Model with interaction

Remark 4.4.8. Actually, condition (4.45) is obviously fulfilled for ν = 2 (compare Theorem 4.4.3). For the case ν = 3 the bound of l.h.s. of condition (4.45) is 1 and, hence always fulfilled.

to a model with intercept by removing the interaction term x₁x₂. As it was pointed out in Remark4.4.5the functionf_β(x) is invariant w.r.t. simultaneously scaling of x, i.e., f_β(λx) =f_β(x). Let λ= 1/(x₁x₂) then we obtain

f_β(x) =β₁x₁+β₂x₂+β₃x₁x₂⁻¹











x₁ x₂ x1x2











(4.48)

=β₁t₂+β₂t₁+β₃⁻¹











t₂ t₁ 1











=f^◦_β(t) (4.49)

where t = t1, t2

>

, tj = 1/x_j, j = 1,2. The range of t = t(x), as x ranges over X = [a, b]² is a cube given by T = ^h(1/b), (1/a)ⁱ² with f^◦(t) = (t₂, t₁,1)^>. One can rearrange the terms of (4.49) by making use of the 3×3 anti-diagonal transformation matrixQ. So we havef˜(t) = Qf^◦(t) = (1, t₁, t₂)^> and ˜β =Q^>−1β =β₃, β₂, β₁^>. Hence, f˜β˜(t) = f˜^>(t) ˜β⁻¹f˜(t) and rewrites as

f˜β˜(t) = β₃+β₂t₁+β₁t₂⁻¹











1 t1

t₂











, t ∈^h(1/b), (1/a)ⁱ². (4.50)

Since (4.50) coincides with that for a gamma model with intercept the D-criterion is equivariant (see Radloff and Schwabe (2016)) with respect to a one-to-one transforma-tion fromT =^h(1/b), (1/a)ⁱ² to Z = [0,1]² where

t_j →z_j = 1

(1/a)−(1/b)t_j− 1/b

(1/a)−(1/b), j = 1,2. (4.51) For a given transformation matrix

B=













1 0 0

−(1/b) (1/a)−(1/b)

1

(1/a)−(1/b) 0

−(1/b)

(1/a)−(1/b) 0 (1/a)−(1/b)¹













with B⁻¹ =













1 0 0

1 b

1

a− ¹_b 0

1

b 0 _a¹ −¹_b













we have f˜(z) =Bf˜(t) = (1, z₁, z2)^> withβ˜˜=B^>−1β˜ = (β˜˜0, β˜˜1, β˜˜2)^> where

˜˜

β0 = β3 + (1/b)(β₁ +β2) , β˜˜1 = β2((1/a)−(1/b)) and β˜˜2 = β1((1/a)−(1/b)). It follows thatf˜β˜˜(z) =f˜^>(z)β˜˜⁻¹f˜(z) which rewrites as

f˜β˜˜(z) = β˜˜₀+β˜˜₁z₁+ β˜˜₂z₂)⁻¹











1 z₁ z2











,z∈[0,1]². (4.52)

Let M(x,β) = f_β(x)f^>_β(x), M˜(t,β) =˜ f˜β˜(t)f˜^>β˜(t) and M˜˜(z,β) =˜˜ f˜β˜˜(z)f˜^>β^˜˜(z) be the information matrices for the models which corresponding to (4.48), (4.50) and (4.52), respectively. It is easy to observe that

M(x,β) = Q⁻¹M˜(t,β)Q˜ ⁻¹ =B⁻¹Q⁻¹M˜˜(z,β)Q˜˜ ⁻¹B⁻¹,

thus the derived D-optimal designs on X, T and Z, respectively are equivariant. Ac-cording to the mapping of x to t in the line following (4.49) and the mapping from t toz in (4.51) each component is mapped separately: x_j →t_j →z_j without permuting them. Therefore, one modifies the direct one-to-one transformationg :X → Z where

xj →zj = 1/x_j

(1/a)−(1/b) − 1/b

(1/a)−(1/b), j = 1,2. (4.53) Letξ_g^∗ be a design defined onZ that assigns the weights ξ(x) to the mapped support points g(x), x ∈ supp(ξ^∗). In fact, ξ^∗ on X is locally D-optimal (at β) if and only if ξ_g^∗ onZ is locally D-optimal (at β). Note that the function˜˜ f˜β˜˜(z) is injective and thus the optimal design on Z is only supported by the vertices (cp. Theorem 4.2.1). It is worth noting by transformation (4.53) we obtain

(b, b)^>→(0,0)^>, (b, a)^> →(1,0)^>, (a, b)^>→(0,1)^>, (a, a)^>→(1,1)^>.

