Uncertain Resources and Designing for Additional Information

Rainer Schwabe

Preprint Nr. A-17-95

Freie Universität Berlin, Fachbereich Mathematik und Informatik,

1. Mathematisches Institut, Arnimallee 2–6, D-14 195 Berlin

Mai 1995 – Serie A – Mathematik

Uncertain Resources and Designing for Additional Information

Rainer Schwabe Freie Universität Berlin, 1. Mathematisches Institut, Arnimallee 2–6, D-14195 Berlin, Germany

ABSTRACT: In the present paper the concept of uncertain resources in experimental design is extended to situations in which the desire for additional information increases with the amount of experiments available. For a number of simple but relevant examples different strategies are compared and optimal designs are developed in the scope of the theory of generalized designs.

Key words: linear model, interactions, main effects, D-optimality, weighted optimality criteria, generalized design, marginal model.

AMS Subject Classification: 62K05

1. Introduction Starting from the work of Mandal, Shah and Sinha (1991) the concept of uncertain

resources has got increasing importance in the field of experimental design. The idea behind that concept can be described as follows: At the beginning of an experiment it is only sure that a comparatively small number of experiments can be performed and that there is a specified chance that additional experiments may be available on a second step. These additional experiments constitute the uncertain resources.

At the first glance there is no need for particular investigations on uncertain resources in the theory of approximate or generalized designs, because all experimental units can be subdivided in a suitable way, such that both the starting design as well as the design augmented with the additional resources can be chosen in an optimal way.

However, the quality of the inference based on any design is a direct function of the amount of experimental material available in it. Thus it is quite natural to start with a small amount of information desired in the starting design and increase the qualitative amount of information simultaneously with the amount of experiments available. This means that the underlying model can be investigated more precisely if the additional resources are present.

We will consider a general linear model

E(X(t)) = f(t)^{T} θ

Research supported by grant Ku719/2-1 of the Deutsche Forschungsgemeinschaft.

for the outcome X(t) of an experiment based on the setting t = (t_{1}, ..., t_{K}) for the adjustable factors. This includes models with both qualitative and quantitative factors of influence (see Kurotschka (1984)). Based on the starting design of size (resource) R_{1} we are interested only in a subset β_{1} of the whole unknown parameter vector θ whereas additional experiments can be used to investigate a larger subset β_{2} including β_{1}, e. g. β_{2} = θ. These additional uncertain resources of size R_{2} −R_{1} may become available with a prespecified probability α giving an augmented design of size R_{2}, while with probability 1 − α only the experiments from the starting design are available.

In the setting of generalized designs we are looking for design pairs (δ_{1}, δ_{2}) under the side condition R_{1}δ_{1} ≤ R_{2}δ_{2} - with the interpretation that R_{1}δ_{1} is the starting design, R_{2}δ_{2} is the augmented design, and R_{2}δ_{2} − R_{1}δ_{1} is the design for the supplementary experiments. Here, as usual, δ_{1} and δ_{2} are supposed to be discrete designs with finite support and weights summing up to 1.

There are two appealing strategies proposed by Mandal, Shah and Sinha (1991). Strategy I starts with an optimal starting design δ_{1}^{∗} for β_{1} and looks for an augmentation \overline{δ}_{2} optimal for β_{2}, whereas strategy II is oriented backwards and starts from an optimal design δ_{2}^{∗} for β_{2} in the augmented experiment and looks for a reduced design \overline{δ}_{1} which is optimal for β_{1} (see Section 3). An early attempt to those strategies can be also found in Smith and Schmoyer (1982) and an iterative version of a strategy I type procedure has been considered e. g. by Chow (1992). An application to the setting of computer experiments has been given by Abt and Sinha (1993).

The present design problem leads to weighted optimality criteria such as those introduced by Läuter (1974) for the related situation of uncertain modelling (see also Bunke and Bunke (1986) and Bandemer et al. (1977) for a general treatment). The criteria are redefined in Section 2 to fit them to the present situation of uncertain resources and designing for additional information. Section 3 deals with the strategies I and II in relation to the optimality criteria. The implications are illustrated in the subsequent sections.

