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Abstract
In the first year of this project, model robust designs for experiments with mixtures are investigated, where a mixture experiment is an experiment in which the q-ingredients are nonnegative and subject to the simplex restriction Pqi=1xi = 1 on the (q − 1)-dimensional probability simplex Sq−1. The polynomial regression
models for mixture experiments introduced by Scheffe(1958, 1963) are considered. The literature about designing experiments with concerns about uncertainty of the models dates back to Box and Draper (1959), where it was demonstrated that large bias may be introduced in estimation when the model is assumed to be a simple linear polynomial and the true regression function may be quadratic. In this project we consider during the design stage about uncertainty with linear or quadratic polynomial type being the two possible candidate models for the mixture experiments with either Scheff´e . A model robust type of D-optimality criterion as defined in Dette (1990) has been used here to find the model robust D-optimal designs for the mixture experiments. Similarly model robust type of A-optimality designs for mixture experiments will be defined accordingly and be used to find optimal designs. The model robust A- and D- optimal designs for the Scheff´e model under different classes of candidate models are obtained respectively.
In the second year of this work, a complete class of designs under the Kiefer ordering for linear log contrast model with mixture experiment is presented. Based on the completeness result, φp-optimal designs for p,−∞ ≤ p < 1, including D- and A-optimality are obtained, where the eigenvalues of the design moment
matrix are used. By using the approach presented here, it gives us some insights about how these φp-optimal
designs behave. In the end,φp-optimal designs for some p,−∞ ≤ p < 1, for a quadratic log contrast model
are also discussed, where some numerical solutions are obtained .
Keywords : A-optimality, complete classes, convex combination, D-optimality, efficiency, equivalence theorem, invariant symmetric block matrices, Kiefer ordering, linear and quadratic models, minimax robust criteria.
Part 1: Model robust D- and A- optimal designs for Scheff´e model
1
Introduction and prerequisites
A mixture experiment is an experiment in which the q factors xi are nonnegative and subject to the simplex
restrictionPqi=1xi= 1, that is, the factors represent relative proportions of all q ingredients in the mixture. Thus, in a mixture experiment with q components, the experimental conditions are represented by points in the (q − 1)-dimensional probability simplex Sq−1 := {x ∈ [0, 1]q : x01
q = 1}, with 1q := (1, . . . , 1)0 ∈ Rq.
Cornell (1990) has numerous examples and applications of mixture experiments.
Various types of models such as polynomial models of degree d ≤ 3, models with inverse terms, or log-contrast models have been proposed. In this paper, we investigate mixture experiments where there is uncertainty as to whether Scheff´e’s (1958) first-degree or second-degree polynomial mixture model is appropriate, and we explore experimental designs which are robust towards this type of uncertainty. The regression functions specifying Scheff´e’s (1958) first-degree and second-degree mixture models are
f1: Sq−1→ Rq, x 7→ x, (1) f2: Sq−1→ R( q+1 2 ), x = (x1, . . . , xq)07→ (x0, (xixj)1≤i<j≤q)0, (2) and lead to E[y(x)] = θ0f1(x) = q X i=1 θixi, (3) E[y(x)] = β0f2(x) = q X i=1 βixi+ q X i, j = 1 i < j βijxixj (4)
with unknown parameter vectors θ ∈ Rq and β ∈ R(q+12 ), respectively. In both models, all observations taken
in an experiment are assumed to be uncorrelated and to have common unknown variance σ2> 0.
An experimental design ξ for a mixture model is a probability measure on Sq−1 with finite support,
|supp ξ| < ∞. The statistical properties a design ξ displays in the two mixture models specified by (1) and
(2) are captured by its information matrices
Mi(ξ) := Z
Sq−1fi(x)f
0
i(x) dξ(x)for i = 1, 2.
The design ξ is called feasible in a given model if and only if its information matrix is regular. Associated with any feasible design in the two models (1) and (2), we have the dispersion functions
di(., ξ) : Sq−1→ [0, ∞], x 7→ fi0(x)Mi−1(ξ)fi(x)
A design ξDis called as D-optimal if it maximizes the determinant of M (ξ) among all feasible designs defined
on Sq−1. For models with a continuous regression function and a compact design space, by the well known
equivalence theorem, ξD minimizes the maximum of d(x, ξ) among all feasible designs ξ, in other words the D-optimal design and the minimax design are equivalent, and maxxd(x, ξD) ≤ q for any point x ∈ Sq−1 and
equality holds at the design points.
Besides D-optimal design, another emphasis in this paper is A-optimal design. A design ξAis called A-optimal
for a model f (x) if and only if
f0(x)M−2(ξA)f (x) ≤ Tr M−1(ξA), for any point x ∈ Sq−1, or
Tr (M (ξ)M−2(ξA)) ≤ Tr M−1(ξA),
for ξ is an arbitrary design on Sq−1, with equality if and only if x or ξ is assigned to the design points
(Pukelsheim (1993, p.221)).
Draper and Pukelsheim (1999), Draper et al. (2000) prove a useful complete class result which makes the research of an optimal design under very general criteria much more easier. Draper and Pukelsheim (1999) shows that the vertex points design is the unique optimal design under the Kiefer ordering in the first-degree mixture model. For the second-degree mixture model with two or three ingredients, complete class results under the Kiefer ordering are also derived there. Draper et al. (2000) show that the set of weighted centroid designs constitutes a convex complete class under Kiefer ordering for the second-degree mixture models. For four ingredients, the class is minimal complete. With the complete class result, we are able to focus on finding an optimal design in that class. Klein (2003) has that in the complete class of weighted centroid designs, a method to prove equivalence theorem. It has analyzed a quadratic subspace of block matrices which are invariant under the action of a group H arising from the design of mixture experiments. Klein (2003) uses these results to find the D-,A- and E- optimal designs respectively.
About the mixture experiments the corresponding optimal designs have been investigated by Kiefer (1961), Galil and Kiefer (1977), Mikaeili (1989, 1993), Lim (1990), Guan and Chao (1987), and Klein (2003). Chan (2000) gives a comprehensive overview of existing results.
Concern about uncertainty of the models for the experiments dates back to Box and Draper (1959), where it was observed that if we use simple linear function to estimate the expected values of the responses when the true model is quadratic, it would result in a large bias term for estimation. Since then while designing an experiment for regression models, robustness has always been an important issue. Many papers thereafter have addressed this issue and provided different design strategies, for some examples see Stigler (1971), Atkinson and Cox (1974), Huber (1975), Studden (1982), Sacks and Ylvisaker (1984), Huang and Studden (1988), Dette (1990, 1991, 1993, 1994 1995), Pukelsheim and Rosenberger (1993). For more details about different types of optimal design criteria see Pukelsheim (1993).
2
Model robust D-optimal designs
As the D-optimal designs for the nested models seem to have the characteristics that the supports of the
D-optimal design for the simpler model are in the subset of the optimal supports for the more complicate
models, it is conjectured that a suitable convex combination of those D-optimal designs. Therefore in this work we will first look for designs from the convex combinations of those D-optimal designs, if there is not an optimal one in that class we will relax to find optimal designs from those designs with the optimal supports for the largest model. We illustrate this idea with two candidate models, i.e. first and second-degree models for mixture experiments by Scheff´e, as follows.
