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PREFERENCES FOR ENVIRONMENTAL QUALITY UNDER UNCERTAINTY AND THE VALUE OF PRECISION NITROGEN APPLICATION By DAVID CARLOS ROBERTS Bachelor of Arts in Spanish The University of Tennessee Knoxville, Tennessee 2003 Master of Science in Agricultural Economics The University of Tennessee Knoxville, Tennessee 2006 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY July, 2009 PREFERENCES FOR ENVIRONMENTAL QUALITY UNDER UNCERTAINTY AND THE VALUE OF PRECISION NITROGEN APPLICATION Dissertation approved: Dr. B. Wade Brorsen Dissertation Adviser Dr. Tracy A. Boyer Dr. Francis M. Epplin Dr. William R. Raun Dr. John B. Solie Dr. A. Gordon Emslie Dean of the Graduate College iii ACKNOWLEDGEMENTS While words fail to describe the full measure of my gratitude, I am obliged herein to acknowledge the contributions of my graduate committee members toward my success. I am grateful for their willingness to invest their time and efforts to assist me in developing my professional capital, and insuring the integrity of the work presented in this dissertation. I therefore extend my heartfelt appreciation to Dr. B. Wade Brorsen, Dr. Tracy A. Boyer, Dr. Francis M. Epplin, Dr. William R. Raun, and Dr. John B. Solie. Many thanks also to Dr. Jayson L. Lusk, who, though not on my committee, made similar contributions to my success in this endeavor. I am grateful also to Gracie Teague for her crucial and very thorough assistance in formatting this document. I also wish to express my gratitude to the Agricultural and Applied Economics Association, the Southern Agricultural Economics Association, the 9th International Conference on Precision Agriculture, and the 8th Annual Heartland Environmental and Resource Economics Workshop for providing opportunities to vet my research before an audience of my professional peers. My sincere thanks are also extended to the editors and the anonymous peer reviewers at the journal Ecological Economics for their valuable input in the peerreview and publication process. Chapter III of this dissertation was published in Ecological Economics, and is included herein with the journal’s permission. I acknowledge the instrumental assistance of my coauthors Tracy A. Boyer and Jayson L. iv Lusk, and their efforts to ensure the statistical and prosaic integrity of the aforementioned chapter. Funding for this research was provided by the Sitlington Foundation, the Oklahoma Department of Wildlife Conservation, the Targeted Initiative Program managed through Oklahoma State University’s Division of Agricultural Sciences and Natural Resources, and the Jean and Patsy Neustadt Chair. Their funding was crucial in conducting and disseminating this research. I am also very pleased to acknowledge the undying devotion, honor, and respect I owe to my wife (Bonny D. Roberts) and my daughter (Naomi S. Roberts). Though they did not take the courses, collect the data, or write the dissertation, this accomplishment is as much theirs as it is mine. They made the requisite sacrifices with me, and they will reap the rewards with me also. Thanks to my father and mother, who have encouraged me in my educational endeavors as long as I can remember. Thanks also to all my kith and kin, who have not only influenced me for good but also appreciated me for who I am. Many thanks also to my office mates and fellow graduate students at Oklahoma State University. And thanks again to Gracie Teague. v TABLE OF CONTENTS Chapter Page I. PREDICTION UNCERTAINTY AND THE VALUE OF INCREASINGLY SPATIALLY PRECISE NITROGEN NEEDS INFORMATION......................................................................................................1 Abstract ............................................................................................................ 1 Introduction ...................................................................................................... 3 Theory .............................................................................................................. 6 Brief Example: Expected Profit Maximization when the Plateau Yield Is Predicted with Error .......................................................................7 The Producer’s Decision Problem: Choosing the Expected Profit Nitrogen Application System .....................................................................10 How Do Nitrogen Needs Vary Spatially, and What Are the Implications? ..............................................................................................11 SubPaper 1: Spatial Variability, Repeatability and Noise in Predictions Made by the Nitrogen Fertilizer Optimization Algorithm and the Ramped Strip ................................................................................................................ 13 Data ............................................................................................................13 Procedures ..................................................................................................15 Results ........................................................................................................19 Conclusions ................................................................................................30 SubPaper 2: Prediction Uncertainty and the Value of Increasingly Spatially Precise Sampling of Optical Reflectance Data .............................................. 34 Data ............................................................................................................34 Procedures ..................................................................................................37 Variability of Nitrogen Needs by Year and Location ......................... 37 Defining a Predictive Relationship ..................................................... 38 The Perfect Prediction Nitrogen Rate ................................................. 41 Nitrogen Needs Predictions by SiteYear ........................................... 42 Nitrogen Needs Predictions by RegionYear ...................................... 44 Calculation of Expected Yield and Expected Profit ........................... 45 Testing for Differences in Expected Profit, Expected Yield, and Nitrogen Application Rates................................................................. 46 Results ........................................................................................................48 Conclusions ................................................................................................67 II. THE EFFECT OF PARAMETER UNCERTAINTY ON NITROGEN RECOMMENDATIONS FROM NITROGENRICH STRIPS AND RAMPED STRIPS IN WINTER WHEAT............................................................71 Chapter Page vi Abstract .......................................................................................................... 71 Introduction .................................................................................................... 73 Theory ............................................................................................................ 79 Data ................................................................................................................ 80 Procedures ...................................................................................................... 85 SpaceTime Variability of Crop Response ................................................85 Expected Profit Maximizing Application Rates from Perfect Predictors ...................................................................................................86 Predicted Expected Profit Maximizing Application Rates from Ramped Strip Predictors ............................................................................89 Predicted Expected Profit Maximizing Application Rates from Nitrogen Rich Strip Predictors ...................................................................90 Calculation of Expected Yields and Expected Returns .............................96 Results............................................................................................................ 98 Conclusions.................................................................................................. 120 III. PREFERENCES FOR ENVIRONMENTAL QUALITY UNDER UNCERTAINTY .................................................................................................127 Abstract ........................................................................................................ 127 Introduction .................................................................................................. 127 Background .................................................................................................. 132 Methods ....................................................................................................... 136 Results.......................................................................................................... 141 Conclusions.................................................................................................. 151 IV. REFERENCES ....................................................................................................154 V. APPENDICES .....................................................................................................163 Appendix A: Institutional Review Board Approval Letter.......................... 164 vii LIST OF TABLES Table Page Table I1. Number of Observations, Mean Ramped Strip Nitrogen Recommendation, Mean Nitrogen Recommendations, and Mean Predicted Plateau Yield by County for Dataset Two .................................14 Table I2. Planting Date and Sensing Dates for Each Field in Dataset Three ............16 Table I3. Ramped Strip and Nitrogen Fertilizer Optimization Algorithm Recommendations as Functions of FarmerPractice Preplant Nitrogen Rate and County .........................................................................................20 Table I4. Mean Ramped Strip Recommendation, with and without Fixed Effects for Strip Pair and Field ...............................................................................22 Table I5. Mean Nitrogen Fertilizer Optimization Algorithm Recommendation, with and without Fixed Effects for Strip Pair and Field ............................25 Table I6. Locations, Years, Soil Types, and Nitrogen Levels, and Replications for Experiments in Dataset One.......................................................................35 Table I7. Partial Budget for Creation and Use of Three Ramped Strips in a 63 ha field ............................................................................................................37 Table I8. Wheat Yield as a Function of Nitrogen Application with Site and Year Specific Effects on the Intercept and Plateau Yields .................................48 Table I9. Estimated Wheat Yield Response to Nitrogen by SiteYear .....................51 Table I10. Estimated Wheat Optical Reflectance Response to Nitrogen by Site Year, Scaled by a Factor of Ten Thousand ...............................................54 Table I11. Estimated Wheat Optical Reflectance Response to Nitrogen by Year, scaled by a Factor of Ten Thousand ..........................................................57 Table I12. Response of Yield Intercepts, Slopes and Plateaus to Optical Reflectance Intercepts, Slopes and Plateau, Respectively, Estimated by Seemingly Unrelated Regression .................................................................................60 Table I13. Noparametrically Bootstrapped Means of Net Returns, Revenues, NitrogenRelated Costs, Yields and Nitrogen Application Rates for Each Application System, Assuming 32% NitrogenUse Efficiency for Both Topdress and Preplant Nitrogen Applications ...........................................61 Table I14. Nonparametrically Bootstrapped Means of Paired Differences of Expected Profits, Expected Nitrogen Application Rates, and Expected viii Table Page Yields, Assuming 32% NitrogenUse Efficiencies for Both Preplant and Topdress Nitrogen Applications ................................................................62 Table I15. Noparametrically Bootstrapped Means of Net Returns, Revenues, NitrogenRelated Costs, Yields and Nitrogen Application Rates for Each Application System, Assuming 32% and 45% NitrogenUse Efficiency for Preplant and Topdress Nitrogen Applications, Respectively ..............63 Table I16. Nonparametrically Bootstrapped Means of Paired Differences of Expected Profits, Expected Nitrogen Application Rates, and Expected Yields, Assuming 32% and 45% NitrogenUse Efficiencies for Preplant and Topdress Nitrogen Applications, Respectively ...................................64 Table I17. Noparametrically Bootstrapped Means of Net Returns, Revenues, NitrogenRelated Costs, Yields and Nitrogen Application Rates for Each Application System, Assuming 32% and 50% NitrogenUse Efficiency for Preplant and Topdress Nitrogen Applications, Respectively ..............65 Table I18. Nonparametrically Bootstrapped Means of Paired Differences of Expected Profits, Expected Nitrogen Application Rates, and Expected Yields, Assuming 32% and 50% NitrogenUse Efficiencies for Preplant and Topdress Nitrogen Applications, Respectively ...................................66 Table II1. Locations, Years, Soil Types, and Nitrogen Levels, and Replications for Experiments ...............................................................................................81 Table II2. Partial Budget for Creation and Use of a NitrogenRich Strip on a 63 ha Field ...........................................................................................................84 Table II3. Partial Budget for Creation and Use of Three Ramped Strips on a 63 ha Field ...........................................................................................................84 Table II4. Unrestricted and Restricted Linear ResponsePlateau Functions of Wheat Yield as a Function of Nitrogen Application with Random Parameters for SiteYear ..................................................................................................100 Table II5. Estimated Wheat Yield Response to Nitrogen by SiteYear (kg ha1).....101 Table II6. Midseason Predicted Wheat Yield Response to Nitrogen Based on the Ramped Strip Method ..............................................................................106 Table II7. Predicted Production Function Parameters by SiteYear Using Nitrogen Rich Strip Method ....................................................................................109 Table II8. Nonparametrically Bootstrapped Means and Standard Errors of Expected Net Revenue, Expected Yield Revenue, Expected Nitrogen and PrecisionRelated Costs, Nitrogen Application Rates and Yields for Each System Assuming All Preplant Nitrogen from Anhydrous Ammonia ....112 Table II9. Nonparametrically Bootstrapped Means and Standard Errors of the Paired Differences Assuming All Preplant Nitrogen from Anhydrous ix Table Page Ammonia..................................................................................................114 Table II10. Nonparametrically Bootstrapped Means and Standard Errors of Expected Net Revenue, Expected Yield Revenue, Expected Nitrogen and PrecisionRelated Costs, Nitrogen Application Rates and Yields for Each System Assuming All Preplant Nitrogen from Dry Urea ........................117 Table II11. Nonparametrically Bootstrapped Means and Standard Errors of Paired Differences in Profits Given All Preplant Nitrogen from Dry Urea ........118 Table II12. Nonparametrically Bootstrapped Means and Standard Errors of Expected Net Revenue, Expected Yield Revenue, Expected Nitrogen and PrecisionRelated Costs, Nitrogen Application Rates and Yields for Each System Assuming No Increase in NitrogenUse Efficiency ....................121 Table II13. Nonparametrically Bootstrapped Means and Standard Errors of the Paired Differences Assuming No NitrogenUse Efficiency Increase from Midseason Application ...........................................................................122 Table III1. Summary Statistics of Survey Samples ...................................................142 Table III2. Preferences for Algal Bloom and Water Level: Multinomial Logit Estimates ..................................................................................................144 Table III3. WillingnesstoPay Estimates ..................................................................150 x LIST OF FIGURES Figure Page Figure I1. Yield as a linear responseplateau function of nitrogen application. ...........8 Figure I2. Profit as a function of nitrogen application..................................................8 Figure I3. Expected profit maximizing nitrogen application rate vs. standard deviation of the plateau prediction error. .....................................................9 Figure I4. Ramped strip recommendation at one strip vs. ramped strip recommendation from the other strip in the same pair at the same sensing date. ............................................................................................................27 Figure I5. Nitrogen fertilizer optimization algorithm recommendation at one strip vs. nitrogen fertilizer algorithm recommendation from the other strip in the same pair at the same sensing date.......................................................28 Figure I6. Mean ramped strip recommendation from one pair of strips vs. mean ramped strip recommendation from the other pair in the same field at the same sensing date. ......................................................................................29 Figure I7. Mean nitrogen fertilizer optimization algorithm recommendation from one pair of strips vs. mean nitrogen fertilizer optimization algorithm recommendation from the other pair in the same field at the same sensing date. ............................................................................................................30 Figure I8. Ramped strip recommendation from MiMarch vs. ramped strip recommendation from the same strip in MidFebruary. ............................31 Figure I9. Nitrogen fertilizer optimization algorithm recommendation from Mid March vs. nitrogen fertilizer optimization algorithm recommendation from the same strip in MidFebruary. ........................................................32 Figure I10. Ramped strip recommendation vs. nitrogen fertilizer optimization algorithm recommendation from the same strip at the same sensing date.33 Figure I11. Map of experimental locations...................................................................36 Figure I12. Plot of yield data and estimated production function for Lahoma 2007. ...59 Figure II1. Map of experimental locations...................................................................82 Figure II2. Plot of yield data and estimated production function for Lahoma 2007. .105 Figure III1. Choice card from survey without uncertainty. .........................................138 Figure III2. Choice card from survey with uncertainty. ..............................................140 xi Figure Page Figure III3. Estimated probability weighting function. ...............................................148 Figure III4. Willingnesstopay to reduce the probability of an algal bloom..............152 xii NOMENCLATURE ER nitrogen application rate historically recommended by the Oklahoma Cooperative Extension Service (90 kg N ha1) LRP linear responseplateau MNL multinomial logit NDVI normalized difference vegetation index NFOA nitrogen fertilizer optimization algorithm N nitrogen NH3 anhydrous ammonia NRS nitrogenrich strip NRSD deterministic nitrogen needs predictor based on the nitrogenrich strip NRSU stochastic nitrogen needs predictor based on the nitrogenrich strip NUE nitrogenuse efficiency ORI optical reflectance imaging PPD deterministic perfect predictor of actual nitrogen needs based on yield data PPU stochastic perfect predictor of actual nitrogen needs based on yield data xiii RS ramped strip RSD deterministic nitrogen needs predictor based on ramped strip RSU stochastic nitrogen needs predictor based on ramped strip UAN ureaammonium nitrate solution WTP willingnesstopay 1 I. CHAPTER I PREDICTION UNCERTAINTY AND THE VALUE OF INCREASINGLY SPATIALLY PRECISE NITROGEN NEEDS INFORMATION Abstract Nitrogen fertilizer is intensively used in crop agriculture in the United States, and many researchers embrace the goal of improving nitrogenuse efficiency—that is, increasing the proportion of nitrogen fertilizer that is actually used by the crop. This goal can be achieved by applying nitrogen fertilizer to match plant needs as they vary over both time and space. Several different precision agriculture systems have been designed to address this variability of nitrogen needs. Among these innovations are two wholefield systems that use midseason normalized difference vegetation index (NDVI) measures from growing winter wheat to predict the amount of nitrogen the plants require to reach their plateau yield. The nitrogen fertilizer optimization algorithm (NFOA) uses NDVI data from a nitrogenrich strip and a check strip in the same field to determine the rate at which the crop will cease to be responsive to nitrogen. The ramped strip system applies incrementally increasing nitrogen rates in a strip of plots just after planting, and then collects midseason NDVI readings to determine the rate at which crop response ceases. This paper is comprised of two subpapers, the first of which uses datasets from actual ramped strips from onfarm trials. The data used are the outputs from the program 2 Ramp Analyzer 1.2, and include ramped strip recommendations, as well as NFOA recommendations based on these ramped strips. These data are used to determine whether the ramped strip and NFOA recommendations are precise enough to detect spatial variability of nitrogen needs within fields, among fields and among different counties within the state. The results show that the ramped strip recommendation is a noisy measure of nitrogen needs—perhaps too noisy to be unambiguously profitable. The second subpaper uses data from trials at ten experiment station sites throughout the state of Oklahoma. Different preplant nitrogen treatments were applied to replicated plots at these locations between 1998 and 2008, and midseason NDVI and yield data were collected from each plot. These data are used to estimate response of both NDVI and yield to preplant nitrogen as a linear responseplateau. Because the relationship between NDVI and yield is estimated with uncertainty and because the linear responseplateau functional form is nonlinear in parameters, a new methodology is developed using Monte Carlo simulation to predict optimal topdress nitrogen rates based on the NDVI data. This subpaper also determines whether it is necessary to sample NDVI measures from each field, and how much precision—and profit—would be lost by moving from sitespecific (or fieldspecific) NDVI sampling to regionlevel sampling. It is determined that the NDVIbased nitrogen needs predictors developed in this paper are imprecise, with the result that profits from regionlevel sampling and fieldlevel sampling are statistically indistinguishable. Furthermore, it is found that the region and fieldbased sampling systems are no better than breakeven with the historical extension advice to apply preplant anhydrous ammonia at 90 kg ha1. 