Abstract
Experimental designs for field experiments are useful in planning agricultural experiments, environmental studies, etc. Optimal designs depend on the spatial correlation structures of field plots. Without knowing the correlation structures exactly in practice, we can study robust designs. Various neighborhoods of covariance matrices are introduced and discussed. Minimax robust design criteria are proposed, and useful results are derived. The generalized least squares estimator is often more efficient than the least squares estimator if the spatial correlation structure belongs to a small neighborhood of a covariance matrix. Examples are given to compare robust designs with optimal designs.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Becher H (1988) On optimal experimental design under spatial correlation structures for square and nonsquare plot designs. Commun Stat Simul Comput 17: 771–780
Cressie N (1993) Statistics for spatial data. Wiley, New York
Cochran WG, Cox GM (1960) Experimental designs. Wiley, New York
Elliott LJ, Eccleston JA, Martin RJ (1999) An algorithm for the design of factorial experiments when the data are correlated. Stat Comput 9: 195–201
Fang Z, Wiens DP (2000) Integer-valued minimax robust designs for estimation and extrapolation in heteroscedastic, approximately linear models. J Am Stat Assoc 95: 807–818
Fisher RA (1966) The design of experiments, 8th edn. Oliver and Boyd, Edinburgh
Haines LM (1987) The application of the annealing algorithm to the construction of the exact optimal designs for linear regression models. Technometrics 29: 439–447
Horn RA, Johnson CA (1985) Matrix analysis. Cambridge University Press, Cambridge
Kiefer J (1961) Optimum experimental designs. V. With applications to systematic and rotatable designs. In: Proceeding of 4th Berkeley Symposium. Math. Statist. and Prob., vol I, pp 381–405. University of California Press, Berkeley, California
Kiefer J, Wynn HP (1981) Optimum balanced block and Latin square designs for correlated observations. Ann Stat 9: 737–757
Kiefer J, Wynn HP (1984) Optimum and minimax exact treatment designs for one-dimensional autoregressive error processes. Ann Stat 12: 414–450
Martin RJ (1979) A subclass of lattice processes applied to a problem in planar sampling. Biometrika 66: 209–217
Martin RJ (1982) Some aspects of experimental design and analysis when errors are correlated. Biometrika 69: 597–612
Martin RJ (1986) On the design of experiments under spatial correlation. Biometrika 73:247–277 (Correction 75:396, 1988)
Martin RJ (1996) Spatial experimental design. Design and Analysis of Experiments [Handbook of Statistics 13], pp 477–514
Martin RJ, Eccleston JA, Gleeson AC (1993) Robust linear block designs for a suspected LV model. J Stat Plan Inference 34: 433–450
Martin RJ, Eccleston JA (2001) Optimal and near-optimal designs for dependent observations. Stat Appl 3: 101–116
Pukelsheim F (1993) Optimal design of experiments. Wiley, New York
Wiens DP, Zhou J (1996) Minimax regression designs for approximately linear models with autocorrelated errors. J Stat Plan Inference 55: 95–106
Wiens DP, Zhou J (1999) Minimax designs for approximately linear models with AR(1) errors. Can J Stat 27: 781–794
Wiens DP, Zhou J (2008) Robust estimators and designs for field experiments. J Stat Plan Inference 138: 93–104
Zhou J (2001) Integer-valued, minimax robust designs for approximately linear models with correlated errors. Commun Stat Theory Methods 30: 21–39
Author information
Authors and Affiliations
Corresponding author
Additional information
The work was partially supported by research grants from the Natural Science and Engineering Research Council of Canada.
Rights and permissions
About this article
Cite this article
Ou, B., Zhou, J. Minimax robust designs for field experiments. Metrika 69, 45–54 (2009). https://doi.org/10.1007/s00184-008-0173-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-008-0173-8