We first state the D-optimal designs for the Scheff´e first and second-degree models. Let ξD
1 be the D-optimal
design for the Scheff´e first-degree model which assigns a equal weight1
q to each of the points x ↔ (1, 0, · · · , 0).
Let ξD
2 be the D-optimal design for the Scheff´e second-degree model which assigns a weight (q+11
2 )
to each of the points x ↔ (1, 0, · · · , 0) and x ↔ (1
2,12, 0, · · · , 0).
Let ξαbe the convex combination of the two designs ξDi , (i = 1, 2), where mi(i = 1, 2) be the size of the i-th
model’s information matrix and Mi(ξ)(i = 1, 2) be the i-th model’s information matrix, then ξα = αξD
1 + (1 − α)ξ2D, α ∈ [0, 1]
Mi(ξα) = αMi(ξ1D) + (1 − α)Mi(ξ2D) for i = 1, 2.
For a given r ∈ [0, 1], we would like to find a weight αD∈ [0, 1] which maximizes ψr(ξ) := |M1(ξα)| r m1 |M2(ξα)|1−rm2 , (5) with m1= q and m2= ¡q+1 2 ¢
as introduced in Section 1. Taking log on both sides of(5), we obtain the robust
D-optimality criterion ΨD r(ξ) := log ψr(ξ) = r m1log(|M1(ξ)|) + 1 − r m2 log(|M2(ξ)|). (6) A design ξD
r is called robust D-optimal if and only if it satisfies
ΨD
r(ξrD) = max{ΨDr(ξ) | ξ ∈ Ξ with M2(ξ) ∈ PD(m2)}, (7)
where PD(m2) denotes the cone of positive definite m2× m2 matrices. The feasibility condition M2(ξ) ∈
PD(m2) rules out those designs under which there is no linear unbiased estimator for the second-degree
model’s parameter vector. Note that M2(ξ) ∈ PD(m2) implies M1(ξ) ∈ PD(m1). As in Section 1, we
associate the dispersion function
d(., ξ) : Sq−1→ [0, ∞], (8)
x 7→ r
m1d1(x, ξ) +
1 − r
m2 d2(x, ξ), (9)
to a design ξ on Sq−1. In a similar fashion to Dette’s (1990) equivalence theorem, it can be shown that a
design ξD
r is robust D-optimal for a given prior r ∈ [0, 1] if and only if its dispersion function satisfies d(x, ξD
As a preliminary step to solving the design problem stated in (12), we show that the set of competing designs can be restricted to the class of weighted centroid designs.
Definition 2.1. For all 1 ≤ j ≤ m, the j-th elementary centroid design ηj is the uniform distribution on the ¡qj¢ points of the form 1
j
Pj
k=1eπ(k) ∈ Sq−1 with π ∈ Qq. A convex combination η(λ) =
Pq j=1λjηj with λ = (λ1, . . . , λq)0 ∈ Sq−1 is called a weighted centroid design with weight vector λ. We write W :=
{η(λ) | λ ∈ Sq−1} for the set of all weighted centroid designs.
Theorem 2.2. The class W of weighted centroid designs forms a complete class of designs relative to the
robust D-criterion (6).
Corollary 2.3. A design ξD
r is robust D-optimal if and only if its dispersion function satisfies d(x, ξD r ) ≤ 1 for all x ∈ q [ j=1 supp ηj.
The simplified equivalence theorem given above directs us towards evaluating the dispersion function of a candidate design in support points of weighted centroid designs. The following lemma provides a useful representation of such values of the dispersion function when the candidate design is exchangeable.
Lemma 2.4. For every exchangeable design ¯ξ on Sq−1, the dispersion function given in (8) satisfies d(x, ¯ξ) = r qTr (M1(ηj) M −1 1 (¯ξ)) + 1 − r ¡q+1 2 ¢ Tr (M2(ηj) M2−1(¯ξ))
for all x ∈ supp ηj, 1 ≤ j ≤ q.
The following definition is essential to our approach of determining model-robust D-optimal designs. Definition 2.5. Let ξD
1 and ξ2D denote the D-optimal designs for the first- and second-degree mixture model,
respectively. Then we write ξα:= αξD
1 + (1 − α)ξD2 for all α ∈ [0, 1], and we define the convex class of designs
ΞD:= {ξα| α ∈ [0, 1]} = conv {ξD1, ξ2D}.
Note that ΞD ⊆ W. In the following we solve the problem of maximizing the model-robust D-criterion (6)
in two steps. First, we generate an optimality candidate by finding a weight αD
r ∈ [0, 1] such that ξαD r ∈ ΞD
maximizes (6) within ΞD. Second, we prove the candidate design ξαD
r to be optimal among all feasible
designs. The following theorem presents the optimality candidate ξαD r.
Theorem 2.6. For any r ∈ [0, 1], the design ξαD
r ∈ ΞD with weight
αD r =
−2 − r + q (−1 + 2r) +p−8r(−q + r) + (2 + q + r − 2qr)2
2(q − r) ∈ (0, 1)
is the unique model-robust D-optimal design among all designs in ΞD.
In our second step, we will now verify the candidate design ξαD
r to be model-robust D-optimal among all
designs. In order to do so, we use the equivalence theorem (10) and show that d(x, ξαD
r) ≤ 1 holds for all
x ∈ Sq−1. Instead of Kiefer’s (1951) classical way of computing the dispersion function via orthogonal
poly-nomials, we follow Klein’s (2003) approach, which is based on the quadratic subspace properties introduced in Section 1 and yields tractable matrix algebra.
Theorem 2.7. The design ξαD
r ∈ ΞD given in Theorem 2.6 is a model-robust D-optimal design for Scheff´e’s
3
Model robust A-optimal designs
Besides model-robust D-optimality we are interested in model-robust A-optimality. some differences between the A- and D-optimal designs of Scheff´e first and second-degree models. That is the A-optimal designs for Scheff´e second-degree model do not have equal weights on the supports. Guan and Chao (1987) gives that the {q, 2} simplex weighted centroid design with w1
w2 =
√
4q−3
4 is A-optimal for Scheff´e’s second-degree
model with q ≥ 4. Hence, the design ξA
2 assigns equal weights w1 =
√
4q−3
q√4q−3+2q(q−1) to x ↔ (1, 0, ..., 0) and
w2=q√4q−3+2q(q−1)4 to x ↔ (12,12, 0, ..., 0).
By the previous A-optimal designs results of Scheff´e’s linear and quadratic models, we can give three different criteria as follows. ΨAβ(ξ) = β1Tr M1−1(ξ) + β2Tr M2−1(ξ), (11) where β = (β1, β2) = (r, 1 − r), (mr1,1−rm2), ( r Tr M−1 1 (ξ1A) , 1−r Tr M−1 2 (ξA2)
). For a given β = (β1, β2) ∈ [0, 1], a design ξβA is called robust A-optimal if and only
if it satisfies
ΨAβ(ξβA) = min{ΨAβ(ξ) | ξ ∈ Ξ with M2(ξ) ∈ PD(m2)}. (12)
In the following, we just discuss the case with β = (β1, β2) = (r, 1 − r). Therefore, we can obtain the
equivalence theorem for the robust A-optimal designs. Lemma 3.1. For an exchangeable design ξA
r,it is robust A-optimal for the Scheff´e’s linear and quadratic models if and only if
rTr {M1(ηj)M1−2(ξrA)} + (1 − r)Tr {M2(ηj)M2−2(ξrA)} ≤ rTr M−1
1 (ξAr) + (1 − r)Tr M2−1(ξrA) for every centroid design ηj,where j = 1, 2, · · · , q.