In particular, we will demonstrate these concepts in a simple one-dimensional regression model (Section 4) and in the setting of multi-factor models, i. e. models which depend on a number of different factors of influence (Section 5). There, in the starting design only main effects are considered to be of interest whereas in the presence of the additional resources also the interaction terms are to be investigated.

2. Criteria Any reasonable criterion for judging a design pair (δ_{1}, δ_{2}) should be based on the

information or covariance for the starting design R_{1}δ_{1} - weighted by the probability 1 − α of no additional experiments - and on the information or covariance for the supplemented design R_{2}δ_{2} - weighted by the probability α of additional experiments. Apart from the approach by Mandal, Shah and Sinha (1991) which cannot be applied directly to the present situation of increasing interest in additional parameters there is a bulk of criteria which can be regarded as a generalization of the common D-criterion of minimizing the generalized variance (for the related generalization problem in a Bayesian approach see e. g. Atkinson and Donev (1992), p. 214).

Let I_{1}(δ_{1}) be the information matrix of δ_{1} associated with the parameters β_{1} of interest in the starting experiment and let I_{2}(δ_{2}) be the information matrix of δ_{2} associated with the (larger) set of parameters β_{2} which are to be investigated in the supplemented experiment. Particular examples will be given later on. Then, up to a constant factor σ^{2}, \frac{1}{R_{1}} I(δ_{1})^{−1} is the covariance matrix for the best linear unbiased estimator β̂_{1} of β_{1} based on the starting design of size R_{1}, and \frac{1}{R_{2}} I(δ_{2})^{−1} is the covariance matrix for the estimator β̂_{2} of β_{2} based on the supplemented experiment of size R_{2}.

The most reasonable design criterion is the expected generalized variance

EV (δ_{1}, δ_{2}) = \frac{α}{R_{2}} det I_{2}(δ_{2})^{−1/p_{2}} + \frac{1 − α}{R_{1}} det I_{1}(δ_{1})^{−1/p_{1}}

which is to be minimized. Note that in order to make the information matrices I_{1}(δ_{1}) and I_{2}(δ_{2}) comparable their determinants have to be normalized by the power of one over the dimension p_{1} and p_{2} of the corresponding parameters β_{1} and β_{2} respectively. Note also that detI_{1}(R_{1}δ_{1}) = R_{1}^{p_{1}}detI_{1}(δ_{1}) and detI_{2}(R_{2}δ_{2}) = R_{2}^{p_{2}}detI_{2}(δ_{2}).

Alternatively one can consider the expected information

EI(δ_{1}, δ_{2}) = αR_{2} det I_{2}(δ_{2})^{1/p_{2}} + (1 − α)R_{1} det I_{1}(δ_{1})^{1/p_{1}}

which is to be maximized. The latter is related to the C-criterion of Läuter (1974) who also proposed a criterion of the relative efficiency in the information which can be defined after renormalization according to

eff(δ_{1}, δ_{2}) = α (\frac{det I_{2}(δ_{2})}{det I_{2}(δ_{2}^{∗})})^{1/p_{2}} + (1 − α) (\frac{det I_{1}(δ_{1})}{det I_{1}(δ_{1}^{∗})})^{1/p_{1}}

which is to be maximized, where δ_{1}^{∗} and δ_{2}^{∗} are the D-optimal designs for β_{1} and β_{2} respectively. More reasonable again seems to be the minimization problem of the relative deficiency in the generalized variance

def(δ_{1}, δ_{2}) = α (\frac{det I_{2}(δ_{2})}{det I_{2}(δ_{2}^{∗})})^{−1/p_{2}} + (1 − α) (\frac{det I_{1}(δ_{1})}{det I_{1}(δ_{1}^{∗})})^{−1/p_{1}}

Also logarithmic versions of the criteria are of interest. For example, maximizing the expected logarithmic information

E_{log}I(δ_{1}, δ_{2}) = \frac{α}{p_{2}} ln det I_{2}(δ_{2}) + \frac{1 − α}{p_{1}} ln det I_{1}(δ_{1}) ,

which is related to Läuter’s (1974) S-criterion, is equivalent to minimizing the expected logarithmic generalized variance E_{log}V = −E_{log}I. For a recent treatment of the last criterion in the related setting of uncertain modelling see e. g. Dette (1990).

