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ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
A Sequential Approach of
Estimating Two-factor Interactions
by
Bin Zhou and Yue Fang
MIT
Sloan School Working Paper 3614
September 1993
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
A Sequential Approach of
Estimating Two-factor Interactions
by
Bin Zhou and Yue Fang
MIT
Sloan School Working Paper 3614
September 1993
it
M.I.T.
LIBRARIES
A
Sequential Approach Of
Estimating Two-factor Interactions
Bin Zhou
Sloan School of Mangement, MIT, Cambridge,
MA
02139
Yue Fang
Operations Research Center, MIT, Cambridge,
September
MA, 02139
22. 1993
Abstract
Most literature in fractional factorial design is devoted to studying
the main effects of control factors. Recent studies in strategic planning
and concurrent engineering deal with completely different problems.
In designing a parallel process, knowing interactions among factors,
rather than main effects, is even more important. For a large number of control factors, resolution
large
and
difficult to find.
V
fractional factorial design
to estimate all two-factor interactions.
scenario, requires r?
+ 4n -
system
is
available.
often
The
procedure, in the worst
The number of exsome prior knowledge
8 experimental runs.
perimental runs can be reduced substantially
of the
is
This paper suggests a sequential procedure
if
The proposed procedure
utilizes a series of
modified resolution IV designs and provides a systematic approach to
construct high resolution experiment designs for any large systems.
1
Introduction
In designing
and
strategic planning of a
complex system, one often needs
know interactions of many potential factors. For example, in concurrent
product and process design, the objectives are to consider simultaneously all
to
1
elements of the product development cycle. Understanding of the interactive
relationships among design parameters will help in optimizing the product or
knowing the interactive relationships among
In marketing,
process design.
the different features of a line of products will help to reduce the competition
among
the products themselves.
different
ing,
parameters
when
will
In designing,
knowing interactions among
help to creat more parellel designs. In manufactur-
different control factors will help to optimize the calibration
common
among
procedure. The
a system needs constant calibration, knowing interactions
feature of these problems
interactions
is
that understanding of
is
essential for developing
all
significant
an optimal system.
existing research on two-level experiment design concentrate on es-
Most
timating main
When
effects.
ligible, the fractional factorial
popular resolution
III
two and higher order interactions are negand two-level orthogonal P-B designs are two
all
When
designs.
two-factor interactions are present,
main effects. Webb(1968) and
Margolin (1969) proved that the minimum number of runs required for n
factor resolution IV designs is 2n. To estimate some two-factor interactions
in addition to main effects, resolution V designs may be needed. Resolution V designs often require a large number of experimental runs. Box and
resolution IV designs are needed to estimate
Hunter (1961(b))gave the smallest resolution
(7) control factor. Addelman (1965)
2^''"^
17 factors with
resolution
Many
III,
runs.
IV and
V
Some
V
designs for less than seven
constructed a resolution
V
design for
of the known results related to orthogonal
designs are
shown
in
Table
statistical softwares provide resolution III
1.
and IV designs
to a con-
siderably large number of experimental parameters. However, the complex
structure of resolution V designs makes generating resolution V designs for
a large
is
number
of control factors extremely difficult. For large system, there
no general systematic method
V
designs except for
tial
all
full
to construct two-level orthogonal resolution
fractional designs.
This paper proposes a sequen-
design procedure using the available resolution IV design to estimate
two-factor interactions involved one factor at a time. For
n
factors, this
+ 4n -
8 experimental runs to have all interacprocedure needs at most n~
tions estimated. Main effects can also be solved from confounding patterns.
Because it is a sequential method, any knowledge about interaction can be
used to reduce experimental runs.
This paper
olution
V
is
organized as follows.
design to estimate
all
Section 2 describes
how
to use res-
interactions involving one factor.
Section
Table 1:
Designs
Minimum Number
Number
of Factors
of Experimental
Resolution III
Runs Required
Resolution
IV
4
16
10
16
11
(2)
Resolution
Orthogonal
V
'
''
16
16
(1)
in
16
16
16
32
16
64
16
W^
"272r
64
24T2r
128
128
128
For fractional factorial designs only
For fractional factorial designs the number of minimum runs are 32
3 presents a sequential approach of estimating
all two-factor interactions.
Section 4 presents some simulation results of the proposed procedures.
Estimating Two-factor Interactions by Resolution IV Design
2
Consider the following model
Y =
^L
+ 3X + X'^AX +
t
X
= (Xi, A'2, ..., A'„)'-^ with values 1 or -1, and t is a term including
the cumulative efTect of other variables not included in the model and some
random noises. Assume that c is normal distributed with mean zero and
constant variance a^ and the different c's in each experiment are uncorrelated.
where
/i,
vector
main and
and matrix
.4
are the coefficients associated with grand mean,
interaction effects, respectively.
The diagonal
entries of matrix
.4
are zeros.
