MINIMUM
IN THE
G. A.
~J
A.
a.
BIAS ESTIMATION
M.U'L'URE PROBLEM
Manson and L. A. Nelson
Institute of Statistics
Mimeograph Series No. 757
Raleigh - July 1971
iv
LIST OF CONTENTS
Page
LIST OF TABLES . .
v
LIST OF FIGURES
vi
LIST OF APPENDIX TABLES
viii
1.
INTRODUCTION·..··
1
2·
THE ESTIMATOR AND ITS VARIANCE
6
3.
DESIGNS
10
3.1 Designs for the 2-dimensional Simplex:
3.2 Designs for the 3-dimensional Simplex:
(q=3) •
(q=4)
10
25
4.
SUMMARY
..
42
5·
REFERENCES .
44
6.
APPENDIX A
45
A.l Regional Moments
. . . .
A.2 The A Matrices for the Estimator (2.6)
.
A·3 Theorem A. 3
.
A.4 Theorem A.4
A·5 Simultaneous Protection for Lower De~ree Trur
Polynomial Models: (Fitted model y(!) = !i £1) •
A.6 An Expression for the Generalized Inverse of a
Partitioned Matrix
7·
APPENDIX B
.
46
50
54
54
54
55
56
v
LIST OF TABLES
Page
·
·.
·
3·1
Designs for (q=3, d=l, k=l)
3·2
Designs for (q=3, d=l, k=2)
3·3
Designs for (q=3, d=2, k=l)
3·4
Designs for (q=4, d=l, k=l)
3·5
Designs for (q=4, d=l, k=2)
·
3·6
Designs for (2=4, d=2, k=l)
·
16
....
19
25
.....
38
....
40
41
vi
LIST OF FIGURES
Page
Contours of constant V for (3,0)+(3'5)' n0 == 0
(q==3, d=l, k==l, N==6)
13
3·2
Contours of constant V for
(q==3, d=l, k=l, N=7)
14
3·3
Contours of constant V for
(q=3, d=l, k=l, N==8)
3·1
3·4
·.....·········
(3,0)+(3,~), n = 1
·.....·0· ·····
~3:0~+~3:5~' n0 = 2 ·
· · ··· ··
•
15
·····
20
····· · ·······
21
Contours of constant V for (3,0)+(6,0), n = 0
.· ·····0··
(q=3, d=l, k=2, N=9)
..
·
·
3·5
Contours of constant V for (3,0)+(6,0), n0 = 1
(q=3, d=l, k=2, N=10)
· ·
3·6
Contours of constant V for (3,0)+(6,~), n0 == 1
(q=3, d=l, k=2, N=10)
·
·
····
22
Contours of constant V for (3,0)+(6,~), n = 2
0
(q=3, d=l, k=2, N=ll)
·
··
· ·····
23
3·8
Contours of constant V for (3,0)+(6,~), n0 = 2
(q=3, d=2, k=l, N=ll)
·
···
26
3·9
Contours of constant V for (3,0)+(6,0), n = 2
0
(q=3, d=2, k=l, N=ll)
·
3·10
3·7
3·11
3·12
3·13
3·14
3·15
·······
····
·
· · ······
·
·····
C!7
Contours of constant V for (1,2) , n = 0
0
(q=4, d=l, k=l, N=8)
··
········ ·
30
Contours of constant V for (1,2) , n "" 1
0
(q=4, d=l, k=l, N=9)
31
Contours of constant V for (1,3) , n0 = 0
(q=4, d=l, k=l, N==10)
· ·· ·
32
Contours of constant V for (1,3), n0 == 1
(q==4, d==l, k=l, N==ll)
·
·
33
·
····
·
·
···············
·
· ·· ·······
·· ·· ·········
Contours of constant V for (1,3) , n0 == 2
(q=4, d==l, k==l, N=12)
·
·
···
·····
Contours of constant V for (1,3), n0 == 3
(q==4, d=l, k==l, N==13)
·
·
··· ·
··
34
········
35
·
vii
LIST OF FIGURES (Continued)
Page
3·16
3·17
Contours of constant V for (1,3), nO = 4
(q=4, d=l, k=l, N=14) . • . • . . . . . . . . . . . . .
36
Contours of constant V for (1,3), nO = 5
(q=4, d=l, k=l, N=15) . . . . . . . . . . . . . . . . .
37
viii
LIST OF APPENDIX TABLES
Page
..
A.l
Regional Moments for q=3 (R triangular)
A·2
Regional Moments for q=4 (R tetrahedral)
B.l
Designs with V < V* = 3·0:
(q=3, d=l, k=l)
B·2
Designs with V < V* = 6.0:
(q=3, d=2, k=l)
B·3
Designs with V < V* = 4.0:
(q=4, d=l, k=l)
B.4
Designs with V < v* = 10.0:
•
( <t=4, d=2, k=l)
49
..
49
'5r
.
59
63
64
1.
INTRODUCTION
In a q-component mixture, let z.l be the proportion (by volume,
th
weight, moles, etc.) of the i
component in the mixture. Then the
possible mixtures are restricted to the regular (q-l)-dimensional
simplex
q
z.
l
>
-
0,
(i = 1, 2, " ' , q);
L: z. = 1 .
i=l
(1.1)
l
A response surface over the mixture proportions involves a fUnctional relationship
between a response ~*, the z. (i
l
= 1,
2, ... , q) and m parameters
... , e . An experimental design is defined to be the choice
m
of a matrix D, consisting of N rows and q columns, which specifies
the levels of the z. (i
l
= 1,
2, •.. , q) at which the response
~*
is to be measured.
Frequently the form of the true function '1'\* is unknown and is
to be approximated by a polynomial g(~) of low degree.
There are
two main sources of discrepancies between the fitted polynomial
and the true function; namely
1)
that due to sampling error; (called variance error) and
2)
that due to the inadequacy of g(~) to represent exactly
the true response '1'\* (called bias error).
In many pUblished articles, the optimality of designs is based solely
2
on considerations of variance error, with no allowance made for the
possibility of bias errors.
This limitation is not entirely desirable.
In fact, Box and Draper (1959, 1963) and Draper and Lawrence (1965a)
have demonstrated in typical situations the overriding importance of
bias considerations in the choice of response surface designs-
Scheff~ (1958, 1963) introduced the simplex-lattice and simplexcentroid designs for m-tic polynomial regression on the simplexThe proportions used for each component in the (q, m) sinr.plexlattice design have the m+l values, equally spaced on
z.l
= 0,
1
° to 1,
2
-,
-, •.. , 1; and all possible mixtures with these proportions
m m
for each component are used.
Atwood (1969), Kiefer (1961) and
Uranisi (1964) proved separately that the simplex-lattice designs
are G and D-optimal for m = 1, 2, 3 respectively; these being
optimality criteria related to variance error only.
Objections to
the (q, m) simplex-lattice design include the fact that while it is
intended for q-component mixtures, the observations are on mixtures
of at most m components; for example, a (q, 1) simplex-lattice
involves no mixtures.
q
The simplex-centroid design contains 2 -l
points, obtained by making the q permutations of (1, 0, "', 0), the
(~) permutations of (~, ~, 0, .,., 0), the (~) permutations of
111
(3'
3' 3'
. t (1
1 .. " 1).
-, -,
0, ... , 0), ... , an d th e pOln
q q
q
A'
galn on1y
one observation involves all q components.
The problem considered in this paper is the choice of an estimator
for g(~) and a design such that the true response T'\* is closely
approximated in a mean square error sense.
It is assumed that
and T'\* are polynomials of degree d and d+k respectively.
It is
g(~)
3
further supposed that the region of experimeLtal interE:st is the
(1.1),
entire simplex
with equal importance attached tc a11points.
The criterion for choice of estimator and design is to minimize J',
the expected mean square error averaged over the simplex.
Due to the dependence of the proportions in a mixture
(i:z.
~
=
1),
i t is convenient to eliminate one of the variables, say
z , by use of a transformation.
q
equivalent to
(1.1)
If z
q
is eliminated, a new region R,
needs to be defined. in terms of
independent variables x l' x 2'· .. , x q-l·
q.~.l
linearly
All considerations are
henceforth confined to the (xl' x ' ... , xq_l)-space.
2
In this new
space, the true response is denoted by ,,(!) and the approximating
function by "y(!).
Let the true response be a polynomial of degree d+k,
(1.2)
This true response is to be fitted by a polynomial of degree d,
A
y(,X_)
= -1-1
x'b
Let Xl be the matrix of values taken by the variables in !{ for the
N observations in a design, X the matrix of values taken by the
2
variables in
!~
+
y(x.) = "(x.)
'I
_~
_~
and l the vector of observed responses
€., (i
= 1, 2.,
~
E(€.) = 0, E(€~) =
~
~
e ..
l'O
,
N) •
It is assumed that
and E(E.E.) = 0, i
~
J
f
j
,
The integrated mean square error (lMSE), normalizedviith respect to
the number of observations and ~2J is
4
(1.4)
where
n-1 :;:
J'
, R
'
,/ R
~,
2
A
fEy (~) - 1') (~)} d:~ ,
(1.6)
and
The integrand in (1.6) is easily recognized to be the square of tb.e
A
bias in y(~) at the point!..