All of the above criteria depend on the sizes of the resources through the proportion ϱ = \frac{R_{1}}{R_{2}} . In particular, for the efficiency, the deficiency and the logarithmic criterion this dependence comes in only by the constraint δ_{2} ≥ ϱδ_{1}. In the subsequest examples the different influence on the optimality of design pairs (δ_{1}, δ_{2}) will be exhibited for each criterion. However, note that in case that the D-optimal designs δ_{1}^{∗} and δ_{2}^{∗} for the parameters β_{1} and β_{2} respectively satisfy the relation δ_{2}^{∗} ≥ ϱδ_{1}^{∗} then the design pair (δ_{1}^{∗}, δ_{2}^{∗}) with the supplementary design R_{2}δ_{2}^{∗} − R_{1}δ_{1}^{∗} is optimal with respect to each of the above criteria. Therefore the original problem of Mandal, Shah and Sinha (1991) in which β_{2} = β_{1} has a straightforward solution (δ_{1}^{∗}, δ_{1}^{∗}) in the setting of generalized designs.

3. Two Basic Strategies Mandal, Shah and Sinha (1991) proposed to consider design pairs (δ_{1}, δ_{2}) in which

either the starting design δ_{1}^{∗} or the augmented design δ_{2}^{∗} is optimal. In the first case the additional design R_{2}δ_{2} − R_{1}δ_{1}^{∗} is chosen to obtain the best possible design \overline{δ}_{2} (strategy I), whereas in the second case a fraction R_{2}δ_{2}^{∗} − R_{1}δ_{1} is deleted from δ_{2}^{∗} in such a way that the best possible desgin \overline{δ}_{1} remains (strategy II).

Let δ_{1}^{∗} be D-optimal for β_{1}, i. e. δ_{1}^{∗} maximizes det I_{1}(δ_{1}). The strategy I results in a design pair (δ_{1}^{∗}, \overline{δ}_{2}), where \overline{δ}_{2} maximizes det I_{2}(δ_{2}) under the constraint δ_{2} ≥ ϱδ_{1}^{∗}. Given δ_{1}^{∗} the pair (δ_{1}^{∗}, \overline{δ}_{2}) is simultaneously optimal for each of the criteria introduced in the previous section and for each α. Analogoulsy, let δ_{2}^{∗} be D-optimal for β_{2}, i. e. δ_{2}^{∗} maximizes det I_{2}(δ_{2}). Then strategy II yields a design pair (\overline{δ}_{1}, δ_{2}^{∗}), where \overline{δ}_{1} maximizes det I_{1}(δ_{1}) under the constraint δ_{1} ≤ ϱ^{−1}δ_{2}^{∗}. Given δ_{2}^{∗} the pair (\overline{δ}_{1}, δ_{2}^{∗}) is simultaneously optimal for each criterion and each α. (For a recent survey on optimal desings under constraints see Cook and Fedorov (1994).)

Obviously, strategy I is preferable to strategy II if the probability α for the additional experiments is small, whereas strategy II dominates strategy I for large probabilities α. More precisely, in the general situations that neither δ_{1}^{∗} nor δ_{2}^{∗} is optimal for the alternative problem, there will be a critical value α_{0} for each criterion such that strategy I produces a better design than strategy II for α < α_{0}, and vice versa for α > α_{0}.

4. An Example in Polynomial Regression We start with a quite simple illustrating example. Assume that the underlying model

equation is given by a one-dimensional quadratic regression

E(X(t)) = θ_{0} + θ_{1}t + θ_{2}t^{2}

on the standardized interval −1 ≤ t ≤ 1. In the starting experiment we are interested in the slope and the intercept, i. e. β_{1} = (θ_{0}, θ_{1}), whereas in case of additional experiments we are willing to investigate the whole parameter vector β_{2} = (θ_{0}, θ_{1}, θ_{2}). Both the D-optimal designs δ_{1}^{∗} and δ_{2}^{∗} for β_{1} and β_{2} respectively are concentrated on {−1, 0, 1}. δ_{1}^{∗} assigns weight \frac{1}{2} to the midpoint 0 and weights \frac{1}{4} to each of the endpoints −1 and 1, whereas δ_{2}^{∗} assigns equal weights \frac{1}{3} to each of them.