3 Introduction Crop agriculture in the United States and other developed nations intensively uses nitrogen fertilizer (N) to increase yields. Expenditures on N account for 28% and 32% of operating expenses for U.S. producers of wheat and corn, respectively (United States Department of Agriculture, 2005). Many researchers have focused on improving Nuse efficiency (NUE) in agriculture (e.g., Raun and Johnson, 1999; Greenhalgh and Faeth, 2001; Cassman et al., 1998). Raun and Johnson (1999) find that only 33% of N applied to cereal crops worldwide is recovered in grain. Traditionally, N has been applied prior to planting at a uniform rate selected to meet a yield goal based on historical yields. However, Solie, Raun and Stone (1999) show that natural soil N content (inversely related to crop requirements for N application) varies significantly at a spatial scale of approximately 1 m2. Additionally, many studies (e.g., Lobell et al., 2005; Mamo et al., 2003; Washmon et al., 2002) find that crop response to N varies within and between fields over time. In other words, potential yield and N requirements vary temporally and spatially within and between fields. This variability results from weather, topology, and their combined effects on N deposition, mineralization, and volatilization. Precision agriculture focuses on providing information to reduce uncertainty about N needs so producers can improve profit margins by avoiding under or overapplication of N. One innovation in precision agriculture is the sensorbased nitrogen fertilizer optimization algorithm (NFOA) developed by Raun et al. (2002, 2005). The NFOA uses midseason measures of normalized difference vegetation index (NDVI) from growing plants in a nonlimiting, nitrogenrich strip (NRS) to predict the midseason, topdress N application rate required by the crop. Additionally, Raun et al. (2008) have developed a 4 ramped strip (RS) technology to predict optimal N application rates for crops including corn and wheat. This practice involves applying N at incrementally increasing rates to plots arranged in a strip. Such strips can be used to predict, either by visual inspection or by using an optical reflectance sensor, the midseason, topdress N application rate at which crop response to N will cease. The goal of these technologies is to improve NUE— or reduce loss of N inputs to volatilization and runoff—without decreasing yields, so as to improve producer profits. More than one RS or NRS may be used in a single field, but it is recommended that producers place at least one strip in each field each year (Arnall, Edwards and Godsey, 2008). However, is it likely two fields “very close” to each other have similar N requirements? Or what about three such fields? In other words, what is the optimal spatial scale at which to sample NDVI data from experimental strips? Should fields be divided into management zones with a strip in each zone? Is one strip per field sufficient? Or perhaps several strips spread throughout a county could provide an accurate enough prediction for all fields within the county. A countywide system would be especially valuable to producers who grow wheat for both grain and grazing, for whom establishing an experimental strip might be prohibitively costly due to new fencing costs. The answers to questions about the optimal spatial scale of sampling also will be affected by the strength of the relationship between yields and the NDVI data used to predict them. Despite reduction in uncertainty about spatial and temporal variability of crop response, uncertainty remains an issue for the NFOA and RS technologies as a result of prediction error. Babcock (1992) suggests that uncertainty results in the historic producer habit of “overapplying” N at a uniform rate every year. He proposes chronic overapplication 5 indicates that producers assume crop response to N follows a linear responseplateau (LRP) functional form in which the plateau is uncertain. Tembo et al. (2008) similarly address uncertainty about plateau yields among fields and years. They develop an analytical formula to determine the optimal application rate given interannual or interfield variability of plateau yields. Both Babcock (1992) and Tembo et al. (2008) show that the expected profit maximizing strategy given uncertainty about plateau yields is to apply more N than the deterministic solution suggests. Therefore, inclusion of uncertainty—especially prediction error in the relationship between NDVI and yields— may be essential to accurately predicting the expected profit maximizing midseason, topdress N application rate using the NFOA or RS. This means that prediction error in the predicted intercept and slope should be addressed in addition to plateau uncertainty to improve N requirement prediction. The remainder of this paper (following the theory section) is divided into two subpapers, which use different datasets to explore sets of related questions about spatial variability of N requirements. The objectives of the first section are 1a) to determine whether N requirements as predicted by the RS and the NFOA vary by county within a single year and 2a) to determine how consistent (or repeatable) NFOA and RS predictions are over time and space. The objectives of the second section are 1b) to determine whether average plant N requirements for a large region vary by year, 2b) to develop a new process for including prediction error in the RS predictor and 3b) to estimate the relative profitability of four different systems for choosing N application rates. These systems are: 6 a) a perfect predictor system that uses yield data directly to determine the expected profit maximizing topdress N application rate; b) the historical recommendation of 90 kg N ha1 as preplant anhydrous ammonia (NH3); c) a siteyearspecific, NDVIbased predictor of topdress N requirements based on the process developed in objective (2b) above; and d) a regionyearspecific, NDVIbased predictor of topdress N requirements based on the process developed in objective (2b) above. The results will determine whether annual collection of state or countylevel NDVI data—and subsequent dissemination of N recommendations based on these data—has potential value for winter wheat producers in Oklahoma. Such regional N recommendations, if accurate, might be especially beneficial to those who produce wheat for both grazing and grain, who would likely find the cost of fencing off an experimental strip in each field prohibitive. Notably, using a regionbased system would entail more uncertainty about N requirements at any particular site. However, rather than seeking to reduce uncertainty in N requirements predictions, this work seeks to account for remaining uncertainty in the predictors, and thereby to reduce the cost of prediction error. Theory Prior research indicates that output is a function of the most limiting input (e.g., Paris and Knapp, 1989; Berck and Helfand, 1990; Paris, 1992; Chambers and Lichtenberg, 1996; Berck, Geoghegan, and Stohs, 2000; Monod et al., 2002). This functional form is known as a linear responseplateau (LRP). Here, the most limiting input is assumed to be either 7 N or an unspecified input that is represented as a plateau level of output. However, variables determining the intercept and plateau yields—such as N deposition, mineralization and volatilization—are not known in advance at any given site in any particular year (Mamo et al., 2003). Thus, producers face substantial uncertainty in choosing N application rates. Midseason collection of NDVI data from each site each year can reduce uncertainty caused by spatial and temporal variability. However, predicting yields based on NDVI introduces prediction error that has not yet been addressed in the NFOA or RS methods. The following brief example illustrates how prediction error about the plateau (and only the plateau) affects the process of expected profit maximization. Brief Example: Expected Profit Maximization when the Plateau Yield Is Predicted with Error Suppose a LRP function of expected yield response to N has been predicted for a single siteyear based on NDVI data from a RS. For ease of exposition, assume that all parameters besides the plateau are predicted without error—an admittedly unrealistic assumption. Figure I1 illustrates the hypothetical LRP function. Figure I2 illustrates the resulting profit function. These two figures show that, when the plateau yield is known with certainty, the profit maximizing N application rate is 30 kg ha1. Observe the slope of the profit function before and after the optimal rate to see that underapplication is relatively more costly than overapplying by the same amount due to the relative prices of N and wheat. However, because the plateau yield is predicted with error, the costs of under or overapplying are not guaranteed—i.e., there is some probability that applying 8 2000 2050 2100 2150 2200 2250 2300 2350 2400 0 10 20 30 40 50 60 Nitrogen Application Rate (kg ha1) Wheat Yield (kg ha1 ) Figure I1. Yield as a linear responseplateau function of nitrogen application. 300 320 340 360 380 400 420 440 0 10 20 30 40 50 60 Nitrogen Application Rate (kg ha1) Profit ($ ha1 ) Figure I2. Profit as a function of nitrogen application. 9 25 30 35 40 45 50 55 60 0 500 1000 1500 2000 2500 Standard Deviation of Plateau Prediction Error (kg ha1) Expected Profit Maximizing Nitrogen Application Rate (kg ha1 ) Figure I3. Expected profit maximizing nitrogen application rate vs. standard deviation of the plateau prediction error. an additional kg of N will increase profits, and some probability that it will only increase costs. The rate that maximizes expected profit is that at which the probability the crop will use the last kg of N applied is the price of N divided by the price of wheat. This fulfils the necessary condition that expected marginal revenue must equal marginal cost for an expected profit maximum. The N application rate at which this condition is met depends upon the variability of the plateau. In this case, it depends on the prediction error in the plateau parameter. Figure I3 shows the schedule of expected profit maximizing N application rates for varying levels of uncertainty about the plateau based on equation (14) in Tembo et al. (2008). As prediction error in the plateau parameter increases, higher N application rates are required to satisfy the necessary condition that expected marginal revenue equals 10 marginal cost. Note again that this example treats only the error in the predictive relationship between NDVI data and the yield plateau, assuming the other parameters are known with certainty. Prediction error in the intercept, slope and plateau parameters of predicted yield LRP functions will be jointly addressed by Monte Carlo simulation in the procedures section, but consideration of these prediction errors is not conducive to graphical analysis. The Producer’s Decision Problem: Choosing the Expected Profit Nitrogen Application System A producer’s decision problem is to maximize expected profit under uncertainty (from several sources) by choosing an N recommendation system. This problem can be written as: (1) max [ ( ( ) ( ))] k k k k k k E π y N N = F φ , where k π is profit from system k; y is yield; k N is the nitrogen rate recommended by system k; k φ is the information set used by system k in making an N requirement prediction; and k F is the function used by system k to make a prediction based on k φ . An expected profit maximizing producer will abandon information set 1 φ and adopt information set 2 φ only if: (2) [ ( ( ) ( ))] [ ( ( ) ( ))] 1 1 1 1 1 2 2 2 2 2 E π y N N = F φ < E π y N N = F φ . For example, imagine that information set 1 φ provides a more accurate prediction of N needs than information set 2 φ , helping the producer to reduce N costs from overapplication, but that it provides this increased accuracy at a cost that exceeds the expected 11 N savings. In this case, the producer expects more profit from a less accurate predictor due to the high cost of information, and will switch from 1 φ to 2 φ . Thus, improved prediction accuracy attained by using fieldspecific information rather than regionspecific information must be sufficient to offset the cost of the more spatially precise information. In the case of NDVIbased predictors, prediction error will be determined by multiple factors, including the strength of the relationship between midseason NDVI data and yield, measurement and sampling error in collecting the NDVI measures, as well as the spatial scale of the data collected. So the questions arise: How do crop N requirements vary among fields? Do they vary among regions? Are they predictable using NDVI data? How Do Nitrogen Needs Vary Spatially, and What Are the Implications? That Crop N requirements vary temporally and spatially is well established (Lobell et al., 2005; Mamo et al., 2003; Washmon et al., 2002). Both spatial and annual variability in N requirements are related to weather and climate. If spatial variability of N requirements is detectable for different regions (counties, say) within a state, knowledge of this variability could allow somewhat accurate prediction of N requirements for fields within the region. Accounting for both spatial and temporal effects, crop N response is assumed to follow the form: (3) pit pit i t i i t t pit y = min( + N + v + , P + v + + + ) + u 0 1 β β ε ω ε υ , where pit y is the yield on plot p in field i in year t; pit N is the N application rate on plot p of field i in year t; 0 β and P are the estimated intercept and yield plateau, respectively; 1 β is the slope of N response; i v and i ω are random effects for field, shifting the 12 intercept and plateau, respectively; t ε and t υ are random effects for year, also shifting the intercept and plateau, respectively; pit u is a random disturbance from the mean; and i v , i ω , t ε , t υ , and pit u are all independent and normally distributed with means of zero and variances 2 v σ , 2 ω σ , 2 ε σ , 2 υ σ , and 2 u σ , respectively. When the true parameters of equation (3) are known, the uniform profit maximizing N requirement for field i in year t ( it N ) can be expressed as follows: (4) + + − + + − > + + − − = 0, otherwise, ( ) , if ( ) ( ) i t 0 1 c i t 0 i t 0 1 a it P p P P p N ω υ β β ω υ β ω υ β β here c p and a p are the price of the crop and the cost of applying N, respectively, and the remaining symbols are previously defined. Because P , 0 β and 1 β are constant, the only parameters changing N requirement from one siteyear to another are i ω and t υ . If annual effects ( t υ ) on N requirements within a region are significant and large, and if they can be predicted based on some information set—NDVI from RSs at experiment stations, say—producers may find a regional prediction of this annual effect valuable. If the annual effects are large relative to fieldspecific effects ( i ω ) on N requirements, a fieldspecific information set may not significantly improve producer profit relative to a regional information set. Thus regional predictions of N requirements might be preferable. 13 SubPaper 1: Spatial Variability, Repeatability and Noise in Predictions Made by the Nitrogen Fertilizer Optimization Algorithm and the Ramped Strip Data The first dataset used (hereafter called “countylevel data”) is comprised of onfarm trials conducted in 2007. This dataset contains 268 observations from onfarm trials of RSs in 15 counties in Oklahoma. Each observation includes the county in which the trial was located, a RS recommendation, a NFOA recommendation, the predicted yield intercept and plateau from the NFOA, and amounts of N actually applied by the producer prior to planting. The exact location of each strip within the county was not recorded. Table I1 gives the number of observations, mean RS recommendation, mean NFOA recommendation, and the mean predicted yield intercept and plateau from the NFOA by county. All of these measures are outputs of the program Ramp Analyzer 1.2 that fits a linear responseplateau function to the NDVI data to determine the N requirements if N is to be applied at the Feekes 5 growth stage (Raun et al., 2008). The N recommendations in this dataset are used to determine whether the recommendations of the NFOA and RS technologies predict any consistent variability in N requirements among counties. Also, total rainfall data by county are provided from Oklahoma Mesonet stations in or near each county. Rainfall is low for some counties (lowest is 58.29 cm) and high for other counties (highest is 150.80 cm) The second dataset (hereafter called “fieldlevel data”) contains observations from nine onfarm RS trials conducted in Canadian County in 2008. To create these data, two pairs of RSs were applied in each field as topdress ureaammonium nitrate solution 14 Table I1. Number of Observations, Mean Ramped Strip Nitrogen Recommendation, Mean Nitrogen Recommendations, and Mean Predicted Plateau Yield by County for Dataset Two County Trials RS Rate (kg ha1) NFOA Rate (kg ha1) NFOA Intercept (kg ha1) NFOA Plateau (kg ha1) Total Rainfall (cm) Blaine 10 24.53***a (6.01)b c c 3448.70*** (125.87) 132.23 Canadian 44 66.18*** (7.19) 21.51*** (2.23) 2781.62*** (71.76) 3395.95*** (91.94) 135.94 Ellis 5 22.62* (8.81) 5.38** (1.64) 1710.91*** (154.12) 1837.25*** (150.71) 58.29 Grant 20 47.77*** (6.76) 22.68*** (3.80) 2893.63*** (190.74) 3648.15*** (209.50) 103.73 Greer 3 45.17** (7.71) 15.68* (3.88) 1870.40** (266.96) 2273.60** (358.34) 77.13***d (6.42) Jackson 6 64.49** (21.51) 22.40** (6.16) 2619.68*** (130.69) 3178.00*** (246.80) 55.35 Kingfisher 2 83.44 (22.96) 23.52 (5.60) 2701.44 (739.20) 3944.64*** (60.48) 146.46 Muskogee 83 60.75*** (4.77) 26.21*** (2.50) 2925.95*** (70.85) 3599.05*** (92.96) 121.87 Noble 19 67.61*** (11.18) 24.93*** (3.04) 2639.90*** (140.87) 3355.76*** (152.51) 150.80 Nowata 15 58.54*** (6.54) 29.27*** (3.59) 3084.93*** (102.25) 4271.23*** (133.21) 108.43 Okmulgee 5 60.70*** (7.12) 17.47** (4.05) 2870.52*** (162.74) 3316.45*** (241.04) 112.70***e (27.84) Ottowa 33 63.57*** (4.78) 26.57*** (2.43) 2643.20*** (59.50) 3322.53*** (74.31) 121.92 Pawnee 10 77.62*** (20.43) 35.39*** (5.54) 2461.54*** (136.95) 3423.84*** (202.78) 135.08 Payne 5 88.48*** (7.10) 40.77* (16.61) 3240.38*** (352.27) 4359.94*** (262.83) 137.03***f (4.89) Wagoner 8 66.36*** (15.49) 25.06*** (3.77) 2872.80*** (112.59) 3492.30*** (175.43) 111.89 a One, two or three asterisks indicate statistical significance at the 0.10, 0.05 or 0.01 levels, respectively. The null hypothesis is that the means are zero. b Numbers in parentheses are standard errors. c This variable is not available for observations in Blaine County. d This is the average measure from the three closest Mesonet stations. e This is the average measure from the two Mesonet stations in Okmulgee County. f This is the average measure from the three Mesonet stations in Payne County. 