13 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Aliasing (factorial experiments) | 5/7 | https://en.wikipedia.org/wiki/Aliasing_(factorial_experiments) | reference | science, encyclopedia | 2026-05-05T09:48:49.670316+00:00 | kb-cron |
The use of arithmetic modulo 2 explains why the factor levels in such designs are labeled 0 and 1.
=== Example 2 === In a 3-level design, factor levels are denoted 0, 1 and 2, and arithmetic is modulo 3. If there are four factors, say
A
,
B
,
C
{\displaystyle A,B,C}
and
D
{\displaystyle D}
, the effects group
G
{\displaystyle \mathbb {G} }
will have the relations
A
3
=
B
3
=
C
3
=
D
3
=
I
.
{\displaystyle A^{3}=B^{3}=C^{3}=D^{3}=I.}
From these it follows, for example, that
D
4
=
D
{\displaystyle D^{4}=D}
and
D
6
=
I
{\displaystyle D^{6}=I}
.
A defining equation such as
t
1
+
t
2
+
t
3
+
2
t
4
=
0
(
mod
3
)
{\displaystyle t_{1}+t_{2}+t_{3}+2t^{4}=0{\pmod {3}}}
would produce a regular 1/3-fraction of the 81 (=
3
4
{\displaystyle 3^{4}}
) treatment combinations, and the corresponding defining word would be
A
B
C
D
2
{\displaystyle ABCD^{2}}
. Since its powers are
(
A
B
C
D
2
)
2
=
A
2
B
2
C
2
D
{\displaystyle (ABCD^{2})^{2}=A^{2}B^{2}C^{2}D}
and
(
A
B
C
D
2
)
3
=
I
{\displaystyle (ABCD^{2})^{3}=I}
, the defining subgroup
H
{\displaystyle \mathbb {H} }
would be
{
I
,
A
B
C
D
2
,
A
2
B
2
C
2
D
}
{\displaystyle \{I,ABCD^{2},A^{2}B^{2}C^{2}D\}}
, and so the fraction would have defining relation
I
=
A
B
C
D
2
=
A
2
B
2
C
2
D
.
{\displaystyle I=ABCD^{2}=A^{2}B^{2}C^{2}D.}
Multiplying by
A
{\displaystyle A}
, for example, yields the aliases
A
=
A
2
B
C
D
2
=
B
2
C
2
D
.
{\displaystyle A=A^{2}BCD^{2}=B^{2}C^{2}D.}
For reasons explained elsewhere, though, all powers of a defining word represent the same effect, and the convention is to choose that power whose leading exponent is 1. Squaring the latter two expressions does the trick and gives the alias relations
A
=
A
B
2
C
2
D
=
B
C
D
2
.
{\displaystyle A=AB^{2}C^{2}D=BCD^{2}.}
Twelve other sets of three aliased effects are given by Wu and Hamada. Examining all of these reveals that, like
A
{\displaystyle A}
, main effects are unaliased with each other and with two-factor effects, although some two-factor effects are aliased with each other. This means that this fraction has maximum resolution 4, and so is of type
3
I
V
4
−
1
{\displaystyle 3_{IV}^{4-1}}
. The effect
B
C
D
2
{\displaystyle BCD^{2}}
is one of 4 components of the
B
×
C
×
D
{\displaystyle B\times C\times D}
interaction, while
A
B
2
C
2
D
{\displaystyle AB^{2}C^{2}D}
is one of 8 components of the
A
×
B
×
C
×
D
{\displaystyle A\times B\times C\times D}
interaction. In a 3-level design, each component of interaction carries 2 degrees of freedom.
=== Example 3 === A
2
5
−
2
{\displaystyle 2^{5-2}}
design (
1
/
4
{\displaystyle 1/4}
of a
2
5
{\displaystyle 2^{5}}
design) may be created by solving two equations in 5 unknowns, say
{
t
1
+
t
2
+
t
4
=
1
t
1
+
t
3
+
t
5
=
1
{\displaystyle {\begin{cases}t_{1}+t_{2}+t_{4}=1\\t_{1}+t_{3}+t_{5}=1\end{cases}}}
modulo 2. The fraction has eight treatment combinations, such as 10000, 00110 and 11111, and is displayed in the article on fractional factorial designs. Here the coefficients in the two defining equations give defining words
A
B
D
{\displaystyle ABD}
and
A
C
E
{\displaystyle ACE}
. Setting
I
=
A
B
D
{\displaystyle I=ABD}
and multiplying through by
D
{\displaystyle D}
gives the alias relation
D
=
A
B
{\displaystyle D=AB}
. The second defining word similarly gives
E
=
A
C
{\displaystyle E=AC}
. The article uses these two aliases to describe an alternate method of construction of the fraction. The defining subgroup
H
{\displaystyle \mathbb {H} }
has one more element, namely the product
(
A
B
D
)
(
A
C
E
)
{\displaystyle (ABD)(ACE)}
=
B
C
D
E
{\displaystyle =BCDE}
, making use of the fact that
A
2
=
I
{\displaystyle A^{2}=I}
. The extra defining word
B
C
D
E
{\displaystyle BCDE}
is known as the generalized interaction of
A
B
D
{\displaystyle ABD}
and
A
C
E
{\displaystyle ACE}
, and corresponds to the equation
t
2
+
t
3
+
t
4
+
t
5
=
0
(
mod
2
)
{\displaystyle t_{2}+t_{3}+t_{4}+t_{5}=0{\pmod {2}}}
, which is also satisfied by the fraction. With this word included, the full defining relation is
I
=
A
B
D
=
A
C
E
=
B
C
D
E
{\displaystyle I=ABD=ACE=BCDE}
(these are the four elements of the defining subgroup), from which all the alias relations of this fraction can be derived – for example, multiplying through by
D
{\displaystyle D}
yields
D
=
A
B
=
A
C
D
E
=
B
C
E
{\displaystyle D=AB=ACDE=BCE}
. Continuing this process yields six more alias sets, each containing four effects. An examination of these sets reveals that main effects are not aliased with each other, but are aliased with two-factor interactions. This means that this fraction has maximum resolution 3. A quicker way to determine the resolution of a regular fraction is given below. It is notable that the alias relations of the fraction depend only on the left-hand side of the defining equations, not on their constant terms. For this reason, some authors will restrict attention to principal fractions "without loss of generality", although the reduction to the principal case often requires verification.
=== Determining the resolution of a regular fraction === The length of a word in the effects group is defined to be the number of letters in its name, not counting repetition. For example, the length of the word
A
B
2
C
{\displaystyle AB^{2}C}
is 3.
Using this result, one immediately gets the resolution of the preceding examples without computing alias relations: