EXPERIMENTAL DESIGNS FOR ESTIMATING THE SLOPE OF
SECOND ORDER POLYNOMIAL RESPONSE SURFACES
by
SUNG HYUN PARK
Institute of Statistics
Mimeograph Series No. 969
December 1974 - Raleigh
iv
TABLE OF CONTENTS
Page
L
INTRODUCTION....... • • • •
1
2.
ONE FACTOR EXPERIMENTAL DES IGNS
7
201
2.2
2.3
3.
Designs for the Linear Model When the True
Response is Quadratic •
• . • •
Designs for the Quadratic Model When the True
Model is Also Quadratic
•••••••
Designs for the Quadratic Model When the True
Model is Cubic •• . • • • • • •
10
14
MULTIFACTOR EXPERIMENTAL DESIGNS •
3.1
3.2
18
Designs for the Quadratic Model When the True Model
is Also Quadratic
. • • • •
3.1.1
3.1.2
24
Properties of S lope-Rotatability ••
Second Order Slope-Rotatable Designs ••
26
40
Designs for the Quadratic Hodel When the True Model
is Cubic
• • • • • • •
3.2.1
302.2
59
Designs With Points Equally Spaced on a
Single Circle • • • • • • • • •
Designs With Points Equally Spaced on Two
Concentric Circles
••
0
4.
8
0
60
70
•
SELECTION OF EXPERIMENTAL DESIGNS
4.1
4.2
75
The Computer Program • • •
• • • • • • ••
Selection of Optimal and Near-Optimal Designs .
4.2.1
4.2.2
When Rand RO are the Unit Square
When R is the Unit Square and RO is not
Limited . • • •
~
When Rand RO are the Unit Circle
When R is the Unit Circle and RO is not
Limited . • . . • .
~
0
4.2.3
4.2.4
0
•
•
0
•
0
•
0
•
0
•
75
77
80
0
0
•
•
0
0
84
85
•
•
89
A
4.3
Comparison of Hexagonal, y-Rotatable Central
Composite and 3 2 Factorial Designs • • • •
93
5.
SUMMARY
103
6•
REFERENCES •
106
7.
APPENDIX ••
..
108
1.
INTRODUCTION
Tbis thesis considers a problem arising in the design of
experiments for empirically investigacing functional relationships
between a dependent response variable and one or more independent
continuous variables,
Suppose that an experimenter wishes to explore
11,
a functional relationship between a response,
xl' x '
2
independent variables,
••• , x
and several
, over some region of
p
experimental interest in the space of the quantitative factors.
The
relationship could be expressed generally as
'f1 = f (xl' x 2 ' ••• , x p ' 9 l'
where
9.'s
1
e2'
,."
(1)
9 r)
are unknown parameters which must be estimated by
experimental data.
The response relationship is usually interpreted geometrically as
a response surface in a space of dimension
ordinates are the
p
+ 1
p
variables,
explore the response relationship
+
"
1
Q
•
where the co,
x
p
and
'll '
N observations of the response
are taken in the region of experimental operability at a set of
corresponding levels of each
To
N
x. ; these levels constituting an
1,
.-
experimenta I design.
The data are then used to fit an equation, y ,
intended to estimate
'll
over the region of interest.
interest (which will be denoted by
it is feasible to perform tri.als.
RO.
The region of
R) is not necessarily the same
as the region of operability (which will be denoted by
to
'll
Usually
R
RO) in wb.ich
is smaller than or equal
It is convenient to transform the actual experimental
variables so tbat their od,gin
(0, 0,
.,
0
J
0)
is at
th(~
cen.ter of
the reg loon of interest,
R, and so tb.at
dimensional cube defined by
P
2
by
L: x. :5: 1 •
i==1
-1 :;,; x. :;,; 1
~
R
is eit.!ler the p-
or the unit sphere defined
L
If the true function form
is unknmvn, as is oft.en
the case, t.he relationship may be approximated by the low order terms
in the Taylor series expansion of equation (I), which might be
expressed as
p
11
60 + L:
1=1
S.X,
+
.1 1.
p
L:
i:5:j=l
S ij x i x j +
p
I:
isj:5:k=l
S LjkKiX(k +
, , ,
,
A first order approximation is obtained by disregarding all terms beyond the iirse summation sign; a second order approximation i.8
obtained by adding the second summation sign to the first order
approximation, etc,
Box and his co-workers have found that in most
experimental situations the first or the second order approximation
will be adequate for practical purposes,
The coefficients of such
polynomials are theEl the unknown parameters, which are usually estimated
hy least squares methods from experimental data,
Variolls properties
of the resulting fitted polynomials have been discussed; in
particular, t.hose properties which mainly depend on the choi.ce of the
experimental design,
The study o£ design optimality criteri.a bas been of fundamental
interest to researcbers in the area of response surface analysis,
Initially criteria were largely concerned with variance; eichec of the
individual .... oefficien.ts or of the fitted polynomial a': a whole,
One
of the most popular \Jariance criteria at present, alor'.g with tbe Doptimality crLterion, i.8 the concept of Yotatahility due to Box and
3
T."l.e n'LaLable dc'31gn
dl.sc.an.ce fh)m
ever
R
varia'li.:.e
x
slm~1t:.aneDiJSly.
(IY'
independent of
t.o t."e cen'::er of thE' de;,igno
a:-'. a ba"'L: critecJe>r1o
Lu ~):leLh.E:'r
J<::I
i~.:"
'.Ltentat:j,.n In
Bex and Hunter have
Thi.s criteci.cll. c,)o;-;idec:3 bi.aY and
C':'D:3iderat !C,D c' f bias ",eems per t i.ne'lt 1.0
not the cbvi('e of a design can offer pn'per p.u·r.ecLL·,n
aga.in: o f. t<e pu,-s Lbi.l Ley \1£ C'lrvatTre in the r:.cue re'-pon2e
t,~'e
3drfac~' 0
f a aecuCl.d e'rder deBign, une mi.ght. be JoLere:,Led in a cle,~i.gn
(abe
agatnst the exi,'3tence .:)1' i'lbi,(
tt'nn,~
Suppc:"e an experimentEr U,ts a pcolync1m.ial model c,f
t'del'
wLich affcrd:;
pc.~:.e('.l.iJm
integrated VV8r
.J -
R
,"i~_
13
SRF (Y C"')
- Tl (i;) .: 2d'2.
cr
C2
In
-1
'J
J",R
dx
and
cr
.2
hew". f:"'at
LrI
d .
l
the
Ln
4
,I
V + B
where
v _. 21~.
JR.
Var(Y(lf) )d:l£
a
and
B:~~ JR [ECy(~»)
a
Th116
J
l,.;;
Var(Y(lf)
t1:-,e :.:,um of
-
1l(~)J2d~
•
B, int.egrated squared bl.as, and
\', integrated
•
Estimatiull requices calcdlation of the regres;,<l.on c'.·,e.ffi.cient.s from
experiment.al data.
method.
Box and Draper 'used t.he :3tarl.dard lea,3t: squares
Since the mean squared error criterion. was intn,duced many
researchers ,hav'e co''-',centrated on experimental design crIt.eria which
include bias as a consideration.
Several auth0rs (2. 10. 11. 12) have
suggeated dLffen::nJ; est.imators for the regression coefficl.ents to
achieve smaller mean. squared error under some c i.rcumstances.
All of theBe works regarding design criteria. whether they are
variance criented
C'L
bias oriented. ha,:e
0.'1.12
thing
1..<1
c ommo D., namely
that they cuncern t.he estimatioEl of the height (,f t.he respunoe in the
factor space rathe'c tban t.he esti.mation of di.fference,':"
However. th.e main srxeam in stati.::1tical work on
~l:te
of respcnse c"urface df:signs has been
<t'ce~ ••
on the
In thie peIne cf vie\v r:he ,:,tudy
except.lo~lal
emphai'l.zed the t,".t.i.matLun of abEol.ut,e response.
oversigh.t.
re~pcn~e.
dfs:,gn cf
experimen.t.s has been on. t.he cempari20n of treatmen.Ls,
e6r:imatLon (, f t_ceatment (.ontcasts.
i.Cl
si.nce it. [la:', mainly
Tb.i;; ;an he a
~~eriCU:3
Fi't.i.maU.C'.'1 of differences i.n respc!1.se at di.fferent pctnts
5
in the factor space will often be of great importance.
If differences
at points close together are involved, estimation of the local slopes
(the rates of change) of the response surface is of interest.
This
problem, estimation of slopes, occurs frequently in practical
situations.
For instance, there are the cases in which we want to
estimate rates of reaction in chemical experiments, rates of change in
the response of a plant to various fertilizers, rates of disintegration
of radioactive material in an animal, etc.
The problem considered in this thesis is, therefore, that of the
choice of the experimental design so as to achieve maximum information
on the estimated slopes of the response surface.
In designing these
experiments allowance is made for bias, due to an inadequate model.
The mean squared error of the estimated slope integrated over
be the basic measure of information provided by the design.
R will
Only the
least squares estimator of the slope will be considered.
This problem of estimating slopes is relatively unexplored in the
literature.
Several authors have considered the topic, but with
limited discussions.
Atkinson (1) mainly considers 'first order poly-
nomials when the true models are quadratic.
Ott and Mendenhall (15)
discuss a second order polynomial, but they limit the case to a single
variable and to a quadratic polynomial as a true response.
Matteson
(13) and Munsch (14) extend the concept of minimum bias estimation
which was initially suggested by Karson, Manson and Hader (11) to the
estimated slope of response surfaces.
In what follows we will pursue a thorough investigation of the
topic in cases where second order polynomials are fitted for single
6
and for several independent variables when the trlle responses are
either quadrati.c or cubic.
A new desi.gn concept called slope-
rotatability will be introduced.
Designs with points equally spaced
on one, two or three circles will be investigated extens ively.
A
computer search reJUtine will be used to find optimum designs in this
class.
7
2
0
ONE FACTOR EXPERIMENTAL DESIGNS
Suppose the true model is a polynomial of degree
d
Z
in a single variable but for one reason or another the experimenter
fits a polynomial of degree
.£1
where
d
l
S
d
2
'
is the least squares estimate of
error integra~ed over
J =
NO
2
a
iiI.
The mean squared
R for the estimated slope,
JR E(£:
dx
~)2dx
dx
where
0 -1
dy/dx, is
= JR
dx
v + B •
Thus
J
which we want to minimize as the de.sign optimality criterion
is the sum of integrated variance, denoted by
squared bias, denoted by
V , and integrated
B.
In order that the bias portion can be evaluated the bias of the
polynomial coefficients must first be evaluated,
matrix the elements of the
the terms of
x.
-1
for the
squares estimator for
.£1
itl}.
i th
is
Suppose
X
i
is a
row of which are the values taken by
experimental combination
The least
8
where
:L
is the vector of the
N observations.
TheIl the bias of the
coefficients is given by
The extent of the bias is determined by
the matrix
A , the alias matrix.
Designs for one variable are the subject of the following
sections.
In Section 2.1, designs for the linear model protecting
against quadratic effect will be briefly discussed, and then in
Sections 2.2 and 2.3 designs for the quadrati.c model when the Crue
model is either quadratic or cubic will be investigated.
of interest,
2.1
R, is assumed to be
-1
s;
The region
x :s; 1 •
Designs for the Linear Model When
the True Response is Quadrati.c
If the true model is quadratic,
first order model,
y(x)
=:
b + blx , is fitted, how shou1.d experiments
O
be designed so that the estimate of the slope is as precise as
possible?
Suppose the experiment consists of observations at two
fixed factor levels.
Atkinson (1) indicates that
J
is minimized
when the observations are equally divided between t.he two equally
spaced levels from the center of the design.
We will c0ly, thprc[ofc,
discuss the designs with the trials equally divided betwEt?o
equally spaced factor levels from the desi.gn center.
Odr
t!1E'
tlvu
emphasis is
on determining an "optimal:' straight line fit with a small Dnmber of
9
obsenlatiuns rather t.han on detecting departure.s from linearity,
For
the latter purpcse more than two distinct levels would be needed.
Suppose
on each
N
-h
(even number) trials are allocated with
level and
center of the design.
2o'Z
level where
h
trials
is the distance from the
Then one can show that
N Jl
= ._--
h
N/2
E(b
-1
1
2
- 13 ) dx
1
since
Then
v
=
Jl
Var(bl)dx
N Jl
(13 1 - (13 1
N·
2c Z -1
and
-
- 2cr Z
=
-1
4 \[N 13 2 2
3(
cr
)
•
+ ZI3 Zx) 2 dx
=
..EZ J~l.
2cr
(-:213 2x) 2dx
10
C0Dsequently, the mean ::squared error integrated over
R
is
J-'V+B
_. l/h
2
2
+ 4a /.3
where
Of
=:
-2S x , is zero only at the center of
2
Observe that the bia:::.,
B, is independent of
the squared bias,
h"
Also the integrated
V, is a monotonically decreasing function of
variance,
thus minimized for any given
Of
when
h
Rand
h.
is
J
takes on the largest feasible
value.
The region of operability could be larger than
R ; however, in
nearly all practi.cal problems there is obviously Some practical limit
or bound on
h.
minimi.zed at
h
generalizing
R
arbi.trary point
is
-1
~
x
~
If
=:
h
b •
o •
J
is
Atki.nson (1.) di3cusses thiS case extensively
to be the line of length
X
b, then
i.s assumed to be bounded by
2c
centered at an
Atkinson. aseumes tbat the region of operability
1. •
If the fitted model }-,appens to be the crue model, bias is zero"
In this case
J
is equal to
V
and the des ign cricer::L.:rn Is simply
minimize the l!lteg:cated vari.ance of t.he esttmated s1c pe
J
OJe::-
R
0
t.D
11
The estimation of the slope cf the second ordec model requires a
mini.mum of three levels of the independent variable since there are
three parameters in the model,
It is shown by
de la Garza (7) that the precision matrix (variance-covari.ance matrix)
of least squares estimators in the model for a design at more than
three levels of the variable can always be obtained by spacing at just
three judiciously selected le'lels of the variable,
Therefore, i.n what
follows in this section the selection of chree levels of the variable
and the allocation of
N trials ot the three levels, equally spaced
with the center level at zero, will be considered.
N trials are allocated as below:
Suppose
-1:1.
n
I
0
h
r-
n
I
nO
l
l
trials
The same number of trials are allocated to both extreme levels since
one can show that the design criterion is minimized when the both
levels have an equal number of trials"
For the above allocation scheme
the design matrix is
x
x
2
)'"
1
1
X
-h
-h
h;
.
h~
1
.
-h
I
0
1
()
0
0
1
0
0
1
1
h
h
h
-?
l
nn
-
I
~
h
HI trials
h-
trials
2
?
h-
01 trial::,
"2
b
/'
x
involved
jp
1.7 ue,,·d inst.ead of
chi.~
X
One can easily obtain
s~(tioD.
1
2'0
•• - - '1
...
, ,Iv' -
I
-1
o
_---~--.'
'0,1
(.X A)
matrix
1
.)
h~(N-2nl)
o
o
-1
o
From the mode 1,
the estimated slope at
x
is
Thus its variance is
V'ic = Va r
(dy! dx)
The integrated variance over
R
is
where
Lettipg
0- 1
= fl,
-J
dx
f
(3)
13
Equation (3) indicates that to minimize
V,
h
feasible within the region of operability.
minimizing value of
dV/df
=0
f,
V
min
Once
h
is fixed, the
can be obtained by sol ving
U"
, which is
h, Table I shows the corresponding
For \rarious choices of
and
should be as large as
, the minimum value of
aspect that
f~"
quite wide range.
V.
Table 1 provides an interesting
Gt".ays within a narrow range as
Note that for
h
= 1,
which agrees with Atkinson's result.
integer, the optimum
f~'c
fie
f*
h
changes over a
is equal to
In practi.ce, since
0.2847
n
l
is an
can only be approximately achieved unless a
large number of trials are observed.
Another aspect which should be pointed out is that even though
fie
is a fai.rly narrow interval for different values of
h , never-
theless the integrated variance is not terribly sensitive to
a given
h.
For example, with
V. = 7.195 ; however, i f
ml.n
f
=:
h = 1
0.25,
fic
=:
0.2847
V = 7.333.
f
for
with
Therefore, 25%
allocation of the trials at an extreme level instead of 28.47% is
qui.te acceptable as a good design.
Incidently, Ott and Mendenhall
(15) recommended 25% allocation to an extreme levt':'l based on their
criterion, the minimization of the rate of curvature of
N Var(dy!dx)/c/ •
14
f* and Vmin for fixed h
Table 1
h
f*
Vmin
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
0.2695
0.2744
0.2795
0.284]
0.2900
0.2953
0.3005
0.3056
0.3106
260135
15.993
10.452
7.195
5.163
3.835
2.931
2.295
1. 835
2.3
Designs for the Quadratic Model When
the True Model is Cubic
In this section we will discuss second order designs which can
afford protection against the existence of a cubic term in the true
response function,
The class of designs to be considered i.s the same class we studied in
Section 2.2, that is, designs with three equally spaced levels,
0
and
,
h
of the independent variable and allocation of trials,
,
nO
and
Xl
is identical to
n
l
Xl
2
respectively, on each level.
=
X
n
l
trials
n
l
For this class of designs
in the previous section and
... ,
(_h 3 ,
~h,
X
2
0, 0,
no trials
is gi.ven by
••
n
1
0
,
trials
,
15
( X ' X ).-1
I 1
The preci.3ion matri.x,
is equal to
(X'X)-l
in the previous
section, and
o
o
Hence, the alias matrix is
o
h-2
A :=. (.'XlV
1··1 )-l"'lX:=.
'~12
~'-~2
Here, the vector
take~;
o
the form of the single element
13
3
we can obtain
b
O
E(l?l)
=E
hI
b
0
So
!
= ft 1 + AI2.2
=
13
1
13 2
2
+
h
2,
13 3
0
-~.