Definition 3.2. Let ξA
1 and ξ2A denote the A-optimal designs for the first- and second-degree mixture model,
respectively. Then we write ξα:= αξA
1 + (1 − α)ξA2 for all α ∈ [0, 1], and we define the convex class of designs
ΞA:= {ξα| α ∈ [0, 1]} = conv {ξA1, ξ2A}.
For the robust A-optimal designs, we can also have the similar theorem, corollary and lemma like theorem 2.2,corollary 2.3 and lemma 2.4 for the robust D-optimal designs, and the difference is the criterion. Therefore, we just give two theorems for the robust A-optimal designs.
The first theorem is that to find a corresponding αA
r ∈ [0, 1] such that ξαA
r minimizes the function Ψ
A r(ξr).
The second theorem is that ξαA
r is indeed a robust A-optimal design.
Theorem 3.3. For a given prior r ∈ [0, 1], consider a design ξA
α ∈ ΞA for mixture experiments with q ≥ 4 ingredients, there exists an optimal αA
r. Furthermore, αAr ∈ [0, 1] is a function of r and is one-one and onto.
Theorem 3.4. For a given r ∈ [0, 1], the design ξαA
r ∈ ΞA with optimal α
A
4
The efficiencies of model-robust designs
In Section 3, the robust D-optimal designs for a fixed r ∈ [0, 1] are found, but the choice of r remains to be a problem. In Zen and Tsai (2003), following the discussions about different types of robust D-optimal design criteria in Pukelsheim and Rosenberger (1993), a robust D-optimal design criterion combining concerns about model discrimination and estimation all together is proposed, but it can actually be reduced to a special case of the robust D-optimal design criterion as defined in Dette (1990). Although in Zen and Tsai (2003), the problem of choosing a suitable r has also been addressed, where a maximin criterion for choosing r is proposed. In this section, we will adopt a similar maxmin criterion for the robust D-optimal design problems in mixture experiments. Now consider the efficiency of robust D-optimal design for any fixed r0under the criterion with
weighting r. The relative efficiency of a robust D-optimal design ξαD
r0 under Ψ
D
r-robust D-optimal criterion
is defined as one among all feasible designs which maximizes
Dr-eff(ξαD r0) = ΨD r(ξαD r0) ΨD r(ξαD r) , r, r0 ∈ [0, 1]. (1)
Then utilizing a formula in Fedorov (1972, p16), ΨD r(ξαD r) can be simplified as ΨD r(ξαD r) = |M1(ξαDr)| r m1|M2(ξαD r)| 1−r m2 = C q + q2, (2) where C = 2(1−q)(q (3−2r)+r)q(1+q) (1 + q)rq · µ ¡ 1 − αrD ¢−1+q 2 ¡2 − (1 − q) αD r ¢¶2(1−r)1+q · ¡ 2 + q + qαD r ¢(−1+q)r q and αD r = −2−r+q (−1+2 r)+√−8 r (−q+r)+(2+q+r−2 q r)2 2 (q−r) .
Substituting (2) into (1), it can be written that Dr-eff(ξαD
r0) = g(r, r 0)/g(r, r), where g(r, r0) = ¡1 − αDr0 ¢(−1+q)(1−r) 1+q ¡2 + (−1 + q) αD r0 ¢2−2r 1+q · (2 + q + qαDr0) (−1+q)r q . (3)
Then for a given r0, the pattern of the Dr-eff(ξ αD
r0), r ∈ [0, 1] efficiencies can be found and the minimum value
is obtained in the following.
Lemma 4.1. For a given r0 ∈ [0, 1), the minimum value of Dr-eff(ξ αD
r0) for r ∈ [0, 1] is attained at one of
the two end points, i.e. r = 0 or r = 1.
To find the optimal value of r0 which maximizes the minimum of Dr-eff(ξ αD
r0), we investigate the behavior of
p(r0), which is the ratio of D
0-eff(ξαD
r0) and D1-eff(ξα D
r0). We will solve for p(r
which can be expressed in a simplified form p(r0) = D0-eff(ξαDr0) D1-eff(ξαD r0) = (2 + 2q)−1+qq µ ¡ 1 − αD r0 ¢−1+q 2 ¡2 + (−1 + q) αD r0 ¢¶1+q2 21+q2 ¡2 + q + qαD r0 ¢−1+q q .
Now we have the following results.
Theorem 4.2. For any fixed r0 ∈ [0, 1] and q ≥ 2, we have
min 0≤r≤1{Dr-eff(ξαDr0)} = D0-eff(ξαD r0) , if r 0 ≥ rD D1-eff(ξαD r0) , if r 0 < rD , where D0-eff(ξαD
r0) and D1-eff(ξαDr0) are as in (1) with r = 0 and r = 1, respectively, and r
D is the root of p(r0) = 1.
Theorem 4.3. For any q ≥ 2, min
0≤r≤1{Dr-eff(ξαDr0)} is increasing first, then decreasing in r
0and the maximum value of min
0≤r≤1{Dr-eff(ξαDr0)} is attained at r
0 = rD, i.e.
rD= arg{ max
0≤r0≤10≤r≤1min Dr-eff(ξαDr0)},
where rD is the root of p(r0) = 1.
We present Table 1 in the Appendix which gives some numerical results of rD and the corresponding
effi-ciencies of ξαD rD, ξα
D
0 , ξαD0.67, ξαD0.99 and the special case of D0-eff(ξαD0.67) which is the efficiency of ξαD0.67 under
quadratic model.
From Table 1, we see that the values of rD is approximating 0.67 as q becomes large and the value of
min
0≤r≤1Dr-eff(ξαDr0) is approximately 0.68. It is clear that D-optimal designs for the first degree model can
not be used if the model is second-degree and the efficiency of the second degree D-optimal design ξD
2 = ξαD
0
reduces to be only less than 0.58 when q ≥ 10 if the model is first degree, which is not so efficient. The robust
D-optimal designs under maximin robust D-criterion performs fairly well for both first- and second- degree
models, which seems to be more favorable than the individual D-optimal designs. For the robust A-optimal designs, we have three different criteria as follows.
min{β1Tr M1−1(ξαA) + β2Tr M2−1(ξαA)}, where (β1, β2) = (r, 1 − r), (mr1,1−rm2), ( r Tr M−1 1 (ξ1A), 1−r Tr M−1 2 (ξA2)) and ξ A
α is the robust A-optimal designs for the Scheff´e’s first and second-degree
models.
Taking the criteria as (β1, β2) = (r, 1 − r), then the Ar-eff(ξαA
Ar-eff(ξαA r0) = rTr M−1 1 (ξαA r) + (1 − r)Tr M −1 2 (ξαA r) rTr M1−1(ξαA r0) + (1 − r)Tr M −1 2 (ξαA r0) . Hence rA= arg{ max 0≤r0≤10≤r≤1min Ar-eff(ξαAr0)}.
Under such criteria and definition, we have some numerical results about the Ar efficiency in Table 2 given
in the Appendix.