We
introduce Interaction Structure Matrix (ISM) which
by vector
and matrix
is
characterized
.4 in the above model. The ISM is an n x n sym(3
metric matrix that off-diagonal entries {i.j) are a,/s and diagonal entries
(z,0 are l3^s. Estimating all entries in ISM needs at least n{n + l)/2 experiment runs. Our approach is to estimate one row at a time by modifying
Assume
an orthogonal resolution IV design.
main
vestigating the
A'l.that
and Zi
is
=
Xi and
XiXn. We
effect of
A'iX2, A'iA'3, ....
XiXi,i
easily identify
all
=
all
we
that
are interested in in-
two-factor interactions involving
first
redefine these effect as Zi
=
A'l
Applying resolution IV design on Z/s, we can
from all interaction effects of Z,'s. We
'2,...,n.
effect of Z,'s free
noticed that any two-factor interaction of X,'s are either main effect of Zj or
one of two-factor interaction of Zj's. Therefore, we can estimate all effects
of A'l, XiA'2,
from other main and interaction
A'l A'3, .... A'l A'„ free
effects of
Xi's.
minimum
For an n (mod 4) factor,
ett
and Burman design.
requires
It
'2n
resolution IV design
is
foldover Plack-
experiment runs. Therefore, we have
the following theorem:
Theorem
1
There exists a balanced orthogonal design which estimates
dependently the main effect A, and
perimental runs
(
n= mod
4
all
interactions involving
and n < 268). Futhermore.
A',
in-
with 2n ex-
this is a
minimum
balanced orthogonal design.
The proof
is
When n
design.
given in the Appendix.
is
not a module of
For example,
without the
n
column
last
An example
if
(
=
in
11,
4, we use a larger available resolution IV
we use a resolution IV design with 24 runs
design matrix.
see Figure
a design to estimate
A'l.
use a cyclic shifting
method
1
and Figure
A'iX2,
.
.
.,
2)
demonstrates how to construct
A'iA'„ for
to obtain
n
=
12.
The
Hadamard matrix
first
respectively, as in Figure
.Vi
and
all
is
to
and
column of
of order 12
foldover the matrix to get 24 x 12 design matrix. Associate each
the obtained matrix to the main effect
step
interested
1 1
interactions
1.
To obtain the design matrix for all A'/s, we multply each column with
the first one. The resulting design matrix is shown in Figure 2.
Vi
V,
A
3
Approach
Seqential
to Estimate All
Two-
factor Interactions
To estimate
interactions,
all
we use a sequential procedure and estimate inwe focus on estimating
teractions involving one factor at a time. At lih step,
Since
interactions involving zth control factor.
all
Xi with Xj.j <
i
has been estimated
interaction effects of
all
previous steps, we set them as
in
constants in design matrix and use the procedure described in Section 2 for
Xi,XxXi^\,
and
.
.
As
X-i^Xn-
,
.
A'^A'^,^
effect A'l
The main
effect
In case of
n =
>
i
result, all Xi,Xj,j
<
are confounded with
i
can be solved from the confounding structure
12, the
main
can be estimated free from the other interactions.
if
necessary.
step needs 24 runs, and so do the second, third
first
and eighth steps, 16 runs for each are
required. The remaining three steps need 8, 8 and 4 runs. There are total
180 runs. In general, at most ri^ + 4n — S runs are needed to carry out the
and
fourth. In the fifth, sixth, seventh
procedure
for the
Theorem
1,
n-factor experiment.
By conducting
estimates can be made of
less
2
than n-
The proof
In the
is
-I-
4n —
in
a series of resolution
all
main
V design
given in Theomii.
and two-factor interactions with
8 experimental runs.
y\ppendix.
above procedure, main
ing structures.
effects
J
One may argue
effects are
estimated from solving confound-
that the estimation of main effects
is
affected
by the confounding. As we mentioned at the beginning of the paper, we
focus more on interactions than main effects. Besides only few interactions
are significant in practice (Box & Meyer 1985). therefore we shall be able lo
have reasonable estimates of all main effects.
In many cases, engineers may have partial knowledge of interaction. They
may know that some interactions are zero prior to the experiment. Such prior
information can be used
in
our sequential procedure to further reduce exper-
iment runs. In any step, when an interaction
can
set this interaction
some
runs.
In this case,
confounded to main
we may
is
effect.
to
be insignificant, we
Therefore we
may
save
also manipulate the order of steps to avoid
wasting runs due to the constraint of module
explains our strategies:
known
1.
The
following algorithm
Xi
X,
X2
A3
A4
X5
Xe
X7
Xs
Xg
X\o
X\i
Xu
Table
2;
Step
Trace
in
Solving ISM(i;
Table
3:
The numbers
proaches
Factors
of experimental runs needed in the sequential ap-
Table
Sttp
4:
Trace; of the Six-round Design Strategy for
Model(2)
Table
5:
Random
Level-setting of Constant Factors of the
Strategy for .Model(2)
Step
Six-nnmd Design
Table
6:
Simulation Results
Interactions
for
Model(2)
.
