Optimal design of experiments for multivariate response in two-factor linear models*

RAINER SCHWABE

Institute of Mathematics I, Free University, Berlin, Germany

Abstract—The design of experiments aims at optimizing certain charcteristics of the statistical procedures to be used, in dependence on the settings for the experimental conditions chosen. In particular, certain functionals of the (asymptotic) covariance matrix of the least squares estimator are to be minimized. For multivariate observations designing experiments is substantially more complicated than in univariate settings. If there are additionally many factors of influence which may interact in different ways, the situation becomes even worse. However, by reduction principles, we are able to generate optimal designs for broad classes of multivariate observations in multi-factor settings which will be illustrated here in the case of two factors of influence.

1. INTRODUCTION

A multivariate experimental situation is characterized by the fact, that for each experimental conditions t ∈ T the observation X(t) is multivariate, i. e. r-dimensional real valued vectors X(t) = (X^{(1)}(t), ..., X^{(r)}(t)) are observed instead of real valued random variables with unknown mean response μ(t) = E(X(t)) = (E(X^{(1)}(t)), ..., E(X^{(r)}(t))) ∈ IR^{r} and correlation structure cov(X(t)) = Σ(t).

Here, we consider multivariate observations X(t) = (X^{(1)}(t), ..., X^{(r)}(t)) in which the mean response μ(t) = (μ_{1}(t), ..., μ_{r}(t)) can be linearly parametrized for each component μ_{ϱ}(t) = E(X^{(ϱ)}(t)), ϱ = 1, ..., r, in dependence on the setting t ∈ T for the influential factors. More precisely,

X^{(ϱ)}(t) = a_{ϱ}(t)′β_{ϱ} + Z^{(ϱ)}(t), t ∈ T,  (1.1)

where a_{ϱ} : T → IR^{p_{ϱ}} and β_{ϱ} ∈ IR^{p_{ϱ}} are the known regression functions resp. the unknown parameters for the ϱth component. Z(t) = (Z^{(1)}(t), ..., Z^{(r)}(t)) is the random error associated with the observation X(t), E(Z(t)) = 0. Hence, a multivariate observation X(t) is described by the linear model

X(t) = a(t)β + Z(t), t ∈ T,  (1.2)

or, equivalently, by its mean response

E(X(t)) = a(t)β, t ∈ T,  (1.3)

* Invited paper presented at the Sixth Eugene Lukacs Symposium on Multidimensional Statistical Analysis and Theory of Random Matrices, Bowling Green, Ohio, March 29– 30, 1996 where β = (β′_{1}, β′_{2}, ..., β′_{r})′ ∈ IR^{p}, p = ∑_{ϱ=1}^{r} p_{ϱ}, is the vector of all unknown parameters and the matrix a(t) = diag(a_{ϱ}(t)′)_{ϱ=1,...,r} of regression functions is block diagonal.

A design d of size N is given as an N-dimensional vector (t_{1}, ..., t_{N})′ which contains the adjustments of the factors of influence for N different experiments X_{n}(t_{n}) = a(t_{n})β + Z_{n}(t_{n}), n = 1, ..., N.

With X_{d}^{(ϱ)} = (X_{1}^{(ϱ)}(t_{1}), ..., X_{N}^{(ϱ)}(t_{N}))′ and Z_{d}^{(ϱ)} = (Z_{1}^{(ϱ)}(t_{1}), ..., Z_{N}^{(ϱ)}(t_{N}))′ being the vectors of observations and errors, respectively, the observations for the ϱth component can be written in the usual matrix notation

X_{d}^{(ϱ)} = A_{d}^{(ϱ)}β_{ϱ} + Z_{d}^{(ϱ)}  (1.4)

where A_{d}^{(ϱ)} = (a_{ϱ}(t_{1}), a_{ϱ}(t_{2}), ..., a_{ϱ}(t_{N}))′ is the design matrix for the ϱth component. In accordance with the common notations in multivariate analysis we vectorize the model and arrange its components successively. i. e. X_{d} = (X_{d}^{(1)′}, X_{d}^{(2)′}, ..., X_{d}^{(r)′})′ and Z_{d} = (Z_{d}^{(1)′}, Z_{d}^{(2)′}, ..., Z_{d}^{(r)′})′. Hence, we obtain the vectorized linear model equation for multivariate response

X_{d} = A_{d}β + Z_{d}  (1.5)

in matrix form, where the design matrix A_{d} = diag(A_{d}^{(ϱ)})_{ϱ=1,...,r} is block diagonal.