15 (UAN) after plant emergence. Paired strips were made by making two adjacent passes over the field with the RS applicator, so that the rates in the paired strips increase in opposite directions. Each of the four strips was analyzed with a handheld Greenseeker optical sensor three times during the growing season, so three RS recommendations, three NFOA recommendations, and three yield plateaus and intercepts predicted by the NFOA are available from each strip. It should be noted that in this dataset (but not in the countylevel data) the predicted yield plateaus from the NFOA are right censored at 6048 kg ha1 (90 bu ac1) even when the predicted intercept is above this level. Such censoring may mean that the NFOA predicts no N response even when the raw NDVI data clearly show N response. Table I2 lists the planting dates and sensing dates for each field. The amount of N applied by producers prior to sensing was not recorded. These data are used to determine how repeatable NFOA and RS recommendations are over space and through time within fields as a measure of how much noise is present in the predictions. Procedures The important question of whether the NFOA and RS recommended N application rates vary by county within a single year is addressed using the countylevel data. If different counties have significantly different N requirements, and if these can be predicted by the RS or NFOA, a regional N requirement prediction system based on NDVI may have predictive value. To test for countylevel effects, the following Tobit model is estimated: (5) = + + + ≤ + + = + + + > = Σ Σ Σ − = − = − = 0 if 0 if 0 1 1 * 1 1 * 1 1 jk K k jk jk k k jk K k jk jk k k K k jk k k jk r N D N D r N D r α β δ μ α β δ α β δ μ 16 Table I2. Planting Date and Sensing Dates for Each Field in Dataset Three Field Planting Date Sensing Dates AC 11/6/2007 01/31/2008 AM 10/10/2007 02/01/2008 02/19/2008 03/11/2008 DE 10/14/2007 01/31/2008 02/19/2008 03/11/2008 JL 10/12/2007 01/31/2008 02/20/2008 03/11/2008 KM 10/5/2007 01/31/2008 02/19/2008 03/11/2008 LZ 10/9/2007 01/23/2008 01/31/2008 02/19/2008 RZ 10/12/2007 02/04/2008 02/19/2008 03/11/2008 SN 10/12/2007 02/04/2008 02/20/2008 03/11/2008 TZ 10/10/2007 01/31/2008 02/19/2008 03/11/2008 where jk r is the RS recommendation from strip at site j in county k; α is the intercept recommendation; β is the effect of preplant N application on the RS recommendation; jk N is the amount of preplant nitrogen applied at site j in county k; k δ is a fixed effect affecting the mean N recommendation for county k; k D is an indicator variable equal to one when county is k, and zero otherwise; K is the number of counties; * jk r is an index of the crop’s predicted “need” for N at site j in county k; jk μ is a normally distributed random deviation in predicted N requirements at site j in county k, with mean zero and 17 variance 2 μ σ . Based on this model, a likelihood ratio test is used to test the null hypothesis that county level variation in RS recommendations does not exist (i.e., δ k = 0, ∀ k ). A ttest is used to determine whether preplant application of N has any impact on RS recommendations (whether β = 0 ). The estimation is done using PROC QLIM in SAS. The above estimation in equation (5) is repeated using the NFOA recommendations as the dependent variable, and perform the hypothesis tests again to determine whether NFOA recommendations vary by county. The important questions of repeatability of RS and NFOA recommendations across time and space are addressed using the fieldlevel data. Poor repeatability of these recommendations at the same strip over time, or low correlation between recommendations from two adjacent strips would indicate that the RS or NFOA recommendations are too noisy to be useful in predicting N requirements at the singlefield level. Such noise could stem from either measurement error or high spatial variability within the field. To determine whether RS detects significant withinfield variability of N requirements, the following nointercept Tobit model is estimated: (6) = + ≤ = + > = Σ Σ Σ = = = 0 if 0 if 0 1 * 1 * 1 ijt J j ijt j j ijt J j ijt j j J j j j ijt r D D r D r δ ε δ δ ε where ijt r is the predicted optimal N application rate on strip i in pair j on sensing date t; j δ is a fixed effect for pair j; j D is an indicator variable equal to one for pair j, and zero otherwise; * ijt r is a latent variable representing the level of N (including residual and applied N) the plants in strip i in pair j on sensing date t need to reach the predicted 18 plateau yield; ijt ε is a random error term distributed with mean zero and variance 2 ε σ ; and J is the number of strip pairs. The first hypothesis tested is that N requirement predictions from the RS do not vary between pairs located within the same field—i.e., δ = δ δ = δ δJ = δJ 1 2 3 4 −1 , , Κ , . Rejection of this hypothesis would indicate that predicted N requirements from the RS vary consistently by pair within each field. Failure to reject the hypothesis would indicate either 1) that there is little variability of N requirements between locations within a field or 2) that the RS is not precise enough to detect this variability. Next, the model is restricted so that predicted N requirements do not vary by field—i.e., j y j y δ =δ , ∀ , —to determine whether the RS detects significant variability of N requirements between fields. Equation (6) is then reestimated using the NFOA predictions as the dependent variable ( ijt r ) to determine whether the NFOA recommendations vary consistently within and between fields. Additionally, graphical analyses and correlation coefficients are used to determine the strength and significance of the relationships between both RS and NFOA recommendations from 1) strips in the same pair at the same sensing date, 2) different pairs (mean recommendation) in the same field at the same sensing date, and 3) the same strip at the second and third sensing dates. The second and third sensing dates were chosen because the second date is (usually) closest to Feekes 5—the growth stage at which topdress N is normally applied—and because the third sensing date (usually in March) is closest to harvest, and may therefore be the most accurate. The correlation and plot of the relationship between RS and NFOA recommendations at the same strip for the same sensing date are also provided. 19 Results Based on the countylevel data, results from equation (5) are presented in table I3. Here, the predicted optimal topdress application rate (either from the RS or the NFOA) is modeled as a function of 1) the preplant N application rate for the field and 2) the county in which the field is located. Notably, the mean RS recommendation (64.01 kg N ha1) is more than twice the mean recommendation from the NFOA (31.14 kg N ha1). The signs of the β coefficients for the RS models are negative, which is expected because higher preplant N applications reduce the need for topdress N. Student’s ttests, however, indicate that preplant N application has no statistically significant effect on predicted topdress N requirements from the RS method—i.e., the null hypothesis β = 0 cannot be rejected. On the other hand, the β coefficients for the NFOA models are not only negative but are also statistically significant. Assuming NUE of 32% and 50% for preplant and topdress N, respectively, one kg ha1 of preplant N should reduce the need for topdress N by 0.46 kg ha1, but the coefficients are much smaller: estimated reductions of topdress needs range from 0.12 to 0.22 kg ha1 per additional kg ha1 of preplant N, depending on the model. The likelihood ratio statistic to determine whether RS method recommendations vary by county is LR = −2(991.16 − 998.61) = 14.90 , and is distributed chisquare with 13 degrees of freedom. The chisquare critical statistic at the 0.10 level is 19.81, so the test provides no evidence that RS recommendations vary by county. Similarly, no evidence is found to indicate that NFOA recommendations vary by county. The likelihood ratio statistic for this test is LR = −2(845.85 − 852.50) = 13.30 , which is also 20 Table I3. Ramped Strip and Nitrogen Fertilizer Optimization Algorithm Recommendations as Functions of FarmerPractice Preplant Nitrogen Rate and County Ramped Strip Recommendations Nitrogen Fertilizer Optimization Algorithm Recommendations Parameter Definition Unrestricted Restricted Unrestricted Restricted α Intercept 71.29***a (6.61)b 64.01*** (3.69) 27.18*** (3.34) 31.14*** (1.86) β Effect of Preplant Nitrogen 0.13 (0.11) 0.12 (0.10) 0.18*** (0.06) 0.22*** (0.05) 1 δ Effect for County 2 37.88** (18.79)  12.67 (9.48)  2 δ Effect for County 3 17.59* (9.80)  3.91 (4.95)  3 δ Effect for County 4 25.03 (21.61)  9.58 (10.90)  4 δ Effect for County 5 6.00 (16.97)  1.04 (8.69)  5 δ Effect for County 6 19.45 (26.08)  6.68 (13.14)  6 δ Effect for County 7 14.82* (8.21)  6.19 (4.15)  7 δ Effect for County 8 6.88 (10.80)  4.95 (5.45)  8 δ Effect for County 9 10.40 (10.90)  5.42 (5.50)  9 δ Effect for County 10 8.40 (17.06)  6.61 (8.61)  10 δ Effect for County 11 6.58 (9.32)  1.71 (4.71)  11 δ Effect for County 12 9.74 (12.63)  13.05** (6.38)  12 δ Effect for County 13 17.19 (17.36)  13.58 (8.76)  13 δ Effect for County 14 2.30 (13.90)  1.60 (7.02)  2 μ σ Error Variance 35.89*** (1.81) 37.28*** (1.88) 18.10*** (0.92) 18.73*** (0.95) Log Likelihood 991.16 998.61 845.85 852.50 Notes: The unrestricted models allow the mean N recommendation to vary by county, while the restricted models estimate a single mean for all counties. Units are kg ha1. a One, two or three asterisks represent statistical significance at the 0.10, 0.05 or 0.01 confidence levels, respectively. b Numbers in parentheses are standard errors. c No standard error is estimated because the parameter is restricted. 21 distributed chisquare with 13 degrees of freedom. Thus, neither the NFOA nor the RS predicts any statistically significant variability of N requirements by county. This does not mean, however, that actual N requirements do not vary by county, nor does it mean that this variability cannot be predicted using NDVI data—only that it was not predicted by the RS and NFOA methods used in the countylevel data from 2007. The issues of within and betweenfield variability of N requirements are addressed using the fieldlevel data, which includes data from nine fields in Canadian county in 2008. These data are used to estimate equation (6), which models the predicted optimal N application rate (from the RS or NFOA) as a function of the set of paired adjacent strips in which the strip is located. The estimated parameters of this equation for the RS are contained in table I4. The model with pair effects allows the mean predicted N requirement to be unique for each pair of adjacent strips, while the model with field effects is restricted such that pairs in the same field must have the same mean prediction, and the pooled model assumes the same mean N requirement for all strips in the dataset. To determine whether field affects the N recommendation from the RS, the field effects model is tested against the pooled model using a likelihood ratio test. The test statistic (chisquare with 8 degrees of freedom) is LR = −2(−479.26 + 473.43) = 11.66 , but the chisquare critical statistic at the 0.10 level is 13.36, so the test provides no evidence of variation in N requirements predicted by the RS among fields. Because variation in N requirements among fields is well documented (see Lobell et al., 2005; Mamo et al., 2003; Washmon et al., 2002), this result likely indicates that the RS technology is not precise enough to detect this variability. The test to determine whether mean N recommendations vary among pairs of adjacent strips compares the model with 22 Table I4. Mean Ramped Strip Recommendation, with and without Fixed Effects for Strip Pair and Field Model Parameter Definition Pair Effects Field Effectsa Pooledb 1 δ Fixed effect for pair 1 10.08 (21.18) 19.04 (16.07) 35.19*** (3.42) 2 δ Fixed effect for pair 2 28.00 (21.18) 19.04 (16.07) 35.19*** (3.42) 3 δ Fixed effect for pair 3 65.15*** (12.23) 48.91*** (9.28) 35.19*** (3.42) 4 δ Fixed effect for pair 4 32.67 (12.23) *** 48.91*** (9.28) 35.19*** (3.42) 5 δ Fixed effect for pair 5 59.36*** (12.23) 49.75*** (9.28) 35.19*** (3.42) 6 δ Fixed effect for pair 6 40.13*** (12.23) 49.75*** (9.28) 35.19*** (3.42) 7 δ Fixed effect for pair 7 35.47*** (12.23) 23.07** (9.37) 35.19*** (3.42) 8 δ Fixed effect for pair 8 10.40 (12.54) 23.07** (9.37) 35.19*** (3.42) 9 δ Fixed effect for pair 9 26.88** (12.23) 31.08*** (9.28) 35.19*** (3.42) 10 δ Fixed effect for pair 10 35.28*** (12.23) 31.08*** (9.28) 35.19*** (3.42) 11 δ Fixed effect for pair 11 35.47*** (12.23) 34.91*** (9.28) 35.19*** (3.42) 12 δ Fixed effect for pair 12 34.35*** (12.23) 34.91*** (9.28) 35.19*** (3.42) 13 δ Fixed effect for pair 13 16.07 (12.51) 18.61** (9.49) 35.19*** (3.42) 14 δ Fixed effect for pair 14 21.72* (12.48) 18.61*** (9.49) 35.19*** (3.42) 15 δ Fixed effect for pair 15 24.64** (12.23) 33.13*** (9.28) 35.19*** (3.42) 16 δ Fixed effect for pair 16 41.63*** (12.23) 33.13*** (9.28) 35.19*** (3.42) 23 Table I4. Mean Ramped Strip Recommendation, with and without Fixed Effects for Strip Pair and Field Model Parameter Definition Pair Effects Field Effectsa Pooledb 17 δ Fixed effect for pair 17 68.48*** (12.33) 47.25*** (9.34) 35.19*** (3.42) 18 δ Fixed effect for pair 18 26.69** (12.23) 47.25*** (9.34) 35.19*** (3.42) 2 ε σ Variance of error 29.95*** (2.18) 32.15*** (2.34) 34.05*** (2.48) Log Likelihood 466.79 473.43 479.26 Note: Units are kg ha1. a This model is restricted such that 1 2 3 4 5 6 17 18 δ =δ , δ =δ , δ =δ , Κ , δ =δ . b This model is restricted such that 1 2 3 18 δ =δ =δ =, Κ ,= δ . c One, two or three asterisks (*) indicate statistical significance at the 0.10, 0.05 or 0.01 confidence level, respectively. d Numbers in parentheses are standard errors. pair effects to the pooled mean model. The likelihood ratio statistic, which is distributed chisquare with 17 degrees of freedom, is LR = −2(−479.26 + 466.79) = 24.94 . Since the likelihood ratio statistic is slightly greater than the critical value—24.77 at the 0.10 confidence level—the test provides some evidence that mean N recommendations vary among pairs of strips in a consistent way. However, because yield data are not provided, nothing can be said about the economic significance of this finding. What is surprising, though, is that the statistical significance is not stronger. The inference is that recommendations from two adjacent strips in a pair selected at random are only slightly more homogeneous than readings from two randomly selected strips from different pairs—perhaps on opposite sides of Canadian county. The fact that RS predictions of N requirements do not show strong spatial correlation within pairs perhaps indicates that the predictions are imprecise. The lack of precision could be caused by measurement error, 24 such as would occur if the person reading the strip walked at an uneven pace while using the handheld sensor. It should also be noted that the pair effects model does not have a significantly better fit than the field effects model. The likelihood ratio statistic is LR = −2(−473.43 + 466.79) = 13.28 , and is less than 14.68—i.e., the chisquare critical statistic with 9 degrees of freedom at the 0.10 confidence level. This means that the RS detects no within field variability of N requirements. Table I5 shows the mean N application rate recommended by the NFOA with and without fixed effects for strip pair and field. The likelihood ratio test for field effects compares the model with field effects to the pooled model. The likelihood ratio statistic is LR = −2(−358.39 + 301.65) = 113.48 with 8 degrees of freedom, which exceeds the chisquare critical value of 20.09 at the 0.01 level. The likelihood ratio statistic to determine whether pair effects improve the fit of the model relative to field effects alone is LR = −2(−301.65 + 292.12) = 19.06, and is distributed chisquare with 9 degrees of freedom, and is greater than the critical statistic at the 0.10 level (16.92). Thus, the test finds (marginal) evidence that different sets of paired strips within the same field can have significantly different N recommendations—or that recommendations from adjacent strips in the same pair are more homogeneous than two randomly selected strips from different pairs but within the same field. However, the economic significance of this finding is unknown because yield data are unavailable to verify prediction accuracy. Figures I4 and I5 show plots and correlations of the recommendations from strips in the same pair at the same sensing date for the RS and NFOA, respectively. Note that the correlation between RS recommendations from adjacent strips in figure I4 is 25 Table I5. Mean Nitrogen Fertilizer Optimization Algorithm Recommendation, with and without Fixed Effects for Strip Pair and Field Model Parameter Definition Pair Effects Field Effectsa Pooledb 1 δ Fixed effect for pair 1 7.28 (19.12) 10.64 (15.47) 16.11*** (6.24) 2 δ Fixed effect for pair 2 14.00 (19.12) 10.64 (15.47) 16.11*** (6.24) 3 δ Fixed effect for pair 3 156.84 (0.00) 46.00*** (17.01) 16.11*** (6.24) 4 δ Fixed effect for pair 4 30.53* (16.83) 46.00*** (17.01) 16.11*** (6.24) 5 δ Fixed effect for pair 5 63.65*** (11.04) 63.00*** (8.93) 16.11*** (6.24) 6 δ Fixed effect for pair 6 62.35*** (11.04) 63.00*** (8.93) 16.11*** (6.24) 7 δ Fixed effect for pair 7 156.84 (0.00) 186.66 (0.00) 16.11*** (6.24) 8 δ Fixed effect for pair 8 156.84 (0.00) 186.66 (0.00) 16.11*** (6.24) 9 δ Fixed effect for pair 9 107.71*** (11.04) 81.48*** (8.93) 16.11*** (6.24) 10 δ Fixed effect for pair 10 55.25*** (11.04) 81.48*** (8.93) 16.11*** (6.24) 11 δ Fixed effect for pair 11 43.12*** (11.04) 26.81*** (9.20) 16.11*** (6.24) 12 δ Fixed effect for pair 12 9.86 (12.00) 26.81*** (9.20) 16.11*** (6.24) 13 δ Fixed effect for pair 13 26.74** (11.23) 39.94*** (9.06) 16.11*** (6.24) 14 δ Fixed effect for pair 14 53.94*** (11.15) 39.94*** (9.06) 16.11*** (6.24) 15 δ Fixed effect for pair 15 29.28* (16.63) 45.06*** (16.81) 16.11*** (6.24) 16 δ Fixed effect for pair 16 156.84 (0.00) 45.06*** (16.81) 16.11*** (6.24) 26 Table I5. Mean Nitrogen Fertilizer Optimization Algorithm Recommendation, with and without Fixed Effects for Strip Pair and Field Model Parameter Definition Pair Effects Field Effectsa Pooledb 17 δ Fixed effect for pair 17 45.36*** (11.17) 40.42*** (8.99) 16.11*** (6.24) 18 δ Fixed effect for pair 18 36.03*** (11.04) 40.42*** (8.99) 16.11*** (6.24) 2 ε σ Variance of error 27.04*** (2.53) 30.93*** (2.89) 55.42*** (5.56) Log Likelihood 292.12 301.65 358.39 Note: Units are kg ha1. a This model is restricted such that 1 2 3 4 5 6 17 18 δ =δ , δ =δ , δ =δ , Κ , δ =δ . b This model is restricted such that 1 2 3 18 δ =δ =δ =, Κ ,= δ . c One, two or three asterisks (*) indicate statistical significance at the 0.10, 0.05 or 0.01 confidence level, respectively. d Numbers in parentheses are standard errors. slightly negative, though not significant (p = 0.61). This result indicates that the RS is a noisy predictor of N requirements. On the other hand, the correlation between NFOA recommendations from adjacent strips in figure I5 is 0.56, and is statistically significant (p < 0.01). Figure I6 shows the mean RS recommendation from one pair of strips plotted against the mean RS recommendation from the other pair of strips in the same field at the same sensing date, while figure I7 plots the NFOA recommendations in the same manner. The mean RS recommendations from pairs in the same field have low correlation (0.01) that it is not statistically significant (p = 0.98). However, the mean NFOA recommendations