Corollary 4.5.2. Consider f(x) = x₁, x₂, x₁x₂^> on X = [a, b]², 0 < a < b. De-note the vertices by v₁ = (b, b)^>, v₂ = (b, a)^>, v₃ = (a, b)^>, v₄ = (a, a)^>. Let β= (β₁, β₂, β₃)^> be a parameter point satisfying (4.4). Then the unique locally D-optimal design ξ^∗ (at β) is as follows.

(i) If β₃²+_b¹₂(β₁²+β₂²) + (_b¹₂ −_a¹₂ +_{a b}² )β₁β₂+²_bβ₃(β₁+β₂)≤0 then ξ^∗ assigns equal weights 1/3 to v₁,v₂,v₃.

(ii) If β₃²+_b¹2β₁²+_a¹2β₂²+²_bβ₃β₁+²_aβ₃β₂+ (_b¹2 +_a¹2)β₁β₂ ≤0 then ξ^∗ assigns equal weights 1/3 to v₁,v₂,v₄.

(iii) If β₃²+_b¹2β₂²+_a¹2β₁²+²_bβ₃β₂+²_aβ₃β₁+ (_b¹2 +_a¹2)β₁β₂ ≤0 then ξ^∗ assigns equal weights 1/3 to v₁,v₃,v₄.

(iv) If β₃²+_a¹2(β₁²+β₂²) + (_a¹2−_b¹2 +_ab²)β₁β₂+_a²β₃(β₁+β₂)≤0 then ξ^∗ assigns equal weights 1/3 to v₂,v₃,v₄.

(v) If none of the cases (i) – (iv) applies then ξ^∗ is supported by the four vertices

ξ^∗ =





v₁ v₂ v₃ v₄ ω₁^∗ ω₂^∗ ω₃^∗ ω^∗₄



, where ω^∗<sub>`</sub> >0 (1≤`≤4), ^P4_{`=1</sub>ω<sub>`}^∗ = 1.

Proof. The proof is obtained by verifying the D-optimality ofξ_g^∗ on Z = [0,1]² under transformationg given by (4.53). The regression vectorf˜β˜˜(z) given by (4.52) coincides with that for the two-factor gamma model with intercept onZ = [0,1]² whose intensity function is defined asuβ˜˜(z) = (β˜˜0+β˜˜1z1+ β˜˜2z2)⁻² for all z∈ Z. Denote

c₁ =uβ˜˜((0,0)^>) = β˜˜₀⁻² = (β₃+ 1

b(β₁+β₂))⁻², c₂ =uβ˜˜((1,0)^>) = (β˜˜₀+β˜˜₁)⁻² = (β₃ +β₁1

b +β₂1 a)⁻², c₃ =uβ˜˜((0,1)^>) = (β˜˜₀+β˜˜₂)⁻² = (β₃ +β₁1

a +β₂1 b)⁻², c₄ =uβ˜˜((1,1)^>) = (β˜˜₀+β˜˜₁+β˜˜₂)⁻² = (β₃+ 1

a(β₁+β₂))⁻².

Leth, i, j, k ∈ {1,2,3,4} are pairwise distinct such that c_k = min{c₁, c₂, c₃, c₄} then it follows from Theorem 3.3.1 that if c⁻¹_k ≥ c⁻¹_h +c⁻¹_i +c⁻¹_j then ξ^∗ is a three-point design supported by the three vertices v_h, v_i, v_j, with equal weights 1/3. Hence, straightforward computations show that the condition in case (i) of the corollary is equivalent toc⁻¹₄ ≥c⁻¹₁ +c⁻¹₂ +c⁻¹₃ . Analogous verifying is obtained for the cases (ii), (iii), (iv). For case (v) the four-point design according to Theorem 3.3.1 is locally D-optimal if c⁻¹_k < c⁻¹_h +c⁻¹_i +c⁻¹_j which applies implicitly if non of the conditions (i) – (iv) of saturated designs is fulfilled by a given β (cp. Remark 2.2.4).

It is noted that the optimality conditions provided in parts (i)–(iv) of Corollary 4.5.2 depend on the values of a and b. Changing these values might affect the D-optimality of a design. To see that, more specifically, let a= 1 and b = 2, i.e., the experimental region is X = [1,2]² and define γ₁ = β₁/β₃ and γ₂ =β₂/β₃, β₃ 6= 0.

Here, the parameter space which is depicted in Panel (a) of Figure4.7is characterized by γ₂ +γ₁ > −1, 2γ₂ +γ₁ > −2 and γ₂ + 2γ₁ > −2. It is observed from Panel (a) of Figure 4.7 that the design given by part (i) of Corollary 4.5.2 is not locally D-optimal because the corresponding optimality condition in part (i) of the corollary;

1

4(γ₁²+γ₂²) + ¹₄γ₁γ₂+γ₁+γ₂ ≤ −1 can not be satisfied.

