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Biometrika 1986 73(2):247-277; doi:10.1093/biomet/73.2.247
© 1986 by Biometrika Trust
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On the design of experiments under spatial correlation

R. J. MARTIN

Department of Probability and Statistics, University of Sheffield Sheffield S3 7RH, U.K.

The implications for experimental design when the errors are spatially correlated are investigated. The first part of the paper considers the case when treatment effects are estimated by ordinary least-squares. If only treatment labels are randomized, then A-efficient designs are reasonably independent of the correlation structure, but this is not so for other efficiency criteria. If randomization of designs is considered within one of the classical design sets, then restricted randomizations may be chosen to reduce the variations in efficiency. The distributions of the usual sums of squares are also discussed. The second part considers the case when the analysis to be used is generalized least-squares using a known correlation matrix. Then for a particular correlation structure the efficiency orderings are relatively stable over different criteria, but there are important differences between long-range and short-range correlations whatever the criterion. The efficiency gain over ordinary least-squares is shown to be least for long-range structures. Predictions made from theoretical results on the torus (Martin, 1982) are shown to have some validity in the plane. In both parts numerical evaluations provide illustrations.

Key Words: Correlated errors • Efficiency • Experimental design • Generalized least-squares • Lattice process • Optimal design • Ordinary least-squares • Restricted randomization


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