In case that the additional resource R_{2} − R_{1} exceeds \frac{1}{2}R_{1} we observe that R_{1}δ_{1}^{∗} ≤ R_{2}δ_{2}^{∗} and the design pair (δ_{1}^{∗}, δ_{2}^{∗}) is optimal. So, we can start with a D-optimal design R_{1}δ_{1}^{∗} for β_{1} and augment it to a D-optimal design R_{2}δ_{2}^{∗} for β_{2} in case of availability of the additional resources.

R_{1} = 24  R_{1} = 28
−1  0  1 −1  0  1
strategy I: R_{1}δ_{1}^{∗} 6  12  6 7  14  7
R_{2}\overline{δ}_{2} 9  12  9 8  14  8
strategy II: R_{2}δ_{2}^{∗} 10  10  10 10  10  10
R_{1}\overline{δ}_{1} 7  10  7 9  10  9

Table 1: design pairs (δ_{1}^{∗}, \overline{δ}_{2}) and (\overline{δ}_{1}, δ_{2}^{∗}), R_{1} = 24(28), R_{2} = 30

criterion ϱ → \frac{2}{3} ϱ = \frac{24}{30} ϱ = \frac{28}{30} ϱ = 1
EV  0.7605 0.6636 0.5695 0.5242
EI  0.5575 0.5288 0.5056 0.4953
eff  0.6667 0.5975 0.5370 0.5095
def  0.6667 0.5986 0.5385 0.5101
E_{log}I  0.6667 0.5981 0.5377 0.5098

Table 2: critical values α_{0}

In the more interesting case R_{2} − R_{1} < \frac{1}{2}R_{1}, i. e. ϱ = \frac{R_{1}}{R_{2}} > \frac{2}{3}, such an augmentation is no longer possible. For strategy I only experiments at the endpoints −1 and 1 are to be added to R_{1}δ_{1}^{∗} and the design \overline{δ}_{2}, which assigns weights \frac{1}{2}ϱ to 0 and \frac{1}{4}(2 − ϱ) to −1 and 1 each, is D-optimal for β_{2} under the constraint R_{2}δ_{2} ≥ R_{1}δ_{1}^{∗}. Alternatively , for strategy II, only experiments at the endpoints −1 and 1 are to be deleted to obtain the D-optimal design \overline{δ}_{1} under the constraint R_{1}δ_{1} ≤ R_{2}δ_{2}^{∗}. Here, \overline{δ}_{1} assigns weights \frac{1}{3}ϱ^{−1} to 0 and \frac{1}{6}(3 − ϱ^{−1}) to −1 and 1 each. For the example R_{1} = 24(28) and R_{2} = 30 the corresponding designs are presented in Table 1.

In Table 2 we exhibit critical values α_{0} for each criterion for which the strategy I design (δ_{1}^{∗}, \overline{δ_{2}}) is preferable in case α < α_{0} and the strategy II design (\overline{δ}_{1}, δ_{2}^{∗}) performs better in case α > α_{0}. The second and third column present values of α_{0} for the example R_{1} = 24(28) and R_{2} = 30 of Table 1.

Despite their appealing simplicity both strategies do not produce an optimal design pair (δ_{1}, δ_{2})^{∗} in general. In Table 3 optimal design pairs (δ_{1}, δ_{2})^{∗} concentrated on {−1, 0, 1} are given by their weights δ_{1}(0) = w^{∗} and δ_{2}(0) = ϱw^{∗} for different values of α. The corresponding weights on ±1 equal \frac{1}{2}(1 − w^{∗}) and \frac{1}{2}(1 − ϱw^{∗}) respectively. Recall that w^{∗} = \frac{1}{2} for α = 0 and w^{∗} = ϱ^{−1} \frac{1}{3} for α = 1.

Table 4 exhibits integer approximations to the optimal design pairs (δ_{1}, δ_{2})^{∗} in case of probabilities α = 0.25, 0.5 and 0.75 for the availibility of additional experiments.

