Optimum Designs for Multi-Factor Models

In every experimental situation the outcome of an experiment depends on a number of factors of influence lie temperature, pressure, different treatments or varieties. This dependence can be described by a functional relationship, the response function μ, which quantifies the effect of the particular experimental condition t = (t₁,…, t _K ) The observation X of an experiment is subject to a random error Z. Hence, an experimental situation will be formalized by the relationship X(t) = μ(t) + Z(t). As the response function μ describes the mean outcome of the experiment the observation has to be centered and it is natural to require that E(X(t)) = μ(t) or, equivalently, that the error Z has zero expectation, E(Z(t)) = 0. The distribution of the random error may depend on the experimental conditions. In case of complete ignorance on the structure of the response function it will be impossible to make any inference. Therefore, we will consider the rather general situation in which the response is described by a linear model in which the response function μ can be finitely parametrized in a linear way as introduced in Subsection 1.1. The performance of the statistical inference depends on the experimental conditions for the different observations, and it is one of the challenging tasks to design an experiment in such a way that the outcome is most reliable. This concept will be explained in Subsection 1.2.

In this section we present the definition of various optimality criteria (Subsection 2.1) and collect the corresponding equivalence theorems which are useful tools for checking optimality (Subsection 2.2). We omit the proofs of these standard results since they have been presented extensively in the literature. In particular, we refer to the monographs by Fedorov (1972), Bandemer et al. (1977, 1980), Ermakov et al. (1983, 1987), Silvey (1980), Pázman (1986) and Pukelsheim (1993).

In general, the set of all competing designs is rather large. Therefore, it is necessary to use tools which simplify the characterization of optimum designs and which permit to search for an optimum design in a substantially smaller subclass. The first part of this section is devoted to tools which deal with reductions to subsystems of regression functions and, hence, to subsystems of parameters. In the second part a further inherent structure is assumed in the underlying model which allows for reduction by invariance with respect to suitable transformations on the design region.

In the previous Sections 4 and 5 we have treated experimental situations either with complete product-type interactions or with no interactions and discovered that the same product designs are D-optimum in both cases. This result can be extended to the intermediate interaction structure of complete M-factor interactions which will be shown in Subsection 6.1. In the second subsection we treat the frequently occuring situation that the model associated with one of the factors is invariant with respect to a group of transformations which acts transitively on the marginal design region. In this case the restriction to product designs is possible due to the invariance structure. However, in contrast to the previous results the marginals of the optimum designs are not necessarily optimum in the marginal models.

If the structure of the underlying multi-factor model is more complicated, then there is no good reason to expect product designs to be optimum. Here we will sketch the idea of conditional designs in the situation of one qualitative factor completely interacting with the second factor. These complete interaction structures will include the dependence of the conditional design region T₂(t₁) and the regression function a^(2|t₁⁾ on t₁. Some hints on partial interaction structures will be given and, finally, some limitations of the methodology will be made evident.