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Factorial designs are the most common type
of experimental design. In a factorial design
several factors are controlled at two or
more levels, and the effect on the response
is investigated.
There are two main types: Full Factorial
Designs and Fractional Factorial Designs.
In Fractional Factorial designs the amount
of testing is reduced, but the downside
is that some interactions and factors are
aliased. One of the limitations of Fractional
Factorial designs is that the number of
runs is 2n, if there are more
than a few factors this leads to large gaps
in the available options (4, 8, 16, 32,
64, 128, etc.). Plackett-Burman designs
overcome this, but at the expense of confounding.
Other approaches to the Design of Experiments
include Response Surface Methods and Taguchi
Designs.
Used in experimental design. If the results
of a Resolution III fractional factorial
design leave questions unanswered it can
be converted to a Resolution IV design by
additional testing. The additional tests
use the original design matrix but with
all the signs reversed.
| Fractional
Factorial Designs |
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Factorial designs involve
testing several (n) factors at two levels,
high and low. A Fractional Factorial experiment
uses only a half (2n-1), a quarter
(2n-2), or some other division
by a power of two of the number of runs
that would be required for a Full Factorial
Experiment.
The example shows a 23-1 design:
| Run |
A |
B |
C
+ AB |
| 1 |
-1 |
-1 |
+1 |
| 2 |
+1 |
-1 |
-1 |
| 3 |
-1 |
+1 |
-1 |
| 4 |
+1 |
+1 |
+1 |
C is aliased with AB, thus it is a Resolution
III design.
Factorial designs involve
testing several factors at two levels, high
and low. In a Full Factorial experiment
every possible combination of factors and
permutations is tested. If there are n factors
this is 2n levels.
The design matrix shows the combinations
for a 22 design
| Run |
A |
B |
AB
|
| 1 |
-1 |
-1 |
+1 |
| 2 |
+1 |
-1 |
-1 |
| 3 |
-1 |
+1 |
-1 |
| 4 |
+1 |
+1 |
+1 |
A type of design that allows blocking in
two variables:
| |
Operators
(second blocking variable ) |
| Material
Batch (first blocking variable ) |
1 |
2 |
3 |
4 |
5 |
| 1 |
A |
B |
C |
D |
E |
| 2 |
B |
C |
D |
E |
A |
| 3 |
C |
D |
E |
A |
B |
| 4 |
D |
E |
A |
B |
C |
| 5 |
E |
A |
B |
C |
D |
The treatments are represented by the Latin
letters 'A', 'B' etc.
The design is restrictive because the number
of classes of each blocking variable must
equal the number of treatments. It is appropriate
for eg. testing crops because the land can
be organized into the appropriate number
of 'blocks'.
These are Resolution III (screening designs).
The number of runs in a Plackett-Burman
designs is a multiple of 4, thus avoiding
the limitations of factorial and fractional
factorial designs where the number of runs
is 2k.
The generating vectors for some Plackett-Burman
designs are below:
| n=8 |
(+ + + - + - -) |
| n=12 |
(+ + - + + + - -
- + -) |
| n=16 |
(+ + + + - + - +
+ - - + - - -) |
| n=20 |
(+ + - - + + + +
- + - + - - - - + + -) |
| n=24 |
(+ + + + + - + -
+ + - - + + - - + - + - - - -) |
| n=36 |
(- + - + + + - -
- + + + + + - + + + - - + - - - - +
- + - + + -- + -) |
The generating vectors have ‘n-1’
rows. The last row of a Plackett-Burman
design contains only ‘-‘ values.
The design will have 'n-1' columns and
a factor will be allocated to each column:
- put the generating vector into the first
column (column ‘A’)
- copy the last value from column A into
the first row of column B
- slide the rest of the values from A
below that value
- copy the last value from column B into
the first row of column C
- slide the rest of the values from column
B below that value
- continue until all the columns have
been populated
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