Optimum design of experiments for statistical inference

Abstract

One attractive feature of optimum design criteria, such as D- and A-optimality, is that they are directly related to statistically interpretable properties of the designs obtained, such as minimising the volume of a joint confidence region for the parameters. However, the assumed relationships with inferential procedures are valid only if the variance of experimental units is assumed to be known. If the variance is estimated, then the properties of the inferences depend also on the number of degrees of freedom available for estimating the error variance. Modified optimality criteria are defined, which correctly reflect the utility of designs with respect to some common types of inference. For fractional factorial and response surface experiments, the designs obtained are quite different from those which are optimal under the standard criteria, with many more replicate points required to estimate error. The optimality of these designs assumes that inference is the only purpose of running the experiment, but in practice interpretation of the point estimates of parameters and checking for lack of fit of the assumed treatment model are also usually important. Thus, a compromise between the new criteria and others is likely to be more relevant to many practical situations. Compound criteria are developed, which take account of multiple objectives, and are applied to fractional factorial and response surface experiments. The resulting designs are more similar to standard designs, but still have sufficient residual degrees of freedom to allow effective inferences to be carried out. The new procedures developed are applied to three experiments from the food industry to see how the designs used could have been improved and to several illustrative examples. The design optimisation is implemented through a simple exchange algorithm.

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