ϱ = \frac{24}{30}  ϱ = \frac{28}{30}
criterion α = 0.25 α = 0.5 α = 0.75 α = 0.25 α = 0.5 α = 0.75
EV  0.4879 0.4717 0.4493 0.4701 0.4371 0.4001
EI  0.4810 0.4604 0.4388 0.4648 0.4283 0.3919
eff  0.4848 0.4663 0.4437 0.4681 0.4329 0.3954
def  0.4847 0.4662 0.4439 0.4670 0.4328 0.3965
E_{log}I  0.4601 0.4385 0.4254 0.4216 0.3871 0.3685

Table 3: optimal weights w^{∗} for different α and ϱ

R_{1} = 24

α = 0.25  α = 0.5  α = 0.75
criterion  −1  0  1 −1  0  1 −1  0  1
EV  R_{1}δ_{1} 6  12  6 6  12  6 6  11  7
R_{2}δ_{2} 9  12  9 9  12  9 9  11  10
EI  R_{1}δ_{1} 6  12  6 6  11  7 7  10  7
R_{2}δ_{2} 9  12  9 9  11  10 10  10  10
eff  R_{1}δ_{1} 6  12  6 6  11  7 7  10  7
R_{2}δ_{2} 9  12  9 9  11  10 10  10  10
def  R_{1}δ_{1} 6  12  6 6  12  6 6  11  7
R_{2}δ_{2} 9  12  9 9  12  9 9  11  10
E_{log}I  R_{1}δ_{1} 6  12  6 6  12  6 7  10  7
R_{2}δ_{2} 9  12  9 9  12  9 10  10  10
R_{1} = 28
α = 0.25  α = 0.5  α = 0.75
EV  R_{1}δ_{1} 7  14  7 8  12  8 8  11  9
R_{2}δ_{2} 8  14  8 9  12  9 9  11  10
EI  R_{1}δ_{1} 7  13  8 8  12  8 8  11  9
R_{2}δ_{2} 8  13  9 9  12  9 9  11  10
eff  R_{1}δ_{1} 8  12  8 8  12  8 9  10  9
R_{2}δ_{2} 9  12  9 9  12  9 10  10  10
def  R_{1}δ_{1} 7  14  7 8  12  8 8  12  8
R_{2}δ_{2} 8  14  8 9  12  9 9  12  9
E_{log}I  R_{1}δ_{1} 7  13  8 8  12  8 8  11  9
R_{2}δ_{2} 8  13  9 9  12  9 9  11  10

Table 4: “exact” optimal designs (R_{1}δ_{1}, R_{2}δ_{2})^{∗} 5. Two-factor Models

We consider models with complete product type interactions

E(X(t_{1}, t_{2})) = (f_{1}(t_{1}) ⊗ f_{2}(t_{2}))^{T} θ

where ⊗ denotes the Kronecker product. Particular models possessing that structure are two-way layouts

E(X(i, j)) = μ + α_{i}^{(1)} + α_{j}^{(2)} + γ_{ij}

linear regression

E(X(t_{1}, t_{2})) = θ_{0} + θ_{1}t_{1} + θ_{2}t_{2} + θ_{12}t_{1}t_{2}

or, more complicated, intra-class models

E(X(i, t)) = θ_{i0} + θ_{i1}t

(see Kurotschka (1984)). In all those models product type designs δ_{(1)}^{∗} ⊗ δ_{(2)}^{∗} are known to be D-optimal for the whole parameter vector β_{2} = θ if the components δ_{(k)}^{∗} are D-optimal in the corresponding marginal models

E(X_{(k)}(t_{k})) = f_{k}(t_{k})^{T} θ_{(k)}

(see Hoel (1965)). However, product type designs are not necessarily optimal for the subset of parameters β_{1} associated with the main effects. (For exceptional situations see Schwabe (1995).) The construction of design pairs (δ_{1}, δ_{2}) for uncertain resources in twofactor models will be illustrated by means of the above mentioned examples.

Linear Regression.

