Experimental designs for estimating plateau-type production functions and economically optimal input levels

Estimation of nitrogen response functions has a long history and yet there is still considerable uncertainty about how much nitrogen to apply to agricultural crops. Nitrogen recommendations are usually based on estimation of agronomic production functions that typically use data from designed experiments. Nitrogen experiments, for example, often use equally spaced levels of nitrogen. Past agronomic research is mostly supportive of plateau-type functional forms. The question addressed is if one is willing to accept a specific plateau-type functional form as the true model, what experimental design is the best to use for estimating the production function? The objective is to minimize the variance of the estimated expected profit maximizing level of input. Of particular interest is how well does the commonly used equally-spaced design perform in comparison to the optimal design. Mixed effects models for winter wheat (Triticum aestivium L.) yield are estimated for both Mitscherlich and linear plateau functions. With three design points, one should be high enough to be on the plateau and one should be at zero. The choice of the middle design point makes little difference over a wide range of values. The optimal middle design point is lower for the Mitscherlich functional form than it is for the plateau function. Equally spaced designs with more design points have a similar precision and thus the loss from using a nonoptimal experimental design is small.