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[21] Mikaeili, F. (1993). D-optimal Design for full cubic on q-simplex. Journal of statistical planning and
inference 35, 121-130.
[22] Pukelsheim, F. (1993). Optimal design of experiments. Wiley, New York.
[23] Pukelsheim, F., Rosenberger, J.L. (1993). Experimental designs for model discrimination. Journal
of the American Statistical Association, 88, 642-649.
[24] Sacks, J., Ylvisaker, D. (1984). Some model robust designs in regression. The Annals of Statistics, 12, 1324-1348.
[25] Stigler, S. M. (1971). Optimal experimental design for polynomial regression. Journal of the American
Statistical Association 66, 311-318.
[26] Studden, W.J. (1982). Some robust-type D-optimal designs in polynomialregression . Journal of the
American Statistical Association, 77, 916-921.
[27] Zen, M.M., Tsai, M.H. (2003). Criterion-robust designs for model discrimination and parameter estimation for polynomial models. To appear in Sankhya.
Appendix
Table 1: Table of min
0≤r≤1Dr-eff(ξαDr0) for different r
0 q r D min 0r1 D r -e ( D r D ) D 1 -e ( D 0:67 ) D 0 -e ( D 0:67 ) D 1 -e ( D 0 ) D 0 -e ( D 0:999 ) 2 0.679472 0.915523 0.913557 0.919615 0.816497 0.164968 3 0.679609 0.869229 0.866132 0.875693 0.731004 0.063190 4 0.679667 0.839402 0.835551 0.847453 0.681732 0.034628 5 0.679662 0.818324 0.813938 0.827503 0.649731 0.022876 6 0.679612 0.802506 0.797728 0.812514 0.627290 0.016877 7 0.679531 0.790125 0.785055 0.800752 0.610688 0.013366 10 0.679188 0.764876 0.759294 0.77658 0.579539 0.008436 100 0.672929 0.685299 0.682612 0.690824 0.508413 0.002444
Table 2: Table of min
0≤r≤1Ar-eff(ξαAr0) for r 0= rA q rA min 0≤r≤1Ar-eff(ξαArA) 4 0.995860 0.797231 5 0.997397 0.786961 6 0.998195 0.779256 10 0.999327 0.76018 100 0.999991 0.70422
Part 2. φp-optimal designs for the log contrast model
1. Introduction
Many practical problems are associated with the investigation of experiments with mixture ingredients
x1, x2, . . . , xq, with xi≥ 0, and restriction
Pq
i=1xi= 1. Different types of mixture models such as polynomial,
log contrast model, and so on, have been developed to describe responses under the mixture experiments. Cornell(1990) has listed numerous examples and applications of mixture experiments. Furthermore, the A-and D-optimal design problem for the linear log contrast model with different kinds of design spaces are investigated by Chan(1988, 2001). However, the φp-optimal designs of a linear log contrast model has not
been further explored. This paper therefore aims at determining φp-optimal designs with the following linear
log contrast model for mixture experiments introduced by Aitchison and Bacon-Shone(1984), i.e.
E(y) = β0+ k−1 X i=1 βilog(xi xk) (4.1) for x = (x1, x2, · · · , xk)0∈ X (δ) and Sk−1 = {(x 1, x2, · · · , xk)0: X xi= 1, xi≥ 0(i = 1, . . . , k − 1)}, X (δ) = {x = (x1, x2, · · · , xk)0 ∈ Sk−1: δ ≤ xi xj ≤1 δ, i, j = 1, · · · , k, δ ∈ (0, 1)}.
while by setting ti = log(xxki), a = − log δ, the above model can be represented in the following alternative
form (called the first-degree model):
E(yt) = β0+
k−1
X
i=1
βiti= βf0(t) (4.2)
with experimental domain
T (a) = {t = (t1, t2, · · · , tk−1)0 : −a ≤ ti≤ a, −a ≤ ti− tj≤ a, i, j = 1, · · · , k − 1}
where ytis an observation at t ∈ T (a), and β = (β0, β1, · · · , βk−1) is the vector of unknown parameters. The
observations are assumed to be uncorrelated with constant variance.
An experimental design τ is a probability measure on the experimental domain T (a) with a finite number of support points. The information matrix of τ is defined to be
M (τ ) =
Z
T(a)
f (t)f0(t)dτ (t).
An important family of concave, permutationally invariant and Loewner-isotonic functions are the p-means
φp : φp(M (ξ)) = λmin(M (ξ)), if p = −∞ (detM (ξ))1s, if p = 0 (1 straceM (ξ)p) 1 p, if p ∈ (−∞, 1] \ {0}
A design ξ∗is called φ
p-optimal design if it maximizes the matrix means φp(M (ξ)) and the popular T-,D-,A-,
and E-criteria correspond to parameter valuse 1,0,-1,and −∞,respectively.
In the past, the φp-optimal designs for some general linear regression models without intercept have been
discussed. For example, Cheng (1987) has used the equivalence theorem to determine φp-optimal designs for
the no-intercept linear model over the n-dimensional unit cube [0, 1]n for −∞ ≤ p ≤ 1. Cheng(1995) has
further utilized the complete class under both Kiefer ordering and Schur ordering to extend these results to general permutation-invariant design regions. Draper and Pukelsheim (1999) has proved that the vertex points design is the unique optimal design under the Kiefer ordering in the first-degree mixture model. For the second-degree mixture model with two or three ingredients, complete class results under the Kiefer ordering are also derived there. Furthermore, Draper et al. (2000) extends their results to the second-degree mixture model with four or more ingredients. For four ingredients, the class is minimal complete. Later, these results are used to find the D-, A- and E-optimal design respectively by Klein (2004). In the current study, the first-degree model (2) has a similar structure with the one utilized by Cheng (1995). But, the experiment design regions studied here is not only permutation-invariant as in Cheng (1995) but also symmetric in the origin. Therefore, it is helpful to follow the Cheng’s (1995) approach. In Cheng’s first step, the permutation-invariant design is used to reduce the dimensionality of the information matrices so that only two distinct moments need to be investigated. In his second step, by fixing the largest eigenvalue, the maximum of the smallest value can be found since the reduced information matrices have at most two distinct eigenvalues and the set of these eigenvalues is convex. Therefore, the comparison of information matrices is reduced to a one-variable problem. In his third step, the special structure is used to determine the complete class and, as a result, the φp-optimal designs.
The paper is organized as follows: Section 2 discusses the exchangeable and reflected designs in the first-degree models (with intercept). Let ηi be the uniform design on the vertices of T (a) with i coordinates
equivalent to a or −a. It is shown that the complete class of designs under the Kiefer ordering for linear log contrast model is the vertex points design. Section 3 presents the results on φp-optimal design attained by ηj or a convex combination of ηj and ηj+1 with [k2] ≤ j ≤ (k − 1). Furthermore, it demonstrates how the φp-optimal designs of a linear log contrast model for experiments with mixtures are derived. Section 4 is the conclusion.