Consider the unit square 0 ≤ t_{1}, t_{2} ≤ 1 as the underlying desing region. The product type design δ_{2}^{∗} = δ_{(1)}^{∗} ⊗ δ_{(2)}^{∗} which assigns equal weights \frac{1}{4} to the four corners (0, 0), (0, 1), (1, 0) and (1, 1) of the square is D-optimal for the whole parameter vector β_{2} = θ. However, for the main effects β_{1} = (θ_{0}, θ_{1}, θ_{2}) the observations at (1, 1) are of no use and the optimal design δ_{1}^{∗} for β_{1} assigns equal weights \frac{1}{3} to the remaining relevant corners (0, 0), (0, 1) and (1, 0).

In case ϱ = \frac{R_{1}}{R_{2}} > \frac{3}{4} the optimal design R_{1}δ_{1}^{∗} for β_{1} cannot be augmented to an optimal design R_{2}δ_{2}^{∗} for β_{2}^{∗}. Similarly to the example of quadratic regression in the previous section we can construct strategy I and strategy II designs (Table 5) for which the augmented design \overline{δ}_{2} assigns weight 1−ϱ to (1, 1) and the reduced design \overline{δ}_{1} assigns weight ϱ − \frac{3}{4} to (1, 1) and equal weights to the remaining corners respectively. We determine their critical values (Table 6), derive optimal design pairs (Table 7) with δ_{1}(1, 1) = w^{∗}, δ_{2}(1, 1) = 1 − ϱ(1 − w^{∗}) and equal weights \frac{1}{3}(1 − w^{∗}) resp. \frac{1}{3}ϱ(1 − w^{∗}) on (0, 0), (0, 1), (1, 1), and give some integer approximations (Table 8) which coincide for all criteria in the present situation.

R_{1} = 27  R_{1} = 30
(0, 0) (0, 1) (1, 0) (1, 1) (0, 0) (0, 1) (1, 0) (1, 1)
strategy I: R_{1}δ_{1}^{∗} 9  9  9  0  10  10  10  0
R_{2}\overline{δ}_{2} 9  9  9  5  10  10  10  2
strategy II: R_{2}δ_{2}^{∗} 8  8  8  8  8  8  8  8
R_{1}\overline{δ}_{1} 8  8  8  3  8  8  8  6

Table 5: design pairs (δ_{1}^{∗}, \overline{δ}_{2}) and (\overline{δ}_{1}, δ_{2}^{∗}), R_{1} = 27(30), R_{2} = 32

criterion ϱ → \frac{3}{4} ϱ = \frac{27}{32} ϱ = \frac{30}{32} ϱ → 1
EV  0.5323 0.3379 0.0983  0
EI  0.3763 0.2841 0.1190  0
eff  0.4605 0.3328 0.1326  0
def  0.4459 0.2888 0.0879  0
E_{log}I  0.4536 0.3114 0.1093  0

Table 6: critical values α_{0}

ϱ = \frac{27}{32}  ϱ = \frac{30}{32}
criterion α = 0.25 α = 0.5 α = 0.75 α = 0.25 α = 0.5 α = 0.75
EV  0  0  0.0417 0.0182 0.0790 0.1375
EI  0  0  0.0314  0  0.0501 0.1282
eff  0  0  0.0386  0  0.0675 0.1402
def  0  0  0.0357 0.0078 0.0660 0.1271
E_{log}I  0  0  0.0370  0  0.0667 0.1333

Table 7: optimal weights w^{∗} for different α and ϱ

R_{1} = 27  R_{1} = 30
criterion  (0, 0) (0, 1) (1, 0) (1, 1) (0, 0) (0, 1) (1, 0) (1, 1)
α = 0.25 R_{1}δ_{1} 9  9  9  0  10  10  10  0
R_{2}δ_{2} 9  9  9  5  10  10  10  2
α = 0.5 R_{1}δ_{1} 9  9  9  0  10  9  9  2
R_{2}δ_{2} 9  9  9  5  10  9  9  4
α = 0.75 R_{1}δ_{1} 9  9  8  1  9  9  8  4
R_{2}δ_{2} 9  9  8  6  9  9  8  6

Table 8: “exact” optimal designs (R_{1}δ_{1}, R_{2}δ_{2})^{∗}

Note that for a different design region −1 ≤ t_{1}, t_{2} ≤ 1 the product type design δ_{2}^{∗} which assigns equal weights \frac{1}{4} to each corner (−1, −1), (−1, 1), (1, −1) and (1, 1) is simultaneously D-optimal for β_{1} and β_{2} (see e. g. Schwabe (1995)) and (δ_{2}^{∗}, δ_{2}^{∗}) is an optimal design pair.

