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 Theory of Orthogonal Designs

## General Criteria

The criteria for an orthogonally confounded qk 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 qs blocks occur in the design.
• Put E×B in M. This ensures that effects in E are unconfounded with blocks.

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