2. Preliminaries
2.1 Exchangeablility and reflection
Balancedness and symmetry properties are among the most useful and pleasing features that an experimental design can possess. According to the idea of Draper and Pukelsheim (1999), firstly, one can ask for invariant designs to improve balancedness. So we will introduce invariance we need to reduce the dimensionality of the information matrix. Let P erm(m) be the group of all m × m permutation matrices and Imis the m × m
identity matrix. We shall be interested in the following group acting on the experimental domain T (a):
Definition 2.1.1 A design τ is said to be R-invariant when
τR= τ for all R ∈ R,
where τR(t) = τ (R−1t) is the image of τ under R. A design with this invariant property is called an
exchangeable and reflected design.
A matrix R ∈ R leaves the experimental domain T (a) invariant, and commutes with the regression function
f according to f (Rt) = Ã 1 Rt ! = Ã 1 0 0 R !
t ≡ Qf (t) for all t ∈ T (a).
Therefore f is R − Q−equivariant relative to the k × k matrix group
Q = { Ã 1 0 0 R ! : R ∈ R}.
The information matrix of τRis obtained from the information matrix of τ by a congruence transformation M = QM Q0 for all Q ∈ Q,
and an invariant design possesses an invariant moment matrix. However, the converse is not true, see Lemma 2.1 of Draper et al. (1991). So the information matrix M of the first-order model is said to be permutationally and reflected invariant when
M (τR) = QM (τ )Q0.
Note that for an arbitrary design τ , a corresponding exchangeable and reflected design ¯τ can be obtained by
averaging over the group R,
¯ τ = 1 1 + m! X R∈R τR.
If ¯τ = τ , then τ itself is exchangeable and reflected. Moreover, the information matrix of τ is majorized
by the information matrix τ denoted as M (¯τ ) ≺ M (τ ). For a general discussion of invariant design and
majorization see Pukelsheim(1993).
2.2 Complete class
In the above section, it is easy to see that the set of all exchangeable and reflected designs is essentially complete for Kiefer ordering. Although this provides a substantial reduction of the problem, this set is still too large. Therefore, following Draper and Pukelsheim (1999), the second step is an improvement relative to the usual Loewner matrix ordering within the class of designs being exchangeable and reflected. From now on, without loss of generality, all the designs ¯τ are assumed to be exchangeable and reflected.
To begin with, note that the moment matrix of an exchangeable and reflected design ¯τ is a k × k matrix M (¯τ ) = Ã 1 0 0 u2Ik−1+ u11U2 ! with U2 = 1(k−1)10(k−1)− I(k−1) ∈ Sym(k − 1).
The moments are the averages over the corresponding, possibly distinct individual moments of ¯τ , u2 = u2(¯τ ) = Z T(a) t21d¯τ = · · · = Z T(a) t2k−1d¯τ = Z T(a) ( 1 k − 1 k−1 X i=1 t2i)d¯τ , u11 = u11(¯τ ) = Z T(a) t1t2d¯τ = · · · = Z T(a) tk−2tk−1d¯τ = Z T(a) ( 1 (k − 2)(k − 1) k−1 X i=1 X i6=j titj)d¯τ , and u11, u2 satisfy 0 ≤ u2≤ a2, u11≤ a2, 0 ≤ u2− u11≤ [k 2]a2(k − [k2] − 1) (k − 1)(k − 2) .
It turns out that much of our discussion is phrased in terms of certain simple subsets of T (a), which we now define. Definition 2.2.1 Let ηi= t ↔ ( i z }| { a, · · · , a, k−i−1 z }| { 0, · · · , 0), t ↔ ( i z }| { −a, · · · , −a, k−i−1 z }| { 0, · · · , 0) 1 2(k−1i ), 1 2(k−1i ) where t ↔ ( i z }| {
a, a, · · · , a, 0, · · · , 0) denotes the collection of all permutation of the i numbers a.
Note that ηi is the uniform measure on the vertices of T (a) with i coordinates equal to a or −a. The
design ηi is exchangeable and reflected, with moments u2(ηi) = ia
2 k−1, u11(ηi) = i(i−1)a2 (k−1)(k−2), and eigenvalues λ2(ηi) =(ia) 2 k−1, λ3(ηi) = ia2(k−i−1) (k−1)(k−2).
Now we reduce our discussion on the set
u2 u11 k=3 B A u2-u11=c u2 u11 k=5 D B A u2-u11=c
Figure 1. The graphs of the set B for k = 3 and 5. When k = 3, A, B respectively correspond to the design
η1 and η2. When k = 5, A, B, D correspond to η2, η3, η4.
In the following theorem, a class of designs which are convex combinations of the uniform designs ηi, [k
2] ≤
i ≤ k − 1, is proved to be complete under Kiefer ordering for first-order model.
Theorem 2.2.2 Let ¯τ be an exchangeable and reflected design on the experimental domain T (a) with second
order moments u11(¯τ ), u2(¯τ ). Then the design
η = k−1 X j=[k 2] αjηj with weights αj = a2(j+1)(j−k+2)+(k−1)(k−2)(u2(¯τ )−u11(¯τ )) (2j−k+2)a2 αj+1= 1 − αj if (j+1)a 2(k−j−2) (k−1)(k−2) ≤ u2(¯τ ) − u11(¯τ ) ≤ j(k−j−1)a2 (k−1)(k−2) αi = 0, i = [k 2], · · · , j − 1, j + 2, · · · , k − 1 (4.3) satisfies M (η) ≥ M (¯τ ).
3. φ
p-optimal designs
Based on the discussion in Draper and Pukelsheim (1999), stating that the complete class theorems is that any design not in a complete class can be improved upon by a design in a complete class here, we may focus on finding an optimal design in the class obtained above. Moreover, since the moment matrix of an exchangeable and reflected design ¯τ has a similar structure as that of Cheng (1995), similar arguments can
[k 2] ≤ j ≤ k − 1 by g(j) = 1 if j = k − 1 1 − ln(2j−k+22j+1 ) ln(j(k−2)k−j−1), if [ k 2] ≤ j ≤ k − 2, f (j) = 1 if j = k − 1, 1 − ln( 2j−1 2j−k) ln(j(k−2) k−j−1) , if [k 2] + 1 ≤ j ≤ k − 2, −∞ if j = [k 2]. Then −∞ = f ([k 2]) < g([k2]) < f (j) < g(j) < f (j + 1) < g(j + 1) < 1 = f (k − 1) = g(k − 1) for all [k 2] + 1 ≤ j ≤ k − 2. So {[f (j), g(j)]} k−1 j=[k 2] and {[g(j), f (j + 1)]} k−2 j=[k
2] together form a partition of [−∞, 1].
Based on the above partition, it shows that the φp-optimal designs are either ηj or convex combination of ηj
and ηj+1 for some j at the two various intervals respectively in the following theorem.
Theorem 3.1 Suppose j is an integer with [k
2] ≤ j ≤ k − 2.