Two-way layout.

Assume that the settings i = 0 and j = 0 are control levels, which means that α_{0}^{(1)} = α_{0}^{(2)} = 0 and, additionally, for the corresponding interactions γ_{0j} = γ_{i0} = 0 for every setting i and j. The product type design δ_{2} = δ_{(1)}^{∗} ⊗ δ_{(2)}^{∗} which assigns equal weights \frac{1}{(I+1)(J+1)} to each level combination (i, j) is D-optimal for the whole parameter vector

β_{2} = (μ, α_{1}^{(1)}, …, α_{I}^{(1)}, α_{1}^{(2)}, …, α_{J}^{(2)}, γ_{11}, γ_{12}, …, γ_{IJ}) where I and J are the numbers of non-control levels for the first and second factor respectively.

Again, for the main effects β_{1} = (μ, α_{1}^{(1)}, …, α_{I}^{(1)}, α_{1}^{(2)}, …, α_{J}^{(2)}) the D-optimal design δ_{1}^{∗} is concentrated on those settings in which at least one control level is involved. More precisely, δ_{1}^{∗} assigns equal weights \frac{1}{I+J+1} to the settings (0, 0), (0, 1), ..., (0, J), (1, 0), ..., (I, 0).

For \frac{R_{1}}{R_{2}} > \frac{I+J+1}{(I+1)(J+1)} the design R_{1}δ_{1}^{∗} cannot be augmented to R_{2}δ_{2}^{∗} and designing for uncertain resources becomes important. This becomes relevant, for example, in case I = J = 2 if R_{2} < 1.8R_{1}, and in case I = 4 and J = 5 if R_{2} < 3R_{1}. In the simplest case I = J = 1 the results of Tables 5 to 8 carry over directly.

Intra-class model.

Consider the situation where the qualitative factor i acts on two levels 0 and 1 (the “0” denotes again a control level) and the quantitative factor t may vary on the standard interval −1 ≤ t ≤ 1.

The D-optimal design δ_{2}^{∗} for the whole parameter vector β_{2} = θ is given by the product type design δ_{(1)}^{∗} ⊗ δ_{(2)}^{∗} which assigns equal weights \frac{1}{4} to (0, 1), (0, −1), (1, 1) and (1, −1). For the main effects β_{1} = (θ_{00}, θ_{01}, θ_{10}) the D-optimal design δ_{1}^{∗} assigns weights \frac{1}{3} to the control settings (0, 1) and (0, −1) and reduced weights \frac{1}{6} to the treatment settings (1, 1) and (1, −1).

Again, for \frac{R_{1}}{R_{2}} > \frac{3}{4} the design R_{1}δ_{1}^{∗} cannot be augmented to R_{2}δ_{2}^{∗} and designing for uncertain resources becomes relevant.

Remarks.

1. Note that in all examples the D-optimal design δ_{1}^{∗} for the main effects can be written as δ_{1}^{∗} = \frac{1}{p_{(1)}+p_{(2)}−1}(p_{(1)}δ_{(1)}^{∗} ⊗ δ_{(2)}^{∘} + p_{(2)}δ_{(1)}^{∘} ⊗ δ_{(2)}^{∗} − δ_{(1)}^{∘} ⊗ δ_{(2)}^{∘}) where p_{(k)} is the dimension of θ_{(k)} and δ_{(k)}^{∘} is a suitable design which is optimal for the constant term in the kth marginal model, k = 1, 2.

2. In the presence of more than two factors of influence designing for uncertain resources becomes even more important as a substantially larger number of parameters is involved. Moreover, in a complete product type model β_{2} may be a proper subset of the parameters, e. g. β_{2} might collect all two-factor interactions besides the main effects β_{1}. Alternatively also models can be considered in which only interactions up to a certain order occur (see Schwabe (1993) for optimal designs). Also block effects can be investigated more thorouhgly with the help of supplementary experiments.

Acknowledgement.

The author wishes to express his thanks to Professor Bikas K. Sinha (Calcutta) for a discussion on uncertain resources in the design of experiments.

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