The FACTEX Procedure

General Criteria

The criteria for an orthogonally confounded $q^ k$ design reduce to requiring that no generalized interactions in a certain set $\mc{M}$ can be confounded with zero. (See the section Structure of General Factorial Designs for a definition of generalized interaction.) This section presents the general definition of $\mc{M}$. First, define three sets, as follows:

$\mc{E}$

the set of effects that you want to estimate

$\mc{N}$

the set of effects that you do not want to estimate but that have unknown nonzero magnitudes (referred to as nonnegligible effects)

$\mc{B}$

the set of all generalized interactions between block pseudo-factors

Furthermore, for any two sets of effects $\mc{A}$ and $\mc{B}$, denote by $\mc{A}\times \mc{B}$ the set of all generalized interactions between the effects in $\mc{A}$ and the effects in $\mc{B}$.

Then the general rules for creating the set of effects $\mc{M}$ that are not to be confounded with zero are as follows:

  • Put $\mc{E}$ in $\mc{M}$. This ensures that all effects in $\mc{E}$ are estimable.

  • Put $\mc{E}\times \mc{E}$ in $\mc{M}$. This ensures that all pairs of effects in $\mc{E}$ are unconfounded with each other.

  • Put $\mc{E}\times \mc{N}$ in $\mc{M}$. This ensures that effects in $\mc{E}$ are unconfounded with effects in $\mc{N}$.

  • Put $\mc{B}$ in $\mc{M}$. This ensures that all $q^ s$ blocks occur in the design.

  • Put $\mc{E}\times \mc{B}$ in $\mc{M}$. This ensures that effects in $\mc{E}$ are unconfounded with blocks.