Define
1
M.. :;:: N- (X:'X.), i
lJ
1 J
<
j - 1, 2
(l.8)
w.. :;: n /' x.x:dx,
i
<
(1·9)
and
lJ
' R -l-J -
-
j := 1, 2 .
Draper and Lawrence (1965a, b), u.sing a theorem proved by Box and
Draper (1959), obtained for 3 and 4-component mixtures. designs which
minimize B, the contribution of bias error to J'.
They investigated
situations where the true response is a polynomial of degree d+1,
T)(~) :;:: ~{~1 + ~;P2'
This true response was fitted by a polynomial of
A
degree d, y*(~) = ~{.£!, using standard least squares e3timators:
b* :;::
-1
(X'X
)- 1 XlV.
1 1
1~
(1.10)
~.~:,
The theorem by BO:l!i'l;j;nd; >Pirap~r $ta¢~~)'i):b.~ty.:~::p~CWiided w~i exists, a
necessary and sufficient condition for B to be minimized is that
"
.~,
"'.i..
j'
(1.11)
Drap.er and Lawrence used the sufficient condition (which they also
,(.l.,12)
:, L
~ i : ",.t,: l'" ,[<.'1 ,.X' C,,;) J)(7qt~D'1 rU 1\~ (I ,S) ~.\n,UJ'.i ,J U:Yl;:Y1'l i:G
. (C' . i)
They constructed deii'lgns by choosing various point sets vrhich satlsfy
,}.
J.Jc';
.L.~)_: . g
·\i~:U,::::C:::I .. <.<).
1:.~
some of the conditions
~-JJJi-.~~j'.l
(,j'j::'.rLJ
(1.12)
~.;.~.
that all of the conditions
c<IJI91'[
l~ld:::-
O"i~~;\
oj
C·V!~:_:·J~V·.( ':'~:~Ir,
and then combining them in such a way
(1.12)
C1oi:j·Hlsj~nli.r' hl1 --J.O'}.
H "io
noJ::t
t-rill'f'.-·
were satisfied.
".~ecker (1970 )" (WS:014fli~ ~tP~ c~di ~ i<m$! ~ -+'2) together with
'~-";~l
.
,
1-'
;.1.
1-
1.--'
some orthogonality conditions, obtained designs fQr general q when
" ' . i ; :lX) ~; \v;r:rU ""Jon
• CQ ,~fl) .". 'r, ),n.::: l, ,.W:·:W :I] " F,YI.)e"
d=l and k = l . ' · ' · · · 1.'-'
-,
c,l. .LJ.
'>,
~'~IJ!:'\r
.(:'1'.
:,,"-en'
rllJ-'11ilCflf1j
xLbJ[~"ClCrA
1I1~:n.ocHiT :)~)c:
10 (.1'.
;b .l,p.I'!
The approach used in this paper is that developed by Karson,
Manson and Hader (l-e~91;"i":~r$it,_B i~'jdijr'eStlYnm~l1~mi:zed by choice of
~:.:;.-
an
estim~t~r
£1'
',L
~ ..
i
,.... J
,"
~~.-:~
(.,--~
\::l
The,.estimator obtained gives the same minimum B for
1.
l(
J_
•
;:l.,r
V,lU-'
'::,~/J
f:'_'. :'
r.[;~iL
~:~J
{'ltin~ r~\.J·:'{:
l.rf.,i I,. (
any design within a very large class of designs which satisfies a
\
~\i
r,·~1.~~,1 r~I.';_ .ry,J..:(")~-~ ~:,(!.i J.~r~) ';
simple estimability condition. 'A d.esign is then obtained from this
'.\ i'i
:
f:~j\
( '~~,J)
j J5 d ~~.
?'
class which minimizes V, the variance contribution to J,
'--""!
I.
! r~-B~.;;
,L
".':.,'
J;i
',i
6
2.
THE ESTIMATOR AND ITS VARIANCE
A
Let
~(!)
~
and y(!) be as in (1.2) and (1.3).
~
Since E(y(!)} is a
~
= !{E(£l) = !{~,
polynomial of the same degree as y(!), say E(y(!)}
(2.1)
=
~ (~
(1
- f!l) 'W1l (~ - f!l) -
2(~
- f!l)
'WI~2+ f!2W2~)
where W , W and W are the regional moment matrices defined in
I1 12
22
(1.9). Differentiating (2.1) with respect to ~ and equating the
derivative to zero yields the result that, a necessary and sufficient
condition for minimization of B is
where A • [I:
W~iw12]
and
~/. (l!.i' l!.2)·
Note that
q and dj see Theorem A., ot Appendix A. ,.
Wi~
exists for all
The minimum. value of B is
(2·3)
and this minimum. is achieved for any design for which
Note that the condition
Mi~2
=
Wiiw12
(or
AI!.
is estimable.
Mu = Wll , ~2
=W
12 )
guarantees that E(:2.I) =~, since
The condition E(:2.1 ) • Al!.J however, is less restrictive than the condition ~~2 • Willw12 and therefore less restrictive than the Box and
Draper sufficient conditions ~l
= Wll,
M12 = W12 •
In order to develop the estimator
~l'
generalized inverse of a square matrix, S.
it is necessary to define a
There always exists a
matrix S-, called a generalized inverse or g-inverse, with the
property that
A g-inverse as defined by (2.4) is not necessarily unique.
is non-singular, SFor any point!
When S
=S-1,the ordinary inverse.
€
R and subject to the condition that
~
be
estimable, the minimum variance, linear, unbiased estimator of the
linear function !{~ is given by
A()
, ( I )_ ,
y! =!lA X X Xl.'
where X = [~: X ] and (X'X)2
minimized when
~l
is any g-inverse of X'X.
=A X X
(
I
) -
I
Xl..
Hence V is
(2.6)
The estimator ~l is unique even though (X/X)- may not be, since ~l is
linear in l. and unbiased for ~; see for example Rao (1966, p. 181).
Therefore,
(2·7)
and
8
where M =: N-·1(X 'X), and M- is a g-inverse of M.
It is interesting to compare the results obtained above with
those of the standard least squares (Box and Draper) approach.
'"
the standard least squares estimator by y*(~)
:£1
V*
L_
(X{x1) --X{l.'
=:
=:
!L
,JR'
(J2
Var
Let B*
[~(x)}d!
:=
:=
11\1
.~
==
I'R [E[y*(~)]
'" -
,
(M~}lll)
trace
Denote
~{~t where
~)
'I)(~:) ]CC CL3;. and
•
Lemma 1
Let C denote the clas s of all des igns for which
let, X denote the class of all designs for which
~
M~iM12'
and
is estimable.
Then,
i)
ii)
c l;
X, and
for every design D
€
C,
V*
D
V
:=
D
.
Proof
i)
Let
M~~12 = w~iw12
Therefore,
ii)
Write A
:=
Then
E(£i)
:=
-1
I~l + MIlMl~2
C~X
[I:
M~~2]
and M :=
M"1.1
fir
.1
12
I
M
12
M
22
:= A~
9
Using the expression for a generalized inverse of a partttioned matrix,
(see Appendix A.6),
Lemma 2
Let C be the class of all designs which satisfy M11
M12
= W12
Then for every des ign in C, V*
•
= WII ,
= s, the number of
coefficients in the fitted equation.
Proof
V*
= trace (~iwll) = trace (W~iwll) = trace
(Is)
=s
.
The two lemmas above illustrate the advantages of the approach
used here over the approach used by Draper and Lawrence.
C~
~,
designs exist, in general, for which the minimum bias approach
attains min B while the Box-Draper approach cannot.
V*
]'or, since
= V for
Moreover, since
every design in C, it is likely to find designs in
which V < V*
= s,
provided that C and
~
- C are not empty.
~
for
In fact,
for all situations investigated in this paper, min V < s except where
the particular design configuration considered has no elements in C .
·1.0
).
'The search
fOT
de3igw3 :Ls
which ~ i3 estimable.
DESIGNS
:(iLfi:c.ed
;;,('C,.':lt' C.l~i.j3
It can be Sh'-;Wll (see
fliT
of
dc".]
gL" f.-:r
1966..
!<:,Ii.ampJe F",c;
p. 182) that a necessary aud sufficient condi t.icm fer A~ to b,~
estimable is that
A.
Also, a ne(~essary condition. (lJl:~t
est.imable is that ~1 be estimahlt,
proof see Appendix A..
C'Y
eqniv8"L,,':t1y
1Ml1l
;/
for
;,1.
4.
Of particular interest are those designs for which If < sand.
those designs of given type whi(;h minimize V'
3·1
Designs for the 2-dimensional Simplex:
(q==3)
Suppose there are three components in a mixture, with proportions
z. (i::= 1, 2, 3).
The possible mixtures are restricted tc' the
1
triangular region, in the 3-dimensional space
defined by L: z. == 1, z. > 0, (i := 1., 2., 3)·
.
1
1
1
-
(Z1' 2::.:, 6 ),
3
which is
'This region:an b8
transformed into an equilateral triangle in two dimensions via the
transformation
1
- --2
xl
:x
:x
2
3
-
- .["3
b
1
1
2
~3
'6
1
•.1..