So
o.
13 1
2
+ h S3
13 2
Consequently, the bias of the estimated slope at a point
_.
'J.•.1213.
3
-
3• IJQ 3 x 2
x
is
>
Thus
16
and the integrated squared bias over
R
is
where
N
2a2
1
f -1
(h
= r:i 2(,h. 4 -
2h...
= ~~-
2 2
13 3 - 313 3x ) dx
Z
+ lG 8)
where
B is independent of
Note that
h = 1.
2
f
= o/N
Since t.he integrated variance,
= 'if +
=
B
1
[-:_+
2
ot.
2
h,
J
J
4
is the same as that in
R
is
+01
242
(h
+ 1.8) •
- 2h
(5)
3fh (1-2f)
is minimized at
function of
(4)
(J
B
2fh
Since
\{N 13 3
= ---
and achieves minimum at
V,
Section 2.2, the mean squared error over
J
01
h = 1
and
V
will be minimized at
For a fixed value of
h
is a monotonically decreasing
1
~
h < +00
for any given
one can find the minimiz.ing value of
f
fi; , by setting
dJ/df
0
which implies
<-_._----
,3' 2
4\ _ 2
fi,- "" ..\.-::E';.-±-:L-:
6h
\/3'
\ . h2 + 4
2
Thi::; is the same equation for
section, simply because
B
f~'c
as appea.n::d in the previou3
is independent of
f
and
V has the same
17
h* and f* for given a
Table 2
h*
V
f*
J
(when h is
limited to R)
J
B
(Optimum)
0
1
2
3
4
----> 0
- 7 +00
10397
1. 225
L150
1.104
1.000
+00
003054
0.2966
0.2926
0.2902
0.2847
form as before.
minimizes
Q'
J
If
h
h*) is
1
R,
and
-1
f*
~
x
~
1 , the
= 0.2847
00
h
which
for any given
•
hi (
The
and
f'f(
h~'(
and
oJ/oh
=0
.
oj/of
=0
Q'
.
and
The last column in Table 2 shows the values of
is limited to
R, that is, when
h*
=1
and
Several aspects can be observed from Table 2.
decreases approaching
0.2847.
depend on
f*
are obtained by trial-and-error, and were checked by
substitution into the equation found by setting
h*
+
+00
Table 2 is constructed to show how
h
7.195
7.995
10.395
14.395
19.995
4.017
70779
12.568
18.665
+00
is limited to
(call this
_no> 0
0
1.706
4.202
80136
13.566
2.311
3.577
4.432
5.099
7.195
The ra.nge of
1
h*
stays within a narrow range.
as
and
~
f*
f*
= 0.2847
As
Q'
when
J
•
increases,
also decreases approaching
changes is quite Wide, however,
f*
18
3.
MULI'IFACTOR EXPERIMENTAL DESIGNS
The slope of the response surface for a single variable at any
point is a scalar.
However, with more than one variable the slope
depends on the direction of measurement.
method of estimation of slope for
p
In this chapter the general
variables will first be developed
and then the criterion of rotatabili.ty for the vari.ance of estimated
slope will be extensively discussed.
Suppose the
d
l
'y(~)
order pcl1ynomial model,
= ~l.E.l
is
fitted protecting against the bias effect, if any, of the
1l(~)
polynomial model,
.th
elements of the
~-
of
1:-
x.
-~
for the
.th
= ~l~l + ~2~2
•
Let
X.
l
d
2
order
be a matrix, the
row of which are the values taken by the terms
experimental combination.
Also let
and
II
I
oIL
OX
{
D.
J
is a matrix, the
parital differentiation of
,g,(2S,)
p
j
/
/'
where
II
"
:i th
row of which is obtained by the
with respect to
x.,
.L
Tbe elements of
•
are the estimates of the slope of the fitted response surface
and the elements of
along the
p
..::.(~)
factor axes.
are those of the true resp0:lse surface
Therefore, the slope at
x
on the fitted
19
response surface in the direction specified by the
direction cosines,
~,
bination of
k )
P
that is
1'£
If we let
~ =
p x 1
vector of
, is a linear comE(a) - £. , then
s
represents the vector of biases of the estimated slope,; along the
factor axes estimated at the point
p
~.
The mean squared error of the estimated slope in a specified
direction
k
at a point
x
is
(6)
=
where
V*
V~·~
+
B"k
= -k'D 1Var(b.
)D'k
-1 1-
is the variance component and
is the squared bias component.
= s' kk' s
k
were known, experiments would
be designed to minimize equation (6).
If the direction of interest,
1 ,
If
Bi~
is not specified in advance, one might wish to average
Ji(
over
all directions in such a way that the distribution over all directions
is uniform.
The folluwing lemmas will enable uS to average
J'"
over
all directions.
Lemma 1:
The average of
V"k
over all directions is
(7)
20
Proof:
M = DIVar(~l)Di.
Let
The average of
V*
over all
directions in such a way that the distribution over all directions is
uniform is
v*
= Average(~'M~)
k
= Average[tr(~'M~)]
since
k'Mk
is a scalar,
k
= Average[ tr(~')]
k
= tr[Average(Mkk')]
k
= tr[M(Average(kk'»]
k
Therefore, we have to calculate
Average(kk')
= CJ
kk'dA
k
where
C
is a proper average constant and
dA
is the area element of
the hype r s phe re with un it rad ius.
It can be shown (Appendix
7.9)
.that
matrix whose diagonal elements are all equal.
then
Vi~
=
tr(MvI )
= tr(vM)
p
CJ
Let
kk'dA
CJ -kk'
i.s a diagonal
dA
= vI p
21
= vtr(M)
p
= v
L:
i=l
"'1
where
",.'s
of
and is only associated with
v
and
are eigenvalues of
1.
M
suppose
M
",.1. = 1
Notice that
cJ
kk'dA
is the identity matrix.
for all
v~·~ =
M
i
.
Therefore,
.
v
is independent
To find the constant
The average of
v
=
l/p
and
V*ok
is 1
k'Mk
becomes
(1/ p) tr(M)
Q. E. D.
Lemma 2:
The average of
B~'~
=
B"'~
= Average
B*
over all directions is
(l/p}.:?.' ~ •
(8)
Proof:
=~
I
[cJ
= vs's
(~'kk'~)
kk'dAJE..
since
s
is independent of
k
22
= (l/p).£.'~
•
Q. E. D.
=
v
The relation,
way in Lerrnna 2.
B*
= 1.
lip , can be verified differently in the following
Suppose
Hence,
v
=
lip
s' = (1, 1, ••• , 1)
from
B*
= vs's.
is irrnnediately obtained that the average
(]
2
= -p
-1
tr(D (X'X ) ])') +
III
1
J*
Then
E'~
=p
and
From Lemmas I and 2 it
over all directions is
(1/p)E'~ •
(9)
If we do not specify the point of interest at which the slope is
to be estimated, experiments should be designed to minimize the mean
squared error integrated over the region of interest,
~
J
R
ji: dx
The mean
R is
squared error integrated over
J =
R.
where
-1
(2
=
JR dx
(]
=~
(j
JR (V* + B*)d~
=V +
B
(10)
where
23
The derivation of equation (10) is obtained in the following way:
The
integrated variance i.s
(ll)
and the integrated squared bias is
B
= ~-,
J
R B*dx
iJ
=
NQ
2
iJ
since
JR 1.- ps'sdx
---
24
= E(g)
s
- r
(12)
So
B
= NO Z JR
pO'
Iiz(A'D
l - D2>(D1A
-
DZ)1iZd~
(13)
For the case when the first order model is fitted when the true
model is quadratic, a comprehensive study can be found in the paper by
Atkinson (1).
Accordingly, we will only consider the cases where the
second order polynomial models are fitted.
3.1
Designs for the Quadratic Model When
~he
True Model is Also Quadratic
If we fit the quadrati.c model when the true model i.s also
quadratic, there is no bias involved and the variance of the estimated
25
slope is the only concern.
y(~)
=
The quadratic model with
p
variables,
~lE.l ' will have
x'
-1
and
In this section
x,
.£2
and
b
will be used instead of
for notational convenience since
~l
and
x
D
are not needed.
~ =
Notice that
X
2
~2
E(g) - .!. = 0
As we have obtained before, the average variance of the estimated
slopes at
x
=
(xl' x 2 ,
... , x p)
over all directions is
2
Vi, = SL
p
tr(D(X'X)-lD')
which is a function of the coo rd ina te s of the point,
of the design used and also, of course, of
by increasing
N.
The quantity,
C1
2
experimental design
V<,~)
as well as
, i s thus a
standardized measure of precision of estimation.
the variance function of the design.
,
can be reduced
2
= NV~"/C1
V(,?S)
x
V(~)
wi.ll be called
In other words, for any
provides a standardized measure of the
precision of the estimated slopes averaged over all directions at the
point
x
in the space of the variables.
the elements of the precision matrix,
It is a function of
(X'X)
-1
x
and
,and is uniquely de-
fined for every p-dimensional experimental. design.
Suppose we have two poi.nts which are equidistant from the design
center.
If there is a conS iderable difference between the two points
as far as precision of the estimated slopes is concerned, a certain
26
"imbalance" exists in the use of the surface.
Therefore, i.n the study
of response surface design an impor.tant and interesting property i.s
that of rotatability due to Box and Hunter (5).
Chapter 1, for a rotatable design
Var(Y)
As mentioned in
will be the same for any
two points which are the same distance from the design center.
This
concept of rotatabi1i.ty is now introduced here with respect to the
V<.~)
variance function,
, for the slope estimate.
Suppose
V(~)
for a p-dimensional design is a function only of
P
=
P
~
i=l
2
.x i
the distance from the center, so that the variance function contours
in the space of the variables are circles, spheres or hyperspheres
centered at the origin.
A design giving such a variance function will
be called a s lope-rotatable design and rotatability in the Box-Hunter
sense will henceforth be called y-rotatability.
Properties of Slope-Rotatability
3.1.1
As a first step to develop slope-rotatable designs, we propose a
theorem giving the conditions for slope-rotatability.
(i,j)
will mean a combination of
Theorem 1:
and
i
The notation
j •
The necessary and s'.lfficient conditions that a design be
slope-rotatable are that
p
1)
~
j=l
j;ti
2)
2(Cuv(b .. ,b
1.l
e
.)
1.J
Cov(b., b .. )
J
1.J
=0
+ Cov(b .. ,b .. » +
J J.LJ
for all
i
P
~
k::l
k1:i, j
Cov(b.k,b. )
1.
Jk
=0
27
for any
3)
(i,j)
1. "!' j
where
P
+
4Var(b 00)
:L~
•
Var(b 0.)
L:
are equal for all
LJ
j=l
1. •
j#=i
Proof:
It can be shown that
V'k
2
2:._ tr(D(X'X)-l D ,)
=
p
1
P
L:
= -
Var(b.)
p i=1
2
+-
P
L:
P 1=1
1
x.(2Cov(b.,b
~
1
o
.)
~1
p
+
Cov (b. , b .. »
L:
J
j=l
j;&i
P
L:
2
+P
~J
x.x.[2(Cov(b .. ,b .. )
1. J
~~
~J
(i,j)
+
Cov(b .. ,b .. »
J J 1J
i;&j
p
L:
+
k=l
Cov(bik,b jk)]
k"!'i, j
+ -I
P
L:
p i=1
2x.L4Var(b
.. )
~
1~
+
P
L:
j=l
j;&i
Var(b
.. )'] ..
. .lJ .
(14)
Note that the cO'.l:l.diti.ons in the theorem are suffi.e ient for equation
(14) to be a function only of the distance from the center.
conditions are satisfied, then
V'k
can be wri.tten as
If these
28
I
P
1
Var(b.) + -[4Var(b .. ) +
p 1=1
1
P
11
2:
p
p
2
2: Var(b .. ) J 2: x
1J
i=1 i
j =1
(15)
j#i
which is only a function of
P
2:
P =
i=l
2
x..
1.
Suppose at least one of the three conditions are not satisfied.
Then it is easy to show that there exist two points which are the same
distance from the design center, but which give different values of
V~'(.
Therefore, the conditions are also necessary.
Q. E. D.
The three corollaries immediately following are based on the
theorem.
They are useful in constructing slope-rotatable des igns.
Corollary 1:
Proof:
All ~-rotatable designs are slope-rotatable.
Box and Hunter (5) Show that the covariances associated with
conditions I and 2 in theorem I are all zero for ~-rotatable designs,
and, furthermore,
Var(b .. ) are equal for all
also equal for all
i
and
Var(b .. )
1 1 . 1 J
(i,j)
where
i # j.
Therefore, conditions 1,
2 and 3 are obviously satisfied for ;-rotatable designs.
are slope-rotatable.
Corollary 2:
are
Thus, they
Q. E. D.
If all odd moments are zero, then only condition 3 is
necessary and sufficient for slope-rotatability.
Proof:
If all odd moments are zero, then the covariances in
conditions I and 2 are all zero, which implies that the two conditions
are satisfied.
The only remaining condition is 3.
Q. E. D.
29
Corollary 3:
If all odd moments are zero and ,all mixed fourth
mements are equal, that is
.2.2
[l J J
are equal for all
(i,j) ,
i
~
j , then the necessary and sufficient
condition for slope-rotatability is:
Proof:
Equal
Var(b .. )
.
If all mixed fourth
moments are equal, then it can be shown that
(i,J'),
condition.
p
~
i
equal for all
If
i
If all odd moments are zero, then only condi.tion 3 is
necessary and sufficient from Corollary 2.
all
for all
Ll.
i
j.
Var(b.'>
lJ
Therefore, the condition that
are equal for
Var(b .. )
II
are
is the only remaining necessary and sufficient
Q. E. D.
=2
Corollary 3 is particularly useful, since then there
is only one mixed fourth moment,
[12 22 J
mixed fourth moments are always equal.
,
which implies that all
To make the theorem and its
three corollaries more concrete, the two independent variables case
will be considered in detail.
For the case of
following forms:
p
= 2,
x
b
X,
and
D
take the
--1
Xu
x
1
x
x
1
x
12
2
21
Z2
-
.......
2
xlI
x
x
2
1Z
x
2
22
x
x
2
lN
x
Z
2N
XINX2~
x
Z1
x
ll Z1
x
12 22
X
x
lN
2N
I
~
and
"-
.--/
0
1
0
2x
o
1
o
o
l
D
The variance function will be
= ~ tr(D(X'X)-lD')
p
0
x
2
30
31
To be slope-rotatable from Theorem 1, the following conditions should
be satisfied.
and
Notice that the first two equalities come from condition 1 of Theorem
1, the third from condition 2 and the last from condition 3.
condition above is, in fact, equal to
Var(b
lZ
)
cancels out.
Var(b
11
=
)
Var(b
22
)
The last
since
If the conditions hold, then the variance
function becomes
V(~):=
N [Var(b ) + Var(b ) + (4Var(b ) + Var(b »p2]
Z
2
1
12
ll
2cr
(16)
where
2
This function,
P
distance,
V(~)
, depends only on the
p , from the design center, that is, the variance contours
are to be circles surrounding the design center.
For example, suppose the nine points of the 3
design with three levels,
-1, 0
and
1
2
symmetric factorial
for each variable were used.
For thi.s part.icu1ar symmetric design, it is easy to show that all odd
mements are zero,
Var(b )
l
=:
Var(b ) = ( / /6,
2
Var(b
ll
) = Var(b
22
)
=
32
2
= CJ /2
and
Var(b
slope-rotatable.
lZ
):;
CJ
2
/4.
From Corollary 2 or 3 this design is
The variance function for the des ign is thus from
equation (16)
v ("l:;)
=
N
2
V ~'(
3
81
2" + 8""
2
(17)
p
CJ
The variance contours of equation (17) are shown in Figure 1.
---_.
x :;,1
1
\
Figure I
Variance contours for
An interescing point is that this
y-rotatable.
factorial design
3
2
factorial design is not
From Corollary 1 we know that all 'y-rotatable designs
are slope-rotatable.
However, the reverse is not true.
factorial design is an example for the assertion.
The
2
In fact, there are
many more slope-rotatable designs which are not y-rutatable.
designs will appear in subsequent sections.
3
Such
33
Another interesting point for a slope-rotatable design is that
the variance function is a monotonically increasing function of
with a minimum at the design center.
Consider equation (15).
1p
Var(b .. )J
11
2
P
,
1J
is always positive, and consequently
rotatable design.
It is easy to prove this property.
The coefficient of
[4Var(b .. ) +
p
dV(~)/dp
> 0
for a slope-
Note that the variance function of a 'y-rotatable
design is not in general a monotonically increasing function.
If the point at which the slope is to be estimated could be
anywhere within the region of interest,
designed to minimize
V(~)
R, experiments should be
integrated over
R.
The integrated
variance is then
(2 -1 =
where
f R dx
as has been established in equation (11).
As mentioned earlier, with two or more independent variables the
slope at a point
of measurement.
x
on the response surface depends on the direction
If the directi.on,
l,
i.s not specified, experiments
V'k, over all
would be designed to minimize the average variance,
directions at point 2£.
Furthermore, if the pllint
x
1,8
not
34
Vk
specified, the experimenter would want to minimize t:'Je average
R
integrated over
However, as Theorem 2 shows below, it will not
0
Vl~
be necessary, under Some conditions, to average
V,''', and s'~bsequently V
to find
v*
V directly from
evaluate
over all directions
There is a short-cut method to
0
R, is
The region of interest,
0
assumed to be either the unit circle or the unit square centered at
the origin of the desi.gn.
Theorem 2:
all
If a design is slope-rotatable and
are equal for
Var(b" )
1
i , then for the design
dx
= N02 SR
V"'(dx
i -
cr
i
for any
along the
where
V"':(
~
V~'(
is the variance
of the estimated slope
factor axis.
X.
1.
We want to show that under the two conditions -- the design
Proof:
is slope-rotatable and has the same
integrated
for any
i
Vic
over
where
R,
is
Var(b .)
