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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Aliasing (factorial experiments) | 2/7 | https://en.wikipedia.org/wiki/Aliasing_(factorial_experiments) | reference | science, encyclopedia | 2026-05-05T09:48:49.670316+00:00 | kb-cron |
== Definitions == The factors in the design are allowed to have different numbers of levels, as in a
2
×
3
×
3
{\displaystyle 2\times 3\times 3}
factorial experiment (an asymmetric or mixed-level experiment). Fix a fraction of a full factorial design. Let
U
{\displaystyle U}
be a set of contrast vectors representing an effect (in particular, a main effect or interaction) in the full factorial design, and let
U
~
{\displaystyle {\widetilde {U}}}
consist of the restrictions of those vectors to the fraction. One says that the effect is
preserved in the fraction if
U
~
{\displaystyle {\widetilde {U}}}
consists of contrast vectors; completely lost in the fraction if
U
~
{\displaystyle {\widetilde {U}}}
consists of constant vectors, that is, vectors whose components are equal; and partly lost otherwise. Similarly, let
U
1
{\displaystyle U_{1}}
and
U
2
{\displaystyle U_{2}}
represent two effects and let
U
~
1
{\displaystyle {\widetilde {U}}_{1}}
and
U
~
2
{\displaystyle {\widetilde {U}}_{2}}
be their restrictions to the fraction. The two effects are said to be
unaliased in the fraction if each vector in
U
~
1
{\displaystyle {\widetilde {U}}_{1}}
is orthogonal (perpendicular) to all the vectors in
U
~
2
{\displaystyle {\widetilde {U}}_{2}}
, and vice versa; completely aliased in the fraction if each vector in
U
~
1
{\displaystyle {\widetilde {U}}_{1}}
is a linear combination of vectors in
U
~
2
{\displaystyle {\widetilde {U}}_{2}}
, and vice versa; and partly aliased otherwise. Finney and Bush introduced the terms "lost" and "preserved" in the sense used here. Despite the relatively long history of this topic, though, its terminology is not entirely standardized. The literature often describes lost effects as "not estimable" in a fraction, although estimation is not the only issue at stake. Rao referred to preserved effects as "measurable from" the fraction.
=== Resolution === The extent of aliasing in a given fractional design is measured by the resolution of the fraction, a concept first defined by Box and Hunter:
A fractional factorial design is said to have resolution
R
{\displaystyle R}
if every
p
{\displaystyle p}
-factor effect is unaliased with every effect having fewer than
R
−
p
{\displaystyle R-p}
factors. For example, a design has resolution
R
=
3
{\displaystyle R=3}
if main effects are unaliased with each other (taking
p
=
1
)
{\displaystyle p=1)}
, though it allows main effects to be aliased with two-factor interactions. This is typically the lowest resolution desired for a fraction. It is not hard to see that a fraction of resolution
R
{\displaystyle R}
also has resolution
R
−
1
,
R
−
2
{\displaystyle R-1,R-2}
, etc., so one usually speaks of the maximum resolution of a fraction. The number
p
{\displaystyle p}
in the definition of resolution is usually understood to be a positive integer, but one may consider the effect of the grand mean to be the (unique) effect with no factors (i.e., with
p
=
0
{\displaystyle p=0}
). This effect sometimes appears in analysis of variance tables. It has one degree of freedom, and is represented by a single vector, a column of 1's. With this understanding, an effect is
preserved in a fraction if it is unaliased with the grand mean, and completely lost in a fraction if it is completely aliased with the grand mean. A fraction then has resolution
R
=
2
{\displaystyle R=2}
if all main effects are preserved in the fraction. If it has resolution
R
=
3
{\displaystyle R=3}
then two-factor interactions are also preserved.
=== Computation === The definitions above require some computations with vectors, illustrated in the examples that follow. For certain fractional designs (the regular ones), a simple algebraic technique can be used that bypasses these procedures and gives a simple way to determine resolution. This is discussed below.
== Examples ==
=== The 2 × 3 experiment === The fraction {11, 12, 13} of this experiment was described above along with its restricted vectors. It is repeated here along with the complementary fraction {21, 22, 23}:
In both fractions, the
A
{\displaystyle A}
effect is completely lost (the
A
{\displaystyle A}
column is constant) while the
B
{\displaystyle B}
and interaction effects are preserved (each 3 × 1 column is a contrast vector as its components sum to 0). In addition, the
B
{\displaystyle B}
and interaction effects are completely aliased in each fraction: In the first fraction, the vectors for
B
{\displaystyle B}
are linear combinations of those for
A
×
B
{\displaystyle A\times B}
, viz.,
[
1
−
1
0
]
=
[
1
−
1
0
]
+
0
[
1
0
−
1
]
{\displaystyle {\begin{bmatrix}1\\-1\\0\end{bmatrix}}={\begin{bmatrix}1\\-1\\0\end{bmatrix}}+0{\begin{bmatrix}1\\0\\-1\end{bmatrix}}}
and
[
0
1
−
1
]
=
[
1
0
−
1
]
−
[
1
−
1
0
]
{\displaystyle {\begin{bmatrix}0\\1\\-1\end{bmatrix}}={\begin{bmatrix}1\\0\\-1\end{bmatrix}}-{\begin{bmatrix}1\\-1\\0\end{bmatrix}}}
; in the reverse direction, the vectors for
A
×
B
{\displaystyle A\times B}
can be written similarly in terms of those representing
B
{\displaystyle B}
. The argument in the second fraction is analogous. These fractions have maximum resolution 1. The fact that the main effect of
A
{\displaystyle A}
is lost makes both of these fractions undesirable in practice. It turns out that in a 2 × 3 experiment (or in any a × b experiment in which a and b are relatively prime) there is no fraction that preserves both main effects -- that is, no fraction has resolution 2.