Let us consider another experimental region with a higher length by fixinga= 1 and takingb= 4, i.e., the experimental region isX = [1,4]². The parameter space which is

depicted in Panel (b) of Figure4.7is characterized byγ₂+γ₁ >−1, 4γ₂+γ₁ >−4 and γ₂+ 4γ₁ >−4. In this case all designs given by Corollary4.5.2 are locally D-optimal at particular values of γ₂ and γ₁ as it is observed from the figure. It is obvious that along the diagonal dashed line, γ2 = γ1, there exist at most three different types of locally D-optimal designs.

(a) (b)

Figure 4.7: Dependence of locally D-optimal designs on γ₁=β₁/β₃ and γ2=β2/β3 where for Panel (a) X = [1,2]² and for Panel (b) X = [1,4]². The diagonal dashed line represents the caseγ₂=γ₁. Note that supp(ξ_ijk^∗ ) ={v_i,v_j,v_k} ⊂ {v₁,v₂,v₃,v₄}and

supp(ξ^∗₁₂₃₄) ={v₁,v2,v3,v4}.

.

For arbitrary values of a and b, 0 < a < b let us restrict to case γ₂ =γ₁ = γ, i.e., β₁ =β₂ =β, β₃ 6= 0 and the next corollary is immediate.

Corollary 4.5.3. Consider f(x) = (x₁, x₂, x₁x₂)^> on an arbitrary square X = [a, b]², 0< a < bin the positive quadrant. Letβ = (β₁, β₂, β₃)^>be a parameter point satisfying (4.4) with β₁ =β₂ =β and β₃ 6= 0. Define γ = _β^β

3. Then the locally D-optimal design ξ^∗ (at β) is as follows.

(i) If −^a₂ < γ ≤ −_3b−a^ab , then ξ^∗ assigns equal weights 1/3 to v2,v3, v4.

(ii) If b−3a >0 and γ ≥ _b−3a^ab , then ξ^∗ assigns equal weights1/3 to v₁,v₂,v₃. (iii) Ifb−3a >0and−_3b−a^ab < γ < _b−3a^ab then the designξ^∗ is supported byv₁,v₂,v₃,v₄

with optimal weights given by

ω₁^∗ = ab−(a−3b)γ

4b(a+ 2γ) , ω₂^∗ =ω^∗₃ =

ab+ (a+b)γ²

4ab(b+ 2γ)(a+ 2γ), ω₄^∗ = ab−(b−3a)γ 4a(b+ 2γ) .

Proof. Consider the experimental region X = [a, b]², 0 < a < b. By assumption β₁ =β₂ =β, β₃ 6= 0 the range ofγ = _β^β

3 is given by (−a/2,∞). Assumptionb−3a >0 implies that −^a₂ <−_3b−a^ab < _b−3a^ab . According to Corollary 4.5.2 we show the following under the assumptions of Corollary 4.5.3. Both conditions provided in parts (ii) and (iii) of Corollary 4.5.2are not fulfilled by any parameter point thus the corresponding designs are not D-optimal. In contrast, the designξ^∗ given in (i) of Corollary 4.5.3 is locally D-optimal if the condition provided in part (iv) of Corollary 4.5.2 holds true.

That condition is equivalent to

(3b²+ 2ab−a²)γ²+ 4ab²γ+a²b² ≤0.

The l.h.s. of above inequality is polynomial inγ of degree 2 and thus the inequality is fulfilled by −^a₂ < γ ≤ −_3b−a^ab .

Similarly, the design ξ^∗ in (ii) of Corollary 4.5.3 is locally D-optimal if the condition provided in part (i) of Corollary 4.5.2 holds true. That condition is equivalent to

(3a²+ 2ab−b²)γ²+ 4a²bγ+a²b² ≤0.

The l.h.s. of above inequality is polynomial inγ of degree 2 and thus the inequality is fulfilled by γ ≥ _b−3a^ab if b−3a >0.

The four-point design given in (iii) has positive weights on −_3b−a^ab < γ < _b−3a^ab if b−3a >0 and hence it is locally D-optimal in view of Remark 2.2.4.

Remark 4.5.1. One may note that from Corollary 4.5.3 when β = 0 the uniform design on the vertices v₁,v₂,v₃,v₄ is locally D-optimal.

4.5.3 Model of complete product-type interactions

Schwabe (1996b) developed an approach to construct optimal designs for linear models of complete product-type interactions by making use of optimal designs under marginal models. This approach is independent of the actual structure of the influence of the single factors and, hence, covers models with both qualitative and quantitative factors as well as purely qualitative or purely quantitative models (see Schwabe (1996b), p.35).

