Randomization :: SAS/QC(R) 12.3 User's Guide

Randomization

In many experiments, proper randomization is crucial to the validity of the conclusions. Randomization neutralizes the effects of systematic biases that might be involved in implementing the design and provides a basis for the assumptions underlying the analysis. Refer to Kempthorne (1975) for a discussion.

The way in which randomization is handled depends on whether the design involves blocking:

For designs without block factors, proper randomization consists of randomly permuting the overall order of the runs and randomly assigning the actual levels of each factor to the theoretical levels it has for the purpose of constructing the design.
For designs with block factors, proper randomization calls for first performing separate random permutations for the runs within each block, and then randomly permuting the order in which the blocks are run.

For example, suppose you generate a full factorial design for three 2-level factors A, B, and C in eight runs. The following steps are involved in randomizing this design:

Randomly permute the order of the runs:

$\displaystyle \mbox{Runs:}\ \{ 1,2,3,4,5,6,7,8\}$

$\displaystyle \rightarrow$

$\displaystyle \{ 3,8,1,2,4,7,6,5\}$

Randomly assign the actual levels to the theoretical levels for each factor:

$\displaystyle \mbox{Factor A levels:}\ \{ 0,1\}$	$\displaystyle \rightarrow$	$\displaystyle \{ 1,-1\}$
$\displaystyle \mbox{Factor B levels:}\ \{ 0,1\}$	$\displaystyle \rightarrow$	$\displaystyle \{ 1,-1\}$
$\displaystyle \mbox{Factor C levels:}\ \{ 0,1\}$	$\displaystyle \rightarrow$	$\displaystyle \{ -1, 1\}$

Thus, the effect of the randomization is to transform the original design, as follows:

Run	A	B	C
1	0	0	0
2	0	0	1
3	0	1	0
4	0	1	1
5	1	0	0
6	1	0	1
7	1	1	0
8	1	1	1

$\longrightarrow$

Run	A	B	C
3	1	–1	–1
8	–1	–1	1
1	1	1	–1
2	1	1	1
4	1	–1	1
7	–1	–1	–1
6	–1	1	1
5	–1	1	–1

If the original design is in two blocks, then the first step is replaced with the following two steps:

Randomly permute the order of the runs within each block:

$\displaystyle \mbox{Block 1 runs:}\ \{ 1,2,3,4\}$	$\displaystyle \rightarrow$	$\displaystyle \{ 4,1,2,3\}$
$\displaystyle \mbox{Block 2 runs:}\ \{ 5,6,7,8\}$	$\displaystyle \rightarrow$	$\displaystyle \{ 8,7,6,5\}$

Randomly permute the order of the blocks:

$\displaystyle \mbox{Block levels:}\ \{ 1,2\}$

$\displaystyle \rightarrow$

$\displaystyle \{ 2,1\}$

The resulting transformation is shown in the following:

Run	Block	A	B	C
1	1	0	0	0
2	1	0	1	1
3	1	1	0	1
4	1	1	1	0
5	2	0	0	1
6	2	0	1	0
7	2	1	0	0
8	2	1	1	1

$\longrightarrow$

Run	Block	A	B	C
8	2	–1	–1	1
7	2	–1	1	–1
6	2	1	–1	–1
5	2	1	1	1
4	1	–1	–1	–1
1	1	1	1	–1
2	1	1	–1	1
3	1	–1	1	1

If you use the RANDOMIZE option in the OUTPUT statement, the output data set contains a randomized design. In some cases, it is appropriate to randomize the run order but not the assignment of theoretical factor levels to actual levels. In these cases, specify both the NOVALRAN and RANDOMIZE options in the OUTPUT statement.