Next, we assume a homogeneous error structure Σ(t) = Σ for which Σ is positive definite to avoid degenerated cases, and uncorrelated observations, cov(Z_{m}(t_{m}), Z_{n}(t_{n})) = 0 for m ≠ n. This results in a covariance structure cov(Z_{d}) = Σ⊗E_{N} for the vectorized errors Z_{d}, where E_{N} is the N ×N identity matrix and ⊗ denotes the Kronecker product.

In the linear model (1.5) the general (weighted) least squares estimator β̂_{d} = (A′_{d}(Σ⊗ E_{N})^{−1}A_{d})^{−1}A′_{d}(Σ ⊗ E_{N})^{−1}X_{d} is the best linear unbiased estimator of the unknown parameters β. We will consider only such designs for which β is linearly identifiable (estimable), i. e. for which A_{d} has full column rank. The quality of the estimator β̂_{d} is based on the design d and, hence, the quality of the design d itself will be measured by the covariance matrix cov(β̂_{d}) = (A′_{d}(Σ ⊗ E_{N})^{−1}A_{d})^{−1}.

The information matrix I(d) = A′_{d}(Σ ⊗ E_{N})^{−1}A_{d} of a design d of size N is defined as the inverse of the corresponding covariance matrix cov(β̂_{d}). By rearranging terms it can be seen that the information matrix is the sum of the informations contained in each single setting, I(d) = ∑_{n=1}^{N} a(t_{n})′Σ^{−1}a(t_{n}).

If there are only few different settings and, hence, many replications in an elementary design d it is more convenient to write

d = ( \begin{matrix}t_{1} & t_{2} & ⋯ & t_{J} \\ N_{1} & N_{2} & ⋯ & N_{J} \end{matrix} )  (1.6)

where the settings t_{j} are mutually different and the N_{j} are the numbers of replications at the settings t_{j}, ∑_{j=1}^{J} N_{j} = N. Hence, we obtain for the information matrix I(d) = ∑_{j=1}^{J} N_{j}a(t_{j})′Σ^{−1}a(t_{j}). As the sample size N is fixed, a design d is determined by its relative frequencies w_{j} = \frac{N_{j}}{N} at the actual settings. Thus, d may be identified with a dicrete measure δ_{d} on a finite support, which assigns weights w_{j} to its support points t_{j}.

To solve the integer optimization problem for N_{j} is often a difficult task. Therefore it is reasonable to embed the optimization problem in a continuous setting and to drop the requirement of w_{j} to be an integer multiple of \frac{1}{N}. Hence, as a (generalized) design δ we define any measure on T, normalized to one, which is concentrated on a finite number of supporting points. In accordance with hte above considerations we define the information matrix

I(δ) = ∫ a(t)′Σ^{−1}a(t) δ(dt),  (1.7)

such that, for any design d of sample size N, I(δ_{d}) = \frac{1}{N} I(d) is a normalized version of the information matrix I(d).