from the different pairs are highly (0.74) and significantly correlated (p < 0.01). Figures I8 and I9 show plots of recommendations at the same strip at the second sensing date (usually February) and the third sensing date (usually March) for the RS and 27 Correlation = 0.08 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 Ramped Strip Recommendation (kg N ha1) Paired Ramped Strip Recommendation (kg N ha1 ) Figure I4. Ramped strip recommendation at one strip vs. ramped strip recommendation from the other strip in the same pair at the same sensing date. NFOA, respectively. For the RS measures, the correlation is only 0.10, and is not statistically significant (p = 0.57). The correlation for the NFOA recommendations is 0.56, and is significant at the 0.01 confidence level. The plots and correlations in figures I4 through I9 indicate that the RS recommendations are not stable over time and space within the same growing season. This result likely indicates that RS recommendations in the fieldlevel data do not very accurately represent actual N requirements. However, the relative spatial and temporal stability of the NFOA recommendations does not necessarily mean that NFOA recommendations are any more accurate than the RS predictions. To explicitly determine whether NFOA predictions are accurate, production 28 Correlation = 0.56 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 Nitrogen Fertilizer Optimization Algorithm Recommendation (kg N ha1) Paired Nitrogen Fertilizer Optimization Algorithm Recommendation (kg N ha1 ) Figure I5. Nitrogen fertilizer optimization algorithm recommendation at one strip vs. nitrogen fertilizer algorithm recommendation from the other strip in the same pair at the same sensing date. functions would have to be estimated using yield response data (which were not recorded) from the fields in the fieldlevel dataset. One reason why the NFOA recommendations show higher spatial relatedness may be the NFOA’s propensity to predict optimal rates of zero kg ha1. The NFOA, as used in the fieldlevel dataset, restricts the predicted plateau yield for each strip to be no greater that 6048 kg ha1. Thus, in cases where the NFOA predicts a yield intercept greater than 6048 kg ha1 the predicted plateau yield is still no greater than 6048 kg ha1, without regard to NDVI response to N. However, if NDVI is a noisy predictor of yield—i.e., if the relationship between NDVI data and yields varies among fields or by wheat variety— 29 Correlation = 0.01 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 Mean Ramped Strip Recommendation (kg N ha1) Mean Ramped Strip Recommendation (kg N ha1 ) Figure I6. Mean ramped strip recommendation from one pair of strips vs. mean ramped strip recommendation from the other pair in the same field at the same sensing date. then imposing this restriction on the plateau yield could bias the NFOA to predict that no N should be applied when, in fact, it would be optimal to apply N in some quantity. Figure I10 shows a plot of NFOA recommendations against RS recommendations from the same strip at the same sensing date. Note that the NFOA often recommends no application while the RS recommends some positive application rate (36 of 100 observations). This means that even when NDVI data indicate an N response— i.e., the average NDVI reading at one end of the strip is different from the average NDVI reading at the other end—the NFOA still assumes no N response by assuming that the relationship between NDVI and yields is estimated without error. However, the error 30 Correlation = 0.74 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 Mean Nitrogen Fertilizer Optimization Algorithm Recommendation (kg N ha1) Mean Nitrogen Fertilizer Optimization Algorithm Recommendation (kg N ha1) Figure I7. Mean nitrogen fertilizer optimization algorithm recommendation from one pair of strips vs. mean nitrogen fertilizer optimization algorithm recommendation from the other pair in the same field at the same sensing date. variance may be large, or may be heteroskedastic such that it increases for higher NDVI readings, or may be unique to each field. Thus, imposing this type of restriction on a plateau predicted with error may bias the NFOA predictions toward zero. Perhaps this problem could be solved by explicitly introducing this error variance into the NFOA. Conclusions First and foremost, the results indicate that the RS technique for N requirements prediction in growing winter wheat is likely too noisy to be useful in terms of accurately 31 Correlation = 0.10 0 20 40 60 80 100 120 140 0 50 100 150 200 250 Ramped Strip Recommendation, MidFebruary (kg N ha1) Ramped Strip Recommendation, MidMarch (kg N ha1 ) Figure I8. Ramped strip recommendation from MiMarch vs. ramped strip recommendation from the same strip in MidFebruary. and consistently predicting optimal N application levels. For example, the RS does not detect any significant, consistent variability of N requirements between counties, between fields, or within fields (tables I3 and I4, respectively). Furthermore, RS recommendations are neither 1) significantly correlated with RS recommendations from nearby strips (figure I4) nor steady across sensing dates (figure I8). These facts together indicate that the RS technology requires continuing development to address the sources of noise that adversely affect the consistency of its predictions. 32 Correlation = 0.56 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 Nitrogen Fertilizer Optimization Algorithm Recommendation, MidFebruary (kg N ha1) Nitrogen Fertilizer Optimization Algorithm Recommendation, MidMarch (kg N ha1 ) Figure I9. Nitrogen fertilizer optimization algorithm recommendation from Mid March vs. nitrogen fertilizer optimization algorithm recommendation from the same strip in MidFebruary. The NFOA recommendations (as opposed to the RS recommendations) seem more consistent with expectations about variability of N requirements between and within fields (table I5 and associated hypothesis tests). NFOA recommendations are also significantly correlated within pairs (figure I5), within fields (figure I7) and across time within the growing season (figure I9). However, the reason for this high correlation may be the restriction on the plateau yield predicted by the NFOA. Because the plateau and intercept are predicted based on the estimated (with error) relationship between NDVI data and yields, the predictions are uncertain. Because of this estimation error, the NFOA 33 Correlation = 0.11 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 Nitrogen Fertilizer Optimization Algorithm Recommendation (kg N ha1) Ramped Strip Recommendation (kg N ha1) Figure I10. Ramped strip recommendation vs. nitrogen fertilizer optimization algorithm recommendation from the same strip at the same sensing date. occasionally predicts crop yields will be unresponsive to N (by capping the predicted yield plateau) even when NDVI data are N responsive. Ultimately, both the NFOA and RS methods used to create these data are too noisy to accurately predict crop N requirements. However, these techniques have been— and continue to be—used by producers (Raun et al., 2008). Producers using the RS and NFOA technologies do so because they believe it is profitable. Perhaps these producers are not using the technology precisely as intended. For example, they may be integrating farmer intuition into the process of choosing an N application rate—using the NFOA or RS in addition to rules of thumb they have always used. It may be optimal to use a combined information set that includes the old (farmer practice) and new (NFOA or RS) 34 decision tools in the choice of N application rates. The results in this paper suggest several potential avenues of related research, including: 1) the creation of a formal Bayesian framework that will allow producers to input a set of fieldspecific rules of thumb, say, into the NFOA and RS methodologies, 2) the development of a framework for including uncertainty (such as the error variance of the relationship between NDVI and yield) in the NFOA or RS methodologies, 3) use of improved estimation methods for the ramped strip linear responseplateau functions and 4) development of more accurate measurement techniques for collecting NDVI data (as opposed to walking with a handheld sensor. Any of these pursuits (or several jointly) might improve the accuracy of midseason N requirements predictions based on the RS and NFOA. SubPaper 2: Prediction Uncertainty and the Value of Increasingly Spatially Precise Sampling of Optical Reflectance Data Data The dataset used in this subpaper consists of experiments conducted at ten sites throughout the state of Oklahoma between 1998 and 2008. The ten sites are located at the Efaw, Haskell, Hennessey, Lahoma, Lake Carl Blackwell, Perkins, Stillwater, and Tipton agricultural experiment stations. Table I6 contains the specifics about N treatment levels, replications, soil types, and dates for each location, while the map in figure I11 shows the locations of the sites. Each siteyear had at least three different levels of N treatment, which differed across sites, and occasionally between years at the same location. The number of replications at each N application rate varies by siteyear. NDVI measures for 35 Table I6. Locations, Years, Soil Types, and Nitrogen Levels, and Replications for Experiments in Dataset One Experiment Station Years Soil Type Nitrogen Treatment Levels (kg ha1) Efaw 1 19992006 Easpur loam 0 (3) 45 (3) 90 (3) 179 (3) 269 (3) 538a (3)b Efaw 2 19992003 Easpur loam 0 (3) 56 (6) 90 (6) 123 (6) Haskell 19992002 Taloka silt loam 0 (8) 112 (16) 168 (4) Hennessey 20002003 Shellabarger sandy loam 0 (3) 56 (5) 90 (6) 123 (6) Lahoma 19992008 Grant silt loam 0 (8) 22 (4) 45 (4) 67 (4) 90 (4) 112 (4) Lake C.B. 2004, 2006 Port silt loam 0 (4) 50 (4) 100 (4) Perkins 1c 19982006 Teller sandy loam 0 (3) 56 (3) 112 (3) 168 (3) Perkins 2 1998 Teller sandy loam 0 (9) 56 (9) 112 (9) 168 (9) Stillwater 19992006, 2008 Norge silt loam 0 (8) 45 (4) 90 (4) 134d (4) Tipton 1998 Tipton silt loam 0 (12) 56 (12) 112 (12) 168 (12) a Rate not available in 2000. b Numbers in parentheses are the number of replications at each rate each year. c Numbers of replications are the same in 1998 as at Perkins 2. d Rate not available in 2004, 2005, 2008. each observation were collected around Feekes growth stage 5, and yield was measured at harvest. These data are used to 1) determine whether year has a significant impact on N requirements across locations throughout the state of Oklahoma, 2) determine the relationship between NDVI information and the parameters of the LRP functions yield response to N, 3) create a framework for introducing the uncertainty about this relationship into a RStype N requirements prediction techniques, and 4) estimate the relative profitability of the different N requirement prediction systems described in the introduction. 36 Figure I11. Map of experimental locations. Based on local cooperative prices on February 14, 2009, assumed prices of N from UAN and NH3 are $1.10 kg1 and $0.57 kg1, respectively. Custom application costs for UAN are assumed to be $9.71 ha1 and custom application of NH3 is $20.49 ha1 (Doye, Sahs and Kletke, 2007). The wheat price is $0.24 kg1 RS application is assumed to take place early after planting as topdress UAN. Producers are advised to apply as many as 3 strips per field, each measuring 3 m by 55 m (0.0165 ha), starting at an application rate of 0 kg ha1, and increasing the application rate in increments of 14.56 kg ha1, until reaching the maximum rate that could possibly be used by the plants (assumed to be 134 kg ha1). Thus, the average N application rate in the three RSs is 67 kg ha1. It is also assumed that because the RSs are applied separately 37 Table I7. Partial Budget for Creation and Use of Three Ramped Strips in a 63 ha field Operating Input Units Price Quantity Cost UAN kg 1.10 3.31 3.64 Road Time km 4.12 8 32.96 Coop Labor hr 17.50 2.50 43.75 Sensor ha 1.08 63.00 68.04 Producer Labor hr 17.50 2.50 43.75 Total Cost of RS Field 192.14 Total Cost of RS ha 3.04 from preplant N, the producer pays road time totaling eight km per field at $4.12 km1 for delivery of the RS applicator. It is assumed that the custom application of the strips takes 2.5 hours of custom labor, and that the producer later spends 2.5 hours reading the three strips with his own Greenseeker® sensor. Thus, the total cost of creating and using three RSs is $192.14 per field, or $3.04 ha1 for a 63 ha field on a 1000 ha farm, where the cost of the sensor is spread over the entire farm. Table I7 is a partial budget for the creation and use of the strips. Procedures Variability of Nitrogen Needs by Year and Location One objective of this subpaper is to quantify variation in N requirements by year and experimental site. This is of interest because, for a regional N requirement prediction system to be of value, annual effects on N needs within the region must exist and be predictable with some accuracy. Thus, tests for fieldspecific and year specific effects on N needs are conducted based on the following model: (7) t t pit N i i i N i t i i N i pit pit i i y = + N +Σv D + P +Σv D +Σ D + + + u − = − = − = min( , ) 1 1 1 1 1 1 0 1 β β ε ω ε υ , 38 pit y is yield on plot p of field i in year t; 0 β is the intercept yield; 1 β is the crop N response rate; P is the expected yield plateau; i v and i ω are fixed effects for field i, shifting the intercept and plateau, respectively; i D is an indicator variable equal to one for field i and zero otherwise; t ε and t υ are random effects for year t, also shifting the intercept and plateau, respectively; pit u is a random disturbance; and t ε , t υ and pit u are from independent normal distributions with means zero and variances 2 ε σ , 2 υ σ , and 2 u σ , respectively. The determination of whether plateau yield shifts randomly (and independently of intercept yield) by year is made using a likelihood ratio test with one degree of freedom to test the restriction 2 = 0 υ σ . Rejection of this restriction would be evidence that accurate predictions of annual effects could be valuable information to producers making a choice of N application rate in the region for which the prediction was made. The restriction , , 0 1 2 1 = = = = N− ω ω Κ ω is also tested, where N −1 is the number of estimated fieldspecific plateau fixed effects in the model, to determine whether the mean yield plateau varies by site (at the fieldlevel, say). If this type of variation of plateau yield is found, it will indicate that N requirements also vary by field, which is expected on the basis of the literature (Lobell et al., 2005; Mamo et al., 2003; Washmon et al., 2002). Defining a Predictive Relationship This paper develops an N requirements prediction system based on the RS methodology that accounts for two types of uncertainty: 1) estimation uncertainty—or uncertainty about the value of the parameters of NDVI response to N—and 2) prediction 39 uncertainty—or uncertainty in the predictive relationship between NDVI response and yield response to N. To this end, the following equations are estimated for each siteyear: (8) it t t it t it y = min( + N , P ) + u 0 1 β β and (9) it t t it t it insey = min(α +α N , φ ) +η 0 1 , where it y is the measured yield on plot i in fieldyear t; 0t β is the intercept yield for fieldyear t; 1t β is the yield response to N in fieldyear t; t P is the plateau yield in fieldyear t; it u is a normally distributed disturbance with mean zero and variance u t σ 2 for plot i in fieldyear t; it insey is the measured NDVI on plot i in fieldyear t; 0t α is the NDVI intercept for fieldyear t; 1t α is the NDVI response to N for fieldyear t; t φ is the NDVI plateau for fieldyear t; and it η is a normally distributed disturbance with mean zero and variance 2 t η σ for plot i in fieldyear t. Of paramount interest is the accuracy with which the parameters of equation (8) can be predicted by the parameters of equation (9). In other words, how do the LRP functions of NDVI compare with the LRP functions of actual yields? To answer this question, seemingly unrelated regression is used in SAS PROC MODEL to estimate the following: (10) t t t β = λ +λ α +ε 0 0 1 0 ˆ ˆ (11) t t t = + + r 1 0 1 1 βˆ γ γ αˆ , and (12) t t t Pˆ = ρ + ρ φˆ + e 0 1 , where 0t ˆβ , 1t ˆβ and t P ˆ are the estimated parameters of the LRP response of yield to N application from equation (8); 0t αˆ , 1t αˆ and t φˆ are the estimated parameters of the LRP 40 response of NDVI to N application from equation (9); 1 λ and 2 λ are the intercept and slope, respectively, of the relationship between the NDVI intercepts and the yield intercepts; 1 γ and 2 γ are the respective intercept and slope of the relationship between the responses of yield and NDVI to N application; 0 ρ and 1 ρ are the intercept and slope of the relationship between the NDVI plateau and yield plateau; and t ε , t r and t e are random, correlated error terms with means zero and variancecovariance matrix: (13) I Σ I σ σ σ σ σ σ σ σ σ Σ ⊗ = ⊗ = c e er ee r rr er r e ε ε εε ε ε , where σεε is the n by n variancecovariance matrix for equation (10); where σrr is the n by n variancecovariance matrix for equation (11); where σee is the n by n variancecovariance matrix for equation (12); the offdiagonal elements are nonzero crossmodel correlation matrices of the contemporaneous error terms, and I is an n by n identity matrix. The parameters estimated in equations (8) through (13) are used to determine the optimal N application rates for 1) the fieldlevel perfect predictor, 2) the fieldlevel NDVIbased predictor, and 3) the regional NDVIbased predictor, as well as to calculate the net returns above Nrelated costs1 (hereafter simply called “net returns”) for each prediction system. 1 Nrelated costs as defined here include 1) the cost of purchasing N, 2) the cost of custom application of N and 3) the cost of any technology and/or experimental strip required any given system for predicting the optimal N application rate. 41 The Perfect Prediction Nitrogen Rate The perfect prediction N application rate for each siteyear is determined based on the yield data. This rate produces the maximum possible expected profit for a topdressonly N application system. However, because the true parameters of the LRP functions estimated in equation (8) are unknown, and because the LRP functional form is nonlinear in parameters, the true optimal N application rate cannot be calculated deterministically (Babcock, 1992). Babcock (1992) and Tembo et al. (2008), however, do not consider uncertainty in all parameters of the LRP functional form—only in the yield plateau. The solution derived here is different, in that it accounts for uncertainty in the intercept, slope and plateau parameters. This work also differs from that of the preceding authors by considering parameter estimation uncertainty, rather than uncertainty caused by annual variability of the plateau. To account for estimation uncertainty, ten thousand Monte Carlo observations are used to determine the expected profit maximizing N application rate for each siteyear. These simulated observations are obtained by the process: (14) jt t t j βˆ = βˆ +Q 'z and QtQt '= t , where jt β ˆ is the jth simulated 4 by 1 vector of LRP parameters for siteyear t based on the estimation of equation (8)—i.e., 0 jt ˆβ , 1 jt ˆβ , jt P ˆ , and u jt σˆ 2 ; t β ˆ is the 3 by 1 vector of LRP parameter estimates for siteyear t from equation (8); ' t Q is the 3 by 3 lower triangular Cholesky decomposition matrix of t , which is the 3 by 3 variancecovariance matrix of parameter estimates for siteyear t; j z is the jth 3 by 1 vector of random deviates from a standard normal distribution; j = 1, Κ , J ; and J is ten thousand. 