Optimum designs for optimum mixtures

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Statistics & Probability Letters 76 (2006) 1369–1379

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Optimum designs for optimum mixtures

Manisha Pal, Nripes K. Mandal�

Department of Statistics, University of Calcutta, Kolkata 700 019, India

Received 29 April 2005; received in revised form 5 January 2006

Available online 10 March 2006

Abstract

In a mixture experiment, the measured response is assumed to depend only on the relative proportion of ingredients or

components present in the mixture. Scheffe [1958. Experiments with mixtures. J. Roy. Statist. Soc. B 20, 344–360; 1963.

Simplex—centroid design for experiments with mixtures. J. Roy. Statist. Soc. B 25, 235–263] first systematically considered

this problem and introduced different models and designs suitable in such situations. Optimum designs for the estimation

of parameters of different mixture models are available in the literature. However, in a mixture experiment, often one is

more interested in the optimum proportion of ingredients. In this paper, we try to find optimum designs for the estimation

of optimum mixture combination on the assumption that the response function is quadratic concave over the simplex

region.

r 2006 Elsevier B.V. All rights reserved.

MSC: 62K99; 62J05

Keywords: Mixture experiments; Second-order models; Non-linear function; Asymptotic efficiency; Weighted centroid designs; Optimum

designs

1. Introduction

In many production processes, the response depends on the proportions xi; . . . ;xq of a number ofingredients/components satisfying:

xiX0; i ¼ 1; 2; . . . ; q;X

i

xi ¼ 1. (1.1)

Scheffe (1958) introduced following models in canonical forms of different degrees to represent the responsefunction Zx:

Linear:

Zx ¼X

i

bixi.

e front matter r 2006 Elsevier B.V. All rights reserved.

l.2006.02.007

ing author.

ess: [email protected] (N.K. Mandal).

www.elsevier.com/locate/stapro

ARTICLE IN PRESSM. Pal, N.K. Mandal / Statistics & Probability Letters 76 (2006) 1369–13791370

Quadratic:

Zx ¼X

i

bixi þXioj

bijxixj.

Special cubic:

Zx ¼X

i

bixi þXioj

bijxixj þX

iojok

bijkxixjxk.

Full cubic:

Zx ¼X

i

bixi þXioj

bijxixj þX

iojok

bijkxixjxk þXioj

dijxixjðxi � xjÞ.

The experimental region and the response functions in mixture experiments differ from the ordinaryresponse surface problem because of the constraint (1.1). Scheffe (1958, 1963) also introduced simplex latticedesigns and Simplex centroid designs appropriate in such situations. Kiefer (1961) established D-optimality ofsimplex lattice designs for the estimation of parameters of the models. Draper and Pukelsheim (1998) gave analternative representation of the response function based on Kronecker product algebra. In a later paper,Draper and Pukelsheim (1999) discussed the problem of improving a given moment matrix in terms ofsymmetry as well as obtaining larger moment matrix under Loewner order. The two together constitute theKiefer ordering. For the first-degree model, they showed optimality of vertex points design. For the second-degree model, complete class results relative to the Kiefer ordering were derived for the case of two and threefactors. Other contributions on the optimality in mixture models can be found in Kiefer (1975), Galil andKiefer (1977), Liu and Neudecker (1995) and others.

Box and Wilson (1951) first systematically considered the problem of determining the optimum factorcombination in a quantitative multi-factor experiment. Afterwards, a number of contributions is reported inthe literature e.g. Mandal (1978), Silvey (1980), Chatterjee and Mandal (1981), Mandal (1986), Mandal andHeiligers (1992), Fedorov and Muller (1997), Muller and Pazman (1998), Cheng et al. (2000), Melas et al.(2003).

In this paper, we are interested in estimating the optimum mixture combination as accurately as possible bya proper choice of design. For this, we have assumed the response function to be quadratic concave. Anoutline of the paper is as follows: In Section 2, we formulate the problem and study its properties. Optimumdesigns are derived for the cases q ¼ 2 and 3 in Sections 3 and 4, respectively. Finally, some conclusions aregiven in the last section, Section 5.

2. The problem and the perspective

Consider a mixture experiment with q ingredients:

xiX0; i ¼ 1; 2; . . . ; q;X

i

xi ¼ 1.