In this subsection we extend that approach for gamma models of complete product-type interactions. We will show that locally optimal designs for gamma models of complete product-type interactions can be obtained from locally optimal designs under the marginal counterparts.

We consider K marginal models each is containing ν_k factors, k = 1, . . . , K. The marginal ν_k-factor model is defined with a power link as in (4.2) where

µ^ρ_k(x_k) = f^(k)>(x_k)β^(k), x_k = (x_k1, . . . , x_kνk)^> ∈ X_k ⊆R^νk (4.54) with exponentρ∈R, ρ >0. So all marginal gamma models are having equal exponent ρ, or, equivalently, all link functions of the marginal models are the same. The positivity assumption (4.4) of the expected meanµ_k is to be satisfied, i.e., f^(k)>(x_k)β^(k)>0 for allx_k= (x_k1, . . . , x_kνk)^>∈ X_k where

f^(k):X_k →R^pk, β^(k)= (β₁^(k), . . . , β_p^(k)

k )^> ∈R^pk (1≤k ≤K).

For each k, the marginal model (4.54) has intensity function

u_k(x_k,β^(k)) =f^(k)>(x_k)β^(k)−2, x_k ∈ X_k (1≤k ≤K). (4.55) Letνdenotes the total number of factors in all marginal models, i.e., ν=^PK_k=1ν_k. The resulting ν-factor gamma model of complete product-type interactions with exponent ρis thus defined by

µ^ρ(x) =f^>(x)β

=

p1

X

i1=1

· · ·

pK

X

iK=1

f_i⁽¹⁾₁ (x₁)·. . .·f_i^(K)

K (x_K)β_i1,...,iK (4.56) where x = (x^>₁, . . . ,x^>_K)^> ∈ X = X₁× · · · × X_K, in which f(x) collects all K−fold products of the componentsf_i^(k)(x_k) which belong to the regression functionsf^(k)(x_k), k = 1, . . . , K and x = (x₁₁, . . . , x_1ν1, . . . , x_K1, . . . , x_KνK)^> is a ν-tuple. The un-known parameter β_i1,...,iK is equal to ^QK_k=1β_i^(k)_k and β is a p-dimensional parameter vector, i.e., β ∈ R^p where p = ^QK_k=1p_k. Note that β collects the parameters β_i1,...,iK, i_k= 1, . . . , p_k, k = 1, . . . , K and in lexicographic order β rewrites as

β= (β_1,...,1,1, β_1,...,1,2, . . . , β_1,...,1,pK, β_1,...,2,1, . . . , β_{p1,...,pK−1,pK})^>.

Note that f(x) : X →R^p and can be described by Kronecker products “⊗” as in the following;

f(x) = f⁽¹⁾(x₁)⊗ · · · ⊗f^(K)(x_K) =

K

O

k=1

f^(k)(x_k), (4.57) with β =β⁽¹⁾⊗ · · · ⊗β^(K)=

K

O

k=1

β^(k). (4.58)

Therefore, model (4.56) rewrites as µ^ρ(x) =

K

O

k=1

f^(k)(x_k)

> K

O

k=1

β^(k)

(4.59) and of coursef^>(x)β>0 for all x∈ X. The latter positivity assumption is obtained from that in the marginal models; that is because

f^>(x)β =

K

O

k=1

f^(k)(x_k)

> K

O

k=1

β^(k)

=

K

Y

k=1

f^(k)>(x_k)β^(k) >0.

The intensity function u(x,β) for model (4.56) is determined by the product of the intensity functions (4.55) in the marginal ν_k-factor models (4.54).

Lemma 4.5.1. The intensity function for model (4.56) is given by u(x,β) =

K

Y

k=1

u_k(x_k,β^(k)).

Proof. In general, the intensity function of gamma models with linear predictorf^>(x)β is defined byu(x,β) = (f^>(x)β)⁻². In view of (4.57) and (4.58) it follows that

u(x,β) =



 K

O

k=1

f^(k)(x_k)

> K

O

k=1

β^(k)





−2

=





K

O

k=1

f^(k)>(x_k)β^(k)





−2

=

K

O

k=1

f^(k)>(x_k)β^(k)

−2

=

K

Y

k=1

u_k(x_k,β^(k)).