42 The true (or perfect prediction) application rate that maximizes expected profit for siteyear t is then calculated based on the Monte Carlo observations generated in equation (14) using the following maximization problem: (15) n t a t J j c jt jt t jt jt t N p N p J p N P E N t δ β β β π − − + =Σ= 1 0 1 0 max(min( ˆ ˆ , ˆ ), ˆ ) max ( ( )) , where π is profit; t N is the uniform N application rate for siteyear t; c p is the wheat price; 0 jt ˆβ is the jth simulated intercept coefficient for siteyear t; 1 jt ˆβ is the jth simulated slope coefficient for siteyear t; jt P ˆ is the jth simulated plateau coefficient for siteyear t; n p is the price of N from UAN solution; a p is the custom application cost for UAN solution; t δ is an indicator variable equal to one if > 0 t N ; J is ten thousand; and the max function ensures that yield is always greater than or equal to the intercept yield. Nitrogen Needs Predictions by SiteYear Next, the predicted economically optimal N application rate must be predicted for each siteyear based on the available NDVI data. The methods use to predict these application rates differ from those of Raun et al. (2008) by accounting for estimation uncertainty about the estimated parameters in equation (9) and of the parameters estimated in equations (10) through (12). To begin the prediction process, ten thousand sets of Monte Carlo simulated parameters are generated for each siteyear based on the parameter estimates from equation (9). The process for generating these Monte Carlo simulations is the same as that described in equation (14), and is used (as before) to account for parameter estimation uncertainty. However, these simulated parameters cannot be 43 directly used to predict the expected profit maximizing N application rate because they represent the response of expected NDVI measures (rather than yields) to N application. To predict the economically optimal N application rate based on these LRP functions of NDVI, the Monte Carlo simulated parameters based on equations (9) and (14) must be converted to expected yield parameters. This transformation is made using the seemingly unrelated regression parameters estimated in equations (10), (11) and (12), where the parameters of the expected yield functions depend on the parameters of the NDVI functions. However, the parameters describing the relationships between the LRP functions of NDVI and yield data are also estimated with error. Thus, Monte Carlo simulation is again used to generate ten thousand vectors of simulated parameters based on the joint normal distributions of the parameters estimated in equations (10), (11), and (12). These vectors are generated as follows: (16) j j λˆ = λˆ +Q'z , and QQ'= (X'Σ−1X)−1 where j λ ˆ is the jth simulated 6 by 1 vector of parameter estimates based on the estimated system in equations (10), (11) and (12)—i.e., 0 j ˆλ , 1 j ˆλ , 0 j γ , 1 j γ , 0 j ρ , and j 1 ρ ; λ ˆ is the 6 by 1 vector of estimated parameters from equations (10), (11) and (12); Q' is the lower triangular Cholesky decomposition of (X'Σ−1X)−1 , which is the 6 by 6 variancecovariance matrix of the parameters in λ ˆ , where: (17) = 3 2 1 0 0 X 0 X 0 X 0 0 X and Σ−1 = Σ−1 ⊗I c , such that 1 X , 2 X and 3 X are the n by 2 matrices with n 1s and n N recommendations from equations (10), (11) and (12); and Σc and I are defined in equation (13). 44 Then, using the Monte Carlo simulated parameters from equation (16), the simulated parameters of the LRP functions of NDVI for each siteyear—see equations (9) and (14)—are transformed from NDVI parameters to expected yield LRP parameters as follows: (18) 0 jt 0 j 1 j 0 jt ˆ ˆ ˆ ~ β = λ +λ α , (19) 1 jt 0 j 1 j 1 jt ˆ ˆ ˆ ~ β = γ +γ α , and (20) jt j j jt P ρˆ ρˆ φˆ ~ 0 1 = + where 0 jt ~β , 1 jt ~β and jt P ~ are, respectively, the jth simulated intercept, slope and plateau coefficients of the predicted expected yield LRP function for siteyear t; 0 jt αˆ , 1 jt αˆ and jt φˆ are the jth simulated intercept, slope and plateau coefficients, respectively, of the LRP function of NDVI measures for siteyear t; 0 j ˆλ , 1 j ˆλ , 0 j γˆ , 1 j γˆ , 0 j ρˆ and 1 j ρˆ comprise the jth simulated set of parameters relating LRP functions of yield and NDVI. 0 jt ~β , 1 jt ~β and jt P ~ in place of 0 jt ˆβ , 1 jt ˆβ and jt P ˆ in equation (15) to calculate the predicted expected profit maximizing N application rate. Nitrogen Needs Predictions by RegionYear The process for making regionyear predictions of the economically optimal N application rate is similar to the process for obtaining siteyear predictions. To begin, data from all sites in a given year are pooled to estimate: (21) iy y y iy y iy insey = min(α +α N , φ ) +ε 0 1 , 45 where iy insey is the NDVI measure on plot i in year y; 0 y α , y 1 α , y φ are, respectively, the intercept, slope, and plateau of the LRP response of NDVI measures to N application in year y; iy N is the N application rate on plot i in year y; and iy ε is a stochastic error term with mean zero and variance 2 ε σ . The parameter estimates from equation (21) are then used with their estimated variancecovariance matrix to simulate ten thousand 4 by 1 vectors of parameters. These simulated parameters are transformed to parameters of expected yield LRP functions of N using the process described in equations (18), (19) and (20). Finally, these simulated parameters are used to predict the optimal topdress N application rate for the statewide region in year y using the maximization problem in equation (15). Calculation of Expected Yield and Expected Profit Next, because one of the major objectives of this paper is to estimate the differences in relative profitability between the perfect predictor, the siteyearspecific predictor, the regionyear predictor and the historically recommended extension rate, the expected yield and expected profit are calculated for each system in each siteyear as follows: (22) Σ= + = J j jt jt kt jt kt J N P E y N 1 0 1 min( ˆ ˆ , ˆ ) [ ( )] β β , (23) n kt a kt k J j c jt jt kt jt kt p N p p J p N P E N − − − + =Σ= δ β β π 1 0 1 min( ˆ ˆ , ˆ ) [ ( )] , where y is yield; kt N is the N application rate prescribed by system k for siteyear t; 0 jt ˆβ is the jth simulated intercept coefficient of the yield response function for siteyear t; 1 jt ˆβ 46 is the jth simulated slope coefficient of the production function in siteyear t; jt P ˆ is the jth simulated yield plateau for siteyear t; kt δ is an indicator variable equal to one if > 0 kt N and zero otherwise; k p is the cost of acquiring and using the information set for system k; k is either the regionyear, siteyear, historical extension rate, or perfect prediction system; and all other symbols are previously defined. Testing for Differences in Expected Profit, Expected Yield, and Nitrogen Application Rates Based on the calculations of expected yields and profits in equations (23) and (24), and the predicted economically optimal N application rates for each system and siteyear, paired differences tests are used to determine whether any statistically significant differences exists between three systems in terms of yields, profitability and N use. These paired differences are calculated as: (24) D E y N E y N q k qt kt y qkt = [ ( )] − [ ( )], ≠ (25) D E N E N q k qkt qt kt π = [π ( )] − [π ( )], ≠ , and (26) D N N q k qt kt N qkt = ( ) − ( ), ≠ , where y qkt D is the difference between the expected yield for methods q and k in siteyear t; qt N is the amount of N prescribed by system q in siteyear t; kt N is the N application rate prescribed by method k in siteyear t; π qkt D is the difference of expected profit from methods q and k for siteyear t; N qkt D is the difference of the N application rates prescribed by methods q and k for siteyear t; methods q and k are two N application 47 recommendation systems selected from the siteyear, regionyear, historical extension, and perfect predictor systems; and all other symbols are previously defined. Because the student’s t test relies on normality of the data, nonparametric bootstrapping of these differences is performed to test the null hypothesis that the mean paired differences of profits, yields and N application rates are zero. This is done by random sampling with replacement from the original sample of observations on the 52 siteyears2 to create ten thousand random samples of 52 siteyears each. Using the means of the sample means and simulated standard errors (i.e., standard deviations of the sample means), t tests are conducted to determine whether the siteyear, regionyear, or historical extension recommendation system should be recommended for expected profit maximization. Sensitivity analysis is performed to determine whether the results are sensitive to assumptions about NUE from topdress N application, as compared with preplant N application. NUE levels assumed for the purpose of sensitivity analysis are 32%, 45% and 50%—with 32% being the average NUE for preplant N applications (Roberts, 2009; Raun et al., 1999), 50% being the NUE for topdress applications assumed by Raun et al. (2005) and 45% being an intermediate level of NUE. These assumed levels of NUE correspond to multiplying Monte Carlo simulated slope parameters for topdress systems by 1, 1.41 and 1.56, respectively, before solving for the optimal N application rates and proceeding with the calculation of expected profits and yields. 2 The original sample contains 53 siteyears; however, experiments were only conducted at one location in 2007. As a result, a regionyear N application could not be calculated for the Lahoma site in 2007. 48 Table I8. Wheat Yield as a Function of Nitrogen Application with Site and Year Specific Effects on the Intercept and Plateau Yields Model Parameter Definition Unrestricted No Plateau Random Effects No Plateau Fixed Effects 0 β Expected intercept yield 4144.99***a (196.52)b 3699.72*** (182.26) 4327.57*** (172.44) 1 β Crop response to nitrogen 19.41*** (1.70) 19.30*** (1.79) 65.45*** (15.98) P Expected yield plateau 5522.14*** (188.54) 5535.95*** (175.22) 5243.81*** (154.43) 2 u σ Variance of error term 631321.00*** (26599.00) 677913.00*** (28651.00) 729510.00*** (30954.00) 2 ε σ Variance of intercept random effects for year 167307.00*** (23791.00) 188344.00*** (20907.00) 144457.00*** (31221.00) 2 υ σ Variance of plateau random effects for year 170663.00*** (33424.00)  208312.00*** (40004.00) 2 v σ Variance of intercept fixed effects for site 51143301.00*** (9613634.00) 38989411.00*** (7422990.00) 61587700.00*** (8685127.00) 2 ω σ Variance of plateau fixed effects for site 13124989.00*** (1610810.00) 14357724.00*** (4374240.00)  Log Likelihood 9187.50 9203.50 9251.50 a Three asterisks (*) indicate statistical significance at the 0.01 confidence level. b Numbers in parentheses are standard errors. Results Table I8 contains estimates of the parameters from equation (5), where wheat yield is a function of site, year and the preplant N application rate. The unrestricted model allows plateau and intercept yields to vary by site and year—i.e., 2 , 2 , 2 , 2 > 0 ε υ ω σ σ σ σ v . The model with no plateau random effects is restricted such that 2 = 0 υ σ , limiting the model so that the average plateau yield across all locations does not vary by year. The likelihood ratio statistic—LR = −2(−9203.50 + 9187.5) = 32.00—is distributed chisquare with one 49 degree of freedom, and exceeds the critical value at the 0.01 confidence level (6.64). This result indicates that the average plateau yield for the entire state of Oklahoma varies consistently by year. The implication is that if these annual effects on the plateau yield are predictable over a large region, NDVI data from locations (experiment stations, for example) dispersed throughout the region would provide valuable information to all producers therein. However, at the statewide level, the variability represented in the annual plateau effects is only about 10% of the total plateau variability—i.e., /( + ) = 0.10 υ υ ω σ σ σ . At the state level, annual effects have relatively little (albeit statistically significant) influence on N requirements; however, this may not be trivial. For example, if a perfect predictor—accounting for both field and annual effects— improves profit above the current practice by $7.01 ha1(Roberts, 2009), a system that perfectly predicts annual effects would improve profits by about $0.70 ha1. It is also possible that at smaller spatial resolutions, such as at the county level, annual effects might play a relatively larger role in variation of N requirements. The model with no plateau fixed effects is restricted such that , i i ω = 0 ∀ — meaning that there is no individual effect on the average plateau yield for site i within the state of Oklahoma. Given = 0 i ω , the restriction may also be expressed as 2 = 0 ω σ ; however, because the model is estimated with fixed effects for site, the likelihood ratio statistic has nine degrees of freedom—i.e., the number of fixed effects estimated. The likelihood ratio statistic for this test is LR = −2(−9251.50 + 9187.50) = 128.00 , which exceeds the chisquare critical value of 21.67, indicating that the average plateau yield over all years varies from site to site. Farmers who have fieldspecific experience and expectations could then adjust their expectations (and topdress N applications) annually 50 based on midseason regional NDVI data collected at agricultural experiment stations and disseminated by the Cooperative Extension Service. Tables I9 and I10 contain the estimated LRP parameters for yield and NDVI data as functions of preplant N for each siteyear from equations (8) and (9), respectively. Table I11 displays the results from the annual, statewide estimation of the LRP function of NDVI as a function of N from equation (21). As noted in these tables, some of the estimated parameters have no standard errors. This occurs because the data for some siteyears do not reach a plateau. In these cases, PROC NLMIXED estimated a linear model, but generated a plateau equal to the expected yield at the maximum rate applied in the data for these siteyears. These estimates without standard errors are biased downward, because they tell us only that the plateau is expected to be greater than or equal to the estimate. This is also the case for estimates of the slope given without standard errors. At the Lahoma site in 2007, for instance, it appears “likely” that no data points are found on the slope of the production function. Figure I12 illustrates this type of data limitation. In such instances, the estimate is a lower bound on the expected value of the slope parameter. The dashed lines show how the true production function might deviate from the estimated function, but exactly how the true slope deviates from the parameter estimated in PROC NLMIXED is uncertain. Additionally, for the Perkins 1 site in 2001 there are no standard errors for the intercept or plateau parameters. In this case, PROC NLMIXED estimated the mean yield for the siteyear, but failed to provide standard errors because of data constraints. The fact that all points occur on the plateau means Monte Carlo simulation to account for estimation uncertainty is unnecessary because the 51 Table I9. Estimated Wheat Yield Response to Nitrogen by SiteYear Site Year Intercept Slope Plateau Perkins 1 1998 1134.16***a (132.79)b 8.30*** (1.80) 2102.70*** (131.46) Perkins 2 1998 1316.98*** (94.25) 1.22 (1.30) 1487.41*** (107.84) Tipton 1998 2942.65*** (93.34) 12.46*** (0.43) 5037.68*** (21.57) Efaw 1 1999 1040.52*** (226.84) 5.46*** (1.50) 3068.36*** (323.60) Efaw 2 1999 2169.07*** (192.95) 19.27*** (4.22) 3514.67*** (96.48) Haskell 1999 1767.41*** (288.21) 7.71c 2072.13*** (182.28) Lahoma 1999 1515.22*** (116.66) 26.28*** (2.28) 4443.08*** (181.36) Perkins 1 1999 1077.20*** (177.94) 12.71** (4.49) 2431.26*** (125.83) Stillwater 1999 856.12*** (103.51) 10.90** (4.00) 1712.27*** (110.65) Efaw 1 2000 911.11** (380.28) 26.84*** (6.57) 3384.06*** (294.56) Efaw 2 2000 2246.40*** (579.52) 1.53 (6.18) 2160.87*** (415.54) Haskell 2000 4262.17*** (212.53) 13.77*** (1.20) 2719.13*** (212.53) Hennessey 2000 3833.55*** (453.84) 0.29 (4.84) 3817.26*** (324.55) Lahoma 2000 1944.08*** (152.73) 25.03*** (6.09) 3515.75*** (130.79) Perkins 1 2000 2599.85*** (714.43) 6.55 (14.72) 3333.56*** (319.59) Stillwater 2000 1120.71*** (83.13) 17.05*** (1.34) 3414.03*** (96.79) Efaw 1 2001 921.82*** (215.47) 15.52** (6.80) 2024.16*** (112.53) Efaw 2 2001 2693.37*** (285.19) 8.80 (6.23) 3301.97*** (142.60) Haskell 2001 3669.98** (1368.34) 6.77 (10.92) 3121.59*** (387.02) 52 Table I9. Estimated Wheat Yield Response to Nitrogen by SiteYear Site Year Intercept Slope Plateau Hennessey 2001 1951.38*** (184.75) 7.01*** (0.76) 2815.16*** (91.34) Lahoma 2001 1495.54*** (201.16) 3.48 (17.18) 1651.35*** (142.25) Perkins 1 2001 2602.15d 1.35 (1.09) 2602.15d Stillwater 2001 1054.21*** (142.89) 12.70** (5.52) 1636.39*** (142.89) Efaw 1 2002 732.37** (325.25) 30.95*** (10.26) 2705.91*** (178.15) Efaw 2 2002 1811.65*** (305.03) 19.95*** (6.67) 3575.11*** (152.52) Haskell 2002 3500.96*** (938.17) 13.98* (1.45) 3112.43*** (262.23) Hennessey 2002 3898.07*** (28.52) 10.17*** (2.44) 2986.17*** (189.00) Lahoma 2002 2711.28*** (194.42) 16.54c 3075.88*** (122.96) Perkins 1 2002 2711.83*** (192.26) 1.55*** (0.18) 2971.97*** (161.91) Stillwater 2002 961.60*** (77.43) 16.03*** (1.54) 2987.25*** (114.85) Efaw 1 2003 1077.11** (477.42) 24.02*** (8.25) 3996.63*** (320.26) Efaw 2 2003 2792.10*** (403.20) 20.31*** (6.03) 4950.90*** (312.61) Hennessey 2003 2337.13*** (256.09) 14.67*** (3.65) 3760.42*** (166.31) Lahoma 2003 2760.86*** (209.35) 46.43*** (8.30) 5716.37*** (177.55) Perkins 1 2003 2796.69*** (190.99) 12.81** (4.82) 3779.32*** (135.05) Stillwater 2003 1136.43*** (176.83) 19.88*** (6.86) 2473.30*** (144.36) Efaw 1 2004 2079.37*** (570.45) 22.90 (18.01) 4132.65*** (285.13) Lahoma 2004 1871.40*** (313.47) 29.23c 2526.56*** (198.26) 53 Table I9. Estimated Wheat Yield Response to Nitrogen by SiteYear Site Year Intercept Slope Plateau Lake C.B. 2004 2227.34*** (248.19) 18.21*** (2.14) 4063.86*** (32.38) Perkins 1 2004 1936.34*** (393.48) 19.77* (9.93) 3399.90*** (278.24) Stillwater 2004 2080.99 (2250.37) 2.77 (28.29) 1895.02*** (220.59) Efaw 1 2005 1164.41*** (210.37) 4.56*** (1.39) 2845.72*** (300.10) Lahoma 2005 1754.09*** (188.07) 18.44** (7.27) 2683.34*** (151.63) Perkins 1 2005 3494.44*** (267.04) 9.84c 4021.48*** (178.03) Stillwater 2005 1764.35*** (145.62) 15.36 2223.53*** (118.90) Efaw 1 2006 1081.14*** (275.92) 8.05 (4.77) 2291.79*** (174.51) Lahoma 2006 2229.78*** (199.48) 4.03 (3.18) 2680.96c Lake C.B. 2006 1277.42*** (291.04) 37.69*** (8.16) 4377.41*** (291.04) Perkins 1 2006 917.24*** (113.69) 12.33*** (2.87) 2053.63*** (80.39) Stillwater 2006 1333.57*** (0.17) 5.64*** (0.68) 772.77*** (40.72) Lahoma 2007 2540.65*** (177.01) 28.81c 3162.98*** (129.27) Lahoma 2008 2761.46*** (294.09) 59.55*** (11.73) 5525.64*** (251.85) Stillwater 2008 1381.12 (174.25) 15.99*** (4.31) 2697.59*** (251.12) Mean for all siteyears 2004.70*** (124.73) 13.19*** (1.92) 3071.95*** (139.93) Note: Units are kg ha1. a One, two, or three asterisks (*) indicate statistical significance at the 0.10, 0.05 or 0.01 level, respectively. b Numbers in parentheses are standard errors. c Standard error cannot be estimated due to lack of data points on the slope or plateau. The estimated parameter is biased downward. d Standard errors for the intercept and plateau are not estimated because all available data are on the plateau. 54 Table I10. Estimated Wheat Optical Reflectance Response to Nitrogen by SiteYear, Scaled by a Factor of Ten Thousand Site Year Intercept Slope Plateau Perkins 1 1998 595.56***a (24.82)b 1.60*** (0.34) 804.24*** (25.90) Perkins 2 1998 571.95*** (23.12) 0.87 (0.58) 663.34*** (16.35) Tipton 1998 693.77*** (8.83) 1.18*** (2.23) 804.08*** (6.20) Efaw 1 1999 383.81*** (32.08) 3.34*** (1.01) 618.43*** (16.04) Efaw 2 1999 693.32*** (17.56) 1.44*** (0.38) 783.52*** (8.71) Haskell 1999 619.21*** (23.99) 2.29c 669.79*** (15.17) Lahoma 1999 615.96*** (13.21) 2.04*** (0.35) 785.83*** (14.55) Perkins 1 1999 466.16*** (23.44) 1.93*** (0.59) 591.37*** (16.48) Stillwater 1999 553.76*** (32.94) 3.52c 634.99*** (26.90) Efaw 1 2000 702.43*** (93.12) 7.82*** (1.61) 1488.30*** (72.13) Efaw 2 2000 864.23*** (38.39) 7.21c 891.76*** (15.67) Haskell 2000 600.48*** (37.85) 2.10c 625.01*** (23.94) Hennessey 2000 961.20*** (2.36) 0.14 (0.22) 978.53*** (25.06) Lahoma 2000 784.50*** (18.58) 5.10*** (0.74) 1092.05*** (15.92) Perkins 1 2000 652.11*** (53.91) 3.99c 770.82*** (31.12) Stillwater 2000 558.14*** (21.62) 7.19*** (0.84) 935.22*** (21.62) Efaw 1 2001 627.63*** (37.81) 2.46*** (0.65) 876.16*** (25.37) Efaw 2 2001 896.45*** (24.13) 0.21 (0.36) 922.09*** (18.71) Haskell 2001 674.80*** (32.57) 0.36 (0.31) 822.37 55 Table I10. Estimated Wheat Optical Reflectance Response to Nitrogen by SiteYear, Scaled by a Factor of Ten Thousand Site Year Intercept Slope Plateau Hennessey 2001 726.26*** (48.56) 1.29*** (0.56) 912.51c Lahoma 2001 774.70*** (35.75) 0.82 (2.78) 805.71*** (25.28) Perkins 1 2001 834.69d 0c 834.69d Stillwater 2001 677.06*** (42.71) 2.76 (1.65) 824.16*** (42.71) Efaw 1 2002 537.19*** (87.72) 2.39 (2.77) 649.67*** (43.86) Efaw 2 2002 638.31*** (14.87) 1.79*** (0.33) 742.28*** (7.43) Haskell 2002 517.16*** (7.65) 0.92 (1.25) 672.40*** (202.60) Hennessey 2002 652.30*** (0.01) 0.39 (0.41) 616.92*** (29.23) Lahoma 2002 753.81*** (48.93) 4.24c 843.41*** (30.94) Perkins 1 2002 721.90*** (13.34) 0.35** (0.13) 834.69c Stillwater 2002 448.76*** (13.06) 3.71*** (0.50) 692.68*** (13.03) Efaw 1 2003 346.68*** (37.06) 1.55*** (0.36) 670.93*** (33.83) Efaw 2 2003 652.54*** (36.81) 1.38*** (0.55) 816.61*** (28.54) Hennessey 2003 876.40*** (70.44) 1.69 (1.05) 1073.04*** (54.63) Lahoma 2003 570.00*** (1.39) 9.00*** (1.39) 860.00*** (11.94) Perkins 1 2003 496.14*** (19.24) 1.20*** (0.18) 684.12c Stillwater 2003 391.54*** (25.24) 3.71*** (0.62) 648.51*** (25.72) Efaw 1 2004 478.65*** (81.33) 3.40 (2.57) 781.76*** (40.64) Lahoma 2004 598.04 (85.90) 10.32c 757.59*** (54.33) 56 Table I10. Estimated Wheat Optical Reflectance Response to Nitrogen by SiteYear, Scaled by a Factor of Ten Thousand Site Year Intercept Slope Plateau Lake C.B. 2004 418.32*** (46.44) 2.20 (9.16) 639.75 (877.19) Perkins 1 2004 480.82*** (14.34) 1.17*** (0.20) 617.06*** (15.70) Stillwater 2004 727.06* (407.27) 2.45 (5.12) 564.32*** (44.51) Efaw 1 2005 497.75*** (33.02) 2.29*** (0.57) 763.08*** (20.87) Lahoma 2005 543.05*** (15.09) 3.20*** (0.59) 735.75*** (12.11) Perkins 1 2005 471.63*** (21.72) 1.33*** (0.32) 669.34*** (26.93) Stillwater 2005 550.07*** (2.92) 1.66*** (0.46) 699.18*** (38.05) Efaw 1 2006 306.18*** (50.30) 2.05** (0.87) 527.35*** (31.81) Lahoma 2006 484.41*** (30.91) 4.78c 564.29*** (19.55) Lake C.B. 2006 501.91*** (4.79) 1.03 (0.80) 606.19*** (76.37) Perkins 1 2006 268.27*** (24.38) 2.20*** (0.62) 476.83*** (17.25) Stillwater 2006 354.27*** (1.31) 1.00*** (0.29) 488.87*** (37.81) Lahoma 2007 513.11*** (12.08) 3.22*** (0.93) 597.96*** (8.54) Lahoma 2008 508.56*** (19.71) 5.36*** (0.53) 912.38*** (21.66) Stillwater 2008 690.54*** (68.31) 1.82c 771.92*** (55.78) Mean for all siteyears 594.78*** (21.08) 2.56*** (0.33) 756.87*** (23.49) a One, two, or three asterisks (*) indicate statistical significance at the 0.10, 0.05 or 0.01 level, respectively. b Numbers in parentheses are standard errors. c Standard error cannot be estimated due to lack of data points on the slope or plateau. The estimated parameter is biased downward. d Standard errors for the intercept and plateau are not estimated because all available data are on the plateau. 