On-farm experimentation has become more and more popular due to advancements in technology. These experiments are not as costly as before, as current machinery can allocate different levels of treatment to specific plots. The main goal of this kind of experiment is to obtain a site-specific nutrient level. The yield behavior is different based on the researcher’s treatment. One unanswered question for on-farm experimentation is how the treatments should be allocated in the first place such that the appropriate model can be estimated precisely. Poursina and Brorsen (2022) obtained the nearly Ds-Optimal allocation design of the experiment for the linear in parameters model with SVC. Poursina and Brorsen showed that their optimal allocations are more informative than the standard design of experiments such as strip plot or randomly assigned design which is usually used for on-farm experiments. They use the Ds-Optimality criterion that maximizes the determinant of the Fisher information matrix for a subset of parameters. In some cases, however, the linear plateau with spatially varying coefficients (SVC) is an appropriate modeling scheme and Poursina and Brorsen’s linear in parameters assumption would not apply.
The linear plateau model creates two problems. Firstly, since it is a non-linear non-differentiable model, the information matrix cannot be derived directly from the likelihood function of the model. Secondly, the Fisher information matrix depends on the model’s unknown parameters. This paper uses a two-step approximation to obtain the Fisher information matrix. The linear plateau model is approximated with a differentiable model at the first step and then this function is linearized to find the Fisher information matrix. We employ the local optimal design approach for the second problem that considers the parameters' best initial guess. We also examine the robustness of the obtained design against the misspecification in the parameters’ true values.
The obtained optimal designs are far more informative than the standard experimental designs and do not impose any extra cost on the system. Inefficient designs like strip plots made sense when machinery was not able to easily apply nutrients and seeds at different levels within a field. Completely random designs only have 50% efficiency. We can learn much more from our experiments simply by putting more care into designing the location of each treatment.