0
2,J3
61
z1
2
7:
2
')
C~ol.1)
11
Note that x
= 1 always, and so we need work only with xl and x '
2
3
the (xl' x ) coordinate system, the new region of experimental
2
In
interest, R, is the equilateral triangle of side one, centered at
the origin and with base parallel to the Xl-axis.
Designs are constructed by combining regular n-gons which are
concentric about the origin.
Each n-gon is described by (n,e),
where n is the number of vertices and
e
is the counterclockwise
rotation of a reference vertex from the positive x -axis.
2
For example
a (3,0) will denote a 3-gon (an equilateral triangle) centered at the
origin with one vertex on the positive x -axis and with base parallel
2
to the Xl-axis; while a (3'5) will denote another 3-gon centered at
the origin and with the line connecting one of the vertices to the
origin rotated through an angle of ~ radians, counterclockwise from
the positive x -axis.
2
The radius of the i
be specified by the parameter
p.'
l
th
n-gon in a design will
The number of observations taken
at the origin will be denoted by nO'
Different combinations of n-gons, in various relative orientan
7.
tions, have been investigated for 3
~
with smallest V will be tabulated.
(It should be noted that the class
~
However, only designs
of designs, constructed from n-gons, by no means exhausts the class
of designs which satisfy (3.1».
(a)
(q=3, d=l, k=l):
The true model is
12
with
!..i .: :
§.~ .::: (t3:U
(1, Xl' x 2 ).1
!:.;
2
x . .'
c:.
f3 22' f3 l2 )·
'
Tte fitted equation is
A
y(~) ~ ~~E.l
where 'b,
-1
--,' (b , b , b ) I := A\.-'y'X),
-v_"_v
.>.
., J
0
1
2
"
as
.
.III
gl
ver~,
(2.6).
'The
appropriate A matrix is given in Appendix A.2.
estimable.
However these give reIativ,,,,:ly hig'h
v~l'ue~,
c·rv.
,Fnr
example, ~ is estimable for the (5,0) design plus nne 1:?:Tt'::r pcipt
when p == 2 f3 , but then V := 8.6875.
1
arbitrary orientation,
V
= 4.23.
~
For the 5-gon (nO
is estimable when
p -
,:c
0) in B.ny
but then
It should be noted however that no such c!.esigns, built
up from n-gons, exist for
N=4
and
5
which satisfy the Box and Draper
conditions (1.12).
A variety of designs whic:h use six or more points can be obtained
by combining two or more n-gons.
These, for almost every choice of
the p. I s, have X'X non-singular and. hence satisfy the estim3:bility
l
condition
(3.1).
A great deal of design flexibility is therefore
obtained which may be used te") minimize B or t;o s9.tisfy other design
criteria.
Table
3·1
show's those designs with smallest
v·
13
1------------...,.----
kZ- (3,2) -lattice
design
0·52
0.48
0.40
Draper and Lawrence
Design
0·32
0.12
Figure 3·1
0.16
0.20
Contours of constant V for
(Q=3, d=l, k=l, N=6)
(3,0)+(3'5)'
no
=
0
14
1------------r---r--.. . . .~
Simplex Centroid
V = 2.651527
0·52
2.60~
0.36
Draper and Lawrence
Design
0·32
0.12
0.23
0.20
P2
Figure 3.2
Contours of constant V for (3,0)+(3,~), nO
(Q=3, d=l, k=l, N=7)
=1
15
0·52
2.8
~2·5
0.48
P1
0.44
0.40
~.
Draper and Lawrence
Design
0·32
0.12
Figure 3·3
0.16
0·20
0.24
Contours of constant V for (3,0)+(3,~), nO
(Q=3, d=l, k=l, N=8)
=
2
16
Table 3.1
Designs for (Q==3, d==l, k==l)
N
Designs
nO
PI
P2
P3
6
(3,0)+(3,~)
0
0.4190
0.2Z71
------
2·7397
7
(3,0)+(3,~)
1
0.4994
0.2377
------
2· 5971
8
(3,0)+(3,~)
2
0·5150
0.2386
------
2·5556
9
(3,0)+(3,~)+(3,~)
0
0·5773
0.2359
0.2386
2· 4789
10
(3,0)+(3,~)+(3,~)
1
0·5773
0.2386
0.2386
2.4297
*
V
For other designs with V < 3·0, see Table B·l of Appendix B.
From Lemma 2, if conditions (1.12) are satisfied, then V* == 3·0.
It is evident that designs in Table 3.1, together with the appropriate
estimator (2.6), give smaller IMSE than those designs which satisfy
conditions (1.12).
This is obvious since every design in Table 3.1 has
J == min B + V < min B + 3.0 .
Even more important is the fact that for each design with V < 3.0,
there exists a simply connected region which includes the tabulated
design and within which
V < 3·0.
~
is always estimable, min B is attained, and
Examples of such simply connected regions are shown in
Figures 3·1, 3·2 and 3·3 for the designs (3,0)+(3'3) with nO == 0, 1
and 2 center points, respectively.
Designs which satisfy conditions
(1.12) always lie on the V == 3.0 contour as expected by Lemmas 1 and
2; these are denoted in Figures 3.1, 3.2 and 3·3 by a dot enclosed in a
circle (0).
17
Observe that the designs with smallest V are those composed of
combinations of the n-gons (3,0) and (3,~) only.
These n-g()lls are
contractions of the permuted Scheffe sets (1, 0, 0) and
respectively, under the transformation given in (3.1.1).
1
1
(2' 2'
0)
The n-gon
(3,0) was found to be necessary in order to obtain reasonable values
of V.
This phenomenon can be attributed to the fact that without
the 3-gon (3,0), sUbregions adjacent to the vertices of R would be
void of design points.
(b)
(q- 3, d=l, k=2):
~~ = (~ll' ~22' ~12' ~lll, ~222' ~112' ~122)·
The fitted equation is
A
y(~) = ~{£l' where
£1 =
(b ' b , b )
O l
2
I
= A(X/X)-X/~, as given in (2.6).
The appropriate A matrix is given in Appendix A.2·
Combinations of n-gons with N < 8 do not provide designs for
which
~
is estimable.
For each N
designs exist for which
A~
~
9 however, an infinite number of
is estimable.
Designs with smallest V and
designs with nearly smallest V are listed in Table 3.2.
tions of n-gons with 9
~
N
~
12 have been considered.
notation is exactly as was used in Table 3·1.
Only combinaThe design
This model situation
has not been treated in the literature for mixture problems.
However,
from Lemma 2, if designs exist which satisfy the condition (1.12),
they would have V* = 3.0.
18
x'x is singular (rank == 9) for aJ.:l.. desig'cs listed,
,?xce~;t:.
+;bose
involving the combinations (3, C) +(6,~) and (3,C) +( 4, ()) +(4,*).
\
Figures 3· 4, 3·5, 3· 6 and 3·7 show, in the '
~Pl' P2)plane, (:ontuJ.rs of
° ar"d 1 center points,
constant Y for the designs (3,0)+(6,0) plus nO : :.:
and
(3,0)+(6,g)
plus nO == 1 and 2 center points respectively.
It is shown in Appendix A.5 that for fixed d and any design for
which
~
is estimable, the same estimator which attains min B for
k :::: k will also attain min B for k == k < k .
l
l
2
Designs of this
section therefore give simultaneous protection against the trUt,' model
being quadratic or cubic; but not without a price.
A slightly larger
value of Y is obtained when a linear approximation is made to a true
model which is cubic than is obtained when the true model is quadratic.
Note again the high frequency of the specific n-gons (3, n),
(3,~), (6,~) and (6,0) in Table 3.2·
The first three are the equiva-
,
lents in R of contractions of permutations of the Scheffe sets
(1, 0, 0),
1 1
(2'
2'
0) and
12.
(3'
3' 0) respectlvely.
combination of the first two with equal radii.
The fourth is a
Once more, the n·-gon
(3,0), although not necessary for ~ to be estimable, seems essential
for reasonable values of y.
(c)
(q=3, d==2, k=l):
The true model is ~(~) :::: ~{~ + ~~2' with
,
22)'
332
~l :::: (1, Xl' x 2' Xl' x 2' Xlx 2 ' ~2 :::: (xl' x 2' Xlx 2 '
A
The fitted equation is y(~) :::: ~{£l' where
2
x.I x 2 ),
19
Designs for (q=2, d=l, k=2)
Table 3· 2
N
Designs
nO
9
C~,0)+(6,0)
9
V
P
Pl
P2
0
0.4448
0.2321
_ _ ,_ _ _ uoo
2.8090
(3,0)+(3,0)+(3,~)
0
0.4401
o· 2179
0.2389
2.8023
10
(3,0)+(6.0)
1
0.4364
0.2623
--_._--
2·7917
10
(3,0)+(6,i)
1
0.4396
0.2548
-,.,----
2.8098
10
(3,0)+(3,0)+(3,~)
1
0.4389
0.2334
0.2550
2·7822
11
(3,0)+(6,i)
2
0.4311
0.2317
--_._--
2,8436
11
(3,0)+(6.0)
2
0.4275
0.2386
-_._---
2.8237
11
(3,0)+(3,0)+(3,~)
2
0.4164
0·3499
0·2746
2·7633
11
(3,0)+(4,0)+(4,*)
0
0.4568
0.2304
0.2304
2.8333
12
(3,0)+(6,~)
3
0·4220
0·3034
------
2·9182
12
(3·0)+(6,0)
3
0.4238
0.2386
------
2·93°5
12
(3,0)+(3,0)+(3,~)
3
0.4033
0·3624
0.2386
2.8063
12
(3,0)+(4,0)+(4,*)
1
0.4516
o· 25.32
0.2532
2,8496
3
£1
(b ' b , b , b , b , b )' =: A(X'X) -X'l. ' as given in (2.6).