1. If f (j) ≤ p ≤ g(j), the φp-optimal design for model (2) with the experimental domain T (a) is η(αφp) = ηj.
2. If j < k − 1 and g(j) < p < f (j + 1), the φp-optimal design for model (2) with the experimental domain T (a) is η(αφp) = αη j+ (1 − α)ηj+1, where α = (j + 1) 2(k − 2) − (j + 1)(k − j − 2)(2j−k+2 2j+1 ) 1 p−1 (1 + 2j)(k − 2) + (2j − k + 2)(2j−k+22j+1 )p−11 . (4.4)
In Table 1 the φp-optimal designs constructed from Theorem 3.1 are presented. From the table we discover
the φp-optimal designs have a very nice structure: as p increases from −∞ to 1, the optimal design shifts from η[k
2] to ηk−1. It can be seen that the α defined in (4) is a strictly decreasing function of p ∈ (g(j), f (j + 1))
and that as p moves from g(j) to f (j + 1), the α’s cover all of (0, 1). More explicitly, each ηj is φp-optimal for
the p-values in an interval (f (j), g(j)), while each proper convex combination αηj+ (1 − α)ηj+1is φp-optimal
for one single p. Corollary 3.2
1. Suppose k = 2m + 1, m ≥ 2 is an odd integer. If p ≤ 1 − ln k
ln(k−2), the φp-optimal design for model (2)
on the experimental domain T (a) is
η(αφp) = η
[k
2. Suppose k = 2m, m > 1 is an even integer. If p ≤ 1 − ln(k+1)−ln 2ln k , the φp-optimal design for model (2) on the experimental domain T (a) is
η(αφp) = η
[k
2].
Now let us return to the original linear log contrast model (1) and expressed the φp-optimal in the original
design space X (δ).
Theorem 3.3 Suppose j is an integer with [k2] ≤ j ≤ k − 2.
1. If f (j) ≤ p ≤ g(j), the φp-optimal design for linear log contrast model (1) on the experimental domain X (δ) is
ξ(αφp) = ξ
j
2. If j < k − 1 and g(j) < p < f (j + 1), the φp-optimal design for linear log contrast model (1) on the
experimental domain X (δ) is ξ(αφp) = αξ j+ (1 − α)ξj+1 where α = (j + 1) 2(k − 2) − (j + 1)(k − j − 2)(2j−k+2 2j+1 ) 1 p−1 (1 + 2j)(k − 2) + (2j − k + 2)(2j−k+22j+1 )p−11 .
The preceding theorem shows that when k is an even integer, the design ξ[k
2] is φp-optimal for p ∈ [−∞, 1 −
ln(k+1)−ln 2
ln k ]. While k is an odd integer, the design ξ[k
2] is φp-optimal for p ∈ [−∞, 1 −
ln k
ln(k−2)]. Therefore,
when k is an even integer, the design ξ[k
2] is simultaneously D- and A-optimal designs. However, ξ[k2] is
A-optimal design when k is an odd integer. Any convex combination ξ[k
2] and ξ[k2]+1 is D-optimal design.
4. Conclusion
According to the structure of the second order moments u2, u11and eigenvalues ( show as Figure 1), it is clear
that for a fixed value c which is equal to λ3(¯ξ) for a certain ¯ξ, the maximum value of λ2(¯ξ) occurs on a point
of the boundary which is a convex hull of {(u2(ηj), u11(ηj)), [k2] ≤ j ≤ k − 1} and a polytope on the vertices
(u2(ηj), u11(ηj)), [k2] ≤ j ≤ k − 1 as well. Obviously, the designs which second order moments correspond to
the part of the boundary form a complete class under Keifer ordering. In other words, each design in the complete class is a ηj for some j or a convex combination of two ηj and ηj+1 with [k2] ≤ j ≤ k − 1. Each ηj
is φp-optimal for the p-values in some interval, while each proper convex combination αηj+ (1 − α)ηj+1 is φp-optimal for one single p. It is because on the boundary of polytope, the slope changes only at the vertices, which correspond to the design ηj, [k
2] ≤ j ≤ k − 1.
As far as for the quadratic log contrast model
E(y) = β0+ k−1 X i=1 βilog(xi xk) + k−1 X i=1 k−1 X j=1 rijlog(xi xk) log( xj xk)
for x = (x1, x2, · · · , xk) ∈ X , we also have some numerical results for some of the φp-optimal designs, for
example, k = 3, a = 1, p = −0.3, −0.5, −1, −2, etc, where the analytic results will be investigated more explicitly in the future.
Table 1. φp-optimal design for model (2) with the experimental domain T (a),k = 3, 4, 5, 6, 7 k p η(αφp) α 1 α2 α3 α4 α5 α6 3 1 η2 1 0.8 α1η1+ α2η2 0.0163 0.9837 0 α1η1+ α2η2 23 13 -1 α1η1+ α2η2 0.8453 0.1547 -50 α1η1+ α2η2 0.9945 0.0054 4 1 η3 1 0.8 α2η2+ α3η3 0.0877 0.9123 0.5 α2η2+ α3η3 0.8 0.2 p ≤ 0.33906 η2 1 5 1 η4 1 0.8 α3η3+ α4η4 0.210 0.7899 0.7 α3η3+ α4η4 0.6708 0.3292 0.2675≤ p ≤ 0.61437 η3 1 0.1 α2η2+ α3η3 0.8249 0.1751 p ≤ -0.46497 η2 1 6 1 η5 1 0.8 α4η4+ α5η5 0.37501 0.62499 0.54816≤ p ≤ 0.7075 η4 1 0.5 α3η3+ α4η4 0.2857 0.7143 0.4 α3η3+ α4η4 0.7188 0.2812 p ≤ 0.30082 η3 1 7 1 η6 1 0.9 α5η5+ α6η6 0.0135 0.9865 0.8 α5η5+ α6η6 0.5757 0.4243 0.65869≤ p ≤ 0.75505 η5 1 0.6 α4η4+ α5η5 0.5128 0.4872 0.1549 ≤ p ≤ 0.52287 η4 1 0 α3η3+ α4η4 0.5714 0.4286 p ≤ -0.209 η3 1
References
[1] Aitchison, J. and Bacon-Shone, J. (1984). Log contrast models for experiments with mixtures.
Bio-metrika 71, 323-330.
[2] Chan, L. Y. (1988). Optimal design for a linear log contrast model for experiments with mixtures.
[3] Chan, L. Y. and Guan, Y. N. (2001). A- and D-optimal designs for a log contrast model for experi-ments with mixtures. Journal of Applied Statistics 28, 537-546.
[4] Cornell, J. A. (1990). Experiments with mixtures. Design, Models and Analysis of mixture data. 2nd ed. Wiley, New York.
[5] Cheng, C. S. (1987). An application of the Kiefer-Wolfowitz equivalence theorem to a problem in Hadamard transform optics. The Annals of Statistics 15, 1593-1603.
[6] Cheng, C. S. (1995). Complete class results for the moment matrices of designs over permutation-invariant sets. The Annals of Statistics 23, 41-54.
[7] Draper, N. R. Gaffke, N. and Pukelsheim, F. (1991). First and second order rotatability of experi-mental designs, moment matrices, and information surfaces. Metrika, 38, 129-161.
[8] Draper, N. R. and Pukelsheim, F. (1999). Kiefer order of simplex designs for second-degree mixture models with four or more ingredients. Journal of Statistical Planning and Inference, 79, 325-348. [9] Draper, N. R., Heiligers, B. and Pukelsheim, F. (2000). Kiefer order of simplex designs for firs- and
second-degree mixture models. The Annals of Statistics, Vol. 28, No. 2, 578-590.
[10] Klein, T. (2004) Optimal designs for second-degree Kronecker model mixture experiments. Journal
of Statistical Planning and Inference 123, 117-131.
