Fractional factorial design

Builds on Factorial experiment Finite field Subgroup

Definition

Let $k > p \ge 1$ , and code the levels of $k$ two-level factors as $\pm 1$ . Then the runs of the full $2^k$ factorial experiment are the points of $\set{-1, 1}^k$ . For a subset $w \subseteq \set{1, \dots, k}$ , called a word, write $x^w = \prod_{j \in w} x_j$ for $x \in \set{-1, 1}^k$ . Identify words with vectors of $\F_2^k$ , with the symmetric difference as addition. A regular $2^{k-p}$ fractional factorial design is a set of runs $F = \set{x \in \set{-1, 1}^k : x^{w} = \epsilon_w \text{ for all } w \in H},$ where:

$H$ is a subgroup of $\F_2^k$ of order $2^p$ , the defining relation, with no word of size $1$ ;
the signs $\epsilon_w \in \set{-1, 1}$ satisfy $\epsilon_{\emptyset} = 1$ and $\epsilon_{w \triangle w'} = \epsilon_w \epsilon_{w'}$ , so that $F$ is not empty.

Then $F$ has $2^{k-p}$ runs. Two effects with words $u$ and $v$ are aliased if $u \triangle v \in H$ . The resolution of the design is the smallest size of a nonempty word of $H$ .

Remarks

On $F$ , the contrast columns $x^u$ and $x^v$ of aliased effects agree up to the sign $\epsilon_{u \triangle v}$ , so their effects cannot be separated. Here $\F_2$ is the field with two elements.

Example: $k = 3$ , $p = 1$ , $H = \set{\emptyset, \set{1,2,3}}$ with $\epsilon = +1$ , that is, the generator $C = AB$ and the defining relation $I = ABC$ . The runs are $(-1,-1,+1)$ , $(+1,-1,-1)$ , $(-1,+1,-1)$ , $(+1,+1,+1)$ , four of the eight. The main effect $A$ , with word $\set{1}$ , is aliased with $BC$ , since $\set{1} \triangle \set{1,2,3} = \set{2,3}$ . The design has resolution III.

Non-example: the four runs with $x_1 = -1$ form the set $\set{x : x^{\set{1}} = -1}$ , which uses the word $\set{1}$ of size $1$ , so clause 1 fails. In these runs factor $A$ never changes, so its main effect cannot be estimated at all.

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Tag	`10WTA1QZXD`
URI	`m2://10WTA1QZXD`
Kind	definition
Writing	W0 stub
MSC	62K15
Defines	Fractional factorial design
Uses	Factorial experiment, Finite field, Subgroup

Fractional factorial design

Definition

Remarks

Reading path

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