Let us assume that the response function can be approximated by a second-degree polynomial inx1; x2; . . . ;xq:

Eðy=xÞ ¼ Zx ¼ b0 þX

i

bixi þXipj

bijxixj. (2.1)

Because of the constraintP

xi ¼ 1, we can represent (2.1) by the following two equivalent forms:

Zx ¼X

i

bixi þXioj

bijxixj , (2.2)

Zx ¼X

i

bijx2i þ

Xioj

bijxixj. (2.3)

ARTICLE IN PRESSM. Pal, N.K. Mandal / Statistics & Probability Letters 76 (2006) 1369–1379 1371

Further, we assume that the response function is concave and there is a finite maximum in the interior of theexperimental region fðx1;x2; . . . ;xqÞjxiX0; i ¼ 1; 2; . . . ; q;

Pxi ¼ 1g: In this paper, we will work with (2.3) and

we express it in the form

Zx ¼ x0Bx, (2.4)

where x0 ¼ ðx1;x2; . . . ; xqÞ and B ¼ ðð1þ dijÞbij=2Þ, where dij is the Kronecker delta with

dij ¼ 1 if i ¼ j,

¼ 0 if iaj.

Now, subject toP

xi ¼ 1, (2.4) is maximized at

g ¼ d�1B�11, (2.5)

where 1 ¼ 1ðq� 1Þ ¼ ð1; 1; . . . ; 1Þ0 and d ¼ ð10B�11Þ. The problem is to find a (continuous) design in thesimplex (1.1) so that g given by (2.5) can be estimated with maximum accuracy.

Given a design:

x ¼ fx1;x2; . . . ;xn;w1;w2; . . . ;wng, (2.6)

with weights w1;w2; . . . ;wn;wiX0;P

wi ¼ 1 at points x1;x2; . . . ;xn, we can estimate the parameters of themodel (2.3) and hence an estimate of g using (2.5):

g ¼ d�1B�11. (2.7)

It is clear that g given by (2.5) is a non-linear function of the parameters of the model (2.3). To findEðg� gÞðg� gÞ0 in large samples we adopt the d-method:

Eðg� gÞðg� gÞ0 ¼ AM�1A0, (2.8)

where

A ¼ ðqg=qb11; qg=qb22; . . . ; qg=qbqq; qg=qb12; . . . ; qg=qbq�1;qÞ

and M is the information (moment) matrix of the design for the model (2.3):

M ¼X

wifðxiÞfðxiÞ0,

with fðxiÞ ¼ ðx2i1;x

2i2; . . . ;x

2iq;xi1xi2; . . . ;xi;q�1xiqÞ

0.It can be shown that

A ¼ B�1G, (2.9)

where G is given by

G ¼

2ðg21 � g1Þ 2g22 . . . 2g1g2 � g2 . . . 2gq�1gq

2g21 2ðg22 � g2Þ . . . 2g1g2 � g1 . . . 2gq�1gq

. . . . . . . . . . . .

2g21 2g22 . . . 2g1g2 . . . 2gq�1gq � gq

2g21 2g22 . . . 2g1g2 . . . 2gq�1gq � gq�1

0BBBBBBB@

1CCCCCCCA

(see Appendix A).Expressing bij ’s in terms of bij ’s, where B�1 ¼ ðbij

Þ; in Bg ¼ d�11; it can be easily checked that in order thatEq. (2.5) holds, the elements of B�1 ¼ ðbij

Þ and g must satisfy

bij¼ d2gi þ ðq� 1Þd if i ¼ j,

¼ dgigj � d if iaj. ð2:10Þ


Here, d is a constant given by

d ¼ ½dqq�2jBj��1=ðq�1Þ.

(see Appendix B).Then, A simplifies to

A ¼

�2ðq� 1Þg1 2g2 . . . gi � ðq� 1Þg2 . . . gq�1 þ gq

2g1 �2ðq� 1Þg2 . . . g2 � ðq� 1Þg1 . . . gq�1 þ gq

2g1 2g2 . . . gi þ g2 . . . gq�1 þ gq

. . . . . . . . . . . .

2g1 2g2 . . . gi þ g2 gq�1 � ðq� 1Þgq

2g1 2g2 . . . gi þ g2 . . . gq � ðq� 1Þgq�1

0BBBBBBBBB@

1CCCCCCCCCA.

