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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Aliasing (factorial experiments) | 7/7 | https://en.wikipedia.org/wiki/Aliasing_(factorial_experiments) | reference | science, encyclopedia | 2026-05-05T09:48:49.670316+00:00 | kb-cron |
== Partial aliasing == In regular
2
k
{\displaystyle 2^{k}}
fractions there is no partial aliasing: Each effect is either preserved or completely lost, and effects are either unaliased or completely aliased. The same holds in regular
s
k
{\displaystyle s^{k}}
experiments with
s
>
2
{\displaystyle s>2}
if one considers only main effects and components of interaction. However, a limited form of partial aliasing occurs in the latter. For example, in the
3
4
−
1
{\displaystyle 3^{4-1}}
design described above the overall
A
×
B
×
C
×
D
{\displaystyle A\times B\times C\times D}
interaction is partly lost since its
A
B
C
D
2
{\displaystyle ABCD^{2}}
component is completely lost in the fraction while its other components (such as
A
B
2
C
2
D
{\displaystyle AB^{2}C^{2}D}
) are preserved. Similarly, the main effect of
A
{\displaystyle A}
is partly aliased with the
B
×
C
×
D
{\displaystyle B\times C\times D}
interaction since
A
{\displaystyle A}
is completely aliased with its
B
C
D
2
{\displaystyle BCD^{2}}
component and unaliased with the others. In contrast, partial aliasing is uncontrolled and pervasive in nonregular fractions. In the 12-run Plackett-Burman design described in the previous section, for example, with factors labeled
A
{\displaystyle A}
through
K
{\displaystyle K}
, the only complete aliasing is between "complementary effects" such as
A
{\displaystyle A}
and
B
C
D
E
F
G
H
I
J
K
{\displaystyle BCDEFGHIJK}
or
A
B
C
J
K
{\displaystyle ABCJK}
and
D
E
F
G
H
I
{\displaystyle DEFGHI}
. Here the main effect of factor
A
{\displaystyle A}
is unaliased with the other main effects and with the
A
B
{\displaystyle AB}
interaction, but it is partly aliased with 45 of the 55 two-factor interactions, 120 of the 165 three-factor interactions, and 150 of the 330 four-factor interactions. This phenomenon is generally described as complex aliasing. Similarly, 924 effects are preserved in the fraction, 1122 effects are partly lost, and only one (the top-level interaction
A
B
C
D
E
F
G
H
I
J
K
{\displaystyle ABCDEFGHIJK}
) is completely lost.
== Analysis of variance (ANOVA) == Wu and Hamada analyze a data set collected on the
3
I
V
4
−
1
{\displaystyle 3_{IV}^{4-1}}
fractional design described above. Significance testing in the analysis of variance (ANOVA) requires that the error sum of squares and the degrees of freedom for error be nonzero. In order to insure this, two design decisions have been made:
Interactions of three or four factors have been assumed absent. This decision is consistent with the effect hierarchy principle. Replication (inclusion of repeated observations) is necessary. In this case, three observations were made on each of the 27 treatment combinations in the fraction, for a total of 81 observations. The accompanying table shows just two columns of an ANOVA table for this experiment. Only main effects and components of two-factor interactions are listed, including three pairs of aliases. Aliasing between some two-factor interactions is expected, since the maximum resolution of this design is 4. This experiment studied two response variables. In both cases, some aliased interactions were statistically significant. This poses a challenge of interpretation, since without more information or further assumptions it is impossible to determine which interaction is responsible for significance. In some instances there may be a theoretical basis to make this determination. This example shows one advantage of fractional designs. The full
3
4
{\displaystyle 3^{4}}
factorial experiment has 81 treatment combinations, but taking one observation on each of these would leave no degrees of freedom for error. The fractional design also uses 81 observations, but on just 27 treatment combinations, in such a way that one can make inferences on main effects and on (most) two-factor interactions. This may be sufficient for practical purposes.
== History == The first statistical use of the term "aliasing" in print is the 1945 paper by Finney, which dealt with regular fractions with 2 or 3 levels. The term was imported into signal processing theory a few years later, possibly influenced by its use in factorial experiments; the history of that usage is described in the article on aliasing in signal processing. The 1961 paper in which Box and Hunter introduced the concept of "resolution" dealt with regular two-level designs, but their initial definition makes no reference to lengths of defining words (described earlier) and so can be understood rather generally. Rao actually makes implicit use of resolution in his 1947 paper introducing orthogonal arrays, reflected in an important parameter inequality that he develops. He distinguishes effects in full and fractional designs by using symbols
[
⋯
]
{\displaystyle [\cdots ]}
and
{
⋯
}
{\displaystyle \{\cdots \}}
(corresponding to
U
{\displaystyle U}
and
U
~
{\displaystyle {\widetilde {U}}}
), but makes no mention of aliasing. The term confounded is often used as a synonym for aliased, and so one must read the literature carefully. The former term "is generally reserved for the indistinguishability of a treatment contrast and a block contrast", that is, for confounding with blocks. Kempthorne has shown how confounding with blocks in a
k
{\displaystyle k}
-factor experiment may be viewed as aliasing in a fractional design with
k
+
1
{\displaystyle k+1}
factors, but it is unclear whether one can do the reverse.
== See also == The article on fractional factorial designs discusses examples in two-level experiments.
== Notes ==
== Citations ==
== References ==