Besides this traditional extension of the design problem to general designs, in order to embed the optimization problem into a richer set which is convex, a more substantial reason for considering general designs is evident. Because the supposition of a prespecified number of units with prespecified variation is sometimes rather artificial a more general idea should be associated with the concept of general designs (see Kurotschka, 1988). Assume that we may observe at certain design points t_{1}, ..., t_{J} with intensity proportional to w_{j} > 0, ∑_{j=1}^{J} w_{j} = 1, i. e. cov(X_{j}(t_{j})) = w_{j}^{−1}Σ, j = 1, ..., J, then in this general multivariate linear model the best linear unbiased estimator is given by the weighted least squares estimator β̂_{δ} = (A′_{δ}(Σ^{−1} ⊗ W_{δ})A_{δ})^{−1}A′_{δ}(Σ^{−1} ⊗ W_{δ})X_{δ} where X_{δ} = (X_{δ}^{(ϱ)})_{ϱ=1,...,r} with X_{δ}^{(ϱ)} = (X_{j}^{(ϱ)}(t_{j}))_{j=1,...,J} are the vectorized observations, A_{δ} = diag(A_{δ}^{(ϱ)})_{ϱ=1,...,r} with A_{δ}^{(ϱ)} = (a_{ϱ}(t_{1}), ..., a_{ϱ}(t_{J}))′ is the design matrix associated with the supporting points t_{j} of the design δ and the diagonal matrix W_{δ} = diag(w_{j})_{j=1,...,J} is the intensity matrix of the observations. In this setting the covariance structure of the observations obviously becomes cov(X_{δ}) = Σ ⊗ W_{δ}^{−1}.

We notice that I(δ) = ∑_{j=1}^{J} w_{j}a(t_{j})′Σ^{−1}a(t_{j}) = A′_{δ}(Σ^{−1} ⊗ W_{δ})A_{δ} = cov(β̂_{δ}). Hence, a design δ which assigns weights w_{j} to the supporting points t_{j}, j = 1, ..., J, has to be understood as an instruction which demands that J different observations are to be made at the design points t_{j} with intensity proportional to w_{j}, j = 1, ..., J.

The natural optimality criterion of minizing the covariance cov(β̂_{δ}) in the positive difinite sense is too stringent in general settings. As a compromise different matrix means can be considered which are commonly used in both multivariate analysis and optimum design theory. In the following we will be concerned with the D-criterion of generalized variance which aims at minizing the determinant of the covariance or, equivalently, at maximizing the determinant det I(δ, Σ) of the information matrix. For general statements on generalized designs in multivariate settings we refer to Fedorov (1972). More recently, Krafft and Schaefer (1992) and Bischoff (1993) investigated models in which the complexity increases from component to component. Their results lead to weighted optimality criteria as proposed by Läuter (1974) for model uncertainty in univariate settings. An extensive survey of situations in which a single design is simultaneously optimal for all components will be given in Kurotschka and Schwabe (1996).

In most experiments more than one factor of influence is active and the design problems become substantially more complicated. For univariate observations techniques have been developed in Schwabe (1996) to reduce the design problems for multi-factor experiments to their single-factor counterparts which can be solved by standard methods of optimum design theory. In the present treatise we exhibit that those ideas can be extended to situations with multivariate responses. In particular, the optimality of product designs is established, and the marginals of them satisfy certain optimality conditions within the associated marginal models in which only a single factor is involved.

For notational ease we confine to the case of two factors of influence,

E(X(t_{1}, t_{2})) = a(t_{1}, t_{2})β, t_{1} ∈ T_{1}, t_{2} ∈ T_{2}.  (1.8)

The related higher dimensional problems can be treated by combining several different factors to a new “meta-factor”.

In the following sections different interaction structures are investigated. Section 2 deals with Kronecker product type structures in which all influences related to the single factors factorize. The practically more realistic case of additive modelling without interactions is treated in Section 3. In Section 4 invariance structures are introduced for situations in which, at least, one factor of influence is qualitative, i. e. it acts on a finite number of levels. Finally, the concept of conditioning is exposed for intra-class models with multivariate response and some possible extensions are indicated.