V
1.
= ~ SR V*d~
V~
is the same as the integrated
' •
VoJ"
x.
along the
equation (6) for a specified direction
, the
cr
factor axi.s
~
0
R
over
~
From
k
= kID Var(1?) Di k
For the direction of the
.
0
where
• ,0)
1
x.,
factor axis,
1.
appears 1.'1. the
remaining elements are zero.
kID
d. , the
-I.
.th
l.~-
k
1.;3
kU
"".
(0,0,
. .' -,
0
1
place frnm the left and all
In L':1U; case it is easy to ehCM that
o
matrix.
Then
35
V''<; :;; d. (X i X)
-1
-.1
1.
d
:0' 2 .
-1.
Now we want to show that
under the conditionE.
It can be shown that
P
+
= Var(b.)
1
4Cov(b~,b
1.
.. )x. + 2 L: Cov(b., b . .)X .
11
1
11J
J
j=l
j~i
+ 4
P
X.x.Cov(b .. ,b .. )
L:
1. J
j=l
U.
1J
+
P
L:
2
(k,.R,)
k,.e#i
j;f:i
k~.e
?
+ 4Var(b 1-1
.. ) X":' +
1.
P
L:
2
J
Var(b .. )x.
1J
j=l
j;f:i
Notice that if either
JR x~x~dx
J -
or
q
is odd, then
=0
1.
from the symmetry of
OJR
p
R.
Therefore,
P
V·.'l;~d~ ::::: Var(b.)
+ a[4Var(b.)
+ L:
1.
1. 1.
j~l
j#=i
where
2
a :; 0JR.:X:.'1. dx •
.~.
Var(b..
)l
1J .
(18)
36
From equation (14) and
if
p
or
q
is odd, we can obtain
1
OJR V""dx
:= -
P
L:
[Var(b.)
P 1'--1
+
a(4Var(b .. )
1
11
+
P
L: Var(b .. »J
1J
j= 1
j~i
•
(19)
To show
JRl.-V~~dx
.JR V~"dx.:=
for an arbitrary
i , it is necessary and sufficient to have from
equations (18) and (19) that
VarCb.)
1
+
a[4Var(b.~)
L1
+
P
L:
j:=l
(20)
j~i
are equal for all
i..
Theorem 1 says that if a design is slope-
rotatable,
4Var(b.. )
11
are equal for all
+
i.
Var(h .. )
1J
Additionally, if
i , then equation (20) is true.
"arCb. )
1
are equal for all
Hence, the theorem is proved.
Q. E. D.
Suppose
Q
represent.3 the class of y-rotatable designs and
represents the class of sl()pe-rotatable dE;signs.
a subset of
S
from~~orollary
1.
Now we
c\~'ns
We know that
S
Q
ider an interesting
is
37
class of desi.gns whic1:l contains most of the useful designs known and
which are fai.rly easy to construct.
'Let
C
denote t"e class of
designs which satisfies the following four conditions.
1•
All odd moments are zero.
2.
All pure second moments are equal.
3.
All pure fourth moments are equal.
4.
All mixed fourth moments are equal.
Then the following relationship is true.
Corollary 4:
QcCc S •
T!:1e relationship
Proof:
Qc
C
is i.mmediate since the four
h
conditions fer
C
are simply part of the necessary conditions of y-
rotatable designs,
Q.
The relationship
C
c S
f'Jl1ows from
Corollary 3 since the second and third conditions for
Var(b .. )
are equal for all
~~
C imply that
Q. E. D.
i.
Corollary 4 shows that the class of ~-rotatable designs is a
subset of
C
and
[.
'-'
i.s a subset of the class of sl.ope-rotatable
des igns.
Now we want to i.nvestigate the moment matrix of
properties, partly because
C
i.s an important cla,58 ,',f designs and
partly because the moment matri.x of
Assume
C
.2
[1.
J
is of the form:
and
has net been fcnnd explicir.ly.
S
')
= a ,
C and i.ts
[
,.L,
.1.
'J
J'-'J -- c •
The moment matrix of
38
xl
x
x
2
2 2
x
1 2
x
a
a
2.
p
x x
r
I
I
1 I
- t- "- -- --a
0
a
"k:
--
x
2 xlx}
x
p-l p
I
I a
- "\
0
,
-.
-
,
--
1
'/(.'
I
"--
.
~~~
I
,',
0 I
a
N-1X'X
x
p
l
'!'<,;
J
I
.1
--1
b
:=
c
b
I
c
I
c
b
"k
I
I
,
..J
---
c
--
0
...
c
...
(symmetri.c)
.-_.-
0
0
c
where the asterisks indicate null submatrices.
The inverse matrix can
be obtained from Box and Hunter (5) and the variances and covariBnces
of the estimated coefficients for a design in
N Var(b O)
N Var(b .)
-
~
---~---
cr
2
cr
N Var(b .. )
2
1
a
=
N Var(b .. )
1
_ _ _:!:.L. = ~.
2
c
cr
ll-
cr
are given by
C
2
-a(b-c\
=_~::..L-..
N
Cov(b.
"' b .. )
_____
~_~!:-_Uo.
A
and all the rema:L:>::.ing covariances are zero.
cr
2
2
~
a -('
A
""-'-~-=-
Th'JS all firat and
second degree coeffici.8nts are uncorrela ted except the quadratic
(21)
39
coefficient",.
1.\1. equation (21)
+
A = (b-c)((b-c)
p(c_a
2
p
1.3 the number of variables and
» •
From the fact that
is slope-rotatable the variance
C
C
function for a design in
can be obtained from equation (15) and
(21) ,
'r7~
:= N
2
(J
N
1
2
(J -
[~-
P
P
1
P
L: Var(b.) + ~(4\!ar(b,,) + L: var(b. ,») L:
lJ
P i= 1
1.
pu
j:=t
i=1
j7'i
_. 1 + _4..cb+(p-:?2c-(p-1)a~
L
a
+
pA
E.:l.J
cp
2
x.l
1.'
P
2
1: x.1.
i=l
where
p
2
=
P
1:
i=1
2
x.
1.
and
K
K> 0
_. 4Cb+iP.-:Dc-{£:1)a21 + £-1
pA
. cp
can be shown an.d this indicates that
function of
p •
rotatable and
V(?!)
This result should be true since
t~.evariance
an increasi.ng function of
:1.:.3 an increas ing
is siope-
functiun of any slupe-cctatable design is
p.
If
"
A = (b-c)«b-~-)+p(c-aL_).is zero,
then the ffi<Jment matdx i.s 8ingular and some of the coefficients of the
40
design are not esti.mable.
b - c
=
pCa
2
A
a implies that
b· c
or
- c) , that is
[
o2.2J J
1.
or
(22)
When one constracts a design which belongs to
C, one should check
equation (22) to avoid singularity.
3.102
Second Order Slope:-Rotatable
Desi~~
We have shown the precision matrix conditions which a second order
design must satisfy to obtain constant slope-precision on spheres
centered at the origin of the design.
We now consider the problem of
finding arrangements of experimental points which satisfy these
conditions.
The classes of s lope-rotatable designs we present are by
no means exhaustive but rather are intended to illustrate Some of the
possibilities.
From Corollary 1 any ~-rotatable design is slope-rotatable.
Therefore, the y-rotatable designs which are presented in the paper of
Box and Hunter (5) are slope-rotatable.
are not discussed here.
These slope-rotatable designs
In order to discover some of the slope-
rotatable designs which are not 'y-rotatab Ie, w(:' now ccnsider
arrangement of points which are equally spaced on circles, spberes or
hyperspheres,
~i.~.,
regular figures.
Our study beg in8 with the two
dimensional figures.
Suppose
circle with radius
p •
n
points a re equally [;;paced con a
The coordinates of experimer.1.L91 pC1.n.ts are
41
X
Xl
2
'--'--1
P eeL:;
p sin
ct
I
ct
P cos (8-ta )
p sin
p cos (2e-ta)
p sin (2e+ct )
p
cos «(IJ.-l)e-ta)
p
s in
II
« n - 1) e+cr) j
/'
where
ct
1,s the angle of a point on the circle measured CClunter-
clockwise frlfm t.he pcsitive
[lP-Q2ql, where
order moment:,
integers and
Xl
p
~
axis, and
p
and
Q
e ""
2n In.
Any
th
p~'
are nun-ne.gative
Q , can be expressed as
We may write
[cos 4Le+a) ] p-Q
and
With the a bove
as
.57J b stitutiops
t.h e
t~
p-"~'
or d er lUlJment. may b. e expresse d
42
M.
=:
n-l p-q q
1,,) p(l)"
q I:
"L,
( -1)
-" I(p-q\I",Q,.Li8(p-2r:-2r)
'0''.:,)
~.:
I:
)e rn .2
l
fL =0 t=O r=O
• t . "r
iQ'(p-2t-2,r)
e
0
•
If the term
e ia(p-2t-2r)
..
were written out in a rectangular array with the terms in
laterally and the terms in
t
r running
vertically, the 8ummaU.0n could be
preformed diagonally and th.e result expressed as
:
m="O
where
e~8(p-2m)eiQ'(p-2m)
A
m
m=t:+r,
m :; 0, .L, 2,
o •
0
p , and
,
m
A
m
I:
=:
j=O
Therefore, with the understandi.ng that
(~):= 0
for
Ct
< S.
M
may
be written as
M
=:
K
n-I
I:
p
I:
fL=O m=O
where
(?)
M:;; 0
if
(3)
m
K:: (p/2) POI 1) q In.
is Seen that.
(2)
A e~8 p-~m e1.Q' p- m.
Examini.ng the termS in equat.ion (2.3\ it
P - 2m
:f nI
~
I
an integer, since then
n-l
I: e ifL8 (p-2m). 0
fL '=0
by the property of the summat.ion of roots cf unity.
p - 2m
=0
since rhen
A
m
:=
0 •
M '" 11
aLco if
43
Odd moments table
Table 3
Odd Moments
Degree
p = 1
n=2
n~6
0
0
0
0
[12 J
Xb
0
0
0
0
0
X
0
0
0
X
0
X
0
0
0
X
0
X
0
p
3
3
2
2
[1 3 J,[2 J.[1 2J,[12 J
3
J
p
-
4
[132J,[12
p
-
5
[15],[142J,[1322J,[124J,[25]
vanishes
= not
vanishes
For example, let us fi.nd
q
n=5
Oa
=2
bX
n"'4
[1J,[2J
p
aa -
n=3
=1
and
and
-2
or
-2,
m:= 0, 1
as
and
m becomes
n-I
L:
2.
0, I
[l2J
for
n:= 4.
Notice that.
and
In this case
p - 2m
could be
2, respecti.vely.
When
p ""
2,
2, 0
p - 2m
= 2,
e ~e (p-Zm) = 0
~~O
since
p -
2m of 41
and when
~.
_
M ""
:=
,_])l-j(l), ( 1 ,
•
1 .J
J
.- J
j=O
L,
Consequently,
p - 2m
f)
(.
which indicates
°,
:=
A
m
since
'" 0
-1 + 1
[12J:= O.
:=
a
Q
In the same
fashion we can ccnstruct the above odd moments tabl.e [wm equation (:23)
to show which odd moments do and do not vanish.
Table 3 gi'les
valuable infcrmati'J'1 to construct the moment mat.rix. fur a dt'3 Lgn
whose experimental poi.nts are equally spaced on circles.
,
44
F,Yc
c.'te ELest ELlur il1ustraticrl1s of slope-rotatable des igns we
consider arrangements with
pc·i.nts at the des ign center
nO
points equally spaced on a circle of radius
and
PI
Pz
equally spaced on a different circle of. radius
the angle of a point: on the circle of radius
Xl
6
axis and
Total (,beervations are
[lJ
seen that
1
2
=
n
+
6
l\] -
l
- '3 ,
nO.
[2J = [12J = [13 2J :: [12 3J = O.
x
HZ :: 3
1
N-IX'X
It
.-
2
x
2
x x
0
0
a
a
a
a
0
b
-b
-c
a
-c
c
-b
3d
d
0
3d
()
2
r
and
2
d
where
3
2N
_. :3
c
_.
4N
:3
4N
.)
~)
(~.
\fJ
-1 +
p;)
(p
3
1
(p
.3
3in 30t
1
C(,8
3Qt
3
+ Pz
cus 36 )
:3
+ P2 sin
36)
,
,
and
nO
can be shown
Xl
(symme t ric)
b
i.3
Frum Table :3 Lt is
2
Xl
~
Ot
frull1 the pc,sitive
that the moment matrix for such an arrangement is
a
po Lnts
A,,2TJme
0
HI
is that of the other circle.
Fer the JiI~'3.!:' illustration suppose
arbitrary.
P
D
~
I
is
45
The lnve rse ma'or.Lx can be shown t.o be
!
2d
I .~~~
,
I
o
2
o
2d-a
d
-a
~-~2-"d-Y
_(_ -a )
o
B
d
B
N(X'X)-l
-a_
._._
. _ - ? ~
2(2d-a-)
-b
b
c
2B
2B
B
c
-c
2B
b
2B
B
=
(24)
e
(symme t ric)
f
e
--
where
B - a(d
b
2
3d _ (a-?
e =
-~-~
2
+ c)
a
+
2
b·
+ c :2 )
a
and
From equation (24) i t i.s verified that this anan.gement
supply usable slopE-rotatable designs as long as
f pein's
46
If
o ,
or
2d -
singular.
then t'Je moment matrix 1.s
It can. be shown that
')
b2
(2d - a~)(d _
+
a
2.
c ) ~ 0
if
PI :f: P:z •
Therefore, the ccnclusi,o.n is that the above arrangement prc'1ideBUSeful slope-rotatable de.si.gns for any number of center poLnt8 and any
angles of
S
and
a
~
PI
if
P2.
Nute that such an arrangement
of points does not supply a 'y-rotatable design or a dfsLgn i.n
simply because
band
c
cannot be zero at the same time.
For the _secolle! illustration consi.der
no
is arbitrary.
[12J , [1
3
J ,
[2
3
N
We ha ve
J ,
[1
2
2.J
C ,
=:
, and
8 + nO •
[1 "-")J
n
1
4,
:=
D
2
:: 4
and
From Table 3
[lJ , [2J
are all '<:ero.
The moment
matrix of t.his arrangement is readi.ly shown to be
1
_1
N
~XuX
2
2.
x x
2
Xl
x
0
0
2a
2a
0
2a
0
0
0
0
?a
0
0
(~
b
c
dj2
b
-d/2
Xl
x
2
12
(25)
""
(f'ymmet1.l.c)
c
---where
J
4
PIC!
b
+
/,
CL)S-2.Q')
4
+
p;?(l
+
47
?
cus-'?S)
:= --,_.~-,--~,
~-'--~~-~~--~~~
N
and
4
d
PIs 1.1] 4QI
:=
4
+ P2
8
in 46
=._-_.. _-~-----~._~--~
N
The inverse matrix is given by
....
---..
,..,----
b+c
.",,-,~-,,---------
?
o
o
-2a ..
~ ~ "- ~
b+c-8a<'
o
o
o
o
1
2a
o
o
o
e
f
e
(aymrnetric)
o
~
')
b+c-8a~
b+c-8a -
1
2a
-2a
-_._-?'
t __
+d
__
~ _ ~ ''
,_ ~ ' ~
"':Y~
'2 (c
(b-cl"'-d"),
,,\
"'.::'
b-c
'~'-~<~~"-~'-;i
c (b-c)-\,d-
where
and
48
+
(b
I
')
C)'
c - 8a''') (c(b
d 2 j?\
... ,
_
,
which implies t.hat the slope-rotat.able cvnditi.cnE are satisfied.
important point t.o note here is that if the q::J.antity
cert.ain
the VarLa:lJ.C"'s and covariances are infinit.e
()f
IXiXI
attributed tc
b
+ c -
b
+ c _ 8a 2
1.5
'J
quantity
c) - d-j2
c(b
then
rhi8 can be
Q
being zero, thus yielding a useless design.
This will. occur when
The quantity
z~,ro,
is
T
An
8a
2
-= 0
or
. f
1 .•
PI
zero
is zero,) if
Pz
- S =
~
nO "" 0 , and the
and
TT
7.
I,
I
an integer.
Therefore, we concl1.ide that the above arrangement 8uppi I.e's usable
~ - S = ~ • I,
slope-rot.atable designs unless
PI = P2
in case of
=
DO
0
.
In the moment matrix equati.on (25)
4
+ P2
S
4S
in
d "" 0
for any
.f
1.
.
t ance,
For InS
0
=
and
PI
nothing but that of
If
C.
- S
=
TT
=
4
d
'"
_.
I.<
4
Is zerc' i.E
(I. 1,-'
S "" 45
and
U
P1
o
SIn
4cr
,then
d = 0 , then the moment ma t rLx is
In additiDn to
rotatable configuration is obtained.
~
is an integer, or
I
d
~
0 ,
In Ea(- t,
b "" 3c • a y-
if
i.E
PI:; P 2 '
no > 0 , then this arrangement :=:mpplies a useful cla,"s
and
of y-rotatable designs, namely rotatable central cump""ir.e design"'..
Fur the
n
1
~
5
and
"~h.iE.1,.i.llustration of
n
:::;
2
odd mement.s up tu
3
.
We have
P "" 4
the equiradial anangement ccnsider
N = 3
+
n
l
.
+ nO
are zero in case of
followi.ng mom8P.t matrix fCir this arrangement.
n
~
Nutiei.ng that all
5
,
we "bt.ain the
49
XI
x')
"",
2
Xl
0
0
a
]
N-1XcK
x
2
2
XIX'],
a
a
-----0
0
b
-b
-c
a
-c
c
-b
3d
d
()
:=
(s ynune tric)
3d
o
I
:J
where
3
3Plsin3S
c
=: ......
