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 Theory of Orthogonal Designs

## Design Factors

This section explains how the criteria for a design can be reduced to prescribing that certain generalized interactions are not to be "confounded with zero."

Suitable confounding rules depend on the effects you want to estimate with the design. For example, if you want to estimate the main effects of both A and B, the following rule is inappropriate:

With this rule, the levels of A and B are the same in every run of the design, and the main effects of the two factors cannot be estimated independently of one another. Thus, the first criterion for a suitable confounding rule is that no two effects you want to estimate should be confounded with each other.

Furthermore, an effect you want to estimate should not be confounded with an effect that is nonnegligible. For example, if the interaction between C and D is nonnegligible and you want to estimate the main effect of A, the following confounding rule is inappropriate:

(Recall that this section uses a general linear form for confounding rules instead of the usual multiplicative form. For factors with levels +1 and -1, the preceding rule is equivalent to A=C*D.)

Another kind of confounding involves confounding with zero. If a factor or a generalized interaction F has the same value in every run of the design, then F is confounded with zero. Such confounding is denoted as

Interactions are estimable with the design if and only if they are not confounded with zero. Consequently, another criterion for a suitable confounding rule is that no effect that you want to estimate can be confounded with zero. The confounding rule for two main effects

can be written as a generalized interaction confounded with zero.

The right-hand side of the preceding equation is part of the interaction between A and B. Thus, for any two effects to be unconfounded, it is equivalent to prescribe that no part of their generalized interaction be confounded with zero.

Note that it is not enough to make sure that only the confounding rules themselves satisfy these restrictions. The consequences of the confounding rules must also satisfy the restrictions. For example, suppose you want to make sure that main effects are not confounded with two-factor interactions, and suppose that the confounding rule for factor E is

Then the following rule cannot be used for factor F:

Even though the rule for F does not confound F with a two-factor interaction, this rule forces a generalized interaction between E and F to be aliased with the main effect of D, since

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