Estimating uncertainty of economically optimum fertilizer rates

Optimal N fertilization in agricultural production is necessary to achieve both economical success and to minimize nitrate transport to surface and subsurface waters. The degree of uncertainty associated with economically optimum fertilizer rate (EOR) estimation is generally overlooked, and there are no standard procedures to calculate confidence intervals (CIs) of EORs. In this paper, we present methodologies for the estimation of CIs around ex post EORs. We review four CI methods and illustrate the procedures with a sample dataset consisting of five sites, showing computations involved. Ranges in CIs about the EOR values for sites 1 through 5 (averaged across CI methods) were 15.8, 59.9, 41.7, 62.1, and 203.6 kg N ha^-1, respectively. Ranges in the 90% CI of EOR values at sites 2 to 5 were quite large relative to EOR values themselves, ranging from 29 to 106% of EOR values. Classical symmetrical CIs were not adequate for economical optima of fertilizer response curves. Wald CIs as compared with profile-likelihood based CIs tended to overestimate the lower bound and underestimate the upper bound of the EOR CI. Profile-likelihood based CIs provide a superior, computationally viable methodology for EOR uncertainty estimation in nonlinear fertilizer response models. The bootstrap CI methodologies showed great potential to evaluate the uncertainty of EORs. When compared with the profile-likelihood based CIs in our experimental areas, both bootstrapped CIs were likely to give comparable estimates for the lower bound of the CIs. The upper bounds were similar to or lower than the likelihood-based CIs. In all locations these distribution-free CIs performed better than the classical symmetrical Wald CI.