-...o.~- ~-~-..-~~. _.~
4N
and
t~le
Th,e inverse mat.rix 'has
equation (24).
sider
il
l
.2:
0
ig::ll:'.S~i.llu8t:rathJ!1 of
5,
fi.~PtUi:'j"traU(,n~
Therefore, thi.s arrap,gemen,t pnvid",s cJ,::abJe i-lcpe-
rotatable des igns
As the
same ferm as f.'hat cf t':1e
n,) ""
4
and
nO
the eq;jlxadial
1..3 arbit:rary.
al{an,gemE'~l,t,
:S!,mUarlv,
Lt_
(,:"',-
can be
show that: trd.3 arrangement pruvides ueable ,:.:lcpe-r,-Latable d c,cLgn6o
of two ci,reles and center points.
As the last example f( [
t~
50
independent. vaciabl.e.d, we wil.l consid,,"Y the fclJuwLng
given by tr.E-' de.:oign matri.x with
of center
~
XI
K
a
a
1
J
-a
l
-a
1
8
-a
m
0:
8
+
Ill)
,whe:re
iDtS.
a
D
N
==
3
3
I)
2
2
-a
a
-a
2
2
2
...
8 trials
0
0
a
0
-a
0
0
0
0
o
0
4
4
nO trials
The mument matrix for this arrangement Lo;
c(mfjgu~aticn
DO
L.s t:t,e number
51
:2
2
Xl
x,)
xl
')
o
1
?
4a:2+2a Z
o
_ . - -..
_--~
....
=--~
N
o
o
f)
o
o
o
o
o
cn)
4 _ 4
4a +2a
1
-3
()
-~-_.
N
N
o
(symmetric)
The inverse matrix is simple to find from equaticn (2.7), hence it l.S
not written here.
Var(b
22
)
From Corollary 3,
for slope-rotatability.
Var(b
11
)
Let
I
48
1
~
2
2
2
4
+ 2a
-~---*.~-'--'--.---=
.~
4
De.t
sh0uld be equal to
4
4a +2a
1
3
----=--~
N
I
2 2
1 .=_,
2
<fa a
.....- . - . ' -
_o.a.~
N
(3ymmetric)
From the moment matrix equati.on (/.7),
Lf and
only i f
and
52
which 1.3 simplified ti'
)
')
2
7
2... -
-a IJ
[ 2(a.1--a 2-,') + la
• 3 4"
and
(2.8)
In ut~er words, equati.un (2.8) L~ toe nece30ary and
Det "" O.
E".lfficient cCiJdLtlcn for slope-nJtatabUity foY thLs arrangement.
For eqtJatl"n (28)
L~)
be t'tue for an ar-bit ra ry
N, the
[UU"I,) i
ng
ccndiLi,O'!1,S shcd.d held.
1.
()
,
2.
:3 •
Noti.ce
De t
that~
of designs
;t 0 •
thi~
fo'c a design. Ln this a:rrangement. L) bel. ng tc'
C , eq;ja U."ns (')9)
\_
.~
l~
a l'.so necessary an d
8 11 f
cla8s
f i.c.te nt:,
which can be seen frCllTI t':le moment matrix. f,quati.on (.27) si.nce the
.,
'/
first condUi..:m in eqnation (29)
[1
(aI' a 2 , a 3 , a 4 ) ""
(,
')
usable design beh,ngLng t.o
any de..'-':l.gn in
C
Wbat are tbE'
[1"] - [.r]
and
I
4
J = [ ;2'+J
~\(?
-, _. , 1{5,
t.he next one fr,)m
is obtained f.rum
C.
FDY
1)
(Jr
example , lfiitr .
(1[2 ~
I.,
IjT!2
.1,/ '_.?
Obvic;isly there an·
D
n
I)
de
\(57:i}
i.n f :.n.LL':' n:.llllb\c
C
L" sl(ipe-rut.at.ablf.
c(,:'1.diti.cn,~
needed fer t:hL3 arrangemt".",r
.-
v-
t,· gi,·.'t
)
reta tab ill ty?
[j "J""
l.
J
and
53
I
I ' )
.)
[" ',[,'" ] = ,)[,'!,4~J --, .> '1,"'2,'''-J
,
[.
are required to aC:'1e,,,, "y-rGLltabi.lLty.
There-
fore, we can. 'Jbtain. from eqaatiofl (2,n t.fla t
from
[1
2
J
., ,
.- ['/"J
frum
(30)
and
~
ALso we need
6a
Det 'f 0
a;:?
2
1
for non-sing:.tlarlty of
~:;ati8fiei3
Fer i.n.::itance,
supplie" a ',V-yr,r.atable dt-si,gno
and
NL,r.:i.ce trIat e,quatkn (30)
XiX.
equation (2,9), whhhc,:'lO'Jld be
true s i.nce t:'-'J~ c'la :38 of 'y-rotatable des Lgns Ls a Ei'Jbset c f
Box and Hunter (5) deve10ped a number of
'y-rotatable second order designs in three
dimensi.,~'ll'o.
Amc.ng them three
basic ~Y-riJtat.able designs are: one based on the lC.JbahedT\lr), a
12
poi.nted, 2.0 Si.ded regular fi.gure with additL;nal p,Jl.nt.f at t:e cenUr,
a second based on the dodecahedron which has 20 poi.ntF and ]1 facE2;
and a third design ba2ed un a cube and an cctahE-dr, 'n cC1n,u.c tr I.(.a] 1y
placed.
Box and Hanter showed for each case that':k,me :-'trLct
shi.ps shl)",jld
h(~ld
among t:.he eooLdinates cf experime:nta.l. p inc.,:; in (r:dEer
to give ~Y-rc:t:atabil:i.ty. We n"w want tc ,c.;':li~\h7 t:~,at: ft r t
C
t:~'rH'
',8
d2,signs abl,'ve thE cund.i.t·,~on2 needed for 'y-r,)t:atabLL toy ar.;.-,
necessary fcr sL,pe-re,tatability.
In other wDrds,
WE
C·
nut
'.·1
V.I.!..;
i
t.f:lree design c()nfi.gurat.i.on:~ TA71;1::'ch a::-:e :='lr,pe-r r.tatable but may
,',
y-r-o ta tab Ie.
relath'n~
,1.,., be
54
The
.£ir~3t;
Ltlu.st:ration i.s the design confi.guration based on the
icosahedrLrn, whose design matrix is
x
a
1
3
a:l
1
a
-a
2
-al
a
-a
-a')
a
1
a
0
a
0
a
0
-a
0
-a
I
l
1
1
1
0
0
Z
0
2
-a
2
'-
Z
-a
where there are
x
2
0
Z
0
0
0
0
o
0
c.e.nter points.
The moments of thi.s configuration
are given by
for all
i
for all
i
55
2 2
2 2
[ 1.. J. l. = 4.a1a ZIN
and all odd moments
=
for any
i
and
j
where
i ". j
0 •
The matrix of these moments is exactly the same form as that
of
C, so that this design configuration supplies slope-rotatable
designs for any
should be
a
1.618.
l
and
a
Z
•
To be ~-rotatable the ratio,
a /a ,
1 2
This ratio condition is not necessary for slope-
rotatability.
The second illustration is the design configuration based on
the dodecahedron, whose design matrix, with
given below.
[
nO
center points, is
The moments of this configurations are
.2.2]
J
= 12/N
1.
for any
i
and
j
where
i ". j
and all odd moments = 0 •
The matrix. of these moments is also of the form of
the configuration gives slope-rotatable designs.
c
must be
1.618
rotatability.
C , hence
To be ·y-rotatable,
which is not a necessary condition for slope-
56
xl
D
ill
x
2
x
3
0
l/e
e
0
-lie
e
0
lie
-c
0
-l/e
-e
c
0
lie
-e
0
-lie
-e
0
llc
e
0
-lie
llc
e
0
-lie
-e
0
-l/e
e
0
llc
-e
0
-1
-1
-1
1
-1
-1
-1
1
-1
-1
-1
1
1
-1
1
1
1
-1
-1
1
1
1
1.
1
0
0
0
0
0
0
o
o
=
57
The third illustration is based on a cube and an octahedron con-
---
centrically placed.
The design matrix is given by
Xl
-_.--a
x
x
2
3
-a
-a
a
-a
-a
-a
a
-a
a
a
a
-a
-a
a
a
-a
a
-a
a
a
a
a
a
-b
0
0
b
0
0
0
-b
0
0
b
a
0
0
-b
0
a
b
0
0
0
a
0
0
Cube
D
m
=
Octahedron
Center
o
o
,.-/
The moment matrix is easy to obtain and i t indicates that the design
configuration also gives slope-rotatable designs.
the ratio,
bla, should be
s1ope-rotatability.
design.
3/4
2
Any ratio of
~
To be ~-rotatable,
1.682 , which is not necessary Eor
bla
produces a slope-rotatable
58
More than three dimensional des!gns:
There are few regular figures
in more than three di.mensions which may be practically useful in the
formation of slope-rotatable designs.
2P
They are the hypercube of
points (p-dimensional analogue of the cube), the cross polytope of
points (p-d imens ional analogue of the octahedron) and a Zp + 2p
figure
two.
(.i.~. ,
2.p
pointed
central composite designs) which is a combination of the
Central composite designs have the design matrix
x
Xl
a
a
a
a
p
~---1
-a
I· .
Hypercube
I\
D
m
-a
-a
b
0
-b
0
0
b
0
-b
I!
-a
i
:
=
: ••
o
o
b
o
o
-b
o
o
o
o
o
o
Cross Polytope
• Center
i:
o
o
o
/
where
N = Zp + 2p + nO.
.......)
i
The moment matrix of thi.s a'rrangement
indicates that it supplies slope-rotatable designs for any ratio of
b/a.
Note that
b!a
should be
2
p/4
A
to be y-rotatable.
59
3.2
Model~.
Designs for the Quadratic
the True Model is Cubic
In the previous section the emphasis has been on the use of sloperotatable designs for fitting second order models assuming the fitted
models describe the true response functions properly.
the fitted polynomial is not exactly adequate.
In many cases
Therefore, equally
important i.B the consideration of the bias due to the inadequacy of the
medel.
In thiS section we discuss quadratic polynomial designs, in
several variables, which afford protection against the existence of
cubic terms in the true response function.
model is
in which
~l
and
The equation of the fitted
take the forms in
.£.1
il<'~) = [email protected]
Section 3.1 and the true relationship is
2
xpx _ )
p l
and the vector
contains the coefficients corresponding to terms in
as
13 111 ,13 2 2.2'
+ ~2f?.2 where
'~2
;
~2
terms such
are included.
The effect of bias on designing experiments and its properties
can be explained well by examining two illu,':'trative examples in detail.
Two particular classes of designs for two independent. variables are
studied; one class is the designs whose points are equally spaced on a
circle with center points, and the other clas,3 i.s the designs wit.h two
circles with four points on each circle plus center points.
that for two independent variables
X.
2
and
D
2.
Notice
take tbe forms;
60
x
x
3
x
n
3
12
x
3
21
3
22
x
x
2
x
U 21
2
x
12 22
x
U
x
2
21
2
xl.Zx 22
X =
2
x
3
2N
2
xl~2N
and
2
1
3x
0
2x x
1 2
DZ -
a
The matrices
Xl
and
3x
D
1
2
2
2
xl
are given previously in Section 3.1..
The reasonS why we choose the two illustrations above are as
follows.
First, they are commonly known designs.
In particular, the
first class contains the hexagonal design and the ~-rctatable central
composi.te design, and the latter class contains the
3
2
symmetric
factorial design, all of which are very useful in response surface
experimentation.
Secondly, they are slope-rotatable designs.
Finally, they supply optimal or
near~()ptima1.
designs under some
conditions, which will be discussed in the next chapter.
3.2.1
Designs
W.ll. .l t
Points Equall"Y. SQaced on a S ~..le
C:i,.r5~
For the case of two independent variables, the equation (1£ the
fitted model is
61
and the true response function is
hence,
a.nd
Suppose the experimenter wants to place
points on a c.ircle of radius
If
~l
positive
p , augmented by
n
l
~
6
nO
~
equally spaced
1
center points.
is the rotation of a design point on the circ.le from the
xl-axiS, then the design matrix can be written as
x
xl
2
p cos ~l
p sin ~l
p cos (8-ta 1 )
p sin (8-ta 1 )
p cos (28-ta 1 ) p sin (28+c:r 1)
..
P dos «n 1-1). p sin «(n 1-1).
8+cr- I)
8-ta )
1
62
where
e
= 2rr/nl.
It is of interest now to i.nvestigate t.he moment.
matrix and the precision matrix for such designs.
These matrices are
obtained as below.
1
xl
x
a
a
p 2 f /2
-1 ,
N XIX I .-
2
xl
2
z
x x
l 2
p 2£/2
2 /,)
p f .-
a
a
a
0
0
p2f / 2
a
0
0
p f/8
a
2
4
3 p £/8
X
4
a .
3p 4£/8
( s ynnne t ri c)
p4 f
and
b
-1-f1
b
l
a
2
0
b
-1
(1-f)p
2
2
0
b
u
2
22
bIZ
-1
---2"
0
(l-f)p
0
0
0
0
0
0
p f
N(X'X )-1
1 1
=
2
P
2£
3-2£
·---~4
f(l-£)p
2£-1
-.-- 4
(synnnetric)
o
f(.l.-f)p
3-2.f
·fO-f)-:
o
,
I
~ I
p f
---
I
!
/"
j
63
where
N
= nl +
nO
and
f
= nllN ,
the proportion of trials on the
circle to the total number of trials.
It is obvious from the moment
matrix that these designs are y-rotatable, which implies that they are
slope-rotatable.
As mentioned earlier, two types of regi.on of interest are
generally used; one is the unit square and another one is the unit
circle.
We discuss both cases to evaluate the design criterion
the mean squared error integrated over
The unit square region of interest:
slope at
x
=
(xl' x )
2
2
along the
= ~ [_2_
+
2
N
P f
4x 2 (
I
R.
The variance of the estimated
xl
3-2f)
4
f(l-f)p
factor axis is
+ x 22(-84)J.
P f
Since the equiradial designs are slope-rotatable and
Theorem 2 can be used to evaluate
J,
V.
Var(b )
l
= Var(b 2 )
That is,
where
__2_ + 4 (5-4f)
2
4 •
fp
3f(l-f)p
(31.)
,
64
In order to evaluate the integrated squared bias
matrix should be found.
B, the alias
It is readi~ shown to be
r
13 222
SUI
3 °2
I
_ P
, 4
A = (XiX1) -lXiX2 =
L_
SU2
0
0
0
0
1
13 122
°l
'4P
1
2
0
3 2
4 P
0
0
0
0
0
0
0
0
0
0
0
0
4"P
0
S.R
,
.
----
The integrated squared bias is then
B -- NO
2
2
B~'cdx
iJ
= -m
2
iJ
+
SR-12S',ildx
---
NIl
8CJ2
J-If -1
3
['4
B1c = ~'~/2
since
2
P
13 222 +
from (8)
,
122
"4
P
13 112 - (3x 2S 222 + 2x 1x 2S 122
(32)
65
Sl11:= S 222 := S 112 = S 122 ' then
If we assume
?
NSill
B = - 40'2
=~
~
where
over
R
2
(p
1 1
I-II-I
4
8
3
-
p
= \[N S 11/0'.
2
2
(p
+
2
2 2
- (3x 1 + 2x l x 2 + x 2 )) dx l dx 2
28)
~
(33)
Therefore, the mean squared error integrated
is
2 4
8 2
28
.J=V+B = [ 2- + _4(5-4f)4
- J + ~ (p - 3 p + 9)
fp 2
3f(l-f)p
From equation (34) i t is observed that
creasing function of
p
V= 0
lim
p-++OO
and
p
B
V
B
1.155
~
value of
,
is a convex function of
whose minimum is achieved at
p
it is easy to see that minimum
p < +00 for any given
f
f*,
is a monotonically de-
with
.
=\j4TS, that is, approximately 1.155
and
(34)
~.
J
Onc.e
From the behavior of
V
is achieved for
p
is fixed t.he mi.nimizing
can be obtained by solving
dJ/df = 0 ,
which gives
+ 10 - V;;Z--';-;'o
3p 2 + 8
Obviously the
p
which minimizes
and therefore, So does
~
changes.
f*.
J
(call this
Table 4 shows how
p*) depends on
p*
and
f*
~
,
vary as
66
Table 4
p* and
a
f*
VS,
p*
0
a for the unit square R
f*
V
+ en
-+
B
0
0
-+
J
(Optimum)
-+
0
1
1.577
0,7471
3,371
2.664
60035
2
1.387
0.7374
5,224
6,728
11. 952
3
1.304
0.7332
6.477
13.213
19.690
4
10260
0.7310
7.308
22,368
29.676
+00
1.155
0,7257
9.968
+ co
Notice that as
f'/(
a
gets large,
p*
have observed in Section 2.3.
decreases approaching
0.7257.
also decreases approaching
n
l
CD
1.155 and
This is the Same aspect we
fi~
Also
changes over the enti.re i.nterval.
practi.ce, since
+
changes very little as
pi~
However,
is an integer, the optimum
a
changes widely.
f*
In
can only be
approximately achieved.
If the region of operability is limited to
then the largest possible
shape of
and if
that if
R.
n
n
For instance,
= 7 , it is
l
l
~
p
depends on
if
n
l
V
and
1.006, etc.
B
1
because of the special
= 6 , the largest
6 , the largest possible
Therefore, both
n
R, the unit square,
p
is
1.035
It was found by the computer
p
is less than
1.155.
are minimized at t.he largest possible
when the region of operability is limited to
The unit circular rea..i:.l2!!.- oL interest:
R.