Part 3. Exact D-optimal designs for quadratic Sheff´e model
1
Introduction
Consider a mixture experiment with q nonnegative components, where the proportions of components are subject to the simplex restriction Pqi=1xi = 1, xi ≥ 0. The q proportions can be expressed as a column
vector x = (x1, . . . , xq)0 in Sq−1 where
Sq−1= {(x1, . . . , xq)0∈ [0, 1]q : x1+ · · · + xq= 1, xi≥ 0 i = 1, . . . , q}.
An observation y(x) is obtained at x ∈ Sq−1 with E(y(x)) = β0f (x) and variance σ2 independent of x,
where f (x) is a known function and β is an unknown parameter vector. In Scheff´e’s quadratic model the
expectation is expressed as
E(y(x)) = β1x1+ · · · + βqxq+ β12x1x2+ · · · + β(q−1)qxq−1xq (1)
with regression function f (x) = (x1, . . . , xq, x12, . . . , x(q−1)q)0. An exact design with sample size N is a
probability measure on a design space which puts weight pi > 0 at n distinct support points, n ≤ N such
thatPni=1pi= 1 and N pi, i = 1, . . . , n are integers. An approximate design removes the integer restrictions
on N pi , i = 1, . . . , n. Denote a probability measure ξ for a mixture experiment as follows
ξ = Ã x1 · · · xn p1 · · · pn ! = x1,1 .. . x1,q · · · xn,1 .. . xn,q p1 · · · pn ,
where x1, . . . , xn denote the finite supports with the corresponding weights p1, . . . , pn. The information
matrix is therefore defined by
M (ξ) = n
X
i=1
ξ(xi)f (xi)f0(xi),
and the corresponding dispersion function is given by
d(x, ξ) = f0(x)M−1(ξ)f (x).
According to the Equivalence Theorem (Kiefer and Wolfowitz 1960) a design ξ∗ is called an approximate D-optimal designs if ξ∗ maximizes det(M (ξ)) over all feasible designs on the design space. Kiefer (1961)
showed that an approximate design for the Scheff´e’s quadratic model which assigns measure 2/q(q + 1) to
each point of the (q − 1, 2)-lattice. If an exact design ξ∗
N maximizes det(M (ξN)) over all feasible exact designs
on the design space, then it is called an exact N -points D-optimal design.
For a polynomial model on a closed interval [a, b], Salaevskii (1966) conjectured that an exact design which distributes the weights as even as possible on the support points of the approximate designs. The conjecture
of Salaevskii had been studied by Constantine and Studden (1981), Gaffke and Krafft (1982), Gaffke (1987), Huang (1987) and had verified the Salaevskii conjecture holds for most of the cases. Chang and Chen (2004) discussed the exact D-optimal design problem for multivariate linear polynomial models on a simplex, parallelgram and quadratic polynomial models with or without intercept on the q-ball for some cases, and provided some numerical results.
In this work, we investigate the exact D-optimal design for mixture experiments in Scheff´e’s quadratic models
based on results in Kiefer (1961) and provide some results for models with two and three ingredients and numerical verifications for models with ingredients between four and nine.
2
Preliminaries
Kiefer (1961) proved that an approximate D-optimal design ξ∗ on a simplex is supported equally with
weights p = 2/q(q + 1) on the (q − 1, 2)-lattice with q(q + 1)/2 points, where x∗
i, i = 1, . . . , q are the vertexes
and x∗
ij, 1 ≤ i < j ≤ q are the centers of the sides, i.e.
ξ∗= 1 0 .. . 0 0 1 .. . 0 . . . 0 0 .. . 1 1/2 1/2 .. . 0 1/2 0 1/2 .. . . . . 0 .. . 1/2 1/2 p p . . . p p p . . . p .
For the (q − 1, 2)-lattice, Kiefer (1961) provided a q(q + 1)/2 system of quadratic orthonormal polynomials with respect to the q(q +1)/2 support points in ξ∗such that each of which vanishes at all other support points
expect at one point of the lattice. More explicitly the system consists of the functions [2q(q + 1)]12xi(xi−1
2),
1 ≤ i ≤ q, and [8q(q + 1)]12xixj, 1 ≤ i < j ≤ q, these functions are very useful in expressing the corresponding
dispersion function for design supports on the lattice points.
Now let g(x) denote the vector of the orthonormal polynomials mentioned above where
g(x) = g1(x) g2(x) .. . gq(x) g12(x) .. . g(q−1)q(x) = 2x1(x1−12) 2x2(x2−12) .. . 2xq(xq−12) 4x1x2 .. . 4xq−1xq .
For designs supported on S∗= {x∗
1, . . . , x∗q, x∗12, . . . , x∗(q−1)q}, we may transform the regression function f (x)
in the Scheff´e’s quadratic model into g(x). More explicitly g(x) may be expressed as g(x) = F−1f (x), x ∈ Sq−1
where F = (f (x∗
Furthermore by the celebrated Kiefer-Wolfowitz Equivalence Theorem it can be shown q X i=1 g2 i(x) + X i≤j g2 ij(x) ≤ 1, ∀x ∈ Sq−1.
Similarly as in the Salaevskii conjecture, if the total number of trials is N = kp + t where k = q(q + 1)/2,
p, t ∈ N, t < k, there are ( k
t ) different cnadidate exact designs ξ ∗
N,T with |T | = t, T ⊂ T, originated from ξ∗ and T = {1, . . . , k, 12, . . . , (q − 1)q} is the index set. That is
ξ∗ N,T = ( x∗ 1 . . . x∗q x∗12 . . . x∗(q−1)q n1/N . . . nq/N nq+1/N . . . nk/N ) (2)
where n`= p + 1 if ` ∈ T or n`= p if ` 6∈ T . Then the information matrix of ξN,T∗ can be expressed as M (ξ∗ N,T) = X i∈T pif (xi)f0(x i) = X i∈T piF g(xi)g0(x i)F0 = F p1 · · · 0 .. . . .. ... 0 · · · p(q−1)q F0
and the inverse matrix of M (ξ∗ N,T) is M−1(ξN,T∗ ) = (F−1)0 p−11 · · · 0 .. . . .. ... 0 · · · p−1(q−1)q F−1.
Hence, the dispersion function of ξ∗
N,T is rewritten as d(x, ξ∗ N,T) = f0(x)M−1(ξN,T∗ )f (x) = g0(x)F0(F−1)0 p−1 1 · · · 0 .. . . .. ... 0 · · · p−1(q−1)q F−1F g(x) = X `∈T g2 `(x) p` , where p`= (p + 1)/N if ` ∈ T and p`= p/N , ` /∈ T .
Gaffke and Krafft (1982) proved the exact D-optimality of the above candidate designs for quadratic regression on [a, b], based on the geometric-arithmetic means inequality of the information matrix i.e.
det(M (ξ1)) det(M (ξ2))≤ · 1 kTr(M −1(ξ 2)M (ξ1)) ¸k , (3)
where ξ1and ξ2 are two arbitrary designs defined on [a, b]. Note that
Tr(M−1(ξ2)M (ξ1)) = 1 N N X i=1 f0(xi)M−1(ξ2)f (xi) = 1 N N X i=1 d(xi, ξ2),
where xi, i = 1, . . . , N are the corresponding design points of ξ1.
Gaffke (1987) and Huang (1987) independently verified Salaevskii’s conjecture for the most of the cases of polynomial regression. In the following we provide some lemmas analogous to that in Huang (1987) which will be useful to prove the main results.