2. KRONECKER PRODUCT TYPE STRUCTURES

We consider the class of two-factor models

E(X(t_{1}, t_{2})) = (a_{(1)}(t_{1}) ⊗ a_{(2)}(t_{2}))β, t_{1} ∈ T_{1}, t_{2} ∈ T_{2},  (2.1)

in which the regression function factorizes into components corresponding to the single factors of influence, a_{(k)} : T_{k} → IR^{p_{(k)}×r_{k}}. Here ⊗ denotes the usual Kronecker product. Additionally, we assume that the covariance structure Σ = Σ_{1} ⊗ Σ_{2} factorizes appropriately into r_{k} × r_{k} covariance matrices Σ_{k}. The associated marginal models are given by

E(X_{(k)}(t_{k})) = a_{(k)}(t_{k})β_{(k)}, t_{k} ∈ T_{k},  (2.2)

with covariance structure Σ_{k}. Note that r = r_{1}r_{2} for the dimensions of the responses and p = p_{(1)}p_{(2)} for the dimensions of parameters.

THEOREM 2.1. The product design δ^{∗} = δ_{1}^{∗} ⊗ δ_{2}^{∗} is D-optimal in the Kronecker product type two-factor model (2.1) with covariance structure Σ = Σ_{1} ⊗ Σ_{2}, if the designs δ_{k}^{∗} are D-optimal in the marginal models (2.2) with covariances Σ_{k}, k = 1, 2, respectively.

Proof. Let I_{k}(δ_{k}, Σ_{k}) be the information matrix in the kth martinal model. Then tr(Σ^{−1}a(t_{1}, t_{2})I(δ^{∗}, Σ)^{−1}a(t_{1}, t_{2})′) = ∏_{k=1}^{2} tr(Σ_{k}^{−1}a_{(k)}(t_{k})I_{k}(δ_{k}^{∗}, Σ_{k})^{−1}a_{(k)}(t_{k})′) and the result follows from the multivariate version of the Kiefer-Wolfowitz equivalence theorem (Fedorov, 1972, p. 212).

Example 2.1. Assume that both marginal models have a constant and a linear component a_{(k),1}(t_{k}) = 1 and a_{(k),2}(t_{k}) = (1, t_{k})′ each. Then the Kronecker product type two-factor model has four components, which are either constant, linear in one of the factors, or linear in both of the factors with an interaction term, a_{1}(t_{1}, t_{2}) = 1, a_{2}(t_{1}, t_{2}) = (1, t_{2})′, a_{3}(t_{1}, t_{2}) = (1, t_{1})′ and a_{4}(t_{1}, t_{2}) = (1, t_{1}, t_{2}, t_{1}t_{2})′. For T_{k} a compact interval the D-optimal marginal designs δ_{k}^{∗} assign equal weights \frac{1}{2} to both endpoints. Hence, the D-optimal design δ^{∗} = δ_{1}^{∗} ⊗ δ_{2}^{∗} assigns equal weights \frac{1}{4} to the four corners of the rectangle T_{1} × T_{2}.

Remark 2.1. If a_{2} = E_{r_{2}}, i. e. the second factor is not present, then the Kronecker product generates a model with r_{2} replicates of the first marginal. Hence, by Theorem 2.1 a design δ_{1}^{∗} which is D-optimal for the first marginal model is also D-optimal for the replicated model, if Σ factorizes. This establishes the results of Chang (1994).

3. ADDITIVE MODELS

In the class of additive models without interactions between the factors less is required for the model structure. The additive components

E(X_{ϱ}(t_{1}, t_{2})) = β_{ϱ,0} + f_{ϱ,1}(t_{1})′β_{ϱ,1} + f_{ϱ,2}(t_{2})′β_{ϱ,2}, t_{1} ∈ T_{1}, t_{2} ∈ T_{2}  (3.1)

can be combined to a multivariate additive model

E(X(t_{1}, t_{2})) = β_{0} + f_{1}(t_{1})β_{1} + f_{2}(t_{2})β_{2}, t_{1} ∈ T_{1}, t_{2} ∈ T_{2}  (3.2)

with covariance structure Σ, where f_{k}(t_{k}) = diag(f_{ϱ,k}(t_{k})′)_{ϱ=1,...r}. The associated marginal models are given by

E(X_{(k)}(t_{k})) = β_{0} + f_{k}(t_{k})β_{k}, t_{k} ∈ T_{k}.  (3.3)

LEMMA 3.1. Every design δ is dominated by the product of its marginals δ_{1} and δ_{2}, i. e. det I(δ, Σ) ≤ det I(δ_{1} ⊗ δ_{2}, Σ).