Before proceeding further, we need to settle two things:

(i)
Since g given by (2.5) is non-linear in b’s, any measure of accuracy will depend on unknown parameters.We have to find ways to tackle this.
(ii)
To select the criterion function for comparing different designs.
Problem (i) can be tackled in several ways:

(a)
finding a locally optimum design by putting some specific values to the unknown parameters; (b) finding optimum designs for different segments of the domain of unknown parameters; (c) approaching sequentially; or (d) adopting a Bayesian approach with some prior assumption on the distribution of the unknown
parameters.

In this paper, however, we pursue a pseudo-Bayesian approach. Instead of assuming any specific form of theprior distribution on g; we assume that g is random with

Eðg2i Þ ¼ v; i ¼ 1; 2; . . . ; q; EðgigjÞ ¼ w; iaj ¼ 1; 2; . . . ; q; v40; w40. (2.11)

Clearly, vo1=q since 1 ¼ EðP

gÞ2 ¼ qvþ ðq� 1Þw and w40. If nothing is known about the relativeinfluence of the different components, there is no basis for assuming Eðg2i Þ to be unequal for the components.The same justification can be given for taking EðgigjÞ to be equal for all iaj: Box and Hunter (1957), whileintroducing rotatability as a desirable property of a design, used similar argument. If, however, some priorknowledge about the relative importance of different components are available, one can utilize it to have aprior different from (2.11). But, in that case, there is scope for losing the property of invariance and it might bedifficult to find a closed-form solution of the problem.

To resolve (ii), first note that since g01 ¼ 1;Eðg� gÞðg� gÞ0 is singular. As a measure of comparison ofdesigns, we use

fðxÞ ¼ TraceEfEðg� gÞðg� gÞ0g, (2.12)

where E stands for expectations with respect to the prior. A design is said to be optimum if it minimizes fðxÞgiven by (2.12). Using (2.9), we can write (2.12) as

fðxÞ ¼ TraceM�1EðA0AÞ (2.13)

which is a linear optimality criterion (Fedorov, 1972). In Sections 3 and 4, we first find explicit forms of (2.13)for cases q ¼ 2 and 3, respectively. Then using the invariance property of the problem, we will restrict to theclass of symmetric designs. Draper and Pukelsheim (1999) proved that given any symmetric design x, thereexists a weighted centroid design (WCD) Z, which dominates x in the sense of Loewner order. Using this result,we restrict to the class of WCD in finding optimum designs minimizing (2.13).


3. Two factors

Consider the case of two factors. Here

A ¼ ð2=djBjÞ�g1 g2 ð1

2Þðg1 � g2Þ

g1 �g2 ð�12Þðg1 � g2Þ

!.

Let us write the moment matrix of the design for the model as

M ¼

m40 m22 m31m04 m13

m22

0B@

1CA

and its inverse as

M�1 ¼

m40 m22 m31

m04 m13

m022

0B@

1CA.

It may be noted that m022 and m22 need not be equal.It can be shown that under (2.11), fðxÞ given by (2.13), reduces to

fðxÞ ¼ 2vðm40 þ m04Þ � 2ðv� wÞðm13 þ m31Þ þ ðv� wÞm022 � 4wm22 (3.1)

which is invariant with respect to the components of the mixture. Hence, using the result of Draper andPukelsheim (1999), we restrict to the class of WCD in finding optimum designs. With two factors, the twovertex points design Z1 and the overall centroid design Z2 are given by

Z11

0

� �¼ Z1

0

1

� �¼

1

2; Z2

1=2

1=2

!¼ 1 (3.2)

and a WCD is given by Z ¼ aZ1 þ ð1� aÞZ2;pap1, with mass a=2 to each vertex points ð1; 0Þ0 and ð0; 1Þ0 andmass ð1� aÞ to the centroid point ð1

2; 12Þ0: For such a WCD, the information matrix is given by

MðZÞ ¼ ð1=16Þ

1þ 7a 1� a 1� a

1þ 7a 1� a

1� a

0B@

1CA. (3.3)