57 Table I11. Estimated Wheat Optical Reflectance Response to Nitrogen by Year, scaled by a Factor of Ten Thousand Yeara Intercept Slope Plateau 1998 618.49***b (16.09)c 1.33*** (0.40) 749.34*** (11.19) 1999 576.33*** (16.65) 2.11*** (0.74) 685.08*** (10.91) 2000 745.87*** (1.95) 1.64*** (0.23) 1187.04*** (60.97) 2001 731.74*** (19.02) 1.81** (0.88) 821.66*** (11.80) 2002 646.57*** (18.50) 0.45 (1.96) 767.48*** (80.44) 2003 537.63*** (33.07) 3.49*** (0.82) 792.95*** (23.71) 2004 574.33*** (30.83) 1.21 (0.82) 677.89*** (28.07) 2005 549.72*** (12.66) 1.57*** (0.21) 739.71*** (19.17) 2006 427.00*** (21.20) 0.81** (0.34) 534.13*** (30.34) 2008 597.96*** (30.92) 3.40*** (0.80) 856.63*** (37.95) Note: Units are kg ha1. a A response function for 2007 is not estimated because only one site is available in this year. b Two or three asterisks (*) indicate statistical significance at the 0.05 or 0.01 level, respectively. c Numbers in parentheses are standard errors. mean is linear in parameters. Thus, the lack of standard errors for the plateau and intercept in this siteyear is not problematic. The estimated relationships between the parameters of NDVI and yield response—estimated in equations (10), (11) and (12)—are presented in table I12. Here, the relationship describes how yield LRP function parameters (table I9) depend upon midseason NDVI parameters (table I10). The signs of the estimated coefficients are as expected—i.e., higher NDVI intercepts predict higher yield intercepts; higher NDVI response (slope) predicts higher yield response; and higher NDVI plateaus predict high 58 NDVI plateaus. Note based on the coefficients of variation for these relationships ( R2 in table I12) that these relationships are very noisy. These parameters (and their variancecovariance matrix) are used to convert LRP parameters of NDVI response into expected yield response through Monte Carlo simulation described in equations (18) to (20). Nonparametrically bootstrapped means of the prescribed N application rates, expected yields, and return above Nrelated costs for each system are displayed in table I13, assuming NUE of 32% for both preplant and topdress applications. Notably, mean net revenue is greatest for the historically recommended rate of 90 kg N ha1 from NH3, at $639.92 ha1, but this is only slightly greater than the $638.46 ha1 earned by the perfect predictor. While N purchase costs are much lower for the historical rate—because it uses NH3 rather than UAN—N application costs are much higher for the historical rate. The increased application cost, along with a slight yield boost for the perfect predictor system, nearly cancels out any saving on N purchase for the historical rate system. Additionally, the mean recommended application rates for the field and regionbased N requirements predictors are 88.92 and 94.11 kg ha1, respectively—apparently not much different from the historically recommended rate. The fieldbased system does appear to have some predictive power (though not statistically significant) because it achieves slightly higher yield than the historical rate while applying slightly less N. However, the total costs of N purchase and application for the two NDVIbased systems are, respectively, $107.52 and $113.23 ha1—relatively high compared to the analogous costs of $71.79 ha1 for the historical rate system. This difference is primarily due to the relative prices of topdress UAN ($1.10 kg1) and preplant NH3 ($0.57 kg1). It is possible that the field and region based systems could save substantially on Nrelated costs by 59 0 500 1000 1500 2000 2500 3000 3500 4000 4500 0 20 40 60 80 100 120 Nitrogen (kg ha1) Wheat Yield (kg ha1 ) Yield Data Estimated Function Figure I12. Plot of yield data and estimated production function for Lahoma 2007. using a split application—i.e., some N applied preplant as NH3 and some as topdress UAN if the RS shows that the crop is responsive. The mean UAN rate applied by the perfect predictor system is 65.41 kg ha1, compared with about 90 kg ha1 for either of the NDVIbased systems, meaning that the NDVIbased systems used in this paper overapply N substantially, as expected. Table I14 shows the nonparametrically bootstrapped means of the paired differences of expected profits, expected N application and expected yield generated in equations (24), (25) and (26). These results confirm that the profitability difference between the perfect predictor and the historical rate is statistically insignificant, despite the historical N application rate being on average 24.60 kg ha1 higher than the perfect predictor rate. Recall from table I3 that the “perfect predictor” fails to provide the 60 Table I12. Response of Yield Intercepts, Slopes and Plateaus to Optical Reflectance Intercepts, Slopes and Plateau, Respectively, Estimated by Seemingly Unrelated Regression Parameter Definition Estimate 0 λ Intercept of intercept response 480.85 (369.50)a 1 λ Slope of intercept response 255830.70***b (59091.90) R2 Coefficient of determination 0.15 0 γ Intercept of slope response 7.65*** (2.06) 1 γ Slope of slope response 217088.40*** (48025.00) R2 Coefficient of determination 0.24 0 ρ Intercept of plateau response 1440.23*** (429.20) 1 ρ Slope of plateau response 215697.40*** (53847.50) R2 Coefficient of determination 0.09 a Numbers in parentheses are standard errors. b Three asterisks (*) represent statistical significance at the 0.01 level. maximum profit primarily because it is a topdress system, using expensive UAN in place of cheaper NH3. Additionally, the perfect predictor system is significantly (p < 0.01) more profitable than either the fieldbased or the regionbased predictors by $35.14 ha1 on average. The historically recommended application of 90 kg N ha1 preplant is significantly more profitable than both the field and regionbased predictors by respective averages of $36.60 and $38.41 ha1. Expected profits from the field and regionbased systems are not statistically different. Table I15 displays the nonparametrically bootstrapped means of the prescribed N application rates, expected yields, and return above Nrelated costs for each system, assuming that NUE is 32% for preplant N applications and 45% for midseason topdress applications. Note that under this assumption, the perfect predictor system maximizes expected profit compared to the other systems (in contrast to the results in table I13). 61 Table I13. Noparametrically Bootstrapped Means of Net Returns, Revenues, NitrogenRelated Costs, Yields and Nitrogen Application Rates for Each Application System, Assuming 32% NitrogenUse Efficiency for Both Topdress and Preplant Nitrogen Applications System Revenue/Cost Perfect Predictor Historical Rate FieldBased Predictor Region Based Predictor Net Revenue ($ ha1) 638.46 (33.82) 639.92 (33.40) 603.32 (33.56) 601.51 (33.17) Yield Revenue ($ ha1) 717.51 (35.19) 711.71 (33.40) 713.88 (33.74) 714.73 (33.41) NH3 Cost ($ ha1)  51.30   Mean UAN Cost ($ ha1) 71.85 (6.96)  97.81 (2.73) 103.52 (2.08) NH3 Application Cost ($ ha1)  20.49   Mean UAN Application Cost ($ ha1) 7.10 (0.60)  9.71 (0.00) 9.71 (0.00) Precision System Cost ($ ha1)   3.04  Average Yield (kg ha1) 2989.64 (146.61) 2965.46 (139.19) 2974.51 (140.60) 2978.05 (139.23) Mean UAN Rate (kg ha1) 65.41 (6.33)  88.92 (2.49) 94.11 (1.89) Note: All estimates are significant at the 0.01 confidence level. This is because assuming topdress NUE of 45% substantially increases the marginal product of topdress N, while leaving the marginal product of preplant N unchanged. Under this assumption, the bootstrapped mean N application for each topdress system is substantially reduced relative to those in table I13. This occurs because an increase in the marginal product of N means that not as much N is required to reach the plateau. Also noteworthy is the result that expected yield for the topdress systems has increased, indicating that this increase in the marginal product of N makes UAN application more 62 Table I14. Nonparametrically Bootstrapped Means of Paired Differences of Expected Profits, Expected Nitrogen Application Rates, and Expected Yields, Assuming 32% NitrogenUse Efficiencies for Both Preplant and Topdress Nitrogen Applications Difference Expected Profit ($ ha1) Expected Nitrogen Rate (kg ha1) Expected Yield (kg ha1) Perfect Predictor  Historical Rate 1.46 (5.31)a 24.60*** (6.33) 24.18 (26.33) Perfect Predictor  FieldBased Predictor 35.14***b (5.51) 23.51*** (5.91) 15.13 (26.78) Perfect Predictor  RegionBased Predictor 36.95*** (5.69) 28.70*** (6.74) 11.58 (27.36) Historical Rate  FieldBased Predictor 36.60*** (2.61) 1.08 (2.49) 9.05 (13.15) Historical Rate  RegionBased Predictor 38.41*** (2.45) 4.11** (1.89) 12.60 (12.03) FieldBased Predictor  RegionBased Predictor 1.81 (4.10) 5.18 (3.28) 3.55 (19.72) a Numbers in parentheses are standard errors. b Two or three asterisks (*) indicate statistical significance at the 0.05 or 0.01 level, respectively. profitable than it otherwise would be, specifically in siteyears where the slope of the response to preplant N is small. Table I16 contains the bootstrapped means of the paired differences of expected profit, expected N application rate and expected yield for each system, assuming 32% and 45% NUE for preplant and topdress applications. These results confirm that the profitability difference of $24.98—favoring the perfect predictor system over the historical recommendation—is statistically significant at the 0.01 confidence level. The perfect predictor system continues to be more profitable than the field and regionbased systems. Notably, though the mean profit paired differences between the historical rate system and the field and regionbased systems continue to be significant in favor of the historical rate—$6.91 and $9.73 ha1, respectively—the differences are smaller in magnitude compared to those in table I15. There still is no statistically significant 63 Table I15. Noparametrically Bootstrapped Means of Net Returns, Revenues, NitrogenRelated Costs, Yields and Nitrogen Application Rates for Each Application System, Assuming 32% and 45% NitrogenUse Efficiency for Preplant and Topdress Nitrogen Applications, Respectively System Revenue/Cost Perfect Predictor Historical Rate Field Based Predictor Region Based Predictor Net Revenue ($ ha1) 664.90 (34.32) 639.12 (33.40) 633.01 (33.48) 630.19 (33.14) Yield Revenue ($ ha1) 728.52 (35.51) 711.71 (33.40) 718.80 (33.60) 721.25 (33.48) NH3 Cost ($ ha1)  51.30   Mean UAN Cost ($ ha1) 56.33 (5.57)  73.05 (2.29) 81.35 (2.48) NH3 Application Cost ($ ha1)  20.49   Mean UAN Application Cost ($ ha1) 7.29 (0.58)  9.71 (0.00) 9.71 (0.00) Precision System Cost ($ ha1)   3.04 (0.00)  Average Yield (kg ha1) 3035.49 (147.96) 2965.46 (139.19) 2995.01 (140.00) 3005.19 (139.52) Mean UAN Rate (kg ha1) 51.21 (5.06)  66.41 (2.09) 73.95 (2.25) Note: All estimates are significant at the 0.01 confidence level. difference between the field and regionbased systems in terms of profitability, though the regionbased system applies more N by an average of 7.55 kg ha1 (p < 0.05). The nonparametrically bootstrapped means of prescribed N application rates, expected yields, and return above Nrelated costs for each system assuming 32% and 50% NUE for preplant and topdress N, respectively, are presented in table I17. Here, the field and regionbased predictors have returns (net of Nrelated costs) very similar to the returns from using the historical rate. The mean of expected net revenue is slightly higher for the fieldbased system and slightly lower for the regionbased system. The costs of N 64 Table I16. Nonparametrically Bootstrapped Means of Paired Differences of Expected Profits, Expected Nitrogen Application Rates, and Expected Yields, Assuming 32% and 45% NitrogenUse Efficiencies for Preplant and Topdress Nitrogen Applications, Respectively Difference Expected Profit ($ ha1) Expected Nitrogen Rate (kg ha1) Expected Yield (kg ha1) Perfect Predictor  Historical Rate 24.98***a (5.43)b 38.79*** (5.06) 70.03** (34.25) Perfect Predictor  FieldBased Predictor 31.90*** (5.72) 15.20*** (4.84) 40.48 (32.38) Perfect Predictor  RegionBased Predictor 34.71*** (5.55) 22.75*** (5.80) 30.30 (30.82) Historical Rate  FieldBased Predictor 6.91** (2.96) 23.59*** (2.09) 29.55* (16.56) Historical Rate  RegionBased Predictor 9.73*** (3.42) 16.05*** (2.25) 39.73** (15.89) FieldBased Predictor  RegionBased Predictor 2.82 (4.89) 7.55** (3.39) 10.18 (24.37) a One, two or three asterisks indicate statistical significance at the 0.10, 0.05 or 0.01 confidence level, respectively. b Numbers in parentheses are standard errors. purchase and application for the historical, fieldbased and region based systems are$71.79, $76.89 and $84.54 ha1, respectively. The field and regionbased systems make up for their increased N expenditures (and the cost of the RS, in the case of the fieldbased system) through increased yields resulting from higher NUE. Table I18 presents the nonparametrically bootstrapped means of the paired differences of expected profits, expected N application rates and expected yields between the four systems. Note that the perfect predictor system is expected to be more profitable than all other systems by at least $31.74 ha1, and that these differences are statistically significant at the 0.01 confidence level. Additionally, the historical rate is higher than the mean of any other system by at least 21.79 kg N ha1. One problem with the field and 65 Table I17. Noparametrically Bootstrapped Means of Net Returns, Revenues, NitrogenRelated Costs, Yields and Nitrogen Application Rates for Each Application System, Assuming 32% and 50% NitrogenUse Efficiency for Preplant and Topdress Nitrogen Applications, Respectively System Revenue/Cost Perfect Predictor Historical Rate FieldBased Predictor Region Based Predictor Net Revenue ($ ha1) 671.84 (34.09) 639.12 (33.40) 640.10 (33.54) 638.01 (33.18) Yield Revenue ($ ha1) 734.11 (34.64) 711.71 (33.40) 720.03 (33.66) 722.55 (33.50) NH3 Cost ($ ha1)  51.30   Mean UAN Cost ($ ha1) 54.79 (5.28)  67.18 (2.12) 74.83 (2.29) NH3 Application Cost ($ ha1)  20.49   Mean UAN Application Cost ($ ha1) 7.48 (0.56)  9.71 (0.00) 9.71 (0.00) Precision System Cost ($ ha1)   3.04 (0.00)  Average Yield (kg ha1) 3058.81 (144.33) 2965.46 (139.19) 3000.14 (140.27) 3010.62 (139.57) Mean UAN Rate (kg ha1) 49.81 (4.80)  61.07 (1.93) 68.03 (2.08) Note: All estimates are significant at the 0.01 confidence level. regionbased systems as developed in this paper is that they always recommend some level of N application. This is evident because mean application costs for these systems, regardless of assumptions about NUE are $9.71 ha1 (see tables I13, I15 and I17). As a result, field and regionbased methods used here apply substantial N in cases where the true expected profit maximizing N rate is actually zero. This results in a substantial increase in N costs relative to the perfect predictor system without a commensurate increase in yield (because yield reaches a plateau at many sites at 65 kg N ha1). 66 Table I18. Nonparametrically Bootstrapped Means of Paired Differences of Expected Profits, Expected Nitrogen Application Rates, and Expected Yields, Assuming 32% and 50% NitrogenUse Efficiencies for Preplant and Topdress Nitrogen Applications, Respectively Difference Expected Profit ($ ha1) Expected Nitrogen Rate (kg ha1) Expected Yield (kg ha1) Perfect Predictor  Historical Rate 31.92*** (5.39) 40.18*** (4.80) 93.35*** (34.14) Perfect Predictor  FieldBased Predictor 31.74*** (5.57) 11.26** (4.52) 58.67* (31.07) Perfect Predictor  RegionBased Predictor 33.83*** (5.21) 18.21*** (5.46) 48.19 (29.41) Historical Rate  FieldBased Predictor 0.18 (3.03) 28.93*** (1.93) 34.68** (16.75) Historical Rate  RegionBased Predictor 1.91 (3.48) 21.97*** (2.08) 45.16*** (16.12) FieldBased Predictor  RegionBased Predictor 2.09 (4.85) 6.96** (3.13) 10.48 (24.24) a One, two or three asterisks indicate statistical significance at the 0.10, 0.05 or 0.01 confidence level, respectively. b Numbers in parentheses are standard errors. Also noteworthy is that the value of a perfect predictor system—i.e., the profit difference between the perfect predictor and the second most profitable system—is highly dependent on NUE. If NUE for topdress applications is the same as for preplant applications (32%), a perfect prediction of topdress N requirements has no value (see table I15). On the other hand tables I17 and I19 indicate that the value of a perfect predictor given 45% and 50% NUE is $24.98 or $31.74 ha1, respectively. Thus, the value of a perfect predictor of topdress N requirements is strongly dependent on the true NUE for topdress applications. 67 Conclusions One important finding of this research is that the historical extension recommendation— i.e., 90 kg N ha1 as NH3—is statistically indistinguishable from the “perfect predictor” topdress application system using UAN, primarily resulting from relative costs of UAN ($1.10 kg1) and NH3 ($0.56 kg1). The value of a perfect predictor, or the mean difference between the perfect predictor and
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Title  Preferences for Environmental Quality Under Uncertainty and the Value of Precision Nitrogen Application 
Date  20090701 
Author  Roberts, David Carlos 
Department  Agricultural Economics 
Document Type  
Full Text Type  Open Access 
Abstract  This dissertation is comprised of three essays. The first essay in this models winter wheat producers' choice among potential nitrogen fertilizer application systems that predict nitrogen needs based on optical reflectance data collected from growing plants at different spatial resolutions. Monte Carlo simulation is used to account for the uncertain relationship between optical reflectance data. Expected profits are calculated for each nitrogen application system, and paired differences tests determine which system is most profitable. The second essay addresses winter wheat producers' choice among different fieldspecific, uniform rate application systems that predict nitrogen needs using optical reflectance data collected from different types of experimental strips. Additionally, Monte Carlo simulation is used to determine the effects of parameter uncertainty on the profit maximization process given the linear responseplateau functional form. Paired differences tests are used to determine the effect of parameter uncertainty on profit maximization and to estimate the relative profitability of the different experimental strip techniques. Essay three determines whether (and how) uncertainty about environmental outcomes influences recreationists' willingnesstopay for water quality improvements at Lake Tenkiller. Onsite interviews were conducted with recreationists at the lake, and multinomial logit estimation is used to model the effect of uncertain outcomes on willingnesstopay. The evidence presented suggests that the nitrogen application strategy expected to be most profitable is to apply 90 kg ha1 each year as anhydrous ammonia, rather than use topdress ureaammonium nitrate solution, which is much more expensive. Fieldlevel sampling of predictive optical reflectance data is no more profitable than regional sampling. It is also determined that the ramped strip technology is statistically neither more nor less profitable than the nitrogenrich strip technology. Evidence suggests that, in some cases, accounting for parameter uncertainty improves the predictive accuracy and profitability of optical reflectancebased nitrogen needs predictors. The ramped strip technology is expected to be more profitable when accounting for parameter uncertainty. The Lake Tenkiller study shows that uncertain outcomes affect recreationist willingnesstopay for water quality, suggesting that uncertainty should be explicitly included in survey instruments for valuation of natural resources and environmental amenities. 
