total 
 Sample 
 Size		

5
3
10
6
3
30

5
4
5
4
3
20

6
3
10
5
2
30

6
4
15
10
6
60

6
5
6
5
4
30










 
By default, the maximum number of replications is 1 (MAXREPS=1). Therefore, the design in which $v = 5$, $k = 3$, and $b = 20$ is not listed because it consists of two replications of the design in which $v = 5$, $k = 3$, and $b = 10$, which is listed. Also, the number of blocks (b=v to 30) is specified so that it is never less than the number of treatments. Furthermore, the number of treatments per block (k=3 to v-1) is specified to always be less than the number of treatments. Even more complicated expressions are permitted. For example, to limit the number of treatments per block to no more than half the number of treatments, you could specify 

%mktbsize(v=2 to 10, k=2 to 0.5 * v, b=v to 10)

To limit the number of blocks as a function of the number of treatments, you could specify 

%mktbsize(v=2 to 10, k=2 to v - 1, b=v to 2 * v)

However, if you want to limit the number of treatments as a function of the number of blocks, you need to use the ORDER= argument to ensure that the number-of-blocks loop comes first, as in the following specification: 

%mktbsize(b=2 to 10, v=2 to 0.5 * b, k=2 to v - 1, order=bvk)

The macro reports sizes in which $r = b k / v$ and $l = r (k - 1) / (v - 1)$ are integers, $2 \le k < v$, and $b \ge v$. If $r = b k / v$ and $l = r (k - 1) / (v - 1)$ are integers, and $k = v$ and $b \ge v$, then a complete block design might be possible. This is a necessary but not sufficient condition for the existence of a complete block design. If $r = b k / v$ and $l = r (k - 1) / (v - 1)$ are integers, and $k < v$ and $b \ge v$, then a balanced incomplete block design might be possible. This is a necessary but not sufficient condition for the existence of a BIBD. If you specify OPTIONS=UBD and $r = b k / v$ is an integer, then the %MktBSize macro reports unbalanced block design sizes along with BIBD sizes. For example, if you want a design that has 20 treatments and a block size of 6, you can use the following invocation of the %MktBSize macro to find out how many blocks you need: 

%mktbsize(v=20, k=6, options=ubd)

Figure 4 displays the results. 
  
Figure 4: %MktBSize Output










v         
Number of 
Treatments		

k    
Block
Size 		

b        
Number   
of Blocks		

r        
Treatment
Frequency		

Lambda     
Pairwise   
Frequencies		

n     
Total 
Sample
Size  		

20
6
10
3
0.79
60










 
You can then use the %MktBIBD macro to find a design in which each treatment occurs three times but the treatments do not appear together an equal number of times, as in the following example: 

%mktbibd(v=20, k=6, b=10, seed=104)

Figure 5 shows the treatment-by-treatment frequencies and the design. 
  
Figure 5: %MktBIBD Output









Treatment by Treatment Frequencies

















 
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3













Design






x1
x2
x3
x4
x5
x6

12
10
20
9
15
18

7
1
5
19
2
12

2
9
13
20
17
8

19
2
10
6
4
15

9
3
6
13
5
16

11
4
12
7
8
13

1
18
11
14
20
6

3
17
18
4
16
19

14
16
8
5
10
1

15
14
17
3
7
11