Statistics Agronomy Journal • Volume 100, Issue 5 • 2008 1221 Published in Agron. J. 100:1221–1229 (2008). doi:10.2134/agronj2007.0273 Copyright © 2008 by the American Society of Agronomy, 677 South Segoe Road, Madison, WI 53711. All rights reserved. No part of this periodical may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. O ptimal N fertilization in agricultural production is required to achieve both economical success, and to minimize nitrate N transport to surface and subsurface waters (Randall and Mulla, 2001). Nitrogen guidelines in many mid- western states have shiſted from a yield-goal based strategy to a maximum return to nitrogen (MRTN) strategy based on experimental data obtained in field plots (Laboski et al., 2006; Rehm et al., 2006; Sawyer et al., 2006). ese new recommen- dations allow farmers to account for changes in the cost of N and price of their crop. Economically optimum fertilizer rate is an estimate cal- culated by fitting a regression model with crop yield as the response variable and amount of applied nutrient as the pre- dictor variable. Based on the response function, this EOR is the point where maximum profit is achieved (Neeteson and Wadman, 1987). ese regression models can be linear, non- linear continuous, or nonlinear segmented (with a plateau) models (Wallach and Loisel, 1994). Economically optimum fertilizer rates have been used as benchmarks for fertilizer needs (National Academy of Sciences-National Research Council, 1961; Stanford, 1973; Vanotti and Bundy, 1994a; Vanotti and Bundy, 1994b; Rehm et al., 2006), as well as an indicator of N losses (Andraski et al., 2000; Hong et al., 2007). For many years, soil fertility specialists and statisticians around the globe have studied various aspects of these response functions. Much of the research has focused on finding the model that best fits experimental data (Anderson and Nelson, 1971,1975; Cerrato and Blackmer, 1990; Bullock and Bullock, 1994a; Alivelu et al., 2003) and how these models may under or overestimate N recommendations. Others have justified variable rate N applications by assessing the effect of spatial and temporal variability in crop responses to N fertilizer when esti- mating EORs (Oberle and Keeney, 1990; Blackmer et al., 1992; Malzer et al., 1996; Sogbedji et al., 2001; Mamo et al., 2003; Scharf et al., 2005; Miao et al., 2006; Schmidt et al., 2007). Many of these studies found that EORs are highly variable within the field, among fields and from season to season. In most of the studies that involve EOR estimation, research- ers report EORs as point estimates but oſten fail to report the statistical uncertainty inherent to the parameter estimation (i.e., standard error, CI). Scharf et al. (2005) argued that the degree of uncertainty related with EOR estimation has been ignored, and mentioned that there is not a standard procedure to calculate CIs of EORs. Early work by Neeteson and Wadman (1987) acknowledge the large magnitude for EOR CIs, using a symmetric (Wald type) CI. e same authors questioned the use of N recommen- dations based solely on the estimated optima and suggested that the CI of the optima should be examined before making final N recommendations. Another early work that recognized ABSTRACT Optimal N fertilization in agricultural production is necessary to achieve both economical success and to minimize nitrate trans- port to surface and subsurface waters. e 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 fer- tilizer response models. e 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. e upper bounds were similar to or lower than the likelihood-based CIs. In all loca- tions these distribution-free CIs performed better than the classical symmetrical Wald CI. Dep. of Soil, Water, and Climate, Univ. of Minnesota, 1991 Upper Buford Circle, Saint Paul, MN 55108. Contribution of the Precision Agriculture Center, Univ. of Minnesota. Received 15 Aug. 2007. *Corresponding author (jahernan@umn.edu). Abbreviations: BCa, bias-corrected and accelerated; CI, confidence interval; EOR, economically optimum fertilizer rate; MRTN, maximum return to nitrogen; Y0N, yield at 0 kg nitrogen ha –1 ; YEOR, yield at economically optimum fertilizer rate. Estimating Uncertainty of Economically Optimum Fertilizer Rates Jose A. Hernandez* and David J. Mulla