Our aim is deriving an optimal design for model (4.56) as a product type design which is supported by the cross-product of the finite sets of design points of the designs under marginalν_k-factor models and the weights are given by the product of the weights of those designs. To be more specific, denote byξ_ka design defined onX_kfor a marginal ν_k-factor model (4.54) (1≤k ≤K). We introduce ξ_k as in (2.9);

ξ_k =





x_k1 x_k2 . . . x_krk ωk1 ωk2 . . . ωkr_k



, (4.60)

where ξ_k has r_k design points x_kj and corresponding weights ω_kj,j = 1, . . . , r_k. Then the product type design ξ = ^NK_k=1ξ_k is defined on X = X₁ × · · · × X_K and has r = ^QK_k=1r_k design points x_i1,...,iK = (x_1i1, . . . , x_KiK)^> with corresponding weights ωi1,...,i_K =^QK_k=1ωki_k,ik= 1, . . . , r_k, k= 1, . . . , K.

In order to reduce the problem of locally optimal designs at a given β for model (4.56) to the locally optimal designs for the marginalν_k-factor models at a givenβ_k it is required to use a factorized information matrix. The information matrix and, hence, the variance-covariance matrix of a product type design ξ factorizes into its marginal counterparts as it is given by the next lemma.

Lemma 4.5.2. The information matrix of a product type design ξ = ^NK

k=1

ξ_k for a gamma model of complete product-type interactions (4.56) is

Mξ,β=

K

O

k=1

M_kξ_k,β^(k)

Proof. In general, the information matrix ofξ for the model (4.56) is given by

M(ξ,β) =^R_X u(x,β)f(x)f^>(x)ξ(dx). In view of Lemma4.5.1with (4.57) and (4.58) it follows that

Mξ,β=

Z

Xu(x,β)

K

O

k=1

f^(k)(x_k)

K

O

k=1

f^(k)(x_k)

> K

O

k=1

ξ_k(dx_k)

=

Z

X₁· · ·

Z

X_K

K

Y

k=1

u_k(x_k,β^(k))

K

O

k=1

f^(k)(x_k)f^(k)>(x_k)

K

Y

k=1

ξ_k(dx_k)

=

K

O

k=1

Z

X_ku_k(x_k,β^(k))f^(k)(x_k)f^(k)>(x_k) ξ_k(dx_k)

=

K

O

k=1

Mk

ξk,β^(k).

As a consequence of Lemma 4.5.2 we get

detM(ξ,β)= det

K

O

k=1

M_kξ_k,β^(k)

=

K

Y

k=1

det

M_kξ_k,β^(k)

K

Q

j=1 j6=k

pj

. For example; letξ =ξ₁⊗ξ₂⊗ξ₃ then

det(M(ξ,β)) = det(M₁(ξ₁,β⁽¹⁾))^p2p3det(M₂(ξ₂,β⁽²⁾))^p1p3det(M₃(ξ₃,β⁽³⁾))^p1p2. Therefore, for a given parameter pointβ that is evaluated from the parameter points of the marginal models, i.e.,β = ^NK

k=1

β^(k) the best product type designξ^∗ = ^NK

k=1

ξ_k^∗ with respect to the D-criterion is generated from the locally D-optimal designsξ_k^∗ atβ^(k) for the marginalν_k-factor models (1≤k ≤K).

Theorem 4.5.1. Let ξ_k^∗ be a locally D-optimal design (atβ^(k))for a marginalν_k-factor model (4.54) on the experimental region X_k (1 ≤ k ≤ K). Then ξ^∗ = ^NK

k=1

ξ_k^∗ is a

locally D-optimal design (at β = ^NK

k=1

β^(k)) for model (4.56) on the experimental region X =×^K_k=1X_k.

Proof. The proof is obtained by making use of condition (2.11) of The Equivalence Theorem (Theorem2.2.2). To this end, denotef^(k)_β(k)(x_k) =

q

u_k(x_k,β^(k))f^(k)(x_k) and f_β(x) = ^qu(x,β)f(x). Since ξ_k^∗ is locally D-optimal (at β^(k)) we guarantee that f^(k)>

β^(k)(x_k)M⁻¹_k (ξ_k^∗,β^(k))f^(k)

β^(k)(x_k) ≤ p_k for all x_k ∈ X_k. Thus in view of Lemma 4.5.2 with (4.57) and (4.58) we obtain

f^>_β(x)M⁻¹(ξ^∗,β)f_β(x) =u(x,β)

K

O

k=1

f^(k)(x_k)

> K

O

k=1

M_kξ_k,β^(k)

−1 K

O

k=1

f^(k)(x_k)

=

K

O

k=1

u_k(x_k,β^(k))f^(k)>(x_k)M⁻¹_k ξ_k^∗,β^(k)f^(k)(x_k)

=

K

Y

k=1

f^(k)>

β^(k)(x_k)M⁻¹_k (ξ_k^∗,β^(k))f^(k)

β^(k)(x_k)≤

K

Y

k=1

p_k =p

for all x ∈ X. The Equivalence Theorem, thus, proves the local D-optimality of the product designξ^∗ = ^NK

k=1

ξ_k^∗ at a parameter point β= ^NK

k=1

β^(k).