After a little algebra, for a WCD Z; (3.1) simplifies to

fðZÞ ¼ 2d2½s=aþ t=ð1� aÞ�, (3.4)

where d is a constant independent of the design and s and t are given by

s ¼ 2ð4v� 1Þ þ 12; t ¼ 2ð4v� 1Þ. (3.5)

It is easy to see that optimum choice of a is

aopt ¼ s1=2=ðs1=2 þ t1=2Þ

and the minimum value of fðZÞ is given by

fðZÞ ¼ 2d2ðs1=2 þ t1=2Þ2. (3.6)


4. Three factors

In the case of three factors

A ¼ d

�2g1 g2 g3 ðg1 � 2g2Þ ðg1 � 2g3Þ ðg2 þ g3Þ

g1 �2g2 g3 ðg2 � 2g1Þ ðg1 þ g3Þ ðg2 � 2g3Þ

g1 g2 �2g3 ðg1 þ g2Þ ðg3 � 2g1Þ ðg3 � 2g2Þ

0B@

1CA, (4.1)

where d ¼ ð13djBjÞ1=2. The moment matrix has the form

MðxÞ ¼

m400 m220 m202 m310 m301 m211m040 m022 m130 m121 m031

m004 m112 m103 m013m220 m211 m121

m202 m112m022

0BBBBBBBBB@

1CCCCCCCCCA.

Writing M�1 as

M�1 ¼

m400 m220 m202 m310 m301 m211

m040 m022 m130 m121 m031

m004 m112 m103 m013

m0220 m0211 m0121

m0202 m0112

m0022

0BBBBBBBBB@

1CCCCCCCCCA

and using (4.1) in (2.12), the criterion function fðxÞ reduces to

fðxÞ ¼ 24vðm400 þ m040 þ m004Þ � 24wðm220 þ m202 þ m022Þ

þ 4ð6w� 3vÞðm310 þ m301 þ m013 þ m031 þ m130 þ m103Þ

� 12wðm211 þ m121 þ m112Þ þ ð12v� 6wÞðm0220 þ m0202 þ m0022Þ

� 6vðm0211 þ m0121 þ m0112Þ ð4:2Þ

which is invariant with respect to the components of the mixture. Here again, we mention that we have usedthe notation m0ijk in the lower triangle of M�1 to emphasize that m0ijk is different from mijk:We restrict ourselvesto the class of WCD which consists of Z1 supported by three vertex points, Z2 supported by three midpoints ofthe edges and Z3 supported by the overall centroid point

Z1

1

0

0

0B@

1CA ¼ Z1

0

1

0

0B@

1CA ¼ Z1

0

0

1

0B@

1CA ¼ 1=3,

Z2

1=2

1=2

0

0B@

1CA ¼ Z2

1=2

0

1=2

0B@

1CA ¼ Z2

0

1=2

1=2

0B@

1CA ¼ 1=3; Z3

1=3

1=3

1=3

0B@

1CA ¼ 1. (4.3)

Let the weights attached to Z1; Z2 and Z3 be given by a1; a2 and a3 with aiX0 andP

ai ¼ 1: This means thatsuch a WCD assigns mass

(i)
a1=3 to each vertex points ð1; 0; 0Þ0; ð0; 1; 0Þ0 and ð0; 0; 1Þ0. (ii) a2=3 to each midpoints of the edges ð1
2; 12; 0Þ0; ð1

2; 0; 1

2Þ0 and ð0; 1

2; 12Þ0 and

(iii)
a3 to the overall centroid point ð13; 13; 13Þ0.

ARTICLE IN PRESS

Table 1

Optimal values of a1 and a2 and minimum value of trace for different values of v

v a1 a2 Min trace

0.12 0.2565 0.7434 208.276

0.14 0.3066 0.6934 329.440

0.16 0.3285 0.6715 447.073

0.18 0.3410 0.6590 563.613

0.20 0.3492 0.6508 679.646

0.22 0.3551 0.6449 795.404

0.24 0.3589 0.6411 910.995

0.26 0.3620 0.6380 1026.479

0.28 0.3648 0.6352 1141.888

0.30 0.3670 0.6330 1257.244

0.32 0.3686 0.6314 1430.206

0.33 0.3693 0.6307 1430.206

M. Pal, N.K. Mandal / Statistics & Probability Letters 76 (2006) 1369–1379 1375

Writing a3 ¼ 1� a1 � a2; the moment matrix MðZÞ of the WCD Z comes out to be

MðZÞ ¼

a b b b b c

a b b c b

a c b b

b c c

b c

b

0BBBBBBBB@

1CCCCCCCCA, (4.4)

