Abstract
On-farm experiments are increasingly being used as their costs have decreased with technological advances in collecting, storing, and processing geospatial data. A question that has not been well addressed is what spatial experimental design is best for on-farm experiments when the goal is to estimate a spatially varying coefficients (SVC) model. The focus here is determining the optimal location of treatments to obtain a nearly D-optimal experimental design when estimating a linear plateau model. A pseudo-Bayesian approach is taken here because the field’s site-specific optimal nitrogen value is unknown. Optimal designs are generated, assuming a fixed number of replications for each treatment level. The resulting designs are more efficient than classic Latin square, strip plot, and completely randomized designs. The method consistently produces designs that have 95% efficiency or higher. Random designs had efficiencies varying from 41 to 64% with Latin squares having higher efficiencies and strip plots lower.
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Notes
With rectangular plots, the spatial covariance matrices will no longer be symmetric when using an exponential covariance function, but the procedures to find the optimum design remain the same.
A design is locally optimal when the optimization is done by assuming some or all of the parameters. It is locally optimal since it may not be optimal for other values of parameters.
The information matrix defines the asymptotic distribution of a maximum likelihood or Bayesian estimator. The asymptotic distribution of an estimator \(\widetilde{{\varvec{\theta}}}\) is \(\sqrt{N}\left(\widetilde{{\varvec{\theta}}}-{\varvec{\theta}}\right)\to N\left(0,{\mathcal{F}\left({\varvec{\theta}}\right)}^{-1}\right)\) when regularity conditions hold where \({\varvec{\theta}}\) is the vector of true parameters and \(\mathcal{F}\left({\varvec{\theta}}\right)\) is the Fisher information matrix.
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Funding
This work was supported by the A.J. and Susan Jacques Chair, the Willard R. Sparks Chair in Agribusiness Studies, the Oklahoma Agricultural Experiment Station, and USDA National Institute of Food and Agriculture [Hatch Project number OKL03170].
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Poursina, D., Brorsen, B.W. & Lambert, D.M. Optimal treatment placement for on-farm experiments: pseudo-Bayesian optimal designs with a linear response plateau model. Precision Agric 25, 1067–1085 (2024). https://doi.org/10.1007/s11119-023-10105-w
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DOI: https://doi.org/10.1007/s11119-023-10105-w