12
11
22
O 1
2
The appropriate A matrix is given in Appendix A·2·
=
Again, combinations of n-·gons with N :::; 8 do not provide designs
for which
~
is estimable.
For each N ~ 9 however, an infinite number
of designs exist which satisfy the estimabi1ity condition (3·1).
20
0·50
0.48
0.46
0.42
0.40
0·38
0.22
0.18
Figure 3.4
0.24
Contours of constant V for (3,0)+(6,0), no
(Q=3, d=l, k=2, N=9)
,
.
=°
21
0·52
0·50
0.48
0.46
0.44
0.42
0.40
...._ ....._ ....._ ....._ .....
_...,...-..,...-..,..-..,...-..,..-...,.-.a..,c~=
0.18
0·20
0.24
0·22
P2
Figure
3.5
Contours of constant V for (3,()+(6,o), nO
(Q=3, d=l, k=2, N=10)
1
........
22
0.49
0.47
0.45
0.41
0·39
O. 37 ....~~...,._...-_............_...,._......_...-~~-..,._ ......_....-'--.....,~
0·21
0.23
0.25
0·27
0.29
0·31
0·35
P2
Figure 3·6
Contours of constant V for (3,O)+(6,i), nO "" 1
(Q=3, d=l, k=2, N=10)
23
0.47
0.45
0.43
0.41
0·39
0·37
0.21
Figure 3·7
0.23
0.25
0.29
0·31
Contours of constant V for (3,0)+(6,~), n
(Q=3, d=l, k=2, N=ll)
0·33
e
2
24
X'X is singular for all designs consisting of oLly "LrJe !>·gons (3,0))
(3'5) and (6,0); rank (X'X) == 9·
All other designs co::.:;idered for
which A~ is estimable have X'X non-singular.
9
:s N :s 15,
Table 3':3 shows, for
those designs 'with smallest or nearly smallest V.
Draper and Lawrence (1965a) obtained desig!lS which satisfy the
conditions (1.12) for 12:S N:s 15·
would have V*
= 6.0.
For N
:s 12,
By Lemma 2, all of these designs
they found only approximate designs
in the sense that the conditions (1.12) are not satisfiEd exadly but
are almost satisfied (within 10%).
These designs would not attain
min B, and their variance properties were not presented.
All designs in Table 3· 3 have
min B.
A~
estimable and hence attain.
Moreover, since in each case V < VO/< :::: 6.0, these designs have
smaller IMSE than designs which satisfy the conditions (1.l2).
Figures 3·8 and 3·9 show contours of constant V, in the (P , P2)
l
plane, for the designs (3,0)+(6,~) plus nO
nO == 2 respectively.
==
2 and (3,0)+(6,0) plus
Any (PI' P ) combination inside the V
2
==
6.0
contour gives smaller IMSE than designs which satisfy (1.12)·
As in Tables 3·1 and 3·2, the specific n-gens (3,0), (3'5) and
(6,g) occur with high frequency in the designs of Table .3·,.3.
The
I
correspondence between these n-gons and the compositions of Scheffe's
simplex-lattice designs has been discussed earlier.
of these n-gons over others is seen to have a
of the region of experimental interest R.
The superiority
depe~dence
on the shape
25
Table 3.3
N
Designs for (q=3J
Designs
d~2,
k=l)
nO
Pl
P2
P3
V
9
(3,0)+(3,0)+(3,~)
0
0.4472
0.1859
o. a375
5·3508
10
(3,0)+(3,0)+(3,~)
1
0.46a3
0.2453
o. a375
5· 2a31
11
(3,0)+(6,;)
2
0.4704
0.2976
------
5·2512
11
(3,0)+(3,0)+(3,~)
2
0.4675
0.2891
o. a375
5·2513
11
(3,0)+(6,0)
2
0.4675
0.2875
.....
__ ........
5·2514
12
(3,0)+(3,0)+(6,i)
0
0.4909
0.1375
0,3171
5·1043
13
(3,0)+(4,*)+(6,;
0
0·5031
0.150r
0·3:333
~~.
13
(3,0)+(3,0)+(6,;)
1
0·5128
0.2023
033a3
4 9695
14
(3,0)+(3,0)+(6,i)
2
0·5238
0.2'570
0·3325
4.8977
14
(3,0)+(4,*)+(6,~)
1
0·5067
0.1883
0·3333
1j.·9009
15
(3,0)+(3,0)+(6,i)
3
0·5327
0.2971
0·3333
)+,
15
(3,0)+(4,*)+(6,;)
2
0·5096
0.2232
0·3333
4·9231
*
~~.
9J.30
0
9082
For other designs with V < 6.0, see Table B.2 of Appendix B
3·2 Designs for the 3-dimensional Simplex:
(q=41
For 4-component mixtures with proportions zi (i = 1, 2, 3,
the possible mixtures are restricted to the simplex defined by
L:z.1 = 1, z.1 -> 0, (i = 1, 2, 3, 4).
The region can be transformed
into a tetrahedron in three dimensions via the transformation
~~),
26
0·58
0·56
0·54
0·52
0·50
PI
6.0
0.48
~5.25
5·5
0.46
0.44
./
0·42
0.40
0·23
0.25
0.29
0·31
0·33
P2
Figure
3.8
Contours of constant V for (3,O)+(6,~), nO := 2
(Q=3, d=2, k=I, N=II)
0·53
0·54
0·52
0·50
~5·25
0.46
0.44
0.42
0.40
0.19
0.21
0.23
0·25
0.29
P2
Figure 3·9
Contours of constant V for
(Q=3, d=2, k=l, N=ll)
(3,0)+(6,0), no = 2
Xl
1
1
-1
-1
x
1
-1
1
-1
-1
1
1
-1
1
1
1
1
x
2
=
3
x4
Then x4
=1
Zll
z2
z3
z4
J
.
(3·2.1)
always, and attention may be confined to xl' x
2
and x
3
only.
In the (Xl' x ' x ) coordinate system, the new region of
2
3
experimental interest R is the tetrahedron whose vertices form the
I
half replicate of a ~ factorial arrangement.
3
Designs will be constructed from combinations of the following
= -xl x 2x
point sets:
° < p :::; 1
Set No.1:
(p, p, -p), (p, -p, p), (-p, p, p), (-p, -PI -p);
Set No·2:
(r, r, r), (r, -r, -r), (-r, r, -r), (-r, -r, r); 0 < r :::;
Set No·3:
(±s, 0, 0), (0, ±s, 0), (0, 0, ±s);
Set No. 4:
(h,
h
- -,
3
h
(-,
3
h,
h
- 3)'
h
h
(-,
3'
3
h
- -),
3
h
h
-), (h, 3'
3
(-
h
-,
3
(-h,
h
- -,
3
h
h
h, -),
(- -, -h,
3
3
h
h h h) , (- -,
h) , (- -,
-,
3 3
3
h
- 3)'
(-h,
31
h
h
3)
3' ,
h
h
h
(-, -h, 3)'
- -),
3
3
h
- 3'
h h
-h) , (-,
-h);
3 3'
° < h < 1.
These point
sets are suggested by consideration of the results for the
3-component mixture problem.
Point set numbers 1, 2, 3 and 4
correspond to contractions and permutations of the familiar Scheff~
sets (1, 0, 0, 0),
1 1
(2'
2'
0, 0),
111
(3'
3' 3'
0) and
1 2
(3'
3'
0, 0)
respectively, under the transformation (3.2.1).
corresponds to the point
(a)
1
1
1
1
(4' 4' 4' 4)
A center point in R
.
In the (zl' z2'
Zy
z4) -space.
(q=4, d=l, k=l).
The true model is T}(~) = ~~~l + ~!2' with
A
The fitted equation is y(~) = ~{£l ' where
,
£1 = ( b O' b l , b 2 , b ) ' = A( X,X)- Xl...'
as given in (2. 6) •
3
appropriate A matrix is shown in Appendix A·2·
The
Table 3·4 shows parameter values for designs which give smallest
IMSE for 8
~
N
~
15.
Combinations of point sets which form the designs
are given in parentheses under the column heading "Design Composition".
For example (1,2) denotes a design composed of a single point set
No. 1 and a single point set No.2.
XIX is singular (rank = 8) for designs involving the combination
(1,2).
However, ~ is estimable for all values of the design para-
meters p and r·
,
X X is non-singular for all other designs in
Table 3· 4.
Note that designs which satisfy the Box and Draper sufficient
conditions of (1.12) would have V* = 4.0.
Figures 3·10 through 3.17
show contours of constant V in terms of the parameter values for the
designs in Table 3.4.
