The FACTEX Procedure

Minimum Aberration

As discussed in the section Speeding Up the Search, the FACTEX procedure uses a tree search algorithm to find the confounding rules of a design that matches the size and resolution you specify. There might be more than one solution set of confounding rules, and usually the FACTEX procedure chooses the first one it finds. However, there can still be important differences between designs with the same resolution; to deal with these differences, Fries and Hunter (1980) introduced the concept of aberration in confounded fractional factorial designs. This section defines aberration and discusses how to request minimum aberration designs with the FACTEX procedure.

Recall that a design has resolution r if r is the smallest order of the interactions that are confounded with zero. The idea behind minimum aberration is that a resolution r design that confounds as few rth-order interactions as possible is preferable. Technically, the aberration of a design is the vector $\mb{k} = \{ k_1, k_2, \ldots \}$ , where $k_ i$ is the number of ith-order interactions that are confounded with zero. A design with aberration $\mb{k}$ has minimum aberration if $\mb{k} \leq \mb{k}’$ for any other design with aberration $\mb{k'}$ , in the sense that $k_ i < k’_ i$ for the first i for which $k_ i \neq k’_ i$ .

For example, consider the resolution 4 design for seven 2-level factors in 32 runs ( $2^{7-2}_\mr {IV}$ ) discussed in Example 7.11.

By specifying 5 for the order d for the ALIASING option, you can see how many fourth- and fifth-order interactions are confounded with zero. The default design constructed by the FACTEX procedure confounds two fourth-order interactions and no fifth-order interactions with zero.

0 = A*B*F*G = C*D*E*G

Thus, part of the aberration for this design is

$\begin{eqnarray*} \{ k_3,k_4,k_5,\ldots \} & = & \{ 0,2,0,\ldots \} \end{eqnarray*}$

On the other hand, the design constructed by using the MINABS option confounds only one fourth-order interaction and two fifth-order interactions with zero.

0 = C*D*E*F = A*B*C*F*G = A*B*D*E*G

Thus, part of the aberration for this design is

$\begin{eqnarray*} \{ k’_3,k’_4,k’_5,\ldots \} & = & \{ 0,1,2,\ldots \} \end{eqnarray*}$

Since the two aberrations first differ for $k_4$ and $k’_4$ and since $k’_4 < k_4$ , the aberration for the second design is less than the aberration for the first design.

The definition of aberration requires evaluating the number of ith-order interactions that are confounded with zero for all $i\leq k$ , where k is the number of factors. Since there are $q^ k$ generalized interactions between k q-level factors, this evaluation can be prohibitive if there are many factors. Moreover, it is unnecessary if, as is usually the case, you are interested only in small-order interactions. Therefore, when you specify the MINABS option, by default, the FACTEX procedure evaluates the aberration only up to order d, where d is the same as the default maximum order for listing the aliasing (see the specifications for the EXAMINE statement in the section EXAMINE Statement). You can set d to any level by specifying $(d)$ immediately after the MINABS option.

The discussion so far has dealt only with fractional unblocked designs, but one more point to consider is the definition of aberration for block designs. Define a vector $\mb{b}={b_1,b_2,\ldots }$ similar to the aberration vector $\mb{k}$ , except that $b_ i$ is the number of ith-order interactions that are confounded with blocks. A block design with $\mb{k}$ and $\mb{b}$ has minimum aberration if

$\mb{k}$ is minimum
among all designs with minimum $\mb{k}$ , $\mb{b}$ is minimum