Details of the FACTEX Procedure

## BLOCKS Statement

BLOCKS block-specification ;
You use the BLOCKS statement to specify the number of blocks in the design or the size of each block in the design. By default, the FACTEX procedure constructs designs that do not contain blocks. If you use the BLOCKS statement, you also need to use the MODEL statement or SIZE statement. In particular, if you use the BLOCKS statement and your design is a fractional factorial design, you must use the MODEL statement.

The two simplest explicit block-specifications that you can use are
• NBLOCKS=b, which specifies the number of blocks (b) in the design
• SIZE=k, which specifies the number of runs (k) in each block

Use only one of these two options. In all, there are six mutually exclusive block-specifications that you can use, as described by the following list:

NBLKFACS=s
specifies the number of block pseudo-factors for the design. The design contains a different block for each possible combination of the levels of the block pseudo-factors. Values of s are the integers 1, 2, and so on. See "Block Size Restrictions" for details.

If each factor in the design has q levels, then NBLKFACS=s specifies a design with qs blocks. The size of each block depends on the number of runs in the design, as specified in the SIZE statement. If the design has n runs, then each block has n/qs runs.

The following statement illustrates how to request a two-level factorial design arranged in eight (23) blocks:

` blocks nblkfacs=3;`

For more on pseudo-factors, see "Types of Factors" .

NBLOCKS=b
specifies the number of blocks in the design. The values of b must be a power of q, the number of levels of each factor in the design. See "Block Size Restrictions" for details. The size of each block depends on the number of runs in the design, as specified in the SIZE statement. If the design has n runs, then each block has n/b runs. See "Example of a Full Factorial Design in Two Blocks" for an illustration of this option.

The following statement illustrates how to specify a design arranged in four blocks:

` blocks nblocks=4;`

SIZE=k
specifies the number of runs per block in the design. The value k must be a power of q, the number of levels for each factor in the design. The number of blocks depends on the number of runs in the design, as specified in the SIZE statement. If the design has n runs, then it has n/k blocks.

CAUTION: Do not confuse the SIZE= option in the BLOCKS statement with the SIZE statement, which you use to specify the overall size of the design. See "SIZE Statement" for details of the SIZE statement.

The following statement illustrates how to specify blocks of size two:

` blocks size=2;`

NBLKFACS=MAXIMUM
NBLOCKS=MAXIMUM
SIZE=MINIMUM
constructs a blocked design with the minimum number of runs per block, given all the other characteristics of the design. In other words, the block size is optimized. You cannot specify this option if you specify any of the design size optimization options in the SIZE statement (see DESIGN=MINIMUM).

### Equivalence of Specifications

The three explicit block-specifications are related to each other, as demonstrated by the following example.

Suppose you want to construct a design for 11 two-level factors in 128 runs in blocks of size 8. Since 128/24=128/16=8, three equivalent block specifications are

```   blocks nblkfacs=4;
blocks nblocks=16;
blocks size=8;
```

### Block Size Restrictions

The number of blocks and the number of runs in each block must be less than the total number of runs in the design. Hence, there are some restrictions on the block size.
• If you use SIZE=k or NBLOCKS=b, the numbers you specify for k and b must be less than or equal to the size of the design, as specified in the SIZE statement. Or, if you do not use a SIZE statement, k and b must be less than or equal to the number of runs for a full replication of all possible combinations of the factors.

For example, for a 23 design you cannot specify a design arranged in 8 blocks (NBLOCKS=8). Likewise, you cannot construct a design with block size greater than 8 (SIZE=8).

• If you use NBLKFACS=s, the value of s can be no greater than the number of run-indexing factors, which give the number of runs needed to index the design. For details, see "Types of Factors" and Chapter 16,"Theory of Orthogonal Designs" .