For each design, any combination of parameter
values inside the V = 4.0 contour gives
A~
estimable with V
< 4.0, and
hence smaller IMSE than designs which satisfy the conditions (1.12).
30
0.84
0.80
0·72
p
0.68
4·0
~3.61
0.64
0.60
Draper and Lawrence Design
0·52
0.24
0.26
0.23
0·30
0·32
0·34
r
Figure 3.10
Contours of constant V for (1,2), nO
(q=4, d=l, k=l, N=8)
z
0
31
0.84
0.80
0·72
p
0.68
4.0 3·9
"'02- 3.77
0.64
0.60
0·52
0·30
0.24
0·32
0·34
r
Figure 3.11
Contours of constant V for (1,2), nO
(q=4, d=l, k=l, N=9)
=1
32
0·95
0.85
0·75
p
0·55
Draper and Lawrence
Design
0.45
0·35
0.40
0.60
0·70
0.80
0·90
s
Figure 3·12
=0
Contours of constant V for (1,3), n
(q=4, d=l, k=l, N=10)
0
1.00
33
0·95
0·75
p
0.65
0·55
Draper and Lawrence
Design
0·35
0.40
0·50
0.60
0.'70
0.80
(;·90
s
Figure 3.13
Contours of constant V for (1,3) .. nO == 1
(q=4, d=l, k=l, N==11)
1.00
34
0·95
4.0 3.8
0·75
p
0·55
Draper and Lawrence
Design
0·35
0.60
0·70
0.80
0·90
s
Figure 3.14
Contours of constant V for (1,3),
(q=4, d=l, k=l, N=12)
~
=2
1.00
35
0·95
4.0 3·8 3·4
0·75
3.4
p
0·55
0.45
0·35
0.40
0.60
0·70
0.80
0·90
s
Figure 3.15
Contours of constant V for (1,3), nO = 3
(q=4, d=l, k=l, N=13)
1·00
36
0·95
4·0
3·7.5 3· 5 5· 4
0·75
:P
0.65
0·55
0·35
0.40
0.60
0·70
0.80
1.00
s
Figure 3.16
Contours of constant V for (1,3), nO
(q=4, d=l, k=l, N=14)
=.::
4
37
0·95
4.0 3"75
3,31
0·75
p
0·55
0·35
.......-
.........................-
0.40
0·50
....-
0.60
....-
....-""'-t-......,,--.,.--=t"""'*"".......-
0·70
0.80
(:,90
1·00
s
Figure
3·17
Contours of constant V for (1,3). nO
(q=4, d=l, k=l, N=15)
:=
5
..
When designs which satisfy conditions (1.12) exist, they are indicated
C(:»)
in the figures by a dot, enclosed in a circle
Designs for (q=4, d=l, k=l)
Table 3· 4
--
-s
._-'-'
N
Design Composition
nO
p
r
V
8
(1, 2)
0
0.6460
0·3333
------
3·6093
9
(1, 2)
1
0.6693
0·3333
._--,---
3·7680
10
(1, ~)
0
0 ..
61+09
-,---~" ..<>
0·7122
3· 6583
11
(1, 3)
1
0.6975
---.-_---
0.8209
3· 52Z(
12
(1, 3)
2
0·7671
_..... _-_.-
0·9511
3· 3826
13
(1, 3)
3
0·77z(
-------.
1.0000
3·Z(11
14
(1, 3)
4
0·7723
-_..-_--
1.0000
3·2637
15
(1, 3)
5
0·7787
------
1.0000
*
)- 3148
--
For other designs with V < 4.0, see Table B·3 of Appendix B.
(b)
(q=4, d=l, k=2):
The true model is ~(~)
I
I
= ~~~1
+ ~~2 ' with ~~
222
333
2
=
(1, Xl' x 2'
2
2
~2 = (Xl' x 2' X3 ' x1x 2' x1 x3 ' X2x3 ' Xl' x2 ' x3 ' xl x 2, x l x3 ' x l x 2'
~; = (~11' ~22' ~33' ~12' ~13' ~23' ~111' ~222' ~33J' ~112' ~113'
~122' ~223' ~133' ~233' ~123) •
39
"-
The fitted equation is y(~) = ~l£l
~l = (b ' b , b ,
'
where
A(X/X)-X/~, as given in (2.6).
3
appropriate A matrix is shown in Appendix A.2.
O
l
2
b )1 =
Infinitely many designs exist for which
~
The
is estimable, using
combinations of point sets No.1 through No. 4 with N > 12·
Table 3.5
shows parameter values for those designs which minimize V for
12 < N < 20.
The notation is exactly as was used in Table 3·4.
X'X is singular (rank = 12) for all designs in Table 3.5·
Except perhaps for the combination (1, 1, 2) with no center point,
designs build up from point sets No. 1 through No. 4 will not satisfy
the conditions (1.12) for the model situation being considered.
For,
by Lemmas 1 and 2, if designs exist which satisfy the conditions
(1.12), these would have V = V* = 4.0 < min V; a contradiction.
Designs of Table 3.5 also give simultaneous pl'otection against
the true model being quadr~tic or cubic (see Appendix A.5).
(c)
(q=4, d=2, k=l):
The true model is 11(~) = ~{~l + ~~2 with
2
I
2
2
~l == (1, xl' x 2 ' x 3 ' xl' x 2' x 3 ' x l x 2 ' xlx y
x 2x )
3
2
I
2
2
2
2
2
3
(3
3
~2 = xl' x 2 ' x y x l x 2 ' x l x 3 , xl x 2 ' X2Xy xlx y x 2x y Xl x 2x 3 )
t"
-1
=
,
t'o' t'l' t'2' t'3' t'll' t3 22 , t'33' t'12' t'13' t'23) and
t3' = (t'lll' t'222' t'333' t'112' t'113' t'122' t'223 , t'133' t3
233 , t3 123 ) .
-2
40
Table 3· 5
Designs for (q=c4, d;c:l, k"2)
N Design Composition
Parameter Values
nO
V
----,,--Pl
P2
r
1
Y2
12
(1, 1, 2)
0
0.4496
0·5697
0·3332
------
3·8911
13
(1, 1, 2)
1
0·5348
0·5121
0·3332
------
4.0777
14
(1, 1, 2)
2
0·5324 0·5148
0·3332
------
4·3825
15
(1, 1, 2)
3
0·5148
0·5324
0·3332
------
4.6929
16
(1, 1, 2, 2)
0
0·5748
0.4496 0·3266
0·3266
4·3319
Pl
P2
P3
r
17
(1, 1, 1, 2)
1
0·5318
0·5268
0·5121
0·5332
4.1644
18
(1, 1, 1, 2)
2
0·5121
0·5322
0·5264
0·3332
4.4001
19
(1, 1, 1, 2)
3
0·5268
0·5121
0·5318
0·3332
4.6413
20
(1, 1, 1, 2)
4
0·5121
0·5316
0·5271
0·3332
4.8837
"
The fitted equation is y(~)
=:
~lbl ' where
I
bO' b l , b , b , b ll , b , b , b , b , b ) f =: AI ( XI X) - X~,
as
22
12 13
2
23
3
33
given in (2.6). The appropriate A matrix is given in Appendix A·2.
~1 =:
(
Combinations of point sets No. 1 through No. 4 with N ~ 17 do not
provide designs for which
~
is estimable.
For each N ~ 18 however,
an infinite number of designs exist which satisfy the estimability
condition (3.1).
Designs which minimize V are tabulated in Table 3·6.
The notation is exactly as in Tables 3·4 and 3·5·
For example, the
41
combination (1, 1, 2, 3) denotes a design composed of four point sets,
two of type No. 1 and one each of -types No. 2 and No.3.
I
X X is singular (rank = 17) for designs involving the combination
(1, 1, 2, 3).
Table 3.6.
X'X is of full rank for all other designs listed in
From Lemma 2, designs which satisfy the conditions (1.12)
have V* = 10.0 and hence have larger IMSE than designs of Table 3·6.
Table 3·6
Designs for (q=4, d=2, k=l)
N Design Composition
Parameter Values
nO
PI
P2
r
V
s
18
(1, 1, 2, 3)
0
0·7125
0·3824
0·3332
0·9527
8.6007
19
(1, 1, 2, 3)
1
0·7211
0.4340
0·3332
0·932'3
8.4841
20
(1, 1, 2, 3)
2
0·7232
0.4641
0·3332
0·9191
8·5297
21
(1, 1, 2, 3)
3
0·7232
0.4770
0·3332
0·9146
8.6954
p
r
h
22
(1, 2, 4)
2
0·7543
0·3332
0.8119
------
8.4914
23
(1, 2, 4)
3
0·7564
0·3332
0.8162
------
8.4700
24
(1, 2, 4)
4
0·7582
0·3332
0.8197
------
8.6036
*
For other designs with V < 10.0 see Table B.4 of Appendix B.
42
4.
SUMMARY
The general problem of minimum bias estimation has been reviewed
for polynomial response surface models, and the design of experiments
for this estimation procedure considered for 3 and 4-component
mixtures.
By choice of an estimator £1
= A X X Xl'
(
I
) -
I
the bias contri-
bution B to integrated mean square error (IMSE) , is directly
minimized.
Integration is over the entire simplex.
The estimator
attains the same minimum B Whatever the design, provided only that
the design satisfies the condition that
~
be estimable.