where a ¼ ð 181Þð1þ 26a1 þ 19a2=8Þ; b ¼ ð 181Þð1� a1 þ 11a2=16Þ; c ¼ ð 181Þð1� a1 � a2Þ. After some simplification,

for a WCD, we can write (4.2) as

fðZÞ ¼ 6½ð16a1 þ a2Þð36vð2a1 þ a2Þ � 3a2Þ þ 4f ð1� 9vÞð4a1 þ a2Þ2�=a1a2ð8a1 þ a2Þ, (4.5)

where

f ¼ �½32a1 þ a2 � 2a1ð8a1 � a2Þ � ð4a1 þ a2Þ2�=½16a1 þ a2 � ð4a1 þ a2Þ2�. (4.6)

The problem now is to find a1; a2 (and hence a3) in 0pa1; a2; a1 þ a2p1 minimizing (4.5). It seems difficultto find a closed form solution of it algebraically.

Remark. From Table 1, we observe that the optimum design puts positive mass at the vertices and the mid-points of the edges and zero mass at the overall centroid point. While discussing on the estimation of parametersof the polynomial of the second-degree model Kiefer (1961) also obtained D-optimal designs with support pointsat the vertices and the midpoint of the edges. Laake (1975) obtained the same support points in the seconddegree model using integrated variance criterion. From Table 1, we also observe that the minimum traceincreases with the value of v, which is in agreement with the fact that the more accurate the information on g,the higher is the efficiency of the design. The same is true for the case of two components (cf. Eqs. (3.5)–(3.6)).

5. Conclusion

In selecting the design, we have used the criterion of minimizing

TraceEfEðg� gÞðg� gÞ0g

using the prior assumption (2.10). However, instead of using (2.10), one can alternatively find a designminimizingZ

Trace fEðg� gÞðg� gÞ0gdg, (5.1)


where the domain of integration may be the whole of the simplex or a subspace of it, depending on theknowledge on g. If it is the whole simplex, it is not hard to show that the criterion function (5.1) is againinvariant with respect to the components of the mixture and the optimum design is a member of WCD. Asstated in Section 2, one can also find a design minimizing

maxg

Trace fEðg� gÞðg� gÞ0g, (5.2)

where the maximum is taken with respect to g in the simplex (1.1) or a subspace of it. Such studies, in thecontext of general response surface design for the estimation of the extreme points can be found, amongothers, in Mandal and Heiligers (1992), Muller (1995) and Muller and Pazman (1998). It is interesting to seewhether the optimum design under (5.2) is still a member of WCD.

Acknowledgements

The authors thank the referee for his valuable comments which improved the presentation of the paper. Theauthors are also grateful to Professor Bikas K. Sinha, I. S. I., Kolkata for his interest in the problem and hishelpful suggestions. Thanks are also due to Dr. A. Luoma of the University of Tampere, Finland for helpingwith the computation.

Appendix A

To prove (2.9), first note that A ¼ ððqgi=qbijÞÞ. Now from (2.5), we have

ð10B�11ÞBg ¼ 1 . . . . (A.1)

Taking derivative with respect to b11 on both sides of (A.1) we get

ðq=qb11Þ½ð10B�11ÞBg� ¼ 0.

Or, equivalently

½10ðqB�1=qb11Þ1�Bgþ ½ð10B�11ÞðqBg=qb11Þ ¼ 0. (A.2)

After a little algebra, (A.2) reduces to

Bðqg=qb11Þ ¼

2ðg21 � g1Þ

2g21. . .

. . .

2g21

0BBBBBB@

1CCCCCCA

and, in general,

Bðqg=qbiiÞ ¼

2g2i2g2i. . .

2ðg2i � giÞ

. . .

2g2i

0BBBBBBBBB@

1CCCCCCCCCA


so that

ðqg=qbiiÞ ¼ B�1

2g2i2g2i. . .

2ðg2i � giÞ

. . .