Note  Dissertation 
Rights  © Oklahoma Agricultural and Mechanical Board of Regents 
Transcript  PREFERENCES FOR ENVIRONMENTAL QUALITY UNDER UNCERTAINTY AND THE VALUE OF PRECISION NITROGEN APPLICATION By DAVID CARLOS ROBERTS Bachelor of Arts in Spanish The University of Tennessee Knoxville, Tennessee 2003 Master of Science in Agricultural Economics The University of Tennessee Knoxville, Tennessee 2006 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY July, 2009 PREFERENCES FOR ENVIRONMENTAL QUALITY UNDER UNCERTAINTY AND THE VALUE OF PRECISION NITROGEN APPLICATION Dissertation approved: Dr. B. Wade Brorsen Dissertation Adviser Dr. Tracy A. Boyer Dr. Francis M. Epplin Dr. William R. Raun Dr. John B. Solie Dr. A. Gordon Emslie Dean of the Graduate College iii ACKNOWLEDGEMENTS While words fail to describe the full measure of my gratitude, I am obliged herein to acknowledge the contributions of my graduate committee members toward my success. I am grateful for their willingness to invest their time and efforts to assist me in developing my professional capital, and insuring the integrity of the work presented in this dissertation. I therefore extend my heartfelt appreciation to Dr. B. Wade Brorsen, Dr. Tracy A. Boyer, Dr. Francis M. Epplin, Dr. William R. Raun, and Dr. John B. Solie. Many thanks also to Dr. Jayson L. Lusk, who, though not on my committee, made similar contributions to my success in this endeavor. I am grateful also to Gracie Teague for her crucial and very thorough assistance in formatting this document. I also wish to express my gratitude to the Agricultural and Applied Economics Association, the Southern Agricultural Economics Association, the 9th International Conference on Precision Agriculture, and the 8th Annual Heartland Environmental and Resource Economics Workshop for providing opportunities to vet my research before an audience of my professional peers. My sincere thanks are also extended to the editors and the anonymous peer reviewers at the journal Ecological Economics for their valuable input in the peerreview and publication process. Chapter III of this dissertation was published in Ecological Economics, and is included herein with the journal’s permission. I acknowledge the instrumental assistance of my coauthors Tracy A. Boyer and Jayson L. iv Lusk, and their efforts to ensure the statistical and prosaic integrity of the aforementioned chapter. Funding for this research was provided by the Sitlington Foundation, the Oklahoma Department of Wildlife Conservation, the Targeted Initiative Program managed through Oklahoma State University’s Division of Agricultural Sciences and Natural Resources, and the Jean and Patsy Neustadt Chair. Their funding was crucial in conducting and disseminating this research. I am also very pleased to acknowledge the undying devotion, honor, and respect I owe to my wife (Bonny D. Roberts) and my daughter (Naomi S. Roberts). Though they did not take the courses, collect the data, or write the dissertation, this accomplishment is as much theirs as it is mine. They made the requisite sacrifices with me, and they will reap the rewards with me also. Thanks to my father and mother, who have encouraged me in my educational endeavors as long as I can remember. Thanks also to all my kith and kin, who have not only influenced me for good but also appreciated me for who I am. Many thanks also to my office mates and fellow graduate students at Oklahoma State University. And thanks again to Gracie Teague. v TABLE OF CONTENTS Chapter Page I. PREDICTION UNCERTAINTY AND THE VALUE OF INCREASINGLY SPATIALLY PRECISE NITROGEN NEEDS INFORMATION......................................................................................................1 Abstract ............................................................................................................ 1 Introduction ...................................................................................................... 3 Theory .............................................................................................................. 6 Brief Example: Expected Profit Maximization when the Plateau Yield Is Predicted with Error .......................................................................7 The Producer’s Decision Problem: Choosing the Expected Profit Nitrogen Application System .....................................................................10 How Do Nitrogen Needs Vary Spatially, and What Are the Implications? ..............................................................................................11 SubPaper 1: Spatial Variability, Repeatability and Noise in Predictions Made by the Nitrogen Fertilizer Optimization Algorithm and the Ramped Strip ................................................................................................................ 13 Data ............................................................................................................13 Procedures ..................................................................................................15 Results ........................................................................................................19 Conclusions ................................................................................................30 SubPaper 2: Prediction Uncertainty and the Value of Increasingly Spatially Precise Sampling of Optical Reflectance Data .............................................. 34 Data ............................................................................................................34 Procedures ..................................................................................................37 Variability of Nitrogen Needs by Year and Location ......................... 37 Defining a Predictive Relationship ..................................................... 38 The Perfect Prediction Nitrogen Rate ................................................. 41 Nitrogen Needs Predictions by SiteYear ........................................... 42 Nitrogen Needs Predictions by RegionYear ...................................... 44 Calculation of Expected Yield and Expected Profit ........................... 45 Testing for Differences in Expected Profit, Expected Yield, and Nitrogen Application Rates................................................................. 46 Results ........................................................................................................48 Conclusions ................................................................................................67 II. THE EFFECT OF PARAMETER UNCERTAINTY ON NITROGEN RECOMMENDATIONS FROM NITROGENRICH STRIPS AND RAMPED STRIPS IN WINTER WHEAT............................................................71 Chapter Page vi Abstract .......................................................................................................... 71 Introduction .................................................................................................... 73 Theory ............................................................................................................ 79 Data ................................................................................................................ 80 Procedures ...................................................................................................... 85 SpaceTime Variability of Crop Response ................................................85 Expected Profit Maximizing Application Rates from Perfect Predictors ...................................................................................................86 Predicted Expected Profit Maximizing Application Rates from Ramped Strip Predictors ............................................................................89 Predicted Expected Profit Maximizing Application Rates from Nitrogen Rich Strip Predictors ...................................................................90 Calculation of Expected Yields and Expected Returns .............................96 Results............................................................................................................ 98 Conclusions.................................................................................................. 120 III. PREFERENCES FOR ENVIRONMENTAL QUALITY UNDER UNCERTAINTY .................................................................................................127 Abstract ........................................................................................................ 127 Introduction .................................................................................................. 127 Background .................................................................................................. 132 Methods ....................................................................................................... 136 Results.......................................................................................................... 141 Conclusions.................................................................................................. 151 IV. REFERENCES ....................................................................................................154 V. APPENDICES .....................................................................................................163 Appendix A: Institutional Review Board Approval Letter.......................... 164 vii LIST OF TABLES Table Page Table I1. Number of Observations, Mean Ramped Strip Nitrogen Recommendation, Mean Nitrogen Recommendations, and Mean Predicted Plateau Yield by County for Dataset Two .................................14 Table I2. Planting Date and Sensing Dates for Each Field in Dataset Three ............16 Table I3. Ramped Strip and Nitrogen Fertilizer Optimization Algorithm Recommendations as Functions of FarmerPractice Preplant Nitrogen Rate and County .........................................................................................20 Table I4. Mean Ramped Strip Recommendation, with and without Fixed Effects for Strip Pair and Field ...............................................................................22 Table I5. Mean Nitrogen Fertilizer Optimization Algorithm Recommendation, with and without Fixed Effects for Strip Pair and Field ............................25 Table I6. Locations, Years, Soil Types, and Nitrogen Levels, and Replications for Experiments in Dataset One.......................................................................35 Table I7. Partial Budget for Creation and Use of Three Ramped Strips in a 63 ha field ............................................................................................................37 Table I8. Wheat Yield as a Function of Nitrogen Application with Site and Year Specific Effects on the Intercept and Plateau Yields .................................48 Table I9. Estimated Wheat Yield Response to Nitrogen by SiteYear .....................51 Table I10. Estimated Wheat Optical Reflectance Response to Nitrogen by Site Year, Scaled by a Factor of Ten Thousand ...............................................54 Table I11. Estimated Wheat Optical Reflectance Response to Nitrogen by Year, scaled by a Factor of Ten Thousand ..........................................................57 Table I12. Response of Yield Intercepts, Slopes and Plateaus to Optical Reflectance Intercepts, Slopes and Plateau, Respectively, Estimated by Seemingly Unrelated Regression .................................................................................60 Table I13. Noparametrically Bootstrapped Means of Net Returns, Revenues, NitrogenRelated Costs, Yields and Nitrogen Application Rates for Each Application System, Assuming 32% NitrogenUse Efficiency for Both Topdress and Preplant Nitrogen Applications ...........................................61 Table I14. Nonparametrically Bootstrapped Means of Paired Differences of Expected Profits, Expected Nitrogen Application Rates, and Expected viii Table Page Yields, Assuming 32% NitrogenUse Efficiencies for Both Preplant and Topdress Nitrogen Applications ................................................................62 Table I15. Noparametrically Bootstrapped Means of Net Returns, Revenues, NitrogenRelated Costs, Yields and Nitrogen Application Rates for Each Application System, Assuming 32% and 45% NitrogenUse Efficiency for Preplant and Topdress Nitrogen Applications, Respectively ..............63 Table I16. Nonparametrically Bootstrapped Means of Paired Differences of Expected Profits, Expected Nitrogen Application Rates, and Expected Yields, Assuming 32% and 45% NitrogenUse Efficiencies for Preplant and Topdress Nitrogen Applications, Respectively ...................................64 Table I17. Noparametrically Bootstrapped Means of Net Returns, Revenues, NitrogenRelated Costs, Yields and Nitrogen Application Rates for Each Application System, Assuming 32% and 50% NitrogenUse Efficiency for Preplant and Topdress Nitrogen Applications, Respectively ..............65 Table I18. Nonparametrically Bootstrapped Means of Paired Differences of Expected Profits, Expected Nitrogen Application Rates, and Expected Yields, Assuming 32% and 50% NitrogenUse Efficiencies for Preplant and Topdress Nitrogen Applications, Respectively ...................................66 Table II1. Locations, Years, Soil Types, and Nitrogen Levels, and Replications for Experiments ...............................................................................................81 Table II2. Partial Budget for Creation and Use of a NitrogenRich Strip on a 63 ha Field ...........................................................................................................84 Table II3. Partial Budget for Creation and Use of Three Ramped Strips on a 63 ha Field ...........................................................................................................84 Table II4. Unrestricted and Restricted Linear ResponsePlateau Functions of Wheat Yield as a Function of Nitrogen Application with Random Parameters for SiteYear ..................................................................................................100 Table II5. Estimated Wheat Yield Response to Nitrogen by SiteYear (kg ha1).....101 Table II6. Midseason Predicted Wheat Yield Response to Nitrogen Based on the Ramped Strip Method ..............................................................................106 Table II7. Predicted Production Function Parameters by SiteYear Using Nitrogen Rich Strip Method ....................................................................................109 Table II8. Nonparametrically Bootstrapped Means and Standard Errors of Expected Net Revenue, Expected Yield Revenue, Expected Nitrogen and PrecisionRelated Costs, Nitrogen Application Rates and Yields for Each System Assuming All Preplant Nitrogen from Anhydrous Ammonia ....112 Table II9. Nonparametrically Bootstrapped Means and Standard Errors of the Paired Differences Assuming All Preplant Nitrogen from Anhydrous ix Table Page Ammonia..................................................................................................114 Table II10. Nonparametrically Bootstrapped Means and Standard Errors of Expected Net Revenue, Expected Yield Revenue, Expected Nitrogen and PrecisionRelated Costs, Nitrogen Application Rates and Yields for Each System Assuming All Preplant Nitrogen from Dry Urea ........................117 Table II11. Nonparametrically Bootstrapped Means and Standard Errors of Paired Differences in Profits Given All Preplant Nitrogen from Dry Urea ........118 Table II12. Nonparametrically Bootstrapped Means and Standard Errors of Expected Net Revenue, Expected Yield Revenue, Expected Nitrogen and PrecisionRelated Costs, Nitrogen Application Rates and Yields for Each System Assuming No Increase in NitrogenUse Efficiency ....................121 Table II13. Nonparametrically Bootstrapped Means and Standard Errors of the Paired Differences Assuming No NitrogenUse Efficiency Increase from Midseason Application ...........................................................................122 Table III1. Summary Statistics of Survey Samples ...................................................142 Table III2. Preferences for Algal Bloom and Water Level: Multinomial Logit Estimates ..................................................................................................144 Table III3. WillingnesstoPay Estimates ..................................................................150 x LIST OF FIGURES Figure Page Figure I1. Yield as a linear responseplateau function of nitrogen application. ...........8 Figure I2. Profit as a function of nitrogen application..................................................8 Figure I3. Expected profit maximizing nitrogen application rate vs. standard deviation of the plateau prediction error. .....................................................9 Figure I4. Ramped strip recommendation at one strip vs. ramped strip recommendation from the other strip in the same pair at the same sensing date. ............................................................................................................27 Figure I5. Nitrogen fertilizer optimization algorithm recommendation at one strip vs. nitrogen fertilizer algorithm recommendation from the other strip in the same pair at the same sensing date.......................................................28 Figure I6. Mean ramped strip recommendation from one pair of strips vs. mean ramped strip recommendation from the other pair in the same field at the same sensing date. ......................................................................................29 Figure I7. Mean nitrogen fertilizer optimization algorithm recommendation from one pair of strips vs. mean nitrogen fertilizer optimization algorithm recommendation from the other pair in the same field at the same sensing date. ............................................................................................................30 Figure I8. Ramped strip recommendation from MiMarch vs. ramped strip recommendation from the same strip in MidFebruary. ............................31 Figure I9. Nitrogen fertilizer optimization algorithm recommendation from Mid March vs. nitrogen fertilizer optimization algorithm recommendation from the same strip in MidFebruary. ........................................................32 Figure I10. Ramped strip recommendation vs. nitrogen fertilizer optimization algorithm recommendation from the same strip at the same sensing date.33 Figure I11. Map of experimental locations...................................................................36 Figure I12. Plot of yield data and estimated production function for Lahoma 2007. ...59 Figure II1. Map of experimental locations...................................................................82 Figure II2. Plot of yield data and estimated production function for Lahoma 2007. .105 Figure III1. Choice card from survey without uncertainty. .........................................138 Figure III2. Choice card from survey with uncertainty. ..............................................140 xi Figure Page Figure III3. Estimated probability weighting function. ...............................................148 Figure III4. Willingnesstopay to reduce the probability of an algal bloom..............152 xii NOMENCLATURE ER nitrogen application rate historically recommended by the