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Aliased factorial columns prevent separate coefficient identification

Statement

Let $c_{A}, c_{B} \in R^{F}$ be the columns for distinct factorial effects $A, B \subseteq [k]$ in a regular fraction, and suppose $c_{A} = σ c_{B}$ for a fixed $σ \in {- 1, + 1}$ . Consider the unrestricted fixed-design linear model $Y = X β + ϵ$ , where these columns occur in $X$ and the distribution of the error vector $ϵ$ does not depend on $β$ . A coefficient combination is identifiable if it has the same value for all parameter vectors giving the same distribution of $Y$ . The contribution of these two columns depends only on $σ β_{A} + β_{B}$ ; neither coefficient is separately identifiable while the parameter space remains unrestricted. For equal columns ( $σ = + 1$ ), the relevant combination is the sum; for opposite columns ( $σ = - 1$ ), it is the difference. Additional runs can remove the ambiguity only if they break the column relation, and hierarchy or sparsity restrictions are extra modeling assumptions.

Facts & Assumptions

Given: A regular fraction with distinct effects $A, B$ , design columns $c_{A}, c_{B}$ satisfying $c_{A} = σ c_{B}$ , and the fixed-design model and error-law condition in the Statement.

[F1]

Regular two-level fractions, signed defining relations, and resolution: signed-alias columns satisfy $x_{A} = σ x_{B}$ on every run in the fraction; sign $+ 1$ is equality and sign $- 1$ is opposition.

Proof

technique · exhibit the coefficient combination in the fitted vector and a parameter direction that leaves it unchanged

1.1givenF1algebra

The two terms in the fitted vector combine as $β_{A} c_{A} + β_{B} c_{B} = β_{A} (σ c_{B}) + β_{B} c_{B} = (σ β_{A} + β_{B}) c_{B}$ . Thus the fitted vector depends on this pair only through $σ β_{A} + β_{B}$ , with all other coefficients fixed.

2.1givenstep 1.1algebra∎

For any $t \in R$ , replace $β_{A}$ by $β_{A} + t$ and $β_{B}$ by $β_{B} - σ t$ . Then $σ (β_{A} + t) + (β_{B} - σ t) = σ β_{A} + β_{B}$ , so the combination in step 1.1 and therefore $X β$ are unchanged. Because the error law is independent of $β$ , the resulting response distributions are identical. Taking $t \neq 0$ changes both coefficients, proving that neither is separately identifiable in the unrestricted model.

Sources

Oehlert §18.6 and NIST/SEMATECH §5.3.3.4.3 explain the ambiguity left by aliased effects and the extra role of sparsity assumptions. The parameter nonidentifiability argument is given explicitly above.

Depends on

Regular two-level fractions, signed defining relations, and resolution

Used by

Full and targeted foldovers of a resolution III fraction Example
The alias classes in a 2³⁻¹ fraction Example

Dependency tree · two levels

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Sources

Gary W. Oehlert, A First Course in Design and Analysis of Experiments, Chapter 18, §18.6 (standard reference, not scraped)
NIST/SEMATECH e-Handbook of Statistical Methods, §5.3.3.4.3, Confounding (also called aliasing) (standard reference, not scraped)