Appendix
Proof of Theorem
for
(See,
For n
1:
< 268 {mod
Hadamard matrix has been constructed
Burman (1964), Golomb and Hall
4)
example, Paley(1933),Plackett and
(1962) and Raghavarao (1971)).
<
design for each n
Hence there
exists
an orthogonal resolution IV
268.
and H be the resolution IV design matrix
with dimension 2n x n (Dey(1985)). Without lossing generalization assume = \.
Associate the columns of // to the main effects, say X\ and all two-factor inLet n factors be Xi,
.Y2,
.
A'n
,
.
.
i
teractions involving A'i,that
obtain the uniqe settings of
column by
and A'iA'2,
tions.
is
all
A'iA'3,
.
,
.
Now we show
that this
A^i,
full
.
.
.
,
rank by looking
XiA'2, A'lXa,
.
.
,
.
A'l
at
XiA',i.
., A'l A'n free from X2, X3,
and grand mean, X2, -Y3, ....
.
is
to estimate A'l, A'iA'2, A'l A'3
of
^2,
.
.
,
.
H
Since
X„ by
is
One may
respectively.
doing multplications of the
first
resolution IV design estimates of
A'l
main effects and two-factor interacminimal resolution IV designs. Since the design
X\Xn free from other main and interaction ef-
AiA'n are free of other
fects A'l, A'iA'2, A'lATs,
is
A'l,
other columns.
.
X\X2,X\X2,,...,XiXn,
is
A'n
the
is
of
way
full
of
rank. Also grand mean,A'2, A'3,
how
Recall this design
to construct A'2, X3,.
is
.
.
,
.
.
,
.
Xn
An
from
to estimate X\, A'iA'2, A'iA'3,
Xn. we have 2n vectors X\, X1X2, X\Xy,. .... Xi A'n
A'n are independent. So we at least need 2n experi-
....
.
mental runs.
Proof of Theorem
2: Conduct the modified resolution IV design discussed in above
theorem n times. Each time one obtains the completely estimations for one factor
(including the main effect and all two-factor interactions involving this factor). After each step the number of factors decreases by 1 by letting this factor be constant
Let n = Am + k, where m is an integer and k = (J, 1,2,3. For m > 1. the total
number of experimental rtms needed is equal to 8A;(m-l-l)-l-32(m-l-...-f 2)4-8-1-8-1= Sk{m +\) + 16m(rTi -f 1) - 12 = ri^ + -In - k~ + Ik - 12 < n^ + 4n, - 8.
1
References
[1]
Addelman,
S. (1966),
blocks of 32."
[2]
'The construction of a
Technomein.es
7.
2'''"^
resolution
V
plan
in eight
pp. 439-443.
Baumert, L.D., S.W. Golomb and M. Hall (1962), "Discovery of an Hadamard
matrix of order 92." Bull. Amer. Math. Soc. 68, pp. 237-238.
14
[3]
Box, G.E.P. and J.S. Hunter (1961(a)), "The 2"-p fractional factorial designs
Part
[4]
Box, G.E.P. and
Part
[5]
[6]
[7]
Technomctrics
I"
II."
J.S.
[10]
factorial designs
Dey, A. (1985), Orthogonal fractional factorial designs, Wiley,
New
York.
Draper. X.R. and D.K.J. Lin (1990), "Connections between two-level designs
Hamada, M. and
and l." Technornetrics, 32. pp. 283-288.
C.F.J. \Vu (1992), ''Analysis of designed experiments with
aliasing." Journal of Quality Technology 24. pp. 130-137.
Margolin, B.H. (1969), " Results on fractorial designs of resoltuion IV
2" and 2"3^ series," Technornetrics 11. pp.431-444.
Plackett, P.L.
rial
[11]
Hunter (1961(b)), 'The 2"-p fractional
Box, G.E.P. and R.D. Meyer (1993), "Finding the active factors in fractioned
screening experiments," Journal of Quality Technology, 24, pp. 94-105.
complex
[9]
311-351.
Technornetrics 3. pp. 449-458.
of resolution ///*
[8]
3, pp.
and
J. P.
Burman
(1946),
"The design
of
optimum
for
the
multifracto-
experiments," Biometrika 33, pp. 305-325.
Paley, R.E.A.C. (1933).
"On orthogonal matrices," Journal of Mathematics
and Physics 12, pp. 311-32U.
[12]
Raghavarao, D. (1971), Constructions and combinational problems in design
of experiments, Wiley,
[13]
Webb,
S.
New
York.
(1968), "Xon-orthogonal designs of even resolution," Tcchnometrics,
10. pp. 291-299.
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