*Theory of Orthogonal Designs* |

## General Criteria

The criteria for an orthogonally confounded *q*^{k} design reduce to requiring
that no generalized interactions in a certain set
*M* can be confounded with zero.
(See "Structure of General Factorial Designs"
for a definition of *generalized interaction*.)
This section presents the
general definition of *M*. First, define three sets, as follows:
*E*
- the set of effects that you want to estimate
*N*
- the set of effects you do not want to estimate
but that have unknown nonzero magnitudes
(referred to as
*nonnegligible* effects)
*B*
- the set of all generalized interactions
between block pseudo-factors

Furthermore, for any two sets of effects *A* and *B*, denote by
*A*×*B* the set of all generalized interactions
between the effects in *A* and the effects in *B*.
Then the general rules for creating the set of effects *M* that are
not to be confounded with zero are as follows:

- Put
*E* in *M*. This ensures that all
effects in *E* are estimable.
- Put
*E*×*E* in *M*. This ensures
that all pairs of effects in *E* are unconfounded with
each other.
- Put
*E*×*N* in *M*. This ensures
that effects in *E* are unconfounded with effects in
*N*.
- Put
*B* in *M*. This ensures that all
*q*^{s} blocks occur in the design.
- Put
*E*×*B* in *M*. This ensures
that effects in *E* are unconfounded with blocks.

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