Using t.he same prccedure we
have used for the unit square region of interest, we can obtai.n the
following:
p
67
p* and f* vs. a for the unit circle R
Table 5
a
V
B
-+ 0
0
f*
p*
J
(Opt imum)
0
-+ +00
1
1.493
0.7523
3.342
2.177
5.519
2
1.297
0.7431
5.353
4.528
9.881
3
1. 208
0.7378
6.831
7.898
14.729
4
1,153
0.7346
8.031
12.403
20,434
+00
1,000
0.7257
13.291
+ co
+co
v
= _2_
+
f p2
-+ 0
--..,5_-4.-f_-:f(l-f)p 4
and
B
=a
Consequently,
2
J
(p
4
- 2p
=V
2
+ B
+
5
3)
where
= [~+
fp
2
5-4f 4 J + a (p4
f (l-f)p
Notice the difference from the previous case that
p
=1
J
and minimum
minimizing
f
fie
is achieved for
for a given
=
(35)
B
1 ~ p < +00.
pis,
+
4
p ~'c
as
a
expected, Table 5 Shows simi.lar aspects for
p~'c
and
Table 5 shows the variation of
observed in Table 4.
B
Also the
2
2p + 5
D-L:V
2
2p
then
is minimized at
f~'(
and
f~'c
As
as we have
If the points are restricted to the region
is minimized at the largest possible
which also minimizes
changes.
V
term.
Note that at
p
pic:::
(,i.!:..,
1,
R
p"c = 1 )
fic
= 0.725'1
68
When we calculate the integrated squared bias for the two types
of region of interest, we assumed that
Slll
What happens if the assumption is relaxed?
=
6 222 = I3 11Z = 6 122 .
This is an interesting
problem to study, and we propose the following assertion.
Theorem 3:
For any values of
13111'
13 222 ,
6 112
and
6 122
for
the designs with points equally spaced on a single circle in two
independent variables, the integrated squared bias achieves minimum
at
P
p
=
=
1
1.155
Proof:
where
if the region of interest is the unit circle, and at
if the region of interest is the unit square.
The integrated squared bias is given in equation (3Z) as
B
l
is the first term and
expression of
If
S 111 :f 0
B.
,
then
B
2
is the second term in the
Suppose the region of interest is the unit circle.
B
1
can be written
2
3 2
2.
1
B =9'-.S[(- P +- rIP )
4
l
TT
R 4
(3x
'J
2
2
+
+
2xlxZrZ)]-dxldx2
xZr
l
1
69
be written as
B1
3
(~
=
1
which indicates that
13
~
122
0 , then
B1
where
r
=~
1
[4
JR
.2
~6
,
B
then
l
= 1.
2
+ constant]
p
=
1.
SIll
If
o,
and
2
2
(x z + 2x 1x 2 r 3 )] dx 1dx 2
.2
p
-
~
=
\{N 13 122/a
.
It can be shown that
- 2p 2 + constant)
4
(p
Bl
is minimized at
is independent of
consider this case.
p
- 2p
is minimized at
and
which also implies that
= 0
4
is
3 ::: 131l2/l3l22
B ""
1
~ [p
B1
B
1
2
.22
+ i r 1)
p
,
p = 1
B
is minimized at
If
I3 Ul = 13 122
hence it is not necessary to
B
2
Similarly, we can show that
Therefore.
.
p::: 1
is minimized at
if the region of
interest is the unit circle.
Suppose the region of interest is the unit square.
If
SIll ~ 0 •
then
is
where
then
B]
=~
2 .3
(~+
1
i
2 4
8 2
r 1 ) (p - j p + constant)
70
which shows that
13
~
122
B
1
0 , then
B}
2
B1 : ~6 (p
is minimized at
p = 1.155
1.155.
o
S 111
If
and
can be shown that
4
8
3
-
p
2
which is also minimized at
at
p
+ Constant)
= 1.155.
p
Therefore,
B
(J
Similarly,
is minimi.zed at
region of interest is the unit square.
\[N 13 122
a
where
B
2
is minimized
1.155
p
i f the
Q. E. D.
Theorem 3 gives valuable information to the experimenter.
If he
feels that a significantly large bias exists in the quadratic model
and he wants to use the single circle equiradial designs, then he
might want to take
p
near
R, and
p
near 1.155
interest,
1
for the unit circular region of
R, since
for the unit square
Theorem 3 tells the experimenter that no matter what values
are,
p
B
is minimized at
= 1.155
3.2.2
p = I
for the unit square
for the unit circle
R
13
ijk
I
s
and at
R.
Designs With Points Equally Spaced on Two Concentric Circles
Next consider a design with four equally spaced points on a
circle of radius
and another four equally spaced points on a
different circle of radius
P2' augmented by
a point on the first circle makes an angle
second circle makes an angle
S
Ot
nO
If
and a point on the
with the positive
moment matrix and its inverse matrix were found in
(26).
center points.
xl-axie, then the
equation~
(25) and
We showed also that this class of designs is slope-rotatable.
Now we want to find
properties.
V,
Band
J
explicitly to discuss their
71
The unit square region 0LJnterest:
W
is the unit square,
f R DIDI 1dx-
n =0
R,
is given by
11
0
W
If the region of interest,
0
0
0
0
0
1
0
0
0
0
I
0
0
0
4
3
0
0
4
3
0
=
(symmetric)
2
3
W
By substituting
and
1l
of equation (26) into equation
(11), it can be shown that the integrated variance is
V
where
= 1-
+
b,
c
a
2a
22
1 C4c (b-4a 2
)-d +(b+c-8a )(b-c2]
3
(36)
.
,
2
2·
(b+c-8a )(c(b-c)-~d )
and
d
are given in equation (25).
Also
B
is
obtained
8 (b+c) + 28)
2a
9
B
(37)
3
assuming
The quantity
(b+c)/2a
which is independent of
orientations.
4
PI
2
PI
Note that
+
in
and
QI
B
B
S •
4
4." 2.
2
is equal to (PI + P;t{Pl + P2)'
Therefore,
B
is independent of
is minimized at
4
P2
2
+ P2
4
= -3 •
(38)
72
Relation (3S) can be transformed into
( PI2.
2) 2
-"3
+
(2.
2) 2 _ (Irs 2.
P 2. - 3
-~)
which is a part of the circle with radius
~S/3
(2/3, 2/3)
and
on the Cartesian plane of
and the center
as shown in
Figure 2.
2.
P2
-..:+------------:;:f-----..,..
o
4
(3' 0)
Figure 2.
and
P2
for minimum
Figure 2 shows that any combination of
the solid circle line gives minimum
B.
PI
2
P1
B
and
Pz
which is on
The quantity, mi.nimum
4 ~N 13111 2.
.
)
3
(j
-(
from equation (37) subject to the restriction of
13111 - 13 222
B , is
73
The unit c irc'-llar re.8.ion of interest:
the unit circle,
W
11
is
0
w11
= (2 JR
If the region of interest is
0
0
0
0
ol
1
0
0
0
0
1
0
0
a
DiD 1d~
I
I
I
!
1
0
0
1
a
(symmetric)
1
2
i
l
/'
Similary, it can be shown that
V =
222
1
+ 1 [4c(b-4a ')-d +(b+c-8a )(b-~l
(2)
4
2
2·
(b+c:-8a ) (c(b-c)-~d )
and
In this case
B
is minimized at
4
4
PI + P 2
b+c
=
"2 + 2: = 1
2a
PI
P2
and also independent of the angles,
and
minimum
Pz
which miD.imize
B
Q'
and
6 •
The combinations of
are shown in Figure 3.
B, for t.he unit circular region of interest is
under the assumption,
6 111
=:
6 222
=
6 112 -, 6 122 •
The quantity,
74
2
Pz
1
+
(0,1)
(1,1)
)
./
°
Figure 3
/
2
- - - - - / - + -r'- - " 7~
/ PI
/
(1,0)
and
Pz
for minimum
B
We have discussed two particular classes of designs.
Suppose
one is interested in comparing the two classes of designs.
The
comparison could not be done completely without help of the computer.
Note that, for instance, the minimization of
J
for the latter class
of designs is difficult to accomplish analytically, since
J
is a
function of four unknown parameters, two radii and two angles.
Furthermore, if one wants to include other classes of designs for
comparison, it would be very difficult.
For these reasons, an optimum
seeking computer program is necessary.
In the next chapter, com-
parison of experimental designs and selection of better designs by the
aid of a computer program will be discussed.
75
4.
S ELE'C TION OF
EXPERlME.~\JT4,L
DES IG~"S
The Computer Prc.8.~am
4.1
If we are able to find an optimal design fer any given number of
observations, it would be very useful to an experimenter.
However, it
is extremely difficult to find optimal designs by the completely
analytic appruach.
Therefore, a computer program was used to aid in
the searchin.g fo:1:' optimal designs.
The computer program used i.n this thesis for selection of
experimental designs is a modification of one u.'3ed by Evans
(8).
The
majo!:' differences from his program are the design uptima lity criterion
and the estimatur of the p,--.lynomia1 coefficients.
Evans ccn.sidered
minimization of t!le integrated variance of estimated response
the design criterion using the minimum bias estimato·r.
/.
Y
as
Our design
criterion. is basically minimization of the integrated mean squared
error of esti.mat.ed sl(Jpe based on the least squares estimator.
The class of designs considered fur optimiz.ati.on .is cumbinations
of equiradial designs.
Designs with puints equally spaced on one, two
or three circles whi.ch are concen.tric abuat the urigin will be
investigated to select optimal designs.
Therefore, tne radius of each
circle and the angle of a reference design point on each ci.rcle are
unknown paramet.ers to be optimized.
In this
C()mpT~ter
optimal desLgnswe will restrict ourselves to t!'e ca3e
pen.dent variablE's (i.~.,
The mai.n
follows.
feat:~lre8
Eearch fcn
\.Jf
ttMl)
:i.nde-
p = 2 ).
of the program can be bc.Lefly :::kF r:ched as
Fi.rst:, one lJf three criteria L" chcJsen;
V
B or
.J.
76
The shapes and sizes of the region of j,nterest and the cegion of
operability ace ,.:;elected.
Then possible configurations are con-
structed by cumbining c00.centric circles.
The program t.hen searches
for an opt.imal design usi.ng a simplex search, suggested by Hendrix (9),
over the 'region of uperabili.ty for the constructed configurations.
The simplex search s'clbroutine i.s the cere part of the program.
For
that reason, its function is described below.
The basic E'earch, pattern used is the regular s Lmplex in
dimensions, where
k
k
is the number of unknown parameters currently
under investigation.
The program anchors the original simplex at a
point
~I
:=
(aI' a2.' ••• , a )
k
in the region of operability.
Using
this point as one vertex, a regular simplex is formed and can be
specified by the
(k + 1.) x k
a
cr+a
cq+a
DO
=
cq+a
design matrix
a
1
2
cq+a
2
1
cr+a2.
1
cq+a .,
1
cq+a k I
L
cq+a
cr+a k
2
I
I
./
where
1
~.
k ,'-\[:2. [(k -
1)
',
+
\'k~
V" T ~'J
- 1]
,
77
and
c
=:
a scale factor to determine the size of the simplex.
For each poi.nt represented by row
J ) is computed.
of
d.
1.
a yield
Y.
(V ,
1.
or
B
These yi.elds are ranked from smallest to largest.
Then the rows which give poor yields are replaced by the new rows.
Rules for how to drop rows of a simplex and find the new rows are
explained in detail in Hendrix (9).
Once the new simplex is formed,
the process is repeated until it reaches an optimum.
As the simplex
approaches the optimum, it is reduced in size periodically by reducing the value of
c.
In the program
c
0.1
=:
was used as the
initial scale factor and reduced five times, each time by
4.2
Near-Optima1~igns
Selection of Optimal and
The program was desi.gned to handle any
of the fitted model and the true modeL
searched !=he case of
variables.
d
2
=:
l
The case of'd
1.
and
=2
1/10.
d
and
Z
1
and
d
Z
' the orders
However, we have exclusively
=3
d
d
Z
for two independent
=:
2
has been automatically
considered, because, if the coeffici.ents in the cubic terrrBof
are assumed to be zero, this is nothing but the case of
Two regi.ons of
d
l
=:
d
Z
2
=:
3
and
interest are used; one is the unit square and the
other is the unit circle.
The regi.on of operability is also divided
into two types; one is limited to the region of interest and the other
is not limited at alL
There is obviollsly some practical limit or
bound on the region of operability i.n nearly all design problems.
78
Therefore, the latter case should be interpreted in such a way that
the region of operability is much larger than the region of interest.
Designs have be eo studied for observati.ons,
N
< 6, there are no non-singular designs.
thorough search was conducted, but for
obtained from
N:5: 9
N
were investigated.
6:5: N:5: 9
For
~
6:5: N :5: 12.
For
a
10 , several good patterns
Before getting into the
results of the computer search for optimal and near-optimal designs,
it is important t.o introduc.e some nomenclature.
As mentioned earlier, we consider the confLgurations combining
one, two or three concentric circles.
described by the following way.
and
no
Each configuration 1.8
If there are
center points, it is described by
where n.
.
18
1.
t he
outs i.de circle.
n
k
1
concentric circles
- )12, -
••• - ok - nO
. 1 s on LIe
h·
. 1e counte d .f rom t h e
numb er o.f tr1.a.
1th
- ' c l.re
For example,
5-4-0 i.mplies the configuration with
five points on the outer circle, four points in the inner circle and
no points at the design center.
with
n.
1
=
1
for
1.
~
1.
We do not consider the configurations
However, there is no restriction on
nO •
The best design obtained by the computer search for a given configuration is described by the notation,
(n ,8 ) , where
r
8j
r
n.
J
. t.e
h numb er
1.S
0
(n ,8 )
1 1
+
(0 ,8 ) + ••. +
2
fpO.lnts
·
' .Tth
on the
2
c ire 1e an d
is the counterclockwise rotation of a referen.ce po int on the
circle from the positive
xl-axi.s.
consists of two concentric n-gons.
Thus, the dE:sign
(6,0) + (4,~.)
The fi.rst is a 6-gc'il (hexagon)
centered at the origin with a point on the positive Xl-axis, and the
second is a 4-gon (square) with a line connect.ing the reference vertex
to the origin rotated counterclockwise from the Xl-axis through an
angle of
TT
4
radians.
This design is shDwn in Figure 4.
79
x.
!
*\
x
'X
Figure 4
The radius of the
the parameter
The design
j
t~
+
(4 '4
IT.)
n-gon in a design will be specified by
p. , while the number of trials at the origin will be
J
Note that
denoted by
( 6 , 0)
could be different from
In order to find the best design for a given
N, all possible
configurations shc,uld be examined and then by comparing the optimal
designs obtained from each configuration, the best one is selected for
the given
N.
For example, suppose
N = 6 •
Then all possible con-
figurations are; 5-1,4-2,4-2.-0,2-4-0,3-3-0,3-2.-1,2-3-1, and
2-2-2-0.
Now the computer program finds an opt.imal design for each
configuration, and by comparing those optimal desi.gns, tb.e best design
for
N
=6
is decided.
80
Four different cases have been considered according to the region
of interest,
4.2.1
When
R, and the regi.on of operabili.tyo
Rand
RO
RO.
are the Unlt;.,.S9..uare
We first consirler the case where the region of operability is
limited to the uni.t square region of interest.
cannot reali.stically use the
J
without prior knowledge about
criterion to find an optimal design
Q
f.J
assumed that they ar..:. all equaL
for each
N
From the fact that one
I
for the cubic terms, it is
C'
I.Jok '"
0
Table 6
S[,.(lWS
the best designs found
when
\[N S 111
()(==
(J
=1.
More than one configuration may give the best design li.sted in Table
For instance, both 4-4-2 and 4-4-2-0 give the best design for
6.
N
= 10.
each
N
In Table 6 configuration whi.ch produces the best design for
is given.
The optimal and linear-optimal" designs are given
in Appendix Table 7.5.
If there are more than one configuration which
produce a design, they are also listed in the Appendix.
An impcrtant characteristic is that every optimal design (except
that of
N
=6
) in Table 6 contains one observation in each corner
of the region of interest..
For
N
=6
, the possible configuration
which can give one observation in each cerner is 4-2-0, but this configuration supplies only singular designs.
helps to explain the jump in the value of
The optimal de;3 igns for
N ;;:: 9
This lack of estimabil.:Lty
J
from
N "" 6
have the pa t tE'rn of
to
3
factorial design with mGre than one center point in case of
N
~
7
2
N;;:: 10 •
.
Table 6
N
Optimal designs when
Configuration
Rand
Design
IT
RO
are the unit square
NO
P1
P2
P3
--
--0.839
6
5-1
(5 '-'0)
!.
1
1.012
..,t
4-2-1
(4 , IL)+(2
4' '. , 0),
1
\[2
1
8
4-2-2-0
(4,~)+(2~O)+(2:*)
<+
t.
°
\[2
1
9
4-4-1
(4:~)+(4,0)
1
\[2
1
10
4-4-?
(4,%)+(4,0)
2
\[2
1
11
4-4-3
(4 ,~)+(4 ,0)
3
\[2
1
12
4-4-4
rr
(4'4)+(4,0)
4
\[2
1
-----
B (0'=1)
J
17.568
1.428
18.996
12.541
1.611
14.152
9.634
1.443
1l.On
8.250
1.444
9.694
8.214
1.444
9.658
8.539
1.444
9.983
9.000
1.444
10.444
V
00
f--i
82
Thi.s implies that the
3
2
factlnial design is r.ecommended for a re-
sponse surface second or.der desi.gn when
RO
Rand
are the unit
square centered at the origin.
An interesting question which arises is how many points should
one use in the experimental des ign.
Since
is a measure per
J
observation, it can be used to compare the. efficiency of di.fferent
designs.
From Table 6, it is seen that
efficiency.
to that of
However, the
N
=
N
=
9
gives the best
and
N
=
11
are close
11 •
Table 6 is based on
=1
a
does the different value of
efficiency?
values of
J
N = 10
a
a
If
is different from
1 , how
influence the best des igns and
Appendix Tables 7.1-7.4 summarize results for four
a .
different values of
Appendix Tables 7.1-7.4 indicate that
efficiency for any
a
N = 10
also gives the best
and the best configurations obtained for
in Table 6 are left unchanged for different
(all-variance design) and
01 = +co
a I S , including a
(all-bias design).
a
=0
Nevertheless,
the best designs obtained have slightly different radii for some
accord ing to the change of
for
as
N = 6
a
a.