In the following we focus on the local A-optimality. From Lemma 4.5.2 we obtain the next straightforward factorization for every product type designξ = ^NK

k=1

ξk.

trM⁻¹(ξ,β)= tr



 K

O

k=1

M_kξ_k,β^(k)

−1

=

K

Y

k=1

tr

M⁻¹_k (ξ_k,β^(k))

.

Hence, in analogy to the previous case of local D-optimality for a given parameter point β that is evaluated from the parameter points of the marginal models, i.e., β = ^NK

k=1

β^(k) the best product type design ξ^∗ = ^NK

k=1

ξ_k^∗ with respect to the A-criterion is generated from the locally A-optimal designs ξ_k^∗ at β^(k) for the marginal ν_k-factor models (1≤k ≤K).

Theorem 4.5.2. Let ξ_k^∗ be a locally A-optimal design (atβ^(k))for a marginalν_k-factor model (4.54) on the experimental region X_k (1 ≤ k ≤ K). Then ξ^∗ = ^NK

k=1

ξ_k^∗ is a locally A-optimal design (at β = ^NK

k=1

β^(k)) for model (4.56) on the experimental region X =×^K_k=1Xk.

Proof. The proof is obtained by making use of condition (2.12) of The Equivalence Theorem (Theorem 2.2.2 ). Sinceξ_k^∗ is locally A-optimal (at β^(k)) we guarantee that f^(k)>

β^(k)(x_k)M⁻²_k (ξ_k^∗,β^(k))f^(k)

β^(k)(x_k)≤tr(M⁻¹_k (ξ_k^∗,β^(k))) for all x_k ∈ X_k. Thus in view of

Lemma4.5.2 with (4.57) and (4.58) we obtain f^>_β(x)M⁻²(ξ^∗,β)f_β(x) =u(x,β)

K

O

k=1

f^(k)(x_k)

> K

O

k=1

M_kξ_k,β^(k)

−2 K

O

k=1

f^(k)(x_k)

=

K

O

k=1

u_k(x_k,β^(k))f^(k)>(x_k)M⁻²_k ξ_k^∗,β^(k)f^(k)(x_k)

=

K

Y

k=1

f^(k)>

β^(k)(x_k)M⁻²_k (ξ_k^∗,β^(k))f^(k)

β^(k)(x_k)

≤

K

Y

k=1

tr(M⁻¹_k (ξ_k^∗,β^(k))) = tr

^K O

k=1

Mk

ξ_k^∗,β^(k)

−1

= trM⁻¹(ξ^∗,β)

for all x ∈ X. The Equivalence Theorem, thus, proves the local A-optimality of the product designξ^∗ = ^NK

k=1

ξ_k^∗ at a parameter point β= ^NK

k=1

β^(k). Example 4.5.1. Marginal single-factor models.

Here we consider model (4.54) with one factor, ν_k = 1, and two parameters, p_k = 2,.

We restrict ourselves to the case of two marginal modelsk = 2 where

µ^ρ₁(x₁) = f^(1)>(x₁)β⁽¹⁾ =β₀⁽¹⁾+β₁⁽¹⁾x₁, x₁ ∈ X₁ = [0,1], (4.61) µ^ρ₂(x₂) = f^(2)>(x₂)β⁽²⁾ =β₀⁽²⁾+β₁⁽²⁾x₂, x₂ ∈ X₂ = [0,1] (4.62) such that β₀^(k) > 0, β₀^(k)+β₁^(k) > 0, k = 1,2. The resulting 2-factor gamma model of complete product-type interactions is thus written as

µ^ρ(x) = f^>(x)β=β₀+β₁x₁+β₂x₂ +β₃x₁x₂ (4.63) where x= (x₁, x₂)^> ∈ X =X₁× X₂ = [0,1]²,β ∈R⁴,β=β⁽¹⁾⊗β⁽²⁾, hence

β=

















β₀ β₁ β₂ β₃

















=

















β₀⁽¹⁾β₀⁽²⁾ β₀⁽²⁾β₁⁽¹⁾ β₀⁽¹⁾β₁⁽²⁾ β₁⁽¹⁾β₁⁽²⁾

















, (4.64)

where β satisfies f^>(x)β>0 for all x ∈ X. Note that the 2-factor model with in-teraction (4.63) was considered in Subsection 4.5.1 where locally D- and A-optimal designs were derived. However, from Corollary 4.3.1 the marginal design ξ_k^∗ on X_k = [0,1] with support {0,1} is equally weighted D-optimal (at β^(k)). Thus the design ξ^∗ = ξ₁^∗ ⊗ ξ₂^∗ is locally D-optimal (at β = β⁽¹⁾ ⊗ β⁽²⁾) on the experimental region X = [0,1]² for model (4.63). The design ξ^∗ assigns uniform weights 1/4 to the support (0,0)^>,(1,0)^>,(0,1)^>,(1,1)^>.