At this
point design flexibility may be used to satisfy other design criteria.
In particular, designs in this paper were constructed such that V,
the variance contribution to IMSE is minimized.
These min V designs,
using the minimum bias estimator, have smaller IMSE than the designs
developed by Draper and Lawrence (1965 a, b) which minimize B only.
,
Unlike Scheffe's simplex-lattice and simplex-centroid designs,
the designs of this paper, in general, have most of the design points
interior to the simplex.
Even where design points required for
minimum V are objectionably near the boundaries of the simplex, the
design flexibility can be used to move design points farther inside
the simplex with little change in V and still retain V < s·
For each model situation investigated, the class of
designs contains an infinite number of designs.
~ esti~ble
Sufficient design
flexibility is therefore available to satisfy other types of design
criteria:
~.~.
orthogonality, rotatability, etc.
For example, it is
easy to check that all designs of Tables 3·1 and
I
) _
I
3.4 are first order
orthogonal since they all have A( X X A diagonal;
of
~
~.~.,
is separately estimated with smallest variance.
each element
44
5.
REFERENCES
1.
Atwood, C. L. 1969. Optimal and Efficient Designs of Experiments.
Ann. Math. Statist. 40:1570-1602.
2.
Becker, N. G. 1970. Mixture Designs for a Model Linear in the
Proportions. Biometrika, 57:329-338.
3.
Box, G. E. p. and N· R. Draper. 1959· A Basis for the Selection
of a Response Surface Design. Jour. Amer. stat· Assoc.
54: 622-654.
4.
Box, G. E. p. and N. R. Draper. 1963· The choice of a Second
Order Rotatable Design. Biometrika, 50:335-352·
5.
Draper, N. R. and W. E. Lawrence. 1965a. Mixture Designs for
Three Factors. J. R. Statist. Soc. B, 27:450-465·
6.
Draper, N. R. and W. E. Lawrence. 1965b. Mixture Designs for
Four Factors. J. R. Statist. Soc. B, 27:473-478.
7.
Karson, M. J., A. R. Manson and R. J. Hader. 1969· Minimum
Bias Estimation and Experimental Design for Response
Surfaces. Technometrics, 11:461-475·
8.
Kiefer, J.
1961.
Optimal designs in Regression Problems, II.
Ann. Math. Statist. 32:298-325·
9·
Rao, C. R. 1966. Linear Statistical Inference and its
Applications. John Wiley and Sons, Inc.
10.
Ryshik, I. M. ,and I. S. Gradstein. 1957. 'I'ables of Series,
Products, and Integrals. Berlin: Veb Deutscher Verlag der
Wissenschaften.
11.
Scheffe, H. 1958· Experiments with Mixtures.
Soc. B, 20:344-360.
12.
Scheffe, H. 1963. The Simplex-Centroid Design for Experiments
with Mixtures. J. R. Statist. Soc. B, 25:235-263·
I
I
J. R. Statist.
46
A.l
Regional Moments
The elements of the W.. matrices (l'~"
lJ
of the form
the regional moments) are
(A.l.l)
One could, of course, compute these elements directly from (A.l.l) or
by the moment generating fUnction
Mx(t , t , .•. , t q _ ) = n
l
2
_ l
q-l
1
. . ex}) (
R
I:
i=l
x.t.) dx •
1
1
Note that the value of the integral in (A.l.l) is simply the coefficient of (t
Pl P2
Pq-l
t
t
... t
)/rr p. in (A.l.2).
1 2
q-l
For q ~ 4 however,
finding the limits of integration proves to be a formidable task.
An alternative method is given below which requires no knowledge of
the limits of integration.
Let R* denote the (q-l)-dimensional simplex (1.1), l'~"
z.
1
>
0,
-
I:z.
l.
= 1,
(i = 1, 2, ... , q).
It is known that
dz q- 1
rr· r(p.1 )
RLpi)
1
J°
(I:Pi)-l -th
e
h
dh; p.
°
(A.l. )
1
where the sums and products are over i
= 1,
example Ryshik and Gradstein (1957, p. 219).
particularly useful when t = 0,
>
l'~"
2, •.. , q-l; see for
The integral (A.l.3) is
47
Recalling the transformation from (zl' z2'
zq)-space to
"'J
(xl' x 2' •.• , xq_l)-space, the elements of the Wij can be obtained
from (A.l.4).
Examples follow:
(q=3~
Example 1:
Recall the transformation (3·1.1) with xl
=
~(-Zl
+ z2)'
..[3
..[3
x 2 = ~ (-zl - z2 + 2Z ) = ~ (-3 z1 - 3z 2 + 2), x = zl + z2 + z3 = 1
3
3
and Jacobian
i)
=
..[3.
n- 1
2
Then from (A. 1. 4),
= /' 1dx dx
\ R
1
2
J'
,j'
= ..[3
Id d
..[3
2, R*
zl z2 = ~,
1 hdb
0
=
..[3
~
,
ii)
I\
2
= -3
,
I h3
2 /' 1 h3
I
-4 dh + -4 db - -3
0
3. 0
J,1 -h3
l
0 2
dh
1
='24'
etc.
Regional moments, up through order five are listed in Table A.1.
Example 2:
(q=4)
Recall the transformation (3·2.1) with x
xl
x2
= zl
= zl
+ z2 - z3 - z4 = 2z 1 + 2z2 - 1 ,
- z2 + z3 - z4
= 2z1
+ 2Z
3
- 1 ,
4
=
zl + z2 + z3 + z4 = 1,
48
= -zl + z2 + z3 - z4
3
(A.l.4),
X
i)
ii)
etc.
~
2z2 + 2Z
3
- 1 and Jacobian
=
1
16 . 2
= -16.
Then from
/,1 2
h dh
\ 0
= -8 ,
3
1
= -
5
,
Regional moments, up through order five are listed in Table A.2·
49
Table A.l
Regional Moments for q=3 (R triangular)
Integrand Moment
Integrand Moment Integrand Moment Integrand Moment
x5
2
x3x
l 2
0
x l4x
- ~3
x 2x 2
l 2
1
720
x3x 2
1 2
0
x x3
l 2
0
2
x x3
1 2
x5
1
0
x x4
l 2
0
x3
1
0
x
0
x3
~3
360
2
2
2
xl
1
24
x 2x
l 2
2
2
24
1
x x2
l 2
x x
l 2
0
x
Table A.2
4
xl
1
240
2
- ~3
2520
0
-J3
7560
0
Integrand Moment Integrand Moment rt"ntegrand Moment
x.1
0
2
x.x.
x.2
1
5
x.x.x k
1 J
x.x.
1 J
0
4
x.1
x?
0
3
x.x.
1 J
1
3bo
x4
2
Regional Moments for q=4 (R tetrahedral)
Integrand Moment
1
~3
1512
1
240
xl
1
J
1
2
x.x.x
1 J k
0
3
x.x.x
;L J k
2 2
x.x.
1 J
1
21
3 2
x.x.
1 J
0
L..
x~1
0
2 2
x.x'X k
1 J
0
0
4
X.X.
1 J
0
0
1
- 15
35
- 35
50
A.2
The A Matrices for the Estimator (2.6)
1)
)
(a:
q=3; R triangular
,
I
(d=l, k=l) , !l = (1, xl' x2 ) and !2
=
2
2
(xl' x 2 ' xl x 2 )·
Equation (1.9) and Table A.l,
W11 =
n,
r !l!~~
R
r
w
= n
I
12
\ R !1!2~
1
=
=
0
1
24
0
0
0
1
0
0
24
1
24
1
24
0
0
0
- ../3
3bo
- ..[3 ..[3
0
3bo 360
and
W~}i12
=
1
24
1
24
0
0
0
- ..[3
15
- ..[3 ..[3
15 15
0
Hence,
I
I
A = [I
3
W~iw12]
1
24
1
24
0
0
I
=
I
L
3
I
I
I
I
- ..[3
15
..[3
15
0
- ..[3
15
0
From
51
Other A matrices may be similarly derived.
(b) :
,
x == (1, xl' x ) and
(d==l, k=2) , -1
2
,
2
2
2
3 x3
x x2)
x == (xl' x ' xlx ' xl'
x
x
'
2'
l
l 2
2
2
-2
2
I
1
24
0
0
0
0
- ..[3
15
1
10
I - ..[3
..[3
15
0
0
I
A ==
1
..[3
24
I 1
I
31
I
I 15
(c):
- ..[3
0
0
0
1
30
1
10
1
30
0
3"bo 3"bo
,
2
2
(d==2, k==l) , x == (1, xl' x ' xl' x 2' x l x 2 ) and
-1
2
,
2
2
x == (xi, x32' X 1X ' x l x )
2
-2
2
..,
A ==
1
3
0
- ..[3
315
..[3
315
0
1
14
0
0
1
42
0
1
14
1
42
0
0
0
-2..[3
21
0
0
J3
7
- ..[3
21
0
0
0
- ..[3
7
- ..[3
21
.