= 1
N
For ins tance, the bes t des ign rad i i
remains the same for all
a , but for
N = 7
they change
varies.
It is interesting to study the variation of the best designs by
the change of
N = 9.
p1
=
a.
Consider, for example, the 4-4-1 configuration of
Table 7 ,shows that the best design (aU-bias design) has
1. 244
and
p2
:=
1
if
a:=:
+ co •
83
The variat.ion of the best. design of N
ex
Table 7
=9
by the change of
g-
91
9Z
TT/4
0
8.250
1
TT/4
0
8.250
\[2
1
TT/4
0
8.250
?
QI ~. (
\[2
1
TT/4
0
8.250
QI
1
TT/4
0
11.884
ex
PI
0
\[Z
1
\[2
2
3
Pz
1.244
+00
r
B
V
In equation (38) it was shown that
B
J
0
8.250
1.444
2
9.694
1. 444 )
14.028
(1. 4 44)
21.250
+00
Ql2(4/3)
is minimized at
4
+ Pz
2
+ Pz
4
= -3
which should be satisfied by the radii of the all-bias design.
ex s 3
the best designs found are the same as that. of
variance design.
But as
ex gets large.
ex
=a ,
For
the all-
PI approaches 1.244 while
Pz
is left unchanged.
For the other configurations in Appendix Tables 7.1-7.4, we can construct tables similar to Table 7, which show that the best designs for
each configuration are almost identical for moderate valaes of
Ct
'"
"
"
Since Table 7 was constructed under the awsumpt10n
tnat
a 11,
•
I:l
~.
Ok IS
1.J
, other ratios of 13. ok' s
I.J
should be tested.
13.
1J k
0
's
Table 8 shows the data for three different ratios of
regarding the same 4-4-1 configuration as in Table 7.
84
Table 8
Variation of the best design for
Sa",
Ljk
N
-
9
for different
U
. mr;;
I
~
a
S ijk "
PI
81
P2
..
S 111 :S 222: S 112: S 122
:=
1:1:0:0
1 1
1:1:'3:<3
\[2
+00
\[2
\[2
\[2
\[2
0
1
2
.3
\[2
\[2
\[2
2
3
S 111: S 22.2: S 112: S 122
::::
0
1
\[2
+00 1.327
82
J
_---
1 rr/4
1 rr/4
1 rr/4
1 rr /4
1 rr/ 4
0
0
0
rr/4
rr/4
rr/4
rr/4
rr/ 4
0
1
1
1
1
1
B
V
0
0
0
0
0
0
8.250
8.250
8 250
8.250
9.050
11.450
15.450
0
0.800
a 2 (O.SOO)
8.250 a 2 (O.800)
8.250 a 2 (O.800)
0
8.250
8.250
8.250
8.250
9.802
+00
8.250
9.121
11.734
16.090
0
cO. 8ll
871'
a 2'0
t,.
.)
2(0
871)
a.) .•
+00
a-(O.859)
~
SIll :S 222: S 1.12: S 122
:=
1:1:3:3
0
\[2
1
\[2
1.329
1.270
+00 1.169
2
3
rr/4
rr/4
rr/ 4
1 rr/ 4
I rr/ 4
1
1
1
0
0
0
0
0
8.250
14.850
33.685
62.824
8.250
0
8.250
6.600
9 768 0;(5.979)
11.164 oL(5.740)
14.446 a 2 (5.600)
0
+00
In Chapter 3, slope-rotatability is recommended as a desirable
property.
12
Notice that the optimal designs for
N:= 6, 9,
in Appendix Tables 7.1-7.4 are slope-rotatable.
experimenter may prefer t.o use t.he slope-rotatable
for
la,
N
(5,~O)
11
=7
and
, an
design
from the 5-2 configuration in Appendix Table 7.5 rather than the
optimal design from the 4-2-1 confi.guration.
slope-rotatable
its
4.2.2
J
(4,~) + (4,0)
Al:3o for
N:= 8 ,
the
design may be used, especially since
value is very close to that of the optimal design.
When
R
1:3 t~le Unit Square and. RO _Js No~ LimH:ed
If the region of ope rabilLty is much larger t'al1 the (;nit square
region of interest:, the optimal and near-optimal design:.,:; (AppE:ndix
Table 7.6) are different from the previ.ous case.
The be:3t de.sl.gns
85
v
obtained by assuming
~
~
are equal and
6 ijk "
~
1
are summari.zed in
Table 9.
The best efficiency is obtained from
also give
J
N
=9
values very close to that of the opt Imal des ign.
shown by the cumputer program that as loo.g as
~
, but all other
It is
i.snot large, say
Q'
s: 2 , the optimal configurations in Table 9 remain opti.mal.
i f the ratios of
may change.
6. 'k v s
~J
N's
However,
are different., the optimal configurations
6111:6222:6112:6122:::: 1:1:0:0 , then the
For example, i f
6-2-1 configuration replaces
5-2-2-0
a3 the best for
N
=
9 •
Nonetheless, the computer search shows that selferal configurations
such as
6-2-1,
5-2-2-0
and
6-3-0
designs consistently for any ratios of
4.2.3
When
Rand
RO
for
N
S ijk u b..
=
9
provide very good
•
are the Unit Circle
If the region of operability is identical to the unit circular
region of interest, the optimal desi.gns obtained are all equLradi.al
designs that consists of only boundary points and center points.
The
best designs are shown in Tab'le 10 and other near-optimal de:=;igns are
listed in Appendix Table 7. i.
Table 10 an.d .A.ppendix Table 7.7 are
also based on the assumption that all
s ijk u",-
are equal and
~::::
I •
An important result obtained by the computer search after
examining the best designs for different ,,'alues of
optimal designs appearing in Table 10 are
words, whatever the value of
the best deSigns.
best design for any
~
Q'
~-independent.
t!1 other
is ~ the des igns uf Table 10 remain
For example, the hexagonal design of
~
is that the
.'\l = 9
is the
, which implies that the hexagonal design is the
all-bias design as well as the all-variance design.
Table 9
N
Optimal designs when
Configuration
R
is the unit square and
Design
NO
RO
is not limited
B(Q'=l)
J
Pl
P2
P3
V
3.470
2.743
6.213
3.264
2.600
5.864
3.194
2.546
5.740
6
5-1
(5,0)
1
1.588
--
7
5-2-0
(5, 0)+(2 ,~)
0
1.637
0.606
8
5-2-1
(5 ,0)+(2 ,~)
1
1.674
0.915
----
9
5-2-2-0
2TT, (?-'S
'rT)
("5 ~ 0\,+,(2 ~7)+
0
1. 702
0.806
0.768
3.304
2.315
5.619
10
6-2-2-0
(6,0)+(2, 2~)+(2
'%)
0
1.684
0.755
0.752
3.270
2.387
5.657
11
7-2-2-0
(7 ,0)+(2,~)+(2'~5)
0
1.638
0.962
0.221
3.345
2.427
5.772
12
8-2-2-0
(8,0)+(2, 2~)+(2,~0)
0
1.633
0.756
0.526
3.363
2.446
5.809
00
(J'I
87
Table ]iJ
Optimal designs wlccen
and
.R
RO
are the unit circ18
-
=
N
r~
Design
CDnfigura ti,x1
___
~
'
_
,.:. . ,_ _
·
_ ~ _ . _ ~ _ ~ · " - - - " " - W _ ~ ~ _ '
-
__
B (Q'=1)
V
p
NO
~
~
,
~
~
_
~
.
-
-~
.J
~
.
_
~
~
-
6
5~1
(5,0)
.1
1
14.400
0.667
L5.067
7
5-2
(5,0)
2
1
13 .301)
0.667
13 0967
8
6-2
(6,0)
2
.1
13.333
0.667
14.000
9
6-3
(6,0)
3
1
13.500
0.667
14.167
10
1-3
(7,0)
3
1
13.333
0 0667
14.000
11
8-3
(8,0)
3
1.
13.2.91
0 0667
13.958
1.2.
9-3
(9,0)
3
1
13.333
,j
.667
14.000
_ . . . . . - : ' _ , . . , . . ._
- __
__. a . o ; _ ~ " _ '
,~_~,"'""
For the equiradia1 designs, we have sh0wn in equation (35) that
2~V "" _.
2
+
5-4t
~-...;.,..;.--:-
p f
f(l-f)p
4
and
In this case \.7hen
minimized at
13 .235
is
13.902 if
N
=
p-
.R
and
and
and minimum
01
=
11 ,whose
are the unit circle,
RO
0 7:257
f
B
0
is
for a.0.y
2c//3
J:= V
+ B
is
and its mini.mum
01
Therefore, minimum
J
V
1.3
l o I n Table 10 the best efficiency is obtained from
J
value is
13 0958
and whose
f
is
1).121.
An import.ant peint which must be mention.ed is t.hat. different
ratios of
Q
f-I
ijk
Us
do not seem to alter the best designs in. Table 10 0
Tv show this, Table 11 L; const:ructed for
,'\1:= 9 0
that there i2 no c1-!ange for the best des i.gns.
Tahle U
indicates
Ccmparing Table 11
with Appendix Table 7.7, we can also notice t>,ar: there is no change
in the second best
d~aigno
Table 11
S
Three best designs for different ratios of
.~
, 'k '"
~J
S 111: S 222:S 112: S 122
= 1:1:0:0
S 111 :S 222:S 112: S 122
:;:: 1'1.
. .1
<'.2j
~
J
S 111 :S 222: S 112: S 122
= 1:I.:3:3
Configuration
in case of
S.~J'k' s
Design
N
=9
NO
P1
P2
V
B (0'=1)
6-3-0, 6-3
(6 0)
6-2-1, 3-3-3, 3-3-3-0
'
3
1
--
13.500
0.562
14.062
7-2-0, 7-2
(? ,0)
2
1.
--
13 .500
0.562
14.062
5-2-2, 5-2-2-0
2-5-2-0
(5,0)+(2,0.357)
2
1
.1.
14.664
0.618
15.282
6-3-0, 6-3 .
(6 0)
6-2-1, 3-3-3, 3-3-3-0
"
3
1
--
13 .500
0.546
14.046
7-2-0, 7-2
(7,0)
2
1
--
13 .500
0.546
14.046
5-2-2, 5-2-2-0
2-5-2-0
(5,0)+(2,0.357)
2
1
1
14.664
0.570
15.234
6-3-0, 6-3
(6 0)
6-2-1, 3-3-3, 3-3-3-0
,.
3
1
--
13.500
2.250
15.750
7-2-0, 7-2
(7,0)
2
1
--
13.500
2.250
15.750
5-2-2, 5-2-2-0
2-5-2-0
(5,0)+(2,0.357)
2
1
1
14.664
2.474
17.138
"j
J
CX>
CX>
89
Some of the reasons why the equiradial des igns supply the hest
designs are that
minimized at
Theorem 3.
p
:=:
V
is a decreasing function of
1
for any
Q'
and any ratios of
and
p
B
S.LJ0k us
is
from
Since the region of operability is limited to the unit
circle) the maximum
also minimized.
p
which can be achieved i.s
1 , at which
B
is
Thus the equiradial designs whose boundary points on
the unit circle provide the best designs.
Observe that the best
designs in Table 10 are all slope-rotatable.
4.2!4
When
R
is the Unit Circle and
RO
is Not Limited
In this case, exactly tl:1e same configuration:.; appearing 1.n Table
9 give the best designs as Shown in Table 12.
signs, see Appendix Table 7.8.
of
Q'
S
and
ijk
vs
unit square and
For near-optimal de-
The discussions for different values
are quite similar to the case when
RO
is not limited.
R
is the
Therefore, they are not
expressed here again.
We have shown the optimal and near-optimal designs and examined
their properties for four different cases according to the region of
interest and the region of operability.
For simplicity, let us call
them Cases I, II, Ill, and IV in the order they have appeared
previously.
In other words:
Case I:
When
Rand
Case II:
When
R
RO
are the unit square,
is the unit square and
RO
is not
limited,
Case III:
When
Rand
Case IV:
When
R
limited.
RO
are the unLt circle, and
is the unit circle and
RO
is not
Table 12
Optimal designs when
is the unit circle and
RO
is not limited
=
;:
"'"**
N
R
Configaration
Design
NO
PI
P2
P3
-~
-~
~
V
B (0'=1)
J
3.397
2.268
5.665
3.217
2.126
5.343
3.161
2.071
5.232
-~.~.
1
1.505
{50" +(? 1'L')
5 ""
0
1.555
0.591
5-2-1
TT)
(,'5 , 0\''"'
'+\~"5
1
1.588
0.884
---
9
5~2-2~()
(5,0)+(2, "'~)+(2,~)
0
1.621
0.782
0.752
3.241
1.870
5.111
10
6~2-2.-0
(6 ,'"
0) +
'!i). '2 2TT )
1,"-;4' +', '9'
0
1.603
0.740
0.729
:3 .210
1.937
5.147
11
7-2-2-0
I. 7,
'
0
1.564
0.916
0.323
3.260
1.997
5.257
1"
.to
8-2-2~0
(8,0)+(2, 2~)+(2'~6)
0
1.575
0.840
0.435
3.163
2.117
5.280
6
5-1
(5,0)
'~
I.'
5-2-0
8
\,'
,
~.
" ..... ;;0
')
'<")
0). +~'2 '4)+
TT· (2 'IT "
} I5}
.~=-=----.
\0
o
91
From the data and observations we have had from 4.2.1 to 4.2.4, we can
sUllunarize t.he following properties regarding the optimal a'J.d nearoptimal designs.
1.
If ther is no limitation on the region of operability (Cases
n
and IV), the optimal and near-optimal designs for the
,mit square region of interest are very similar to those for
the unit circular region of interest.
Tbis implies t.hat the
shape of region of interest is not important if limit.ation
not imposed on the region of operabi.lity.
18
However, if the
region of operability is limited (Cases I and III), the
optimal and near-optimal designs for the unit square region
of interest are quite different from those for the unit
circular region of interest.
Therefore, the experimenter
must know what shape his region of i.nterest is to make use
of the optimal designs when the region of operability is
limited.
2.
The
J
values of the optimal and near-optimal designs of
Gases II and IV are closer to each other than those of Cases
I and Ill.
This i.s probably due to the limLted region of
operability for Cases I and Ill.
3.
For all cases, the change of
Ci
does not influence the
optimal configurations as long as the values of
modest-s i.zed, say
Ci:S;
2.
Also the value of
the best efficiency remains the same as
Ci
Ci
remai.n
N which give""
changes.
4.
For all cases, there is very little change in
J
values due
to small rotation of the inner circles of the designs
obtained.
Usually the change is in the fourth or fifth
decimal place.
Thus the angles given Ln the Appendices for
the inner circles are not necessarily exactly cptimal, but
are chosen in that way, partly because it is easy to construct designs, and partly because the chosen angles are
very
5.
close to the optimal angles.
The optimal designs have some patterns.
follow the pattern of the
3
2
Fcc Case I, they
factorial design.
F'.::'r Case
III, they are the equiradial designs which have observati.ons
on the unit circle and at the center ,where the
prlJport ion
of points between the circle and the center should be close
to
0.7257.
Notice that the
3
2
factorial and the equi-
radial designs are slope-rotatable.
For Cases II and IV,
they follow the pattern that several points are allocated
OD
the outside circle (more than half of ob;::;enraticns) and a
few points on the inside circle or at the center such as
5-2-1,
6-2.-2-0, etc.
rotatable.
These Gpti.ma1 designs are not
Therefore, i.f an experimenter
want~
;~
lope-
to use slope-
rotatable designs fGr Cases II and IV, he can consult
AppendLx Tables 7.6 and 7.8 to select
t':1P
designe.
6~2
Fur example, conf iguratiu'1
config:ltation
Since the
each
N
J
5-4-0
for
N:-;: 9
are
Elope-rctatdhlE'
f( r
N""
8
and
3lupe~·~(\catable.
value of the best slope-rotatable design for
i.8 clo,3e to that of the best deaign,
these s],-,pe-
rotatable designs are usable in the sen.,5e of the .I-erHer inD.
93
..comEa!i.?gn l"f Hexa.R9nall .i~Rc,tat~J~~e!1tral
4.3
f~ompo~ite
3
and
2
Faetori.<!!1..Desl,gps
'J
I.t was s~LOwn in the previous section
that t12e
3""
fae tc)rial for
Case I and the !::exagonal fur Case I I I are the best des i.gns f.Jr nine
Another design which was introduced by Bcx and Wilson
observations.
(6) and which has been widely used in response surface designs is the
y-rotatable central composite (r.c. c.) des ign.
are slope-rotatable.
These three des igns
We now want to investigage and cumpare thEm in
more detail.
As a measure of comparison, we consider the following cirteria:
1.
Minimization of
2.
Minimi.zation of the rate of curvat'Jre of t:':J€ variance
J,
function, and
3.
B
Minimization of
where criteria
and
2
are based on a proper standardi.:z.ation of
3
the independent variBbles.
The
J
Suppose all
erited.on has been Hsed thus far as the bas ie critert.on.
Q
; -,
Let
are equal.
f-I.1.J'k '"
usual and let the region of interest be the
expressi.ons for
J(hex)
J
=:
J(r.coc.)
'V-N.
rep:r·e8L~l]
.. . .t
ex
f-IU l. /,..
v
Q
2q~are.
110it
a2
The
for the designs are from Secti.cn 3.2,
3
14
p
p
[2' +
-4]
9
[-~--
4 p 2.
+
+ ex
2
39
~~- ]
2p 4
[p
4
8
-"3
2
+ ex [ p
p
4
2
+
28
«~9 ]
8
2
- ~3 p
8
- "5
2
p
,
+
28.
