The %MktBIBD Macro

The parameters of a BIBD are as follows:

The following BIBD has blocks (rows), treatments, and a block size of (columns):

If and are integers, and and , 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 and are integers, and and , then a balanced incomplete block design might be possible. This is a necessary but not sufficient condition for the existence of a BIBD. You can use the macro %MktBSize to find parameters in which BIBDs might exist. The %MktBIBD macro uses a design catalog, Hadamard matrices, and the OPTEX procedure (which does a computerized search) to find BIBDs. The design catalog includes a BIBD for every combination of , , and that exists. The catalog contains some larger designs as well. PROC OPTEX can easily and reliably find some of the larger designs that are not in the catalog, particularly when is small. For other larger designs that are not in the catalog, the macro might not find a BIBD even when one is known to exist. However, it usually works quite well in finding block designs that are balanced or nearly balanced. The parameters of the first 270 BIBDs in the catalog are listed in Table 1.

%MktBIBD(B=b, K=k, V=v <, optional arguments>)

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

%mktbibd(help)
%mktbibd(?)

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;

The following invocation of the %MktBIBD macro generates a BIBD design that contains 10 blocks, five treatments, and three treatments per block:

%mktbibd(b=10, v=5, k=3, seed=104)

Figure 1 displays the design information. The block design efficiency of 100 indicates that the %MktBIBD macro found a BIBD. In many cases, it finds a design that is close to a BIBD, but each of the pairwise frequencies is not constant. For many marketing research problems, this is good enough.

Figure 2 shows the treatment-by-treatment frequencies. Each of the treatments occurs exactly six times in the design, and each treatment occurs together with every other treatment three times.

When the %MktBIBD macro makes a BIBD, it tries to optimize the treatment-by-position frequencies. In other words, it tries to ensure that each treatment occurs in each of the positions equally often, or at least close to equally often. Figure 3 shows the treatment-by-position frequencies, which are perfect for this design.

Figure 4 shows the design.

Every BIBD also has a binary representation, or incidence matrix, that has rows and columns and ones that indicate which treatments appear in each block. The %MktBIBD macro saves the incidence matrix in the OUTI= data set (which is named Incidence by default). The following statements display the incidence matrix:

proc print data=incidence noobs;
run;

Figure 5 displays the results.

You can also view the BIBD as a block-by-treatment factorial design. By default, the %MktBIBD macro saves the design matrix in a data set named Factorial. The following statements display the first 9 (of 30) observations:

proc print data=factorial(obs=9) noobs;
run;

Figure 6 displays the results.

The following invocation of the %MktBIBD macro generates a BIBD that has nondirectional row-neighbor balance:

%mktbibd(b=14, v=7, k=4, options=neighbor, seed=104)

Figure 7 displays the results.

The resulting design is a BIBD, in which each of the seven treatments appears in each of the four positions within a block exactly twice. Furthermore, the row-neighbor frequencies show that each of the pairs of treatments occurs exactly twice. The pairs in this design are , , , , and so on. The order of the treatments in each pair is ignored. Hence, in this design, with the nondirectional row-neighbor balance, the 2 followed by 5 in the first row of the design is treated the same as the 5 followed by 2 in the second-last row of the design. Usually the macro finds this design easily in a few seconds. The last line of the first table ("Row-Neighbor Frequencies Optimized? Yes") along with the constant row-neighbor frequencies shows that perfect row-neighbor balance was achieved.

The following finds a BIBD that has row-neighbor balance, where the order of the treatments does matter, by specifying OPTIONS=SERIAL:

%mktbibd(b=14, v=7, k=4, options=serial,
         seed=1014116, optiter=100 100 0.1 100 10)

Figure 8 displays the results.

When OPTIONS=SERIAL, the row-neighbor frequencies appear both above and below the diagonal. Each pair of treatments occurs exactly once. In this design, which has the serial (directional) row-neighbor balance, the 2 followed by 5 in the first row of the design is not treated the same as the 5 followed by 2 in the ninth row of the design.

An OPTIONS=SERIAL design is usually much harder to find than an OPTIONS=NEIGHBOR design. In this example, a pseudorandom number seed that is known to produce an optimal design quickly was chosen. You usually have to run the macro more than once or change some options (such as the time value in the POSITER= argument) and iterate for up to a few hours to find an equivalent design.

The following invocation of the %MktBIBD macro requests the same design but specifies a different pseudorandom number seed:

%mktbibd(b=14, v=7, k=4, options=serial, seed=104)

Figure 9 displays the results.

The following requests a design for which it is not possible to have constant frequencies in the row-neighbor frequencies matrix:

%mktbibd(b=7, v=7, k=4, options=serial, seed=104)

Figure 10 displays the results.

The %MktBIBD macro reports that the row-neighbor frequencies are optimized because a mix of zeros and ones with no twos or larger values is optimal for this specification. However, the nonconstant frequencies indicate that row-neighbor balance is not possible for this BIBD.