52
2)
(a) :
q=4; R tetrahedral
I
(d=l, k=l) , x = (1, xl' x ' x ) and
-1
2
3
I
2
2
x = (xi, x ' x ' x x ' x l x ' x x )
l 2
2 3
-2
2
3
3
1
A =
(b) :
1
5
-
1
5
-1
0
0
0
0
0
0
0
0
0
0
0
-3
0
0
0
0
-3
0
0
5
4
0
1
3
1
1
.
I
(d=l, k=2), x = (1, xl' x ' x ) and
-1
2
3
2
2
2
3
3
I
3
x = (xl' x ' x3' x x ' x l x ' x x3' xl' x ' x
2
l 2
2
2
-2
3
3
.
2
2
2
2
2
2
x l x 2 ' x l x 3 ' x l x 2 ' x 2x 3 ' x l x 3 ' x 2x3' x l x 2x 3 )
I
A =
'!
15
1
5
1
5
0
0
I
,0
0
0
0
0
I 0
0
0
0
--
0
I
I 0
0
0
-3
0
0
14 /
I
1
1
3
0
0
0
0
0
0
0
0
0
1
-15
0
0
0
0
5
21
0
.L
0
0
0
.L
0
.L
0
0
0
0
.L
0
0
0
.L
0
.L
0
.L
0
0
0
0
1
-3 .L
21
21
21
21
21
21
21
21
53
(c):
2
2
2
I
X = (1, xl' x ' x3' Xl' x ' x ' x 1 x ' x 1 x3' X2 X )
2
2
-1
3
2
3
(d=2, k=l):
and
I
X
-2
=
2
2
2
2
3 3 3 2
(xl' x 2 ' x3' x 1 x 2 ' x 1 x3' x 1 x 2 ' x 2x3' x 1 x 3'
X X X )
1 2 3
0
0
0
0
0
0
0
0
0
1
21
L
0
0
0
0
1
14
0
1
14
0
0
0
L
0
1
14
0
0
0
0
1
14
0
0
0
L
0
1
14
0
1
14
0
0
0
0
0
0
0
0
0
0
0
0
-4
0
0
0
0
0
0
0
0
0
-4
0
0
0
0
0
0
0
0
0
-4
0
0
0
0
-3"
0
--
0
0
0
0
0
0
1
-2
0
0
0
0
0
0
0
0
0
0
--12
7
A =
1
10
7
7
1
1
2
1
2
1
1
1
1
2
0
0
0
54
A·3 Theorem A·3
W is positive definite for all q and d:
ll
Proof
For any vector of constants £ ~ 0, £'~l ~
R.
° almost everywhere
in
Therefore,
' ('
)2
-1 ,
£!l ~ = n £ WII£ > °
J
\ R
n-1 >
Since
A.4
0, W is positive definite.
ll
Theorem A.4
A necessary condition for
~
to be estimable is that
~l
be
estimable:
Proof
~
A
= TX,
is estimable if and only if there exists a matrix T such that
.!:~.,
therefore, I
=T
Xl and
I~l
is estimable,
follows that a necessary condition for
~
l'~"
~l
is
estimable.
It
to be estimable is that
A.5 Simultaneous Protection for Lower Degree True Polynomial Models:
A
(Fitted model y(!)
= ~i£l)
For fixed d, consider two possibilities for the true model:
~
k
(x)
1
-
= -1-1
x'~
+ x'B
-~2
+ X'A
-31::"3
(A·3·1)
55
a polynomial of degree d+k , and
l
a polynomial of degree d+k ; k < k ·
l
2
2
We can write
(A·3·2a)
Let
_A I
I-'
any~)
=
(A I
A I
A I)
A*
1:.1' t:2' 1:.3 ' 1:.
=
(A I
A I
0 ')
1:.1' 1:.2' _
•
Then A~ estimable (for
implies that A~* is estimable also.
Therefore, the same estimator which attains min B for (A.3.l)
also attains min B for (A·3.2a) and hence for (A.3.2).
A.6
An Expression for the Generalized Inverse of a Partitioned Matrix
If a matrix M is partitioned as
then a generalized inverse of M is given by
where
This expression can be directly verified by forming MM-M and
simplifying.
56
7.
APPENDIX B
57
Table B.l
Designs with V < V* = 3·0:
N
Designs
6
(Q=3, d~l, k=l)
V
nO
PI
P2
P
(3,0) +(3,~)
0
0.4190
0.2271
------
2·7397
7
(3,0)+(3,~
1
0.4994
0.2377
------
2·5971
7
(3,0)+(4,*)
0
0.4003
0.2467
------
2.8184
7
(3,0)+(4,0)
0
0.4003
o· 2467
------
2.8184
8
(3,0)+(3,g)
2
0·5150
0.2386
,"",,-~----
2·5556
8
(3,0)+(4,*)
1
0.4420
0.2944
------
2·7252
8
(3,0)+(4,0)
1
0.4389
o· 2386
..... ,......,.-- ....
2· 727 4
8
(3,0)+(5,arb)
0
0.4429
0.2540
_._----
2· 7590
9
(3,0)+(3,~)
3
0·5290
0.2386
"'0.+-_-- ....
, 2.6225
9
(3,0)+(4,*)
2
0.4616
0.2988
.....
9
(3,0)+(4,0)
2
0.4575
o· 2886
------
2,7585
9
(3,0)+(5,arb)
1
0.4837
0.2386
------
2,6753
9
(3,0)+(6,arb)
0
0.4879
0.2644
_..... _,-- ...
2·7074
9
(3,0)+(6,~)
0
0.4879
0·2644
------
2·7074
9
(3,0)+(3,g)+(3,g)
0
0·5773
0.2359
0.2386
2· 4789
9
(3,0)+(3,0)+(3,~)
0
0·5250
0.1655
0.2386
2·5776
10
(3,0) +(3,~)
4
0·5399
0.2387
------
2·7333
10
(3,0)+(4,*)
3
0.4765
0.2988
------
2.8104
10
(3,0)+(4,0)
3
0.4718
0.2386
------
2.8759
10
(3,0)+(5,arb)
2
0.4916
0.2386
-.._----
2.6910
10
(3,0)+(6,arb)
1
0·5163
0.2386
------
2· 6370
3
-~---
2· 7157
Table B.l
(Continued)
N
Designs
10
nO
PI
P2
P
V
(3,0)+(6,~)
1
0·5773
0·3150
---_._-
2.6215
10
(3,0)+(7,arb)
0
0·5591
0.8323
------
2.6545
10
(3,0)+(3,~)+(3,~)
1
0·5773
0.8386
0.8386
2.4297
10
(3,0)+(3,0)+(3,~)
1
0·5245
0.2356
0.8386
2.6123
10
(3,0)+(3,~)+(4,0)
0
0·5773
0.8386
0< 2468
2·5121
10
(3,0)+(3,~)+(4,*)
0
0·5773
0.8386
0.2468
2·5121
10
(3,0)+(3,0)+(4,0)
0
0.4581
0.1974
0.8386
2·7564
10
(3,0)+(3,0)+(4,*)
0
0.4675
0.1798
0.2988
2· 7268
3
59
Table B·2
N
Designs with V <
Designs
nO
v*
=
6.0:
(Q=3, d=2, k=.l)
PI
P2
P
3
V
9
(3.0)+(3,0)+(3,~)
0
0.4472
0.1859
0.2375
5· 3508
9
(3,0)+(3'5)+(3'3)
0
0.4113
0.1313
0.2641
5·8977
10
0,0)+(6,0)
1
0.4706
0.2375
-_.----
5·3215
10
(3,0)+(6,~)
1
0.4623
0·2782
------
5· 5835
10
(3,0)+(3,0)+(3'3)
1
0.4623
0.2)+53
0.2875
5·2231
11
(3,0)+(6,0)
2
0.4675
0.2375
------
5·2514
11
(3,0)+(6,~)
2
0.4704
0.2976
......... -._- .....
5·2512
11
(3,0) +(7, arb)
1
0.4760
o· 2337
._-----
5·4197
11
(3,0)+(3,0)+(3'5)
2
0.4675
o· 2391
o. 837~
5· 2513
11
(3,0)+(3'3)+(5'3)
0
0.4409
0.2375
0.1781
5·7101
11
(3,0)+(3'3)+(5,0)
0
0.4409
0.2375
0.1781
5·7101
12
0,0)+(6,0)
3
0.4659
0.2375
------
5·4362
12
(3,0)+(6,~)
3
0.4743
0·3053
------
5· 3405
12
0,0) +(7, arb)
2
0.4754
o· 2386
------
5·2134
12
(3,0)+(8,arb)
1
0.4835
0.2332
------
5· 4502
12
(3,0)+(3,0)+(3'5)
3
0.4675
o· 3047
0.2375
5·4196
12
(3,0)+(3'3)+(5'3)
1
0.4425
o· 23'75
0.2047
5·8411
12
(3,0)+(3'3)+(5,0)
1
0.4425
0.2375
0.20)+'7
5·8411
12
(3,0)+(3'5)+(6,~)
0
0.4769
0.1008
0·3094
5· 21'74
12
(3,0)+(3,0)+(6,~)
0
0.4909
0.1375
0·3172
5·1043
12
(3,0)+(3'3)+(6,0)
0
0.4691
0.1203
0.2375
5·3023
60
Table B·2 (Continued)
N
Designs
y'
-'0
Pl
P2
P3
0.4863
0.1578
0.2875
5·2297
0.4835
0.2832
0.2832
5·4502
V
12
(3,0)+(3,0)+(6,0)
12
(3,0)+(4,0)+(4,*)
°
1
13
(3,0) +(7, arb)
3
0.4744
0.2886
---'--'-
5·33°1
13
(3,0)+(8,arb)
2
0.4831
0.2886
------
5·2008
13
(3,0)+(9,arb)
1
0.4909
0.2828
,
_..... _, ..... _.