2g21

0BBBBBBBBB@

1CCCCCCCCCA.

Again, taking derivative with respect to b12 on both sides of (A.1) we get

ðq=qb12Þ½ð10B�11ÞBg� ¼ 0.

Or equivalently,

½�10ðqB�1=qb12Þ1�Bgþ ð10B�11ÞðqBg=qb12Þ ¼ 0

which, after a little algebra, reduces to

Bðqg=qb12Þ ¼

2g1g2 � g22g1g2 � g1

2g1g2. . .

2g1g2

0BBBBBB@

1CCCCCCA.

and hence

ðqg=qb12Þ ¼ B�1

2g1g2 � g22g1g2 � g1

2g1g2. . .

2g1g2

0BBBBBB@

1CCCCCCA.

In a similar way,

ðqg=qbijÞ ¼ B�1

2gigj

2gigj

2g1g2. . .

2gigj � gi

. . .

2gigj

0BBBBBBBBBBB@

1CCCCCCCCCCCA.

Combining all these, we get the desired form of G.

Appendix B

To show that B�1 has the form (2.10), i.e.

B�1 ¼ dgg0 þ dD,

where D ¼ qIq � 110; Iq being an identity matrix of order q; let us first write

Q ¼ dgg0 þ dD.


We note that

BQ ¼ 1g0 þ dðqB� B110Þ,

so that

QBQ ¼ Q1g0 þ dðqIq � 110Þ ¼ Q,

since Q1 ¼ dg.Hence,

QðBQ� IqÞ ¼ O. (B.1)

Now, D has q characteristic roots, namely ðq; q; . . . ; q; 0Þ, i.e. q with multiplicity q� 1 and 0 with multiplicityone, and the corresponding orthonormal characteristic vectors are c1x1; c2x2; . . . ; cqxq, respectively,where ci ¼ ði

2 þ iÞ�1=2; xi ¼ ð1; 1; . . . ; 1;�i; . . . ; 0; 0Þ0; the ði þ 1Þth element being �i; i ¼ 1ð1Þðq� 1Þ; andcq ¼ q�1=2; xq ¼ 1.

Therefore, D ¼ qPq�1

i¼1 c2i xix0i.

So, we can write Q ¼ CC0; where C ¼ ðc1ffiffiffiffiffiffiffiffiffiðdqÞ

px1; c2

ffiffiffiffiffiffiffiffiffiðdqÞ

px2; . . . ; cq�1

ffiffiffiffiffiffiffiffiffiðdqÞ

pxq�1;

ffiffiffidp

gÞ. Then,jQj ¼ dðdqÞq�1f

Qq�1i¼1 cig

2jx1; x2; . . . ; xq�1; gj2.

Now,

jx1; x2; . . . ; xq�1; gj ¼ gq

1 1 1 . . . 1 1

�1 1 1 . . . 1 1

0 �2 1 . . . 1 1

0 0 0 . . . �ðq� 2Þ 1

��

��þ ðqþ 1Þ

1 1 1 . . . 1 g1�1 1 1 . . . 1 g20 �2 1 . . . 1 g30 0 0 . . . �ðq� 2Þ gq�1

��

��To find the determinants we use the following results.

Lemma 1. If G be a n� n matrix given by

G ¼

1 1 1 . . . 1 1

�1 1 1 . . . 1 1

0 �2 1 . . . 1 1

0 0 0 . . . �ðq� 2Þ 1

0BBBB@

1CCCCA,

then jGj ¼ n!

Lemma 2. If H be a n� n matrix given by

H ¼

1 1 1 . . . 1 g1�1 1 1 . . . 1 g20 �2 1 . . . 1 g30 0 0 . . . �ðn� 2Þ gn

0BBBB@

1CCCCA,

then jHj ¼ ðn� 1Þ!ðg1 þ g2 . . .þ gnÞ.

(Lemmas 1 and 2 can be proved by induction.)Hence, noting that

Pqi¼1gi ¼ 1, we get

jx1; x2; . . . ; xq�1; gj ¼ ðq� 1Þ!

Then jQj ¼ dqq�2dq�1.This shows that Q is a non-singular matrix. Hence, it follows from (B.1) that BQ ¼ Iq; which gives B�1 ¼ Q.


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Optimum designs for optimum mixtures