7.8 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Aliasing (factorial experiments) | 3/7 | https://en.wikipedia.org/wiki/Aliasing_(factorial_experiments) | reference | science, encyclopedia | 2026-05-05T09:48:49.670316+00:00 | kb-cron |
=== The 2 × 2 × 2 (or 2³) experiment === This is a "two-level" experiment with factors
A
,
B
{\displaystyle A,B}
and
C
{\displaystyle C}
. In such experiments the factor levels are often denoted by 0 and 1, for reasons explained below. A treatment combination is then denoted by an ordered triple such as 101 (more formally, (1, 0, 1), denoting the cell in which
A
{\displaystyle A}
and
C
{\displaystyle C}
are at level "1" and
B
{\displaystyle B}
is at level "0"). The following table lists the eight cells of the full 2 × 2 × 2 factorial experiment, along with a contrast vector representing each effect, including a three-factor interaction:
Suppose that only the fraction consisting of the cells 000, 011, 101, and 110 is observed. The original contrast vectors, when restricted to these cells, are now 4 × 1, and can be seen by looking at just those four rows of the table. (Sorting the table on
A
B
C
{\displaystyle ABC}
will bring these rows together and make the restricted contrast vectors easier to see. Sorting twice puts them at the top.) The following can be observed concerning these restricted vectors:
The
A
B
C
{\displaystyle ABC}
column consists just of the constant 1 repeated four times. The other columns are contrast vectors, having two 1's and two −1s. The columns for
C
{\displaystyle C}
and
A
B
{\displaystyle AB}
are equal. The same holds for
A
{\displaystyle A}
and
B
C
{\displaystyle BC}
, and for
B
{\displaystyle B}
and
A
C
{\displaystyle AC}
. All other pairs of columns are orthogonal. For example, the column for
A
{\displaystyle A}
is orthogonal to that for
B
{\displaystyle B}
, for
C
{\displaystyle C}
, for
A
B
{\displaystyle AB}
, and for
A
C
{\displaystyle AC}
, as one can see by computing dot products. Thus
the
A
B
C
{\displaystyle ABC}
interaction is completely lost in the fraction; the other effects are preserved in the fraction; the effects
A
{\displaystyle A}
and
B
C
{\displaystyle BC}
are completely aliased with each other, as are
B
{\displaystyle B}
and
A
C
{\displaystyle AC}
, and
C
{\displaystyle C}
and
A
B
{\displaystyle AB}
. all other pairs of effects are unaliased. For example,
A
{\displaystyle A}
is unaliased with both
B
{\displaystyle B}
and
C
{\displaystyle C}
and with the
A
B
{\displaystyle AB}
and
A
C
{\displaystyle AC}
interactions. Now suppose instead that the complementary fraction {001,010,100,111} is observed. The same effects as before are lost or preserved, and the same pairs of effects as before are mutually unaliased. Moreover,
A
{\displaystyle A}
and
B
C
{\displaystyle BC}
are still aliased in this fraction since the
A
{\displaystyle A}
and
B
C
{\displaystyle BC}
vectors are negatives of each other, and similarly for
B
{\displaystyle B}
and
A
C
{\displaystyle AC}
and for
C
{\displaystyle C}
and
A
B
{\displaystyle AB}
. Both of these fractions thus have maximum resolution 3.
== Aliasing in regular fractions == The two half-fractions of a
2
3
{\displaystyle 2^{3}}
factorial experiment described above are of a special kind: Each is the solution set of a linear equation using modular arithmetic. More exactly:
The fraction
{
000
,
011
,
101
,
110
}
{\displaystyle \{000,011,101,110\}}
is the solution set of the equation
t
1
+
t
2
+
t
3
=
0
(
mod
2
)
{\displaystyle t_{1}+t_{2}+t_{3}=0{\pmod {2}}}
. For example,
011
{\displaystyle 011}
is a solution because
0
+
1
+
1
=
0
(
mod
2
)
{\displaystyle 0+1+1=0{\pmod {2}}}
. Similarly, the fraction
{
001
,
010
,
100
,
111
}
{\displaystyle \{001,010,100,111\}}
is the solution set to
t
1
+
t
2
+
t
3
=
1
(
mod
2
)
{\displaystyle t_{1}+t_{2}+t_{3}=1{\pmod {2}}}
Such fractions are said to be regular. This idea applies to fractions of "classical"
s
k
{\displaystyle s^{k}}
designs, that is,
s
k
{\displaystyle s^{k}}
(or "symmetric") factorial designs in which the number of levels,
s
{\displaystyle s}
, of each of the
k
{\displaystyle k}
factors is a prime or the power of a prime.
A fractional factorial design is regular if it is the solution set of a system of one or more equations of the form
a
1
t
1
+
⋯
+
a
k
t
k
=
b
,
{\displaystyle a_{1}t_{1}+\cdots +a_{k}t_{k}=b,}