Submitted to: Precision Agriculture
Publication Type: Peer Reviewed Journal
Publication Acceptance Date: March 24, 2010
Publication Date: April 1, 2011
Citation: Jaynes, D.B. 2011. Confidence bands for measured economically optimal nitrogen rates. Precision Agriculture. 12(2):196-213.
Interpretive Summary: Managing nitrogen (N) fertilizer is one of the many challenges facing farmers today. Establishing the economically optimal nitrogen rate (EONR) typically involves fitting a yield response function to yield vs. N rate data and finding the point on the fitted curve where the profit from an incremental increase in yield just pays for the incremental increase in the cost of the added N fertilizer. While many different functions have been used to fit yield – N rate data, what is almost universally lacking is a measure of the statistical reliability in the computed EONR due to uncertainties in the fitted parameters. This paper illustrates a method that can be used to compute EONR, its probability distribution, and error bands, for many commonly used yield response functions. The approach is demonstrated for a range of published yield response data. The method is easy and should be used by all researchers examining the relationships between farming practices and EONR.
While numerous researchers have computed economically optimal N rate (EONR) values from measured yield – N rate data, nearly all have neglected to compute or estimate the statistical reliability of these EONR values. In this study, a simple method for computing EONR and its confidence bands is described and demonstrated. The method is illustrated for seven yield response functions: the linear plateau, quadratic, quadratic plateau, square root quadratic, spherical plateau, exponential, and exponential plateau. Only the quadratic and square root quadratic functions are linear in their parameters with least squares regression yielding parameters that are normally distributed. The other five functions are nonlinear and give parameter estimates that are non-normal and biased in their distribution when fit by least squares. The nonlinear functions were reparameterized to give fitted parameters that were nearly unbiased and normally distributed before computing EONR distributions. EONR distributions were computed using a Monte Carlo method to generate 1000 realizations of EONR based on the fitted response function parameters. From the 1000 realizations, the expectation, confidence bands, and cumulative probability distributions for EONR were easily computed. Applying the approach to six yield datasets from the literature illustrated that the 68% confidence bands for computed EONR can span several 10’s of kg ha-1. There were considerable differences among the distributions of EONR computed from the seven functions with cumulative probability distributions sometimes not overlapping. Given the limited statistical reliability possible for EONR, it is essential that confidence bands always be reported when EONR values are computed from yield data.