~-9]
'2.8
+X5 J
,
(39)
94
where
p
represents the radius which I.S shown in Figure 5.
'"
If the
\.
\\
~-~--~
J
I
/x·
... /
-~,-,. ~
')
3<' Patte rial
Design
r.c.c. Design
Hexagonal Design
The three designs
Figure 5
region of interest is the unit circle instead of the :.mit
square~
then
they are
J (hex
.
)
J.
~r.c.c.)
[ ,~~2 +
p
21 ~J
2 4
+
0/2[p4 _ 2 p 2 +
P
21
3'
9
111_
2_ 4
2
5[4 i + 8p4J +0/ Lp -!.p +j'J,
n
p
6235 p +5 J
(40)
95
TablES 13 thro.lgh 16 are con.st.ructed in orde.r t., c,'mpare
designs by the
.]
criterion under the
f,JUX
t.1,8
differen.t (aees de.3cribed
earlier.
For Cases II and IV thehE'xagonal design
3
values followed by t.he
However, the
2
be3t in. term:' of
1,'3
J
factorial design fo r any lTai.Jt!
values are not much different.•
J
factorial design i.S much bet.ter than the others, and the .'l8xt best is
ex.
the hexagonal dt::,sign regardless ()f
'y-rotatable central compc3.it.e dt;.~l.gn
design is best and the
next best.
ranking.
For Ca;se 11.:. the hexagonal
1'n this case aI80 the value. of
The
3
2
01
dees net.
j;
t.he
I.nflu~'nce
t.he
factorial des ign is far beh Lnd thE' c,thers.
Summarizing the tables, we notLee t.hat the ht'.xag<:'nal dE'S \gn l.s
best under three cases and the
3
2
factorLal deSIgn .16 Ute best on
one case, mainly because the best design is affer:ted by the chcLcE of
the region of operability.
But an important resul t
hexagonal design domLnates the
I::'
that the
'y-rota ta ble cen.tra I compc,,; i te
dl-'S
Lgn
for all four cases.
The
o§~~
criterion we want to discuss for cDmpar: son is the
varIance function.
The variance function of a seumd
rotatable design for two independent varl.ableE
1::'
crder:'L(jpe~
frem eq,Ja tL,,;: 16)
N
?
- 20 2 [Var(b ) + Var{b ) + d"C4 \/an. h ) + Vat hUn
2
l
l1
where
d
222
:: xl.
+
X
z•
Therefore, for the three
shown t l la t: the va r Lance func t iens are
hexagon~
(.
V I.~) .-
.3 +
'2'
p
dt.. 21,
t
~4)
p
de2i~n~ I '
can hE
Table 13
Case L
Rand
RO
are the unit square
==-=De-'ign2
~
0' :=
0
p""1.035
0'
:=
0'
1
p""1.035
:=
0' =
')
"-
3
0''"''+00
p""1.(135
p=1.035
p=1.035
'l
?
Ht'xagc)!1
tr~v
""lA. 98 7
J"":l4.987+L401
,1=14.987+,/(;.401'1
0
0
l=14 .987+0- £, \ 1.40
=2 7 .596
""20.591
"'16 388
J=l':'. 987+ot- <~ 4 (1)
~ ~ ~ ~ ~ - ,~~~~.~~~~.=~~
P ;,o~l • ()!31
p~L081
p~l.081
p""L081
')
,R.
L
i
,.
~
J:=\:
"":6.235
J~L6 .235+1.,368
0
.j
P""~
f=>""l
J,,-,,T
J""8.25.'\+1.444
""8.25()
""9.694
__
~=~~~_~~-
~"'l
i-'
J=16.235+Q-O .368:\
P""L08l,
q
''<, 5+Q
- ,":. ',J. 368 !',I ~I,_ 6 • /..,
I
-
""28.547
=21. 707
=1/ 593
_=~
r;, .~,.
J=16 .23 5-tQ ~ l. 368)
')
')
J""8.250i'Q'~(1.444)
:14.028
~~-
~_~~~o~===~~'
p"'O.89 4
p=l
?
')
J:=8 .2. 50-/-Q' ~ ( ~, .444::
""21.250
.~..
-~~--
'1"~. 4'''?..I--~
J'"
,,-~.\.I' 1,-'3
~~~~~=~~~
.'"
(J\
1~
Table
Case
De" .tgne
0' '"
TT"
,J;.-••.
_~
~
0
0'
~---~
=
1
Q'
RO
:::;
is net
l~mlted
'2
Q'
""
3
0'=+00
==-~~~
P .... +00
Hexagon.
is tree unit square and
R
~I
~
11....0
p ... 1.387
P "'" 1.576
J '"' 3.471+2.663
~
P "'" 1.304
J. "" 5 ._336'i'Qt 2,
(..... 68/'
'+.'
6.114
P "'" 1.155
')
c!
'"
5.508+Q1"(.:L. 4 68)
-, 4
J .,,; 10 .125TQi "\3
= 19.82{1
'" 12.070
~~=oL"-"""",.=-
p, ...
R.C==.
+oo
p =
P '" 1.614
III
J '" \'-+0
.J "" 3.740+2.947
J '" 6.000+0''''(1.753)
p .... +oo
L,
p
P '" 1..249
~.
"" 3 .735+2.938
'" 6.673
J ,.;
7.652.TQi'"(L.506 i
J
4
J ~ 12.6 56+cl <~ \3>1
'" 2L 106
= 1.095
P
?
J "" \'....0
')
= 13.112.
'}
P "" 1.155
P "" 1.323
?
"" 6.68:'
3
~.41~
J '" 5.937+c/-(1.777)
"" 13 .04'/
-~~~--~-=~~~-
:=
1.025
P = 0.894
')
~ "" 7.550-ta"'CL5(6)
1
~
I"
, 2.4.
-_ L_'! .'+L2'1"Q'
~~,'
3-'
:::; 21.102
~--~~-~=~
,0
Table 15
Case III:
Designs
Ci '"
0
p "" 1
Hexagcn
,1 "" V
Of
p
R.. G.u.
•..-,
=1
J :: V
=:
~
2
1
14.167
P "" 1
f
J"" 16.'875+0.667
16.875
"" 17.542
1
3
=
are the unit circle
P '" 1
,T "" 13.500+0.667
'" 13.500
~j
RO
Rand
1
= 1{2
P ,;" 1{2
P
J ... V
"3 250+ (.25"
,J --.;..
36)
= 23.250
=
23.944
Ci
P
=2
=1
= 13.500-ta""(O.667)
Ci
P
')
J
=:
_ 23 ?50 2,25)
.- -ta ~36'
= 26.028
J -
o
p
=1
') ')
"" 19.500
p
=1
J
=:
?
P
=:
1
') ')
16.875+Q;-(O.667) J "" 16.875ia';'(~)
= 22.875
19.542
1
P = 1{2
=1
Ci=+CO
J "" 13.500-+t:/-CO.667) J "" 130500+Q;<'(1>
"" 16.167
P '" 1
2
J '" 16.8:5-ta (0.667)
=3
1
1
P
= 1{2
P
J
.25,
= 23.250-ta 2 \}6)
J
>=
= \[2
= 23.250+a 2,25)
\36
29.500
----~----~--~~~--~-----~-----------~~-~-~-~--------~-
~
00
Table 16
DeRigns
-
Hexagon
Case IV:
R is the
=0
01
p-'+oo
P '" 1.496
01
J "" V....o
=
circle and
1
J "" 3.439+2.L97
:=
p-++oo
01
J "" \!-+O
+00
J "" V-.O
p-'
3
c_
is not limited
=2
01
01=+00
p
:=
J
=:
P
:=
')
?
J '" 5.397+Qt~(1.150;'
J ,,; 7.063fOt'~([).869)
= 9.997
.5.636
=3
P = 1.204
P = 1.304
P "" 1.531
J :<: 3.625+2.469
P
1
? ')
13, • SOO'
"(~l
- -tOt '. 3 '
.. 14.885
P
= 1.183
J :::: 3.651+2.442
"" 6.093
= 1.330
..
P
2
J -.-. 5 938fOt
'l ?59')
.
~.
,., 6.094
?
RO
=
~ ~
H.• C.<:~8
~nit
"-:
J
:=
:=
1.024
7.810fOt~(O.937)
p = 0.959
2
5.911+cr (1.257)
"" 10.936
J ..
?
1
? ?
J "" 16.87.5fOt"(=-)
3
= 16.243
= 10.976
p
= 1.233
p = 0.775
2
J = 7.549fOt (0.951)
t1 )
J = 16 56?;.c/ 3 /
.~
•
&..
\.
= 16.104
\.0
\.0
100
V(~)"::.32 +
r.c.c.:
d
2
(Ill) ,
4p
4p
\/(~)
=
3
2'
+
d
81
(~.~.)
8p 4
2p
where
?
d~
is the distance of a point
the design radi.us.
(41)
x
from the origin and
p
is
Ie compare the designs, we want to standardize the
radii. (subsequently the coordinates of two variables) of the designs
in such a way that the integrated variances
are all equal.
of the three designs
The designs are then compared on the rates of
curvatures of the variance functions.
average
V
In other words, keeping the
V constant, we want to compare the behavior of the variance
functions.
Certainly, a small rate of curvature is desirable.
Suppose the region of interest is the unit square and
designated.
1.101
From the variance portion of equation (39)
to accomplish
V
= 12
for the hexagonal design.
p
V
= 12
is
should be
The variance
function for the hexagonal design is then given by
V(~)
=<
2..475 + 14.289 d
2
in which the rate of curvature is
14.289.
V(~) ,-
1.640 + 15.540 d
2
V(~) -
1.840
+ 15.240 d
2
Similarly, we can obtain
for
r.c.c.. ,
for
3
and
where the rates of curvatures are
15.540
2
,
and
15.240, respectively.
101
Therefore, at
curvature.
= 12
V
If
1.3 different. from
\f
rate of curvature?
values of
, the hexagonal design has the least rate of
12, which desi.gn has t.he least
Similar comparIson was made for a number of other
V. spanning the interval from
V=:4
to
V =: 18.
For all
values checked, t.he hexagonal design shares the smallest rate of
curvature.
The
lh:h~'!
crUerion to be used to compare the des igns is based on
the integrated squared bias.
B
We adopt the same standardization
0
scheme employed in the second criterion.
designs are not compared on
B.
Suppose the region of interest is the
unit square and t.he predetermined
should be
p
1.159
and subsequently
B
=:
1.367
2
V
is 10.
to accomplish
V = 10
becomes
Of
1.333
2
Then from equation (39)
for the hexagonal design
Similarly,
and
B = 1. 361 Of
2
3
for the
for the given standardization of
V
2
differences are quite smalL
5
s:
V
s: 25
B
.
= 10
1.2.25
and
We can observe that. i f
3
2
If
However, the
In t.he same way Figure 6 is constructed
changes for any choice of
the r.c.c. design.
Therefore,
, the hexagonal design is
V
V
in the range of
< 11.2
from the best is the b p xagonal des ign, the
place and the
=:
factC'rl.a1 des ign.
better than the ·y-rotatable central composit.e design.
to show how
p
for the y-rotatable central composit.e design, and
Of
p = 0.949
B
After standardization the
3
2
,
the order of designs
factorial design and
V > 12.6 • the r.c.c. design takes the first
factorial design is the second one.
Nonetheless,
the experimenter may, most likely, wish t.o standardize tIle designs
with a reasonably small value of
V.
If t.i:lis is tbe case. the con-
elusion is that the hexagonal design is the best one under this
102
B
\
\
i
\\
2.0C/
1. 9Q'
2
Hexago
\\\\
\\\.,.'
\\
\
\
v
5
Figure 6
critericn.
10
15
B as a function of
20
25
V
If the region of interest is the unit cixcle instead of
the unit square, it can be shown that similar results hold true.
We have ccnsidered three different criteria to compare the three
designs.
Tn. conclusion,
the hexagonal design is better than the
'y-
rotatable central composite design under any of these three criteria,
and it is also better than the
3
2
factorial design except for the
case when the criterion is J-minimization undecr the unit square region
of interest and region of operability.
1°3
5.
SUMMt\.RY
This work has concerned itself with extending two main criteria
to experimental designs for estimating the slope of response surfaces
by second order polynomial equations when the true response is quadratic or cubic.
One is the criterion of rotatability advanced by Box
and Hunter (5) and the other one is the criterion of
error
integrat<-~d
J , mean squared
over a region of interest, which was introduced by
Box and Draper (3,4).
The least ffluares estimators of the polynomial
coefficients were used to estimate the s lope of reBponse surfaces.
In cases where there is only one independent variable, the class
of designs with three symmetric levels (x :; - h ,0
nO
and
n
l
trials on each level, respectively) was considered.
was shown that
and
f"'~,
andh, and
h
n
It
should be taken as large as possible to minimize
the optimum
f = n/N , stays within a narrow range as
,
l
V,
h
changes over a quite wide range.
It was also observed in the case when
the true model is cubic that, as
Ct =
creaSeS approaching
0.2847 and
\{N S3/cr
increases,
h*, which minimizes
f,'~
de-
J, also de-
creaseS approaching I •
In the case when the true response is quadratic for more than a
single variable, the concept of the estimated slope variance function
for an eXIE rimental design was advanced.
A desirable criterion was
introduced, which required the variance functiun to depend only on the
distance from the origin of the design.
Such designs insure that the
estimated slope has a constant variance at all points which are the
same distance from the center of the design.
perty were called slope-rotatable designs.
Designs having this proThe necessary and
104
sufficient conditions for sLope-rotatability for a design were
developed and the properties of slope-rotatable design8 were discussed.
It was shown that the class of ;-rotatable designs is a
subset of the class of slope-rotatable designs.
A number of second
order slope-rotatable designs which are not ~-rotatable were presented
and discussed.
1n the case where a second order polynomial is fitted but the
true model is cubic, the cri.terion used for optimi.zing designs was to
minimize
J , the integrated mean squared error of the estimated slope.
This criterion considers not only the variance but also the bias due to
the inadequacy of the fitted polynomi.al model.
For this case two
classes of designs were discussed in detail with emphasis on the integrated squared bias.
One class is the set of designs whose points are
equally spaced on a circle with center poi.nt.s, and the other is the
designs of two circles with four points equally spaced on each circle.
A computer program was written to aid in the search for optimal
designs under the J-criterion.
Optimal and near-optimal designs i.n
two dimensions were constructed through computer search for four
different cases according to the square or circular regi.on of interest,
and the li.mited or unlimited region of operabi.lity.
The resulting
optimal and near-optimal designs were catalogued and compared.
This
comparison showed th.at when the regions of interest and operabil i ty
were the uni.t square the optimal pattern followed the shape cf
3
2
factorial desi.gn and when the regions of interest and operability were
the unit circle, the optimal pattern was the de:.:;ign with pcints equ.ally
spaced on a single circle.
rotatable.
Both opti.mal desi.gn patterns are slcpe-
105
Finally, three commonly known desi.gns for. nine
observati.ons~
the
hexagonal design, the y-rotatable central composite design and the
3
2
factorial design were compared under three different: criteria.
The result indicated that the hexagonal design is better than the
~-
rotatable central comp()si.te desi.gn under all. three cri.terion, and it
')
is also better than the
particular
case~
3-
factorial design except for one
namely the unit square region of interest with design
points restricted to t1:"le region.
106
6,
REFERENCES
1.
Atkinson, A. C. 1970
The design of experiments to estimate the
slope of a response surface. Biometrika 57:319-328.
2.
Atwood, C. L. 1971. Robust procedures for estimating polynomial
regression, J. Amer. Stat. Assoc, 66:855-860.
3.
Box, G. E, P. and N. R. Draper. 1959. A basis for the selection
of a response surface design. J. Amer. Stat. Assoc. 54:622654.
4.
Box, G. E. P. and N. R. Draper. 1963. The choice of a second order
rotatable design. Biometrika 50:335-352.
5.
Box, G. E. P. and J. S. Hunter. 1957. Multi-factor experimental
designs for exploring response surfaces. Ann. Math. Stat.
28:195-241.
6.
Box, G. E. P. and K. B. W~lson. 1951.
attainment of optimum conditions.
Series B. 13:1-45.
7.
De la Garza, A. 1954. Spacing of information in polynomial
regression. Ann. Math. Stat. 25:123-130,
8.
Evans, J. W. 1974. Computer generated D-optimal, V-optimal and
trace optimal minimum bias experimental designs. Mimeograph
Series No. 936, Institute of Statistics, North Carolina State
University, Raleigh, North Carolina,
9.
Hendrix, C. D. 1971. Empirical optimization in research and
development, Second Gordon Research Conf. on Operations Research and Management Science, Holderness, New Hampshire
(Aug. 9-13, 1971).
On the experimental
J. Royal Stat. Soc.,
10. Hoerl, A. E. and R. W. Kennard. 1970. Ridge regression bias
estimation and experimental designs for response surfaces.
Technometrics 11:461-475.
11. Karson, M. Jo, A. R. Manson, and R. J. Hader. 19690 Minimum bias
estimation and experimental design for response surfaces,
Technometrics 11:461-475.
12. Kupper, L. L. and E. F. Meydrech. 1973. A new approach to mean
squared error estimation of response surfaceso Biometrika
60:573-579.
107
13.
Matteson, R. J.
1972. Mi.nimum bias estimation of the slope of a
response surface. Unpulbished M. Sc. thesis. School of
Industrial and Systems Engineering~ Georgia Institute of
Technology, Atlanta. Georgia.
14.
Munsch. R. C.
1971. Mlnimum bias des igns for response surfaces.
Unpublished M. Sc. thesis. School of Industrial and Systems
Engineering. Georgia Institute of Technology. Atlanta.
Georgia.
15.
Ott, L. and W. Mendenhall. 1972. Designs for estimating ti:1e
slope and a second order linear modeL Technometri.cs
14:341-353.