Proof. As in the univariate case we can center the regression functions according to f̃_{k} = f_{k} − ∫ f_{k} dδ_{k} without changing the determinant of the information matrices. Hence,

det I(δ, Σ) = det ( \begin{matrix}Σ^{−1} & 0 & 0 \\ 0 & ∫ f̃′_{1}Σ^{−1}f̃_{1} dδ_{1} & ∫ f̃′_{1}Σ^{−1}f̃_{2} dδ \\ 0 & ∫ f̃′_{2}Σ^{−1}f̃_{1} dδ & ∫ f̃′_{2}Σ^{−1}f̃_{2} dδ_{2} \end{matrix} )

≥ det Σ^{−1} det∫ f̃′_{1}Σ^{−1}f̃_{1} dδ_{1} det∫ f̃′_{2}Σ^{−1}f̃_{2} dδ_{2},

(3.4)

with equality for product designs δ = δ_{1} ⊗ δ_{2} because ∫ f̃′_{1}Σ^{−1}f̃_{2} d(δ_{1} ⊗ δ_{2}) = 0 for the off-diagonal elements.

It remains to find the best product design.

THEOREM 3.2. The product design δ^{∗} = δ_{1}^{∗} ⊗ δ_{2}^{∗} is D-optimal in the additive model (3.2) with covariance Σ, if the designs δ_{k}^{∗} are D-optimal in the marginal models (3.3) with covariances Σ, k = 1, 2, respectively.

Proof. Denote by I_{k}(δ_{k}, Σ) the information matrices in the marginal models. Then, by (3.4), det I(δ_{1} ⊗ δ_{2}, Σ) = det Σ det I_{1}(δ_{1}, Σ) det I_{2}(δ_{2}, Σ) which attains its maximum for the D-optimal marginals δ_{1}^{∗} and δ_{2}^{∗}.

Remark 3.1. As in the case of univariate observations the presence of a constant term in each component is required for the result of Theorem 3.1 to hold.

Example 3.1. In an additive model in which the components are given by their regression functions a_{1}(t_{1}, t_{2}) = (1, t_{1}, t_{1}^{2}, t_{2})′ and a_{2}(t_{1}, t_{2}) = (1, t_{1}, t_{2}, t_{2}^{2})′, respectively, the marginals contain a linear and a quadratic regression each. Hence, in view of the results of Krafft and Schaefer (1992), the D-optimal marginal designs δ_{k}^{∗} assign weights \frac{3}{8} to both endpoints and weight \frac{1}{4} to the midpoint of the interval T_{k}. By Theorem 3.1 the product design δ_{1}^{∗} ⊗δ_{2}^{∗} is D-optimal which assigns weights to the four corners, to the four midpoints of the edges and to the center of the rectangle T_{1} × T_{2} in the proportion 9 to 6 to 4.

Example 3.2. In an additive model with components E(X_{1}(i, u)) = α_{1,i} + β_{1}u and E(X_{2}(i, u)) = α_{2,i} + β_{2,1}u + β_{2,2}u^{2}, i = 1, ..., I, u ∈ U, the first factor is qualitative and its influence is described by a one-way layout. The D-optimal marginal design δ_{1}^{∗} is uniform on the levels 1, ..., I. The second marginal model contains a linear and a quadratic regression and the D-optimal design is δ_{2}^{∗} as given in Example 3.1 (with T_{2} = U). By Theorem 3.1 the product design δ_{1}^{∗}⊗δ_{2}^{∗} is D-optimal which is concentrated on 3I design points with weights \frac{3}{8I} and \frac{1}{4I}, respectively.