.5· 4902
13
(3,0)+(3,~)+(6,~)
1
0.4792
0.1414
0·3172
5·2469
13
(3,0)+(3,0)+(6,~)
1
0·5128
0.2023
0·3328
4·9695
12
(3,0)+(3,~)+(6,0)
1
0.4691
0.1672
0.2875
5·4028
13
(3,0)+(3,0)+(6,0)
1
0·5003
0.2297
0.2875
5·2731
13
(3,0)+(3,~)+(7,arb)
0
0.4769
0.0969
0.2875
5· 2823
13
(3,0)+(3,0)+(7,arb)
0.4863
0.1344
0.28'75
5·2240
13
(3,0)+(4,0)+(4,H)
0.4855
0.2883
0.2988
5·1844
13
(3,0)+(4,0)+(6,0)
0.4718
0.1286
0.2886
5· :;884
13
(3,0)+(4,*)+(6,0)
0.4721
0.1297
0.2875
5·6037
13
(3,0)+(4,0)+(6,~)
°
0.4914
0.1432
0·3254
5·119°
13
(3,0)+(4,*)+(6,~)
0·5°31
0.15°7
0·3333
4·9130
14
(3,0)+(8,arb)
°
3
0.4820
0.2886
------
5· 2741
14
(3,0)+(9,arb)
2
0.4894
0.2875
------
5·2132
14
(3,0)+(3,~)+(6,~)
2
0.4800
0.1719
0·3219
5·3298
14
(3,0)+(3,0)+(6,~)
2
0·5238
0.257°
(L
3328
4.8977
14
(3,0)+(3,~)+(6,0)
2
0.4659
0.2172
0.2875
5·5117
°
2
°
°
..
61
Table B.2 (Continued)
N
Designs
nO
Pl
P2
~-
--
P3
V
14
(3,0)+(3,0)+(6,0)
2
0·5081
0·3047
0.3375
5· 3259
14
(3,0)+(3,~)+(7,arb)
1
0.4753
0.1438
0.2875
5·3903
14
(3,0)+(3,0)+(7,arb)
1
0·5003
0.2078
0.2875
5·2496
14
(3,0)+(3,~)+(8,arb)
0
0.4831
0.0797
0.2875
5·2591
14
(3,0)+(3,0)+(8,arb)
0
0.4925
0.1234
0.3375
5·2134
14
(3,0)+(4,0)+(6,0)
1
0.4756
0.1814
0.2886
5·7723
14
(3,0)+(4,*)+(6,0)
1
0.4769
0.1844
0, ;}375
5·7867
14
(3,0)+(4,*)+(6,g)
1
0·5067
0.1883
0·3333
4·9009
14
(3,0)+(4,0)+(7,arb)
0
0.4847
0.1421
0.2875
5· 3341
14
(3,0)+(4,*)+(7,arb)
0
0.4847
0.1433
0.2875
5·3341
14
(3,0)+(5,0)+(5'3)
1
0.4961
0.2886
0·2741
5·5330
14
(3,0)+(5,0)+(6,0)
0
0.4839
0.1706
0.2886
5· 4586
14
(3,0)+(5,0)+(6,~)
0
0.4946
0.1581
0·3249
5·0950
14
(3,0)+(5'3)+(6,0)
0
0.4839
0.1706
0.2886
5· 4586
14
(3,0)+(5'3)+(6,g)
0
0.4989
0.1601
0·3333
5· 0749
15
(3,0)+(9,arb)
3
0.4894
0.2875
------
5·2559
15
(3,0)+(3,0)+(6,~)
3
0·5327
0.2971
0·3333
4·9081
15
(3,0)+(3,0)+(8,arb)
1
0·5050
0·2000
0.2875
5· 2264
15
(3,0)+(3,~)+(8,arb)
1
0.4831
0.1297
0.2875
5·3631
15
(3,0)+(4,0)+(6.0)
2
0.4801
0·2401
0.2886
5·9139
15
(3,0)+(4,*)+(6,0)
2
0.4816
0.2422
0.2875
5·9255
15
(3,0)+(4,0)+(6,~)
2
0·5016
0·2102
0·3333
5·1228
62
Table B.2 (Continued)
N
Designs
P2
----
I,
_.- _------
nO
PI
..
P3
V
1---_.. -_.._-- •
15
(3,0)+(4,*)+(6,i)
2
0·5096
0.22.')2
0·3333
4·9231
15
(3,0)+(4,0)+(7,arb)
1
0.4878
0,
18L,4
0.23'7 '5
5·4100
15
(3,0)+(4,*)+(7,arb)
1
0.4878
0.1844
0.2375
5·4100
15
(3,0)+(5,0)+(5'3)
2
0.4974
0.2386
0.2886
5·2232
15
(3,0)+(5,0)+(6,0)
1
1).4870
0.2°95
().2886
5- 54 71
15
(3,0)+(5,0)+(6,~)
1
0.4983
0.1890
(l,,3249
5·1115
15
(3,0)+(5'3)+(6,0)
1
0.4870
0.2C95
(). 3386
'5· 5)+71
15
(3.0)+(5'3)+(6,~)
1
0·5°24
0.1897
0·3333
5,°758
15
(3,0)+(5'3)+(7,arb)
°
0.4896
0,1612
o. ;'3386
5,3972
15
(3,0)+(5,0)+(7,arb)
°
0.4896
0.1612
0,2386
5· 3972
15
(3,0)+(6,0)+(6,0)
0.4925
0.19~53
0.2375
5·4883
15
(3,0)+(6,~)+(6,0)
°
0
0·5°24
0·3333
0.1706
5·0629
15
(3,0)+(6,~)+(6,~)
°
0·5023
0.1699
o•.3333
5·0650
1
Designs with V <
Table B· 3
v*
:=
4.0:
(q=4, d=l, k=l)
~
N
Design Composition
Parameter Values
nO
V
8
(1,2)
0
0.6460
0·3333
------
3·6093
9
(1.2)
1
0.6693
0·3333
----._-
3·7680
10
(1,3)
0
0.6409
0·7112
_.... _---
3·6583
11
(1,3)
1
0.6975
0.8209
------
3· 5227
12
(1·3)
2
0·7671
0·9511
.....
_-----
3· 3826
12
(1,1,2)
0
0.6117
o· 4684
0·3333
3·8260
12
(1,2,2)
0
0·7965
0·3333
0·3333
3·4693
13
(1,3)
3
0·7727
1.0000
_._-_ ... -
3· 2711
13
(1,1,2)
1
0·5619
0·5620
(j ..
3333
3·9192
13
(1,2,2)
1
0.8056
0·3333
0·3333
3·6092
14
(1,3)
4
0·7723
1.0000
-_._---
3· 2637
14
(1,2,2)
2
0.8137
0·3333
0·3333
3·7693
14
( 4)
2
0·9265
------
------
3· 7951
15
(1,3)
5
0·7787
1.0000
------
3· 3148
15
(1,2,2)
3
0.8207
0·3333
0·3333
3·9421
15
( 4)
3
1.0000
--_.---
------
3·4955
As an illustration of the entries under the column heading
"Parameter values", the design (1, 1, 2) with nO
:=
1 center point has
parameter values; PI
:=
0·3333·
:=
0·5619, P2
:=
0·5620 and r
Draper and Lawrence (1965b) constructed designs, using the
combination (1,2) with nO
:=
1, 2, 3, 4 which satisfy the conditions
64
(1.12).
These designs are however inadmissible for mixture experiments
since each of them has four points outside R.
Table B.4
Designs with V < V* = 10.0:
(q=4,
d~2,
k~l)
Design Composition
nO
18
(1,1,2,3)
0
0·7125 0·3824 0·3332 0·9527 8.6007
19
(1,1,2,3)
1
0·7211 0.4340 o. :3'332 0·9333 8.48)+1
20
(1,1,2,3)
2
0·7232 0.4641 0·3332 0·9191 8·5297
20
(1,2,4)
0
0·7354 0.2645 0·7951 --_. __ .....
21
(1,1,2,3)
3
0·7232 0.47'70 0·3332 0·9146 8.6954
21
(1,2,4)
1
0·7449 0·3035 0.8070 ------ 8.8231
22
(1,1,2,3)
4
0·7232 0.4826 0·3332 0·9145 8·9317
22
(1,2,4)
2
0·7543 0·3332 0.8119 ------ 8.4914
23
(1,1,2,3)
5
0·7234 0.4855 0·3332 0·9156 9· 2084
23
(1,2,4)
3
0·7564 0·3332 0.8162 ------ 8.4700
24
(1,1,2,3)
6
0·7234 0.4871 0·3332 0·9172 9·5096
24
(1,2,4)
4
0·7582 0·3332 0.8197
N
Parameter Values
V
------
9· 5747
8.6036