Lemma 1. There exists p0∈ N such that for all p ≥ p0
X `∈T g`2(x) ≤ p0 p0+ 1 + R2(x) p0+ 1 ≤ p p + 1+ R2(x) p + 1, (4) where R2(x) = max{g2 `(x), ` ∈ T}, ∀x ∈ Sq−1. (5)
Note that (4) can be rewritten in another form as 1 − R2(x)
1 −Pk`∈Tg2
`(x)
≤ p0+ 1. (6)
The proofs of Lemma 1 for q = 2 and 3 are deferred to Section 3. Let
A`= {x|R2(x) = g2
`(x), x ∈ Sq−1}, ` ∈ T, (7)
then we define the region AT used in Lemma 2 where AT = [
`∈T
A`, T ⊂ T.
Lemma 2. For all p ≥ p0, p ∈ N, let ξ∗N,T be as defined in (2) with T ⊂ T, N = kp + t, then
1 Nd(x, ξ ∗ N,T) ≤ 1 p + 1, ∀x ∈ AT, 1 Nd(x, ξ ∗ N,T) ≤ 1 p, ∀x /∈ AT.
Conjecture 1. For N = kp + t ≥ kp0, where 1 ≤ t ≤ k − 1, and p0 is defined as in (4), each of the ξN,T∗ designs is exact D-optimal.
The key steps of proving Conjecture 1 is based on the fact that if conditions in Lemmas 1 and 2 hold, then for any exact design with N trials supported on {x1, . . . , xN}, there is T ⊂ T such that n1≥ t(p + 1) where
n1 is the amount of trials in AT, 1 ≤ n1≤ N , and for design ξN,T∗ ,
1 N N X i=1 d(xi, ξ∗N,T) ≤ n1 p + 1+ N − n1 p = N (p + 1) − n1 p(p + 1) ≤ N (p + 1) − t(p + 1) p(p + 1) = k.
This implies that det(M (ξ)) ≤ det(M (ξ∗
N,T)) by (3).
Hence, Lemma 2 and Conjecture 1 will hold for Scheff´e’s quadratic models if p0 defined in Lemma 1 can be
In the following we find the exact D-optimal designs for Scheff´e’s quadratic models with two or three
ingre-dients. We will show in two steps. First we will identify the maximum functions R(x) in the design region and use it to partition Sq−1 into several regions according to which g
`(x) is the maximum function. Then
we can find p0 defined in Lemma 1 such that inequality (1) holds. Later, we find exact D-optimal designs
for some sample sizes.
3
Two and three ingredients
In this section we would like to find the minimum sample size such that Conjecture 1 holds for two and three ingredients. Lemma 1 is the main tool to derive the results.
3.1
Two ingredients
We start from experiments with two ingredients that q = 2. Then the model is
E[y(x)] = β1x1+ β2x2+ β12x1x2, x ∈ S1.
For this model, the approximate D-optimal design ξ∗ is given by
ξ∗= Ã 1 0 ! Ã 0 1 ! Ã 1/2 1/2 ! 1/3 1/3 1/3
and the quadratic orthonormal polynomials with respect to ξ∗ are g2
1(x), g22(x) and g122 (x) as defined in
Section 2.
For q = 2, we find R2(x) as defined in (5) and the regions A
` as defined in (7) with respect to g12(x), g22(x)
and g2
12(x). Then Theorem 1 holds for the special case with q = 2 as below.
Theorem 1. The design ξ∗
N,T as defined in (2) is an exact D-optimal design in Scheff´e’s quadratic models with two ingredients where T ⊂ T = {1, 2, 12} if N ≥ 3.
Proof. Firstly, we identify the sets A`, ` ∈ T as defined in Section 2. Let g12(x) ≥ g22(x) and g12(x) ≥ g122 (x)
then we have A1= {x|56 ≤ x1 ≤ 1, x ∈ S1}. Similarly, it can be found that A2 = {x|0 ≤ x1 ≤ 16, x ∈ S1}
and A12= {x|16 ≤ x1≤ 56, x ∈ S1}. Let L(x) = (1 − R2(x))/(1 −
P
`∈Tg2`(x)) and L0(x) and L00(x) denote
the first and second order derivative of L(x). Note that in A1, L(x) = (1 − g12(x))/(1 −
P
`∈Tg2`(x)). Since x2= 1 − x1, then for x ∈ A1we have
L(x) = 1 + x1+ 4x 3 1 6(1 − 2x1)2x1, and L0(x) = 1 − 6x1− 4x21− 8x31 6x2 1(−1 + 2x1)3 .
It is easy to see that L0(x) < 0 by x ∈ A
1, then L(x) is monotone decreasing in A1 and the maximum of
L(x) is 28/15 ≈ 1.8667 on x1= 5/6. Since L(x) is symmetric with x1= x2, the maximum of L(x) in A2 is
also 28/15. Similarly for all x ∈ A12, L(x) = (1 − g122 (x))/(1 −
Pk `∈Tg`2(x)), then L(x) =1 + x1− 4x21 6x1− 6x21 , and L00(x) = −1 + 3x1− 3x 2 1 3(−1 + x1)3x31 .
Also we find that L00(x) > 0, ∀x ∈ A
12, then L(x) is convex in A12 and the maximum of L(x) is 28/15 on
x1= 1/6 and x1= 5/6. From (6) we have p0= 1 and by Lemma 2, Conjecture 1 holds for Scheff´e’s quadratic
models with two ingredients if the total trials N ≥ kp0= 3. ¤
3.2
Three ingredients
Now we discuss the case with ingredients q = 3, the Scheff´e’s model turns to
E[y(x)] = β1x1+ β2x2+ β3x3+ β12x1x2+ β13x1x3+ β23x2x3, x ∈ S2.
Then the approximate D-optimal design is given by
ξ∗= 1 0 0 0 1 0 0 0 1 1 2 1 2 0 1 2 0 1 2 0 1 2 1 2 1/6 1/6 1/6 1/6 1/6 1/6 ,
and the corresponding orthonormal polynomials are {g`(x), ` ∈ T} where g`(x), ` ∈ T are defined as in
Section 2 and T = {1, 2, 3, 12, 13, 23}. Similarly, we must identify the regions of A`for every g`2(x), ` ∈ T. In
order to find each A`, ` ∈ T, we first define the following sets A∗i and A∗ij; for i = 1, . . . , k, A∗ i = © x|x ∈ Sq−1, g2 i(x) ≥ g2j(x), 1 ≤ j ≤ q, j 6= i ª
where gi(x) is the corresponding orthonormal polynomial with gi(xi∗) = 1, and for 1 ≤ i < j ≤ k,
A∗
ij= {x|x ∈ Sq−1, gij2(x) ≥ g2kl(x), g2ij(x) ≥ g2i(x), gij2(x) ≥ gj2(x), 1 ≤ k < l ≤ q, kl 6= ij}
where gij(x) is the corresponding orthonormal polynomial with gij(xij∗) = 1.
For example, we have A∗
1 and A∗12 being the sets that
A∗
1= {x|x ∈ S2, g21(x) ≥ g212(x), g21(x) ≥ g213(x)},
A∗
12= {x|x ∈ S2, g122 (x) ≥ g21(x), g122 (x) ≥ g22(x),