Remark 3.2. The result of Theorem 3.1 remains true even if some of the components f_{ϱ,k} of the marginal regression functions are missing. For example, if in case r = 2 the components E(X_{k}(t_{1}, t_{2})) = β_{k,0} + f_{k}(t_{k})′β_{k} depend on a single factor each, then the product design δ^{∗} = δ_{1}^{∗} ⊗ δ_{2}^{∗} is D-optimal for which the marginals δ_{k}^{∗} are D-optimal in the univariate single-factor models given by the components.

4. INVARIANCE

In this section we consider the class of symmetric multivariate linear models in which one of the factors of influence is qualitative, i. e. it acts on a finite number of levels, and in which the model equations are not affected by permutations of those levels of the qualitative factor.

E(X(i, u)) = f_{0}(u)β_{0} + f_{1}(u)β_{i}, i = 1, ..., I, u ∈ U,  (4.1)

with covariance structure Σ. In the particular case of an additive model we have seen in Example 3.2 that a D-optimal design can be obtained as a product with a uniform marginal for the qualitative factor. This result will be extended to general models (4.1). While the first marginal model is a one-way layout, we have to consider both the complete marginal model

E(X_{(2)}(u)) = f_{0}(u)β_{0} + f_{1}(u)β_{1}, u ∈ U,  (4.2)

and the reduced marginal model

E(X_{(2,1)}(u)) = f_{1}(u)β_{1}, u ∈ U,  (4.3)

for the second factor of influence. Denote by I_{2}(δ_{2}, Σ) and I_{2,1}(δ_{2}, Σ) = ∫ f′_{1}Σ^{−1}f_{1} dδ_{2} the information matrices in these two marginal models, respectively.

THEOREM 4.1. The product design δ^{∗} = δ_{1}^{∗} ⊗ δ_{2}^{∗} is D-optimal in the symmetric model (4.1) with covariance Σ, if the design δ_{1}^{∗} is uniform on {1, ..., I} and the design δ_{2}^{∗} maximizes det I_{2}(δ_{2}, Σ) det I_{2,1}(δ_{2}, Σ)^{I−1}.

Proof. The matrix a of regression functions of the symmetric model can be written as a(i, u) = (f_{0}(u), e′_{i} ⊗ f_{1}(u)) where e_{i} is the ith I-dimensional unit vector. Every permutation of the qualitative levels {1, ..., I} induces a linear transformation on a and, hence, every design δ is dominated by it s symmetrization δ_{1}^{∗} ⊗ δ_{2}, i. e. det I(δ, Σ) ≤ det I(δ_{1}^{∗} ⊗ δ_{2}, Σ), where δ_{2} is the second marginal of δ (for details see Schwabe, 1996, Sections 3.2 and 6.2). It remains to find the best δ_{2}. ¿From the representation

I(δ_{1}^{∗} ⊗ δ_{2}, Σ) = ( \begin{matrix}∫ f′_{0}Σ^{−1}f_{0} dδ_{2} & \frac{1}{I}1′_{I} ⊗ ∫ f′_{0}Σ^{−1}f_{1} dδ_{2} \\ \frac{1}{I}1_{I} ⊗ ∫ f′_{1}Σ^{−1}f_{0} dδ_{2} & \frac{1}{I}E_{I} ⊗ ∫ f′_{1}Σ^{−1}f_{1} dδ_{2} \end{matrix} )  (4.4)

of the information matrix we obtain det I(δ_{1}^{∗} ⊗ δ_{2}, Σ) = I^{−Ip_{(2,1)}} (det ∫ f′_{1}Σ^{−1}f_{1} dδ_{2})^{I} det (∫ f′_{0}Σ^{−1}f_{0} dδ_{2} − ∫ f′_{0}Σ^{−1}f_{1} dδ_{2}(∫ f′_{1}Σ^{−1}f_{1} dδ_{2})^{−1} ∫ f′_{1}Σ^{−1}f_{0} dδ_{2}) by the determinant formula for partitioned matrices, where p_{(2,1)} is the dimension in the reduced second marginal model. By formally letting I = 1 in the above arguments we obtain the related representation of det I_{2}(δ_{2}, Σ) which establishes the optimization problem for δ_{2} stated in the theorem.

