Summary. Multitiered experiments are characterized by involving multiple randomizations, in a sense that we make explicit. We compare and contrast six types of multiple randomizations, using a wide range of examples, and discuss their use in designing experiments. We outline a system of describing the randomizations in terms of sets of objects, their associated tiers and the factor nesting, using randomization diagrams, which give a convenient and readily assimilated summary of an experiment's randomization. We also indicate how to formulate a randomization-based mixed model for the analysis of data from such experiments.

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Partial least squares regression has been widely adopted within some areas as a useful alternative to ordinary least squares regression in the manner of other shrinkage methods such as principal components regression and ridge regression. In this paper we examine the nature of this shrinkage and demonstrate that partial least squares regression exhibits some undesirable properties.

There has been much recent interest in supersaturated designs and their application in factor screening experiments. Supersaturated designs have mainly been constructed by using the E s 2 -optimality criterion originally proposed by Booth and Cox in 1962. However, until now E (s 2 )optimal designs have only been established with certainty for n experimental runs when the number of factors m is a multiple of n À 1, and in adjacent cases where m q(n À 1) r (jr j 4 2, q an integer). A method of constructing E (s 2 )-optimal designs is presented which allows a reasonably complete solution to be found for various numbers of runs n including n 8, 12, 16, 20, 24, 32, 40, 48, 64.

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