109
Appendix Table 7.1
Opcirnal designs when Q' = 0 and the region of
interest and the region of operability are the
unit square
cz=
N
Design
Conf igura Uon
-.-----~~.~~~-'
NO
..
PI
:;g ; ...
P2
P3
. _.
~
. .~-----
~_.
- ~ - ~ =
6
5-1
(5,~())
'7
4-2-1
( 4,
8
4-2-2-0
(4,~)+(2,O)+(2,~)
9
4-4-1
(4,~.)+(4,O)
10
4-4-2
(4,~)+(4,O)
11
4-4-3
12
4-4-4
V
"-""--~
1 1.012
t) +(:2 ,0)
; ;
17.568
.,
1
\[2
.I.
0
\[2
1
\[2
1
8.250
2.
\[2
1
8.214
(4,~)+(4,0)
3
\[2
1
8.539
(4,~)+(4,O)
4
\[2
1
9.000
'+
--~-
L2.541
0.839
----
9.634
--~--_.~. ~
Appendix Table 7.2
N
Optimal designs when ~ = Z and the region of interest and the region of
operability are the unit square
Configuration
Design
NO
PI
Pz
P3
V
B
2
J
6
Sol
(5 !!-i
1
1.012
--
--
17.568
01
(1.428)
23.280
"7
4-2-1
(4,~)+(2,0)
1
1.377
1
_0
12.819
1
52/'
01 2/',.l.
'+)
18.916
8
4·-2-2-CI
'4 ~'+(0 0\+(2 rr)
1.'4"'\'-')'
02'
0
\[2
1
9.634
",2(1. 443)
15.406
9
4-,-<-~
(4
TT, '/ 0)
'4")+V+,
,
1
\[2
1
.L
--
8.250
'" 2( 1.444)
14.028
10
404-2
(4,7)+(4,0)
2
\[2
1
.L
--
8.214
'" 2( 1.4 4'
4)
13.992
1.L~
1
4-4-3
(4,~)+(4,O)
3
\[2
1
--
8.539
",2 (}..444)
14.317
12
4-4-4
('4 $4fT) +. (''+, O')
4
\[2
1
--
9.000
01
, '20'
.L
'+
0.839
2
(1.444)
14.778
Optimal designs when Ci = 3 and the. region of interest and the region of
operability are the unit square
Appendix Table 7.3
N
Configuration
Design
"5
TT
NO
)
1
1.012
--
--
n .568
Ci~(1.428)
30.420
1
--
13 .425
2( .L.,,)
4""-"
01)
2,6.221
7
4-2-1
(/
''-+~4)+ /-,}
('> 0'
1
10316
8
4-2-2-0
"4
IT .. (? 0 )+ (?
IT·
\, '4,1+,~,
~ )2)
0
\[2
.L
9
4-4-1
(4~~)+(4,O)
1
\[2
1
10
4-4-2
(4 ,~)+(4~ 0)
2
1.403
,
11
4-4-3
(4~~)+(4,0)
3
1.395
1
12
4-4-4
(4. '4'
!!.)+(4 ~,
0')
4
1.392
1
IT, .
J
P3
5-1
'20'
B
P2
6
i.
V
Pl
1'-"-
"
.i
')
2
9.63L.
01
(1.443)
22.622
--
8.250
01'""( 1.444)
21.250
----
8.347
CJt 2(1...... 4?9'
''''''''"'~
21.205
8.750
01 (1.418)
9.231
01
0.839
')
2
2
(1.415)
21.515
21.969
......
......
t-'
Appendix. Table 7.4
Optimal designs when ()( = +00 and the regirl1
f
interest and the reg i.Dn of operability are Lb.'
unit square
1J:;;r=::r
~
Configuration
Design
N
P 1.
0
;::r;:=;
P2
P3
re·-~ ~
V
B
==---..>.='-"'-."'"-,('-_• ....---
=.'-"'--='.-==- . ~"'-~
,)
6
5-1
(5,~,O)
1. 1..012
7
4-2-1
(4,!J-)+(2,O)
1 1. 183
1.
15 .751
()(
8
4-2-2,-U
(4 !I.)+CJ 0)+(7 11)
0 1.258
1. 0.839 15 .873
Ct'
9
4-4-1
(~ , TI"J+(
4 , 0)
4'
1. 1 2.44
0
1.
11.884 ()('11.3n\
10
4-4-2
(4 , rr"I+(4
4'
, 0)
2 1.244
1.
}O .852
'+
"4
'-',
'-'2
17 .568 Q'( 1 .4'28)
'2
'/
(1.340
-{
, • "1
.~
1
3
"
'J
()(~(J
.333
,)
11
4-4-3
(4 ,~-)+(4,0)
3 1.244
1
lO.8Y2 ()(k(1.J13)
12
4-4-4
(4,~)+(4,O)
4 1.244
1
11.266 ()(-(1.33J)
'+
?
Appendix Table 7.5
N
Configuration
6
.5~
1
3~3-!)
Optimal and near-optimal designs when the region of intereBt and the region
of operability are the unit square
Design
N
0
PI
(5 c ~O)
1
1.012
(3 TT • - '3 51i" ~ 12)+~ '12)
0
1.035
B (0'=1)
J
17.568
1.428
18.996
-~
29.278
2.075
31.354
1
--
12.541
1.611
14.152
P2
P3
~-
~~
0.677
~
7
4~2~1
(4~~)+(2,0)
1
\[2
5~2-0
(
5'20)
2
1.012
--
--
16.055
1.428
17.483
3-3-1
6 -1.
(6~~2)
1
1.035
--
~-
17.072
1.402
18.474
4:'3:':0
(4,~)+(3,O)
0
\[2
1
~-
16.072
2.979
19.051
4~2.~2-0
(4 rr)+(2 0)+(2 IT)
'4'
>, " '2'
0
\[2
1
9.634
1.443
11.077
4-4-0
(4,~)+(4>0)
0
1{2
--
9.745
1.439
11.184
4-2~2
(4,%)+(2,0)
2
\[2
1-L
--
11.666
1.611
13.277
-6 1i '"
0 I2l
2
1.035
--
-~
14.868
1.402
16.270
(,)
!L" + (5- ;)20"
lL~'i
\~(~~4<'" -
1
\[2
1.012-
=~
15.898
1.414
17.312
-'-
\[~
--
14.979
2.984
l?963
.5-2
-j
8
V
TT'
0.946
0.839
3~3~2~O
3-3-2
61::12~O
'-
6')
~'-
2~5=1
,:,=)=
1
"
4 ,~~l+O
_0'\
4"
.
~
w
V
"
~
~-
v-l
Appendix Table 7.5
N
9
(Continued)
Configuration
Jlesign
NO
PI
P2
v
B (Q'=1)
J
16.657
1.428
18.085
8.250
1.444
9.094
12.094
1.455
13 .549
--
12.231
1.412
13.643
--
12.125
1.611
13.736
P3
5-2-1
5-3
5-3-0
(5, ~O)
3
1.012
4-4-1
(4,!t)+(4,O)
1
\[2
1
4-2-3-0
(4,~)+(2,0)+(3,- 77)
o
\[2
1
4-5-0
(4 ,~)+(5, 0)
o
V- Z
4-2-3
(4;2;)+(2~0)
3
\[2
2-5-2-0
2-5-2
(2 '2;)+(5, 20
)
2
\[2
1.012
--
14.778
1.415
16.193
2-6-1
(2 ,~)+(6 '~2)
1
\[2
1.035
--
14.985
1.397
16.382
(6 , !L,
') ~) i
3
1.035
14.987
1.401
16.388
1
1.081
16.235
1.368
17.593
2
1.006
16.747
1.436
18.183
6-3-0
6-3
3-3-3-0
3-3-3
'-t
IT
IT
IT
0.995
1
-0.716
.... <1..'"
6-2~L
8-~
7-2-0
.~..
I
"'""
c:.;~
IT,
(8 (8)
-- IT
.,
(I, 28:
!-'
!-'
-l>
Appendix Table 7.6
N
6
Optimal and near-optimal designs when the region of interest is the unit
square and the region of operability is not limited
Configura tion
Design
P2
P~
.j
--
---
1.588
'.'J'3'
1
0
1.883
1.244
5-2-U
5-2
3-3-1
6-1.
0'+"')
11',
\<., ~5!
(5;0)
0
2
1.637
1. 575
0.606
0 (3 ~J.!
TT
~)+
1
1.422
1~-2-1
(!.;
0)+('1
!l)
-,
,,'""'"' ~ 4'"
1
I
1.732
1. 596
5-2-1
(5;O)+(2~~)
3-3-2-0
6-2-0
3-3-2
6-2
5-3-:}
5.,"""1.
...,
g
Pl
!l"
5-1
5.0 )
< • n~+'~
3-3-1
8
NO
-i-
.5 -=- ~~. c:3?o- - :.';.
-'0''')'
(~
'~)~' J
1.688
--1.534
1
1.674
0.915
(6 ~')+":'~9
0' ('., "'-TT)
0
1.619
0.521
(6~O)
2
1.575
0.645
r')
\,J
(6~0)
'"
q
--
-----
B (0'=1)
3.470
3 .'~6 5
2.743
3.439
6.2,13
6.904
3.264
3.407
3.414
3.552
3.860
2.600
2.646
2.864
2.808
2.997
5.864
6.053
6.278
6.360
6.857
3.194
2.546
5.740
3.309
2.632
5.941
3.385
2.650
6.035
3.397
3.556
3.64}
2.655
6.:15'2
6./SCi
2..877
6•
J
(5s0)+(3~
0
(5,0)
3
i'-
1
1.634
1.581
1.605
0
1. 702
J.806
:).768
3.3U4
2.3l5
S.E-19
It'\'
2
1.683
1.103
--
3.219
2.537
5.7%
2TT 'j
0
1.634
0.811
--
3.342
2.446
5.788
n
1.678
0.802
3.388
2.649
6.03 '7
I.! )
~lrr
~')!!\
()) U )+v: ~7);-(" ~5,1
r-
....
1<"";
)={--~-;~
~;.
6-:.2-1
\.U,·l)+\,,:;",o··,
5 (;
+;')
,. ", . v" ~ 5;'
/ I"
(1-'
" cO,
i
----
.~
5-::' ~i=1
---------
'\7
5)(})+(4~(})
1..)
--
--
2.694-
r""'lf....
)k:~_'
.-'
VI
Appendix Table 7.6
N
(Continued)
Design
Configuration
3-3-3-0
6-3-0
NO
PI
P2
P3
\l
B (Q'=l)
J
3.389
2.651
6.040
3.389
3.388
3.433
2.664
2.666
2.672
6.053
6.054
6.105
3.471
2.663
6.134
3.638
3.755
3.740
2.863
2.776
2.947
6.501
6.531
6.687
(6,0)+(3 ;0)
0
l.615
0.554
(7 ,O)+(2~~5)
J.581
1. 578
1.659
0.167
(5,0)+(3;0)
0
2
1
3-3-3
6-3
(6,0)
3
1.576
--
------
4-4-1
,',
TTl
v+ , 0 +('
"'+ ;4"
(5,0
1
--
(8,0
1
1.731
1.592
1.614
1.378
4
--
--
7-2-0
7i _2
5-3-1
5-4
8-1
(7,0)
--
0.863
--
--
,....
f-'
0'
Appendix Table 7.7
,Jot
interest and the regtcr;
NO
PI
P2
V
B (Q'=l)
J
(5,
1
1
--
i', .400
0.667
~5.067
3-3-0
13 O','~ TI"
\ , )+1,.5 ' ) /
0
'1
25.955
1.048
27.0C)3
5-2aO
5-2
,5,O}
2
1
--
I3 .300
0.667
13.967
(6;0
1
1
--
15.166
0.667
15.833
t.
(4.... UI
r'\+f') TT,
~'-",T)
..... '"
\.
"-+
1
1
1
L8.375
0.667
19.042
(6,0)
2
1
--
;3.333
0.667
14.000
5,0)
3
1
--
U .866
0.667
14.533
(70 0 )
1
1
--
16.000
0.667
16.667
f4,O)+(Z)!.)
\"
'" 4'
2
1
1
.00G
0.667
17 .667
(6,0.
3
1
--
.. 3.5~)O
0.667
14.167
N
Conf:i.guration
6
5- t
3a3-1
6-1.
4-2-1
8
Optimal and near-optimal designs Wl]en trie regwD
of operability are the unit circle
Design
1
f).643
~
6-2-0
6-2
3-3-Z-G
3-3-2
5-3-0
5-3
5-2 .. 1
-l
~~,2
... 2-r:.
':'~2~2
q
l
j
,- ,. . , r ..
OIA.J-v
')-5
i1 a 2-1
3-3-3-0
3-3-3
,.....
"
Appendix Table 7.7
N
Configuration
(Continued)
Design
NO
PI
P2
V
B (0'=1)
J
7-2-0
7-2
(7,0)
2
1
--
13.500
0.667
14.167
5-2-2-0
5-2-2
2-5-2-0
(5,0)+(2,0.883)
2
1
1
14.663
0.667
15.330
5-4-0
5-4
5-3-1
(5,0)
4
1
--
14.850
0.667
15.517
8-1
4-4-1
(8,0)
1
1
--
16.874
0.667
17.541
t-
oo
Appendix Table 7.8
Optimal and near-optimal designs when the region of interest is the unit circle
and the region of operability is not limited
N
Configuration
6
5-1
3-3-0
(500)
(3,0)+(3;:3)
5-2-0
5-}
3-3-1
6-1
4-2-1
(5~O)+(2,~)
(5,0)
(3,0)+(3
(6,0)
".
(4,0)+(2 '4)
°2
5-2-1
(5,0)+(2,~)
3-3-2-0
6-2-0
7
8
9
Design
NO
p
'';
B (0'=1)
J
Pl
Pz
P3
1.505
1.801
--
--
1.165
--
3.397
3.350
2.268
2.683
5.665
6.033
1. 555
0.591
--
3.217
2.126
5.343
-1.312
--
--
1
1
1
1.493
1.669
1.513
1.602
1.454
2.178
2.336
2.331
2.523
5.546
5.655
5.796
6.269
1
1.588
0.884
----
3.368
3.319
3.465
3.746
3.161
2.071
5.232
(6;0)+(2'9)
0
1.538
0.513
--
3.256
2.158
5.414
3-3-2
6-2
(6,0)
2
1.493
--
--
3.341
2.179
5.520
5-3-0
5-3
(5,0)+(3,0)
(5,0)
°3
0.636
--
--
"7 '
t -.1
(7,0)
1
1.553
1.500
1.522
--
3.351
3.527
3.543
2.185
2.231
2.399
5.536
5.758
5.942
5-2-2-0
"
-5 0, . 2 2". . ("".
> )+( '7)+ "- '5)
0
1.621
0.782
0.752
3.241
1.870
5 .111
5-2-2
""1
\'5 , 0).. + CL,S"
2
1.595
L057
--
3.197
2.065
5.262
6-2-1
(6
(') 2'1i)
, ; 0')+''<''9
1
1.566
0.805
--
3.196
2.072
5.268
•
<
".
'1)
2".
1
0
---
-'
~
~
\0
Appendix Table 7.8
N
(Continued)
Configuration
Design
N
0
p1
Pz
P3
V
B (cv=l)
J
---
3.341
2.181
5.522
3.343
2.182
5.525
---
3.332
2.197
5.529
5-4-0
(5,0)+(4,0)
1.594
0.779
3-3-3-0
6-3-0
°
(6,0)+(3,0)
0
1.534
0.552
7-2-0
(7. ,.0)·
_,T (?
'-, TT
15/,
0
1.500
0.160
"'! "
(7,0)
(5,0)+(3,0)
2
1
1.495
1.576
0.845
--
3.337
3.386
2.193
2.203
5.530
5.589
3-3-3
6-3
(6,0)
3
1.496
--
--
3.439
2.197
5.636
4-4-1
5-4
8-1
(4,O)+(4,J;t)
(5,0)
'-+
(8,0)
1.640
1.512
1.531
1.312
--
3.543
3.728
3.625
2.387
2.320
2.469
5 930
6.048
6.094
-t.,.
5-3-1
,
l.
4
1
--
--
--
--
--
0
.......
h:>
o
121
Appendix 7.9
Derivation of
J kk'dA.
We want to 8how that
elements are all equal.
P
2
L: cos 0'.1.
i=l
:=
CJ~k'dA
0.: VI.
_____-E
Let
is a diagonal matrix whose diagonal
k' = (cos 0'1 cos 0'2 ••• cos O'p) where
1 •
Then,
2.
ens 0'1
kk'
cos 0'.1 cos 0'2
2
cos 0'2,
cos O'lcos 0'3
cos 0'. cos 0'
cos O'2 cOS 0']
2
cos 0'3
cos O'2cOS 0'
J
...
P
p
cos O'3 cOS O'p
=
(s ymme tric)
2
cos 0'
P
1/ shows that the spherical coordinates mapping is defined by
Edwards~-
cos 0'1 = cos CPl
CO2 0'2 = p;
Lo
CPI cos CP2
,
cos 0'3 = sin CPI sin CPz cos CP3
cos 0' 2. = sin CPI sin CPz
pcos 0'
:=
p-l
sin CPl sin CPz
cos O'p = sin CPI sin CPz
...
sin
CPp-3
cos
sin CPp-2. cos
sin
CPp-2
CPp-2
,
e
sin 8
----------~.
1/
- Edwards. C.R. 1973. Advanced Calculus of Several Variables.
Press, New York, New York and London, England.
Acad~miL
122
and maps the i.nterval
BP •
unit ball
dA
=
CPi.
S
[O,rr] ,
e
€
[O,2rr] onto the surface of the
Edwards also shows that
. p-2
s l.n
. p-3
CPl s l.n
CPZ •••
Therefore, it can be readily shown that the off-di.agonal elements of
the matrix,
fkk'dA, are all zero, and the diagonal elements of the
matrix are all equal, that is
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