Remark 4.1. The optimization problem of Theorem 4.1 is another application of the weighted optimality criterion proposed by Läuter (1974).

Remark 4.2. For f_{1} = E_{I} we recover the optimality result of Theorem 3.1 for an additive model with, at least, one qualitative factor. Also in the present setting some components of f_{0} or f_{1} may be missing.

5. INTRA-CLASS MODELS

If f_{0} is completely omitted from the model in the previous section, then there are no common effects and each group i = 1, ..., I can be treated separately. If we allow additionally that the modelling may differ from group to group we get the general class of multivariate intra-class models

E(X(i, u)) = f_{i}(u)β_{i}, i = 1, ..., I, u ∈ U,  (5.1)

with covariance structure Σ. In contrast to the previous sections we have to consider conditional models

E(X_{(2,i)}(u)) = f_{i}(u)β_{(i)}, u ∈ U,  (5.2)

for every level i of the qualitative factor. Using Kurotschka’s (1984) conditional approach we obtain

THEOREM 5.1. The δ_{1}^{∗}-mixture δ^{∗} = δ_{1}^{∗} ⊗ δ_{2}^{∗} of the kernel δ_{2}^{∗} is D-optimal in the intra-class model (5.1) with covariance Σ, if

(i) the conditional desgin δ_{2}^{∗}(i, ·) is D-optimal in the ith marginal model (5.2) with covariance Σ, for every i = 1, ..., I, and

(ii) the design δ_{1}^{∗} assigns weights p_{(2,i)}/ ∑_{j=1}^{I} p_{(2,j)} to the levels i = 1, ..., I which are proportional to the dimension p_{(2,i)} of β_{(i)} in the ith conditional model.

Proof. The result is straightforward from the block diagonal structure of the information matrix I(δ_{1} ⊗ δ_{2}) = diag (δ_{1}(i) ∫ f′_{i}Σ^{−1}f_{i} dδ_{2}(i, ·))_{i=1,...,I} for every δ_{1}-mixture of a kernel δ_{2}.

Remark 5.1. Due to the conditional approach in Theorem 5.1 the common design region U and the common cavariance structure Σ may be replaced by a conditional region U_{i} and a conditional covariance structure Σ_{i}, respectively.

6. ADDITIONAL REMARKS

Remark 6.1. The results on additive and Kronecker product type models (Section 2 and 3) can immediately be extended to an arbitrary number K of factors of influence. Moreover, generalizations to intermediate models with complete M-factor interactions are possible if the covariance Σ = Σ_{1} ⊗ ... ⊗ Σ_{K} factorizes appropriately.

Remark 6.2. The considerations of Sections 2, 4 and 5 are also suitable for other optimality criteria like the commonly used A-criterion of minimizing the trace and the E-criterion of minimizing the largest eigenvalue of the covariance matrix. Special emphasis may also be laid on particular linear aspects of the parameters within and between the components like contrasts and mean effects adjusted for blocks.

Remark 6.3. Usually there is no knowledge available on the actual form of the covariance Σ in advance. Hence, the weighted least squares estimator β̂_{δ} can not be calculated in general and different alternatives have to be considered. One common method is to estimate the covariance Σ by a random matrix Σ̂. Then a D_{Σ}-optimal design should aim at minimzing the determinant of the covariance matrix for the resulting randomly weighted least squares estimator in which Σ is replaced by Σ̂. Asymptotically this will coincide with the D-optimality considered above under mild regularity conditions.

Alternatively, the ordinary least squares estimator may be used which results in the covariance matrix (∫ a′a dδ)^{−1} ∫ a′Σa dδ(∫ a′a dδ)^{−1}. Although the problem of obtaining optimal designs in this situation looks more difficult it can be seen that in many situations this optimization is equivalent to the usual D-optimality.

Acknowledgement. The present work was supported by grants Ku719/2-2 and 477/645/96 of the Deutsche Forschungsgemeinschaft.

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