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Details of 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 k = {k1, k2, ... }, where ki is the number
of ith-order interactions that are confounded with zero.
A design with aberration k has minimum
aberration if for any other design with
aberration k', in the sense that ki < k'i for the first i for
which
.
For example, consider the resolution 4 design for seven two-level factors in 32 runs (27-2IV) discussed in Example 15.11.
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.
Thus, part of the aberration for this design is
The definition of aberration requires evaluating the number of
ith-order interactions that are confounded with zero for all , where k is the number of factors. Since there are qk 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 on
"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 b = b1,b2, ... similar to the aberration vector k, except that bi is the number of ith-order interactions that are confounded with blocks. A block design with k and b has minimum aberration if
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