Two-level Factorial and Fractional Factorial
Designs in Blocks of Size Two
YUYUN JESSIE YANG and NORMAN R.DRAPER
Journal of Quality Technology ,35 , p294 ,2003
報告者:梁凱傑
Introduction
IN many experimental situations, it is desirable to group sets
of experimental runs together in blocks.
The block size is governed by many considerations, and
represents, in most experiments, the number of runs that can
be made without worrying (much) about variation caused by
factors not being studied specifically in the experiment.
Often, a block is some natural interval of time (e.g., a week, a
day, or a work shift), of space (an oven, a greenhouse, a work
bench, or a reactor), of personnel (a research worker or a
research team), and so on.
Introduction
Consider a product that can be made in different ways by
varying a set of input factors, each with two levels.
In making boots, for example, variable factors that may be
considered are the type of leather in the uppers, stiffness of
the leather uppers, type of sole/heel cushioning, type of
insoles, thickness of insoles, flexibility of sole, padded or thin
tongue, overall weight of boot, Velcro or laced closure, and so
forth.
Thus, in making boots, one could perform a two level factorial
design that employed every combination of such levels or
perhaps a subset of these combinations. If each boot of every
pair were made to the same specification and if the boots
were worn and used by testers, the boot results would be
perfectly confounded with the testers.
The Two-Factor,
2
2
Design
The Three Factor, 23
Design
The Four Factor, 2
4
Design
The Five Factor, 2
5
Design
Type 3 : B1 = 4, B2 = 5 , B3 = 13, B4 = 23;
Type 4 : B1 = 3, B2 = 15, B3 = 25, B4 = 45;
Type 3 : B1 = 2, B2 = 5 , B3 = 14, B4 = 34.
Stage 1 :(1, 1, 1, 0, 0) and (0, 0, 1, 1, 0, 1, 1, 1, 1, 0), total 9
Stage 2 :(2, 2, 1, 1, 1) and (0, 1, 1, 1, 1, 1, 1, 2, 2, 0), total 17
Stage 3 :(3, 2, 2, 2, 1) and (1, 1, 1, 2, 2, 2, 1, 2, 3, 1), total 26.
Type 5 : B1 = 15, B2 = 25, B3 = 35, B4 = 45;
Type 3 : B1 = 1 , B2 = 2 , B3 = 35, B4 = 45;
Type 3 : B1 = 1 , B2 = 3 , B3 = 25, B4 = 45;
Type 4 : B1 = 4 , B2 = 15, B3 = 25, B4 = 45
Stage 1 :(1, 1, 1, 1, 1) and (0, 0, 0, 0, 0, 0, 0, 0, 0, 0), total 5
Stage 2 :(1, 1, 2, 2, 2) and (0, 1, 1, 1, 1, 1, 1, 0, 0, 0), total 14
Stage 3 :(1, 2, 2, 3, 3) and (1, 1, 2, 2, 2, 1, 1, 1, 1, 0), total 23
Stage 4 :(2, 3, 3, 3, 4) and (1, 1, 3, 2, 2, 2, 1, 2, 1, 1), total 31.
Example1
4 1
2
Consider a
design defined by I = 1234,
IV
If we are prepared to assume that "> or =3fi=0", the
labels reduce to 1, 2, 3, 12+34, 13+24, 23+14, and
4.
To split apart the 12+34, 13+24, and 23+14
combinations, we need to use the 2 4IV1 design
defined by I = -1234.
This addition results in a design of 48 runs.
Example2
5 1
Consider the
2V design defined by I = 12345. The 32
possible estimates are confounded in 16 pairs
I + 12345, 1 + 2345, 2 + 1345, 3 + 1245,
4 + 1235, 5 + 1234, 12 + 345, 13 + 245,
14 + 235, 15 + 234, 23 + 145, 24 + 135,
25 + 124, 34 + 125, 35 + 124, 45 + 123.
If we are prepared to assume that "> or =3fi=0"
We can use any of the groups of three arrangements shown in Table
9 (taken from Table 8) to estimate all the main effects and 2fis of the
projected four factors.
These designs, divided into blocks of size two, each require a total of
64 runs.
5
2 designs, divided into blocks of size two, we need
If we use
three sets, comprising 96 runs in total,
Conclusion
There are numerous ways to divide factorial and fractional factorial
designs into blocks of size two, and the various possibilities achieve
various objectives in terms of the estimation of effects and
interactions.
Here, we have assumed that only main effects and 2fis are of interest,
and that the blocking is done by conventional methods.
This leads to reductions in the numbers of runs that are needed, and
provides choices that depend on the experimenter's requirements.