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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.

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 that are obtained, such as minimizing 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 that are 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 that are 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 treatment model assumed 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 optimization is implemented through a simple exchange algorithm.We shall refer to this as the DP.α/ criterion. In this paper, we shall use α = 0:05 for illustration and refer to the criterion simply as DP, but the required confidence level should be considered carefully for each experiment. Despite the above quotation, Kiefer (1959) did not suggest this additional step, since he did not separate lack of fit from pure error.Similarly, D S -optimality is intended to minimize the volume of a joint confidence region for a subset of p 2 of the parameters by minimizing |.M −1 / 22 |, where M = X X and .M −1 / 22 is the portion of its inverse corresponding to the subset of the parameters of interest. To take account of pure error estimation correctly, the .DP/ S criterion is to minimizeThis criterion should be used, for example, when a major objective of the experiment is to compare the first-order model with the second-order model. Then the higher order terms will form the subset and minimizing the volume of a confidence region for them will be equivalent to maximizing the power of a test for their existence. Note that if the parameters of interest are the treatment parameters and the nuisance parameter(s) is or are the intercept or the int...

This book is about the statistical principles behind the design of effective experiments and focuses on the practical needs of applied statisticians and experimenters engaged in design, implementation and analysis. Emphasising the logical principles of statistical design, rather than mathematical calculation, the authors demonstrate how all available information can be used to extract the clearest answers to many questions. The principles are illustrated with a wide range of examples drawn from real experiments in medicine, industry, agriculture and many experimental disciplines. Numerous exercises are given to help the reader practise techniques and to appreciate the difference that good design can make to an experimental research project. Based on Roger Mead's excellent Design of Experiments, this new edition is thoroughly revised and updated to include modern methods relevant to applications in industry, engineering and modern biology. It also contains seven new chapters on contemporary topics, including restricted randomisation and fractional replication.

SummaryIn this paper we describe the use of a free duplication, sDp2 (I;f), for the recovery, maintenance, and analysis of mutations defining essential genes in the left third of Linkage Group I of Caenorhabditis elegans. The lethals were induced in a strain of genotype (sDp2) + /dpy-5 + unc-13/ dpy-5 unc-15 +, using either 12 mM ethylmethane sulphonate or 1500 r of gamma radiation. Lethal mutations linked to the dpy-5 unc-13 chromosome were recognized by the absence of Dpy-5 Unc-13 individuals amongst the self progeny and were maintained by isolating Unc-13 hermaphrodites. These strains – which have two mutant alleles of the essential gene and a wild-type allele on the duplication – are balanced, since crossing-over does not occur between sDp2 and the normal homologues. Using this sytem we have recovered 58 EMS-induced mutations. These have been characterized with regard to map position and complementation. Twenty-nine of the EMS-induced mutations lie to the left of dpy-5 and define 20 complementation groups; 3 were inseparable from dpy-5 and define 3 complementation groups; 21 were to the right and define 17 complementation groups. Among a set of 29 gamma radiation-induced lethal mutations, 17 appear to be single gene mutations or are very small deletions. We estimate that we have identified from one-sixth to one-half of the essential genes in the sDp2 region.