Confidence bands for measured economically optimal nitrogen rates

While numerous researchers have computed economically optimal nitrogen 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, namely, the linear plateau, quadratic, quadratic plateau, square root quadratic, spherical plateau, and 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 non-linear and give parameter estimates that are non-normal and biased in their distribution when fit by least squares. Prior to computing EONR distributions, the non-linear functions were reparameterized to give fitted parameters that were nearly unbiased and normally distributed. EONR distributions were computed using a Monte Carlo method to generate 1,000 realizations of EONR based on the fitted response function parameters. From the 1,000 realizations, the expectation, confidence bands, and cumulative probability distributions for EONR were easily computed. Applying the approach to six yield data sets from the literature illustrated that the 68% confidence bands for computed EONR can span several tens of kilograms per hectare and are typically skewed about the expectation. There were considerable differences among the distributions of EONR computed from the seven functions, with the 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.