Moreover, from Corollary 4.3.1 the marginal design ξ_k^∗ onX_k = [0,1] with support {0,1} is A-optimal (at β^(k)) with weightsω_k0^∗ = √

2β₀^(k)/((√

2 + 1)β₀^(k) + β₁^(k)) and ω_k1^∗ = (β₀^(k)+β₁^(k))/((√

2 + 1)β₀^(k)+β₁^(k)). For a given β =β⁽¹⁾⊗β⁽²⁾ from (4.64) the product type designξ^∗ =ξ₁^∗⊗ξ₂^∗on the experimental regionX = [0,1]² for model (4.63) is locally A-optimal and assigns weights ω^∗₁₀ω₂₀^∗ to (0,0)^>, ω^∗₁₁ω₂₀^∗ to (1,0)^>, ω^∗₁₀ω₂₁^∗ to (0,1)^> and ω₁₁^∗ ω₂₁^∗ to (1,1)^>. Clearly, the product type design w.r.t. A-criterion coincides with that from Corollary4.5.1 where for instance; the optimal weightω^∗₁₀ω₂₀^∗ of the point (0,0)^> leads to the identity

2β₀⁽¹⁾β₀⁽²⁾ (3 + 2√

2)β₀⁽¹⁾β₀⁽²⁾+ (1 +√

2)(β₀⁽¹⁾β₁⁽²⁾+β₀⁽²⁾β₁⁽¹⁾) +β₁⁽¹⁾β₁⁽²⁾

= 2β₀ c

wherec= (3 + 2√

2)β₀+ (1 +√

2)(β₁+β₂) +β₃.

Example 4.5.2. Marginal two-factor models.

Here we consider model (4.54) with two factors,ν_k = 2, and three parameters,p_k= 3,.

The case of two marginal modelsk = 2 is also adopted where

µ^ρ₁(x₁) = f^(1)>(x₁)β⁽¹⁾ =β₀⁽¹⁾+β₁⁽¹⁾x₁₁+β₂⁽¹⁾x₁₂, x₁ = (x₁₁, x₁₂)^> (4.65) µ^ρ₂(x₂) = f^(2)>(x₂)β⁽²⁾ =β₀⁽²⁾+β₁⁽²⁾x₂₁+β₂⁽²⁾x₂₂, x₂ = (x₂₁, x₂₂)^> (4.66) where x₁ ∈ X_k = [0,1]², k = 1,2 such that β₀^(k) > 0, β₀^(k)+β₁^(k) > 0, β₀^(k)+β₂^(k) > 0, β₀^(k) +β₁^(k) +β₂^(k) > 0, k = 1,2. The resulting 2-factor gamma model of complete product-type interactions is thus written as

µ^ρ(x) =β₀+β₁x₁₁+β₂x₁₂+β₃x₂₁+β₄x₂₂+β₅x₁₁x₂₁+β₆x₁₁x₂₂

+β₇x₁₂x₂₁+β₈x₁₂x₂₂ (4.67)

where x = (x₁₁, x₁₂, x₂₁, x₂₂)^> ∈ X = [0,1]⁴,β ∈ R⁹,β = β⁽¹⁾ ⊗ β⁽²⁾. Note that β₀ =β₀⁽¹⁾β₀⁽²⁾. From part (i) of Corollary 4.4.1 the design ξ^∗_k on X_k = [0,1]² with support {(0,0)^>,(1,0)^>,(0,1)^>} for the kth marginal model is equally weighted D-optimal (at β^(k)) if and only if (β₀^(k))² ≤ β₁^(k)β₂^(k), k = 1,2. The latter optimality conditions transfer to the product type design ξ^∗ = ξ₁^∗ ⊗ξ^∗₂ which is thus locally D-optimal on the experimental region X = [0,1]⁴ for model (4.67) at β=β⁽¹⁾⊗β⁽²⁾ if (β₀⁽¹⁾β₀⁽²⁾)² ≤(β₁⁽¹⁾β₁⁽²⁾)(β₂⁽¹⁾β₂⁽²⁾), i.e., β₀² ≤β₅β₈ and ξ^∗ assigns weights 1/9 to

{(0,0)^>,(1,0)^>,(0,1)^>} × {(0,0)^>,(1,0)^>,(0,1)^>}=

{(0,0,0,0)^>,(0,0,1,0)^>, (0,0,0,1)^>, (1,0,0,0)^>, (1,0,1,0)^>, (1,0,0,1)^> ,(0,1,0,0)^>, (0,1,1,0)^>, (0,1,0,1)^>}.

