PROC FACTEX: General Criteria

General Criteria

The criteria for an orthogonally confounded $\text{[math]}$ design reduce to requiring that no generalized interactions in a certain set $\text{[math]}$ 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 $\text{[math]}$ . First, define three sets, as follows:

$\text{[math]}$: the set of effects that you want to estimate
$\text{[math]}$: the set of effects that you do not want to estimate but that have unknown nonzero magnitudes (referred to as nonnegligible effects)
$\text{[math]}$: the set of all generalized interactions between block pseudo-factors

Furthermore, for any two sets of effects $\text{[math]}$ and $\text{[math]}$ , denote by $\text{[math]}$ the set of all generalized interactions between the effects in $\text{[math]}$ and the effects in $\text{[math]}$ .

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

Put $\text{[math]}$ in $\text{[math]}$ . This ensures that all effects in $\text{[math]}$ are estimable.
Put $\text{[math]}$ in $\text{[math]}$ . This ensures that all pairs of effects in $\text{[math]}$ are unconfounded with each other.
Put $\text{[math]}$ in $\text{[math]}$ . This ensures that effects in $\text{[math]}$ are unconfounded with effects in $\text{[math]}$ .
Put $\text{[math]}$ in $\text{[math]}$ . This ensures that all $\text{[math]}$ blocks occur in the design.
Put $\text{[math]}$ in $\text{[math]}$ . This ensures that effects in $\text{[math]}$ are unconfounded with blocks.