For A-optimality, from part (iv) of Corollary 4.4.2 the design ξ_k^∗ on X_k = [0,1]² with support {(0,0)^>,(1,0)^>,(0,1)^>} and the following weights

ξ_k^∗(0,0)^>=√

3β₀^(k)/c, ξ_k^∗(1,0)^>= (β₀^(k)+β₁^(k))/c,

ξ_k^∗(0,1)^>= (β₀^(k)+β₂^(k))/c wherec= (√

3 + 2)β₀^(k)+β₁^(k)+β₂^(k), k = 1,2, is locally A-optimal (at β^(k)) if and only if

(1 + 2/√

3)(β₀^(k))²+ (1/√

3)β₀^(k)(β₁^(k)+β₂^(k))−β₁^(k)β₂^(k) ≤0, k= 1,2. Hence, the product type design ξ^∗ = ξ₁^∗⊗ξ₂^∗ which is thus locally A-optimal (at β = β⁽¹⁾ ⊗β⁽²⁾) on the experimental region X = [0,1]⁴ for model (4.67) assigns weights ξ₁^∗(x₁)ξ₂^∗(x₂) to the point (x^>₁,x^>₂)^>∈ {(0,0)^>,(1,0)^>,(0,1)^>} × {(0,0)^>,(1,0)^>,(0,1)^>}.

Another result of this type treats the Kiefer Φ_s-optimality (for notational con-venience we use index s instead of index k) which covers D-optimality at s = 0, A-optimality at s = 1 and E-optimality at s → ∞. Therefore, we obtain the next lemma.

Lemma 4.5.3. For every product type design ξ= ^NK

k=1

ξ_k we have the next factorization

Φ_s(ξ,β) =

K

Y

k=1

Φ_s(ξ_k,β^(k)), (0≤s <∞) Proof. Employing the definition of Kiefer Φs-criterion yields

Φ_s(ξ,β) =

1

ptrM^−s(ξ,β)

¹_s

=





1 ptr

K

O

k=1

M_k(ξ_k,β^(k))

−s



1 s

=





K

Y

k=1

1 p_ktr

M^−s_k (ξ_k,β^(k))





1 s

=

K

Y

k=1

Φ_s(ξ_k,β^(k)).

Theorem 4.5.3. Letξ_k^∗ be a locallyΦ_s-optimal design (atβ^(k))for a marginalν_k-factor model (4.54) on the experimental region X_k (1 ≤ k ≤ K). Then ξ^∗ = ^NK

k=1

ξ_k^∗ is a locallyΦ_s-optimal design (at β = ^NK

k=1

β^(k)) for model (4.56) on the experimental region X =×^K_k=1X_k.

Proof. The proof is obtained by making use of condition (2.13) of The Equivalence Theorem (Theorem 2.2.2 ). Sinceξ_k^∗ is locally Φ_s-optimal (at β^(k)) we guarantee that

f^(k)>

β^(k)(x_k)M^−s−1_k (ξ_k^∗,β^(k))f^(k)

β^(k)(x_k)≤tr(M^−s_k (ξ_k^∗,β^(k))) for all x_k ∈ X_k. Thus in view of Lemma4.5.2 with (4.57) and (4.58) we obtain

f^>_β(x)M^−s−1(ξ^∗,β)f_β(x) =u(x,β)

K

O

k=1

f^(k)(x_k)

> K

O

k=1

M_kξ_k,β^(k)

−s−1 K

O

k=1

f^(k)(x_k)

=

K

O

k=1

u_k(x_k,β^(k))f^(k)>(x_k)M^−s−1_k ξ_k^∗,β^(k)f^(k)(x_k)

=

K

Y

k=1

f^(k)>

β^(k)(x_k)M^−s−1_k (ξ_k^∗,β^(k))f^(k)

β^(k)(x_k)

≤

K

Y

k=1

tr(M^−s_k (ξ^∗_k,β^(k))) = tr

^K O

k=1

M_kξ_k^∗,β^(k)

−s

= trM^−s(ξ^∗,β)

for all x ∈ X. The Equivalence Theorem, thus, proves the local Φ_s-optimality of the product designξ^∗ = ^NK

k=1

ξ_k^∗ at a parameter point β= ^NK

k=1

β^(k).

Im Dokument Locally optimal designs for generalized linear models with applications to gamma models (Seite 82-95)