Oklahoma Cooperative Extension Service (90 kg N ha1) LRP linear responseplateau MNL multinomial logit NDVI normalized difference vegetation index NFOA nitrogen fertilizer optimization algorithm N nitrogen NH3 anhydrous ammonia NRS nitrogenrich strip NRSD deterministic nitrogen needs predictor based on the nitrogenrich strip NRSU stochastic nitrogen needs predictor based on the nitrogenrich strip NUE nitrogenuse efficiency ORI optical reflectance imaging PPD deterministic perfect predictor of actual nitrogen needs based on yield data PPU stochastic perfect predictor of actual nitrogen needs based on yield data xiii RS ramped strip RSD deterministic nitrogen needs predictor based on ramped strip RSU stochastic nitrogen needs predictor based on ramped strip UAN ureaammonium nitrate solution WTP willingnesstopay 1 I. CHAPTER I PREDICTION UNCERTAINTY AND THE VALUE OF INCREASINGLY SPATIALLY PRECISE NITROGEN NEEDS INFORMATION Abstract Nitrogen fertilizer is intensively used in crop agriculture in the United States, and many researchers embrace the goal of improving nitrogenuse efficiency—that is, increasing the proportion of nitrogen fertilizer that is actually used by the crop. This goal can be achieved by applying nitrogen fertilizer to match plant needs as they vary over both time and space. Several different precision agriculture systems have been designed to address this variability of nitrogen needs. Among these innovations are two wholefield systems that use midseason normalized difference vegetation index (NDVI) measures from growing winter wheat to predict the amount of nitrogen the plants require to reach their plateau yield. The nitrogen fertilizer optimization algorithm (NFOA) uses NDVI data from a nitrogenrich strip and a check strip in the same field to determine the rate at which the crop will cease to be responsive to nitrogen. The ramped strip system applies incrementally increasing nitrogen rates in a strip of plots just after planting, and then collects midseason NDVI readings to determine the rate at which crop response ceases. This paper is comprised of two subpapers, the first of which uses datasets from actual ramped strips from onfarm trials. The data used are the outputs from the program 2 Ramp Analyzer 1.2, and include ramped strip recommendations, as well as NFOA recommendations based on these ramped strips. These data are used to determine whether the ramped strip and NFOA recommendations are precise enough to detect spatial variability of nitrogen needs within fields, among fields and among different counties within the state. The results show that the ramped strip recommendation is a noisy measure of nitrogen needs—perhaps too noisy to be unambiguously profitable. The second subpaper uses data from trials at ten experiment station sites throughout the state of Oklahoma. Different preplant nitrogen treatments were applied to replicated plots at these locations between 1998 and 2008, and midseason NDVI and yield data were collected from each plot. These data are used to estimate response of both NDVI and yield to preplant nitrogen as a linear responseplateau. Because the relationship between NDVI and yield is estimated with uncertainty and because the linear responseplateau functional form is nonlinear in parameters, a new methodology is developed using Monte Carlo simulation to predict optimal topdress nitrogen rates based on the NDVI data. This subpaper also determines whether it is necessary to sample NDVI measures from each field, and how much precision—and profit—would be lost by moving from sitespecific (or fieldspecific) NDVI sampling to regionlevel sampling. It is determined that the NDVIbased nitrogen needs predictors developed in this paper are imprecise, with the result that profits from regionlevel sampling and fieldlevel sampling are statistically indistinguishable. Furthermore, it is found that the region and fieldbased sampling systems are no better than breakeven with the historical extension advice to apply preplant anhydrous ammonia at 90 kg ha1. 3 Introduction Crop agriculture in the United States and other developed nations intensively uses nitrogen fertilizer (N) to increase yields. Expenditures on N account for 28% and 32% of operating expenses for U.S. producers of wheat and corn, respectively (United States Department of Agriculture, 2005). Many researchers have focused on improving Nuse efficiency (NUE) in agriculture (e.g., Raun and Johnson, 1999; Greenhalgh and Faeth, 2001; Cassman et al., 1998). Raun and Johnson (1999) find that only 33% of N applied to cereal crops worldwide is recovered in grain. Traditionally, N has been applied prior to planting at a uniform rate selected to meet a yield goal based on historical yields. However, Solie, Raun and Stone (1999) show that natural soil N content (inversely related to crop requirements for N application) varies significantly at a spatial scale of approximately 1 m2. Additionally, many studies (e.g., Lobell et al., 2005; Mamo et al., 2003; Washmon et al., 2002) find that crop response to N varies within and between fields over time. In other words, potential yield and N requirements vary temporally and spatially within and between fields. This variability results from weather, topology, and their combined effects on N deposition, mineralization, and volatilization. Precision agriculture focuses on providing information to reduce uncertainty about N needs so producers can improve profit margins by avoiding under or overapplication of N. One innovation in precision agriculture is the sensorbased nitrogen fertilizer optimization algorithm (NFOA) developed by Raun et al. (2002, 2005). The NFOA uses midseason measures of normalized difference vegetation index (NDVI) from growing plants in a nonlimiting, nitrogenrich strip (NRS) to predict the midseason, topdress N application rate required by the crop. Additionally, Raun et al. (2008) have developed a 4 ramped strip (RS) technology to predict optimal N application rates for crops including corn and wheat. This practice involves applying N at incrementally increasing rates to plots arranged in a strip. Such strips can be used to predict, either by visual inspection or by using an optical reflectance sensor, the midseason, topdress N application rate at which crop response to N will cease. The goal of these technologies is to improve NUE— or reduce loss of N inputs to volatilization and runoff—without decreasing yields, so as to improve producer profits. More than one RS or NRS may be used in a single field, but it is recommended that producers place at least one strip in each field each year (Arnall, Edwards and Godsey, 2008). However, is it likely two fields “very close” to each other have similar N requirements? Or what about three such fields? In other words, what is the optimal spatial scale at which to sample NDVI data from experimental strips? Should fields be divided into management zones with a strip in each zone? Is one strip per field sufficient? Or perhaps several strips spread throughout a county could provide an accurate enough prediction for all fields within the county. A countywide system would be especially valuable to producers who grow wheat for both grain and grazing, for whom establishing an experimental strip might be prohibitively costly due to new fencing costs. The answers to questions about the optimal spatial scale of sampling also will be affected by the strength of the relationship between yields and the NDVI data used to predict them. Despite reduction in uncertainty about spatial and temporal variability of crop response, uncertainty remains an issue for the NFOA and RS technologies as a result of prediction error. Babcock (1992) suggests that uncertainty results in the historic producer habit of “overapplying” N at a uniform rate every year. He proposes chronic overapplication 5 indicates that producers assume crop response to N follows a linear responseplateau (LRP) functional form in which the plateau is uncertain. Tembo et al. (2008) similarly address uncertainty about plateau yields among fields and years. They develop an analytical formula to determine the optimal application rate given interannual or interfield variability of plateau yields. Both Babcock (1992) and Tembo et al. (2008) show that the expected profit maximizing strategy given uncertainty about plateau yields is to apply more N than the deterministic solution suggests. Therefore, inclusion of uncertainty—especially prediction error in the relationship between NDVI and yields— may be essential to accurately predicting the expected profit maximizing midseason, topdress N application rate using the NFOA or RS. This means that prediction error in the predicted intercept and slope should be addressed in addition to plateau uncertainty to improve N requirement prediction. The remainder of this paper (following the theory section) is divided into two subpapers, which use different datasets to explore sets of related questions about spatial variability of N requirements. The objectives of the first section are 1a) to determine whether N requirements as predicted by the RS and the NFOA vary by county within a single year and 2a) to determine how consistent (or repeatable) NFOA and RS predictions are over time and space. The objectives of the second section are 1b) to determine whether average plant N requirements for a large region vary by year, 2b) to develop a new process for including prediction error in the RS predictor and 3b) to estimate the relative profitability of four different systems for choosing N application rates. These systems are: 6 a) a perfect predictor system that uses yield data directly to determine the expected profit maximizing topdress N application rate; b) the historical recommendation of 90 kg N ha1 as preplant anhydrous ammonia (NH3); c) a siteyearspecific, NDVIbased predictor of topdress N requirements based on the process developed in objective (2b) above; and d) a regionyearspecific, NDVIbased predictor of topdress N requirements based on the process developed in objective (2b) above. The results will determine whether annual collection of state or countylevel NDVI data—and subsequent dissemination of N recommendations based on these data—has potential value for winter wheat producers in Oklahoma. Such regional N recommendations, if accurate, might be especially beneficial to those who produce wheat for both grazing and grain, who would likely find the cost of fencing off an experimental strip in each field prohibitive. Notably, using a regionbased system would entail more uncertainty about N requirements at any particular site. However, rather than seeking to reduce uncertainty in N requirements predictions, this work seeks to account for remaining uncertainty in the predictors, and thereby to reduce the cost of prediction error. Theory Prior research indicates that output is a function of the most limiting input (e.g., Paris and Knapp, 1989; Berck and Helfand, 1990; Paris, 1992; Chambers and Lichtenberg, 1996; Berck, Geoghegan, and Stohs, 2000; Monod et al., 2002). This functional form is known as a linear responseplateau (LRP). Here, the most limiting input is assumed to be either 7 N or an unspecified input that is represented as a plateau level of output. However, variables determining the intercept and plateau yields—such as N deposition, mineralization and volatilization—are not known in advance at any given site in any particular year (Mamo et al., 2003). Thus, producers face substantial uncertainty in choosing N application rates. Midseason collection of NDVI data from each site each year can reduce uncertainty caused by spatial and temporal variability. However, predicting yields based on NDVI introduces prediction error that has not yet been addressed in the NFOA or RS methods. The following brief example illustrates how prediction error about the plateau (and only the plateau) affects the process of expected profit maximization. Brief Example: Expected Profit Maximization when the Plateau Yield Is Predicted with Error Suppose a LRP function of expected yield response to N has been predicted for a single siteyear based on NDVI data from a RS. For ease of exposition, assume that all parameters besides the plateau are predicted without error—an admittedly unrealistic assumption. Figure I1 illustrates the hypothetical LRP function. Figure I2 illustrates the resulting profit function. These two figures show that, when the plateau yield is known with certainty, the profit maximizing N application rate is 30 kg ha1. Observe the slope of the profit function before and after the optimal rate to see that underapplication is relatively more costly than overapplying by the same amount due to the relative prices of N and wheat. However, because the plateau yield is predicted with error, the costs of under or overapplying are not guaranteed—i.e., there is some probability that applying 8 2000 2050 2100 2150 2200 2250 2300 2350 2400 0 10 20 30 40 50 60 Nitrogen Application Rate (kg ha1) Wheat Yield (kg ha1 ) Figure I1. Yield as a linear responseplateau function of nitrogen application. 300 320 340 360 380 400 420 440 0 10 20 30 40 50 60 Nitrogen Application Rate (kg ha1) Profit ($ ha1 ) Figure I2. Profit as a function of nitrogen application. 9 25 30 35 40 45 50 55 60 0 500 1000 1500 2000 2500 Standard Deviation of Plateau Prediction Error (kg ha1) Expected Profit Maximizing Nitrogen Application Rate (kg ha1 ) Figure I3. Expected profit maximizing nitrogen application rate vs. standard deviation of the plateau prediction error. an additional kg of N will increase profits, and some probability that it will only increase costs. The rate that maximizes expected profit is that at which the probability the crop will use the last kg of N applied is the price of N divided by the price of wheat. This fulfils the necessary condition that expected marginal revenue must equal marginal cost for an expected profit maximum. The N application rate at which this condition is met depends upon the variability of the plateau. In this case, it depends on the prediction error in the plateau parameter. Figure I3 shows the schedule of expected profit maximizing N application rates for varying levels of uncertainty about the plateau based on equation (14) in Tembo et al. (2008). As prediction error in the plateau parameter increases, higher N application rates are required to satisfy the necessary condition that expected marginal revenue equals 10 marginal cost. Note again that this example treats only the error in the predictive relationship between NDVI data and the yield plateau, assuming the other parameters are known with certainty. Prediction error in the intercept, slope and plateau parameters of predicted yield LRP functions will be jointly addressed by Monte Carlo simulation in the procedures section, but consideration of these prediction errors is not conducive to graphical analysis. The Producer’s Decision Problem: Choosing the Expected Profit Nitrogen Application System A producer’s decision problem is to maximize expected profit under uncertainty (from several sources) by choosing an N recommendation system. This problem can be written as: (1) max [ ( ( ) ( ))] k k k k k k E π y N N = F φ , where k π is profit from system k; y is yield; k N is the nitrogen rate recommended by system k; k φ is the information set used by system k in making an N requirement prediction; and k F is the function used by system k to make a prediction based on k φ . An expected profit maximizing producer will abandon information set 1 φ and adopt information set 2 φ only if: (2) [ ( ( ) ( ))] [ ( ( ) ( ))] 1 1 1 1 1 2 2 2 2 2 E π y N N = F φ < E π y N N = F φ . For example, imagine that information set 1 φ provides a more accurate prediction of N needs than information set 2 φ , helping the producer to reduce N costs from overapplication, but that it provides this increased accuracy at a cost that exceeds the expected 11 N savings. In this case, the producer expects more profit from a less accurate predictor due to the high cost of information, and will switch from 1 φ to 2 φ . Thus, improved prediction accuracy attained by using fieldspecific information rather than regionspecific information must be sufficient to offset the cost of the more spatially precise information. In the case of NDVIbased predictors, prediction error will be determined by multiple factors, including the strength of the relationship between midseason NDVI data and yield, measurement and sampling error in collecting the NDVI measures, as well as the spatial scale of the data collected. So the questions arise: How do crop N requirements vary among fields? Do they vary among regions? Are they predictable using NDVI data? How Do Nitrogen Needs Vary Spatially, and What Are the Implications? That Crop N requirements vary temporally and spatially is well established (Lobell et al., 2005; Mamo et al., 2003; Washmon et al., 2002). Both spatial and annual variability in N requirements are related to weather and climate. If spatial variability of N requirements is detectable for different regions (counties, say) within a state, knowledge of this variability could allow somewhat accurate prediction of N requirements for fields within the region. Accounting for both spatial and temporal effects, crop N response is assumed to follow the form: (3) pit pit i t i i t t pit y = min( + N + v + , P + v + + + ) + u 0 1 β β ε ω ε υ , where pit y is the yield on plot p in field i in year t; pit N is the N application rate on plot p of field i in year t; 0 β and P are the estimated intercept and yield plateau, respectively; 1 β is the slope of N response; i v and i ω are random effects for field, shifting the 12 intercept and plateau, respectively; t ε and t υ are random effects for year, also shifting the intercept and plateau, respectively; pit u is a random disturbance from the mean; and i v , i ω , t ε , t υ , and pit u are all independent and normally distributed with means of zero and variances 2 v σ , 2 ω σ , 2 ε σ , 2 υ σ , and 2 u σ , respectively. When the true parameters of equation (3) are known, the uniform profit maximizing N requirement for field i in year t ( it N ) can be expressed as follows: (4) + + − + + − > + + − − = 0, otherwise, ( ) , if ( ) ( ) i t 0 1 c i t 0 i t 0 1 a it P p P P p N ω υ β β ω υ β ω υ β β here c p and a p are the price of the crop and the cost of applying N, respectively, and the remaining symbols are previously defined. Because P , 0 β and 1 β are constant, the only parameters changing N requirement from one siteyear to another are i ω and t υ . If annual effects ( t υ ) on N requirements within a region are significant and large, and if they can be predicted based on some information set—NDVI from RSs at experiment stations, say—producers may find a regional prediction of this annual effect valuable. If the annual effects are large relative to fieldspecific effects ( i ω ) on N requirements, a fieldspecific information set may not significantly improve producer profit relative to a regional information set. Thus regional predictions of N requirements might be preferable. 13 SubPaper 1: Spatial Variability, Repeatability and Noise in Predictions Made by the Nitrogen Fertilizer Optimization Algorithm and the Ramped Strip Data The first dataset used (hereafter called “countylevel data”) is comprised of onfarm trials conducted in 2007. This dataset contains 268 observations from onfarm trials of RSs in 15 counties in Oklahoma. Each observation includes the county in which the trial was located, a RS recommendation, a NFOA recommendation, the predicted yield intercept and plateau from the NFOA, and amounts of N actually applied by the producer prior to planting. The exact location of each strip within the county was not recorded. Table I1 gives the number of observations, mean RS recommendation, mean NFOA recommendation, and the mean predicted yield intercept and plateau from the NFOA by county. All of these measures are outputs of the program Ramp Analyzer 1.2 that fits a linear responseplateau function to the NDVI data to determine the N requirements if N is to be applied at the Feekes 5 growth stage (Raun et al., 2008). The N recommendations in this dataset are used to determine whether the recommendations of the NFOA and RS technologies predict any consistent variability in N requirements among counties. Also, total rainfall data by county are provided from Oklahoma Mesonet stations in or near each county. Rainfall is low for some counties (lowest is 58.29 cm) and high for other counties (highest is 150.80 cm) The second dataset (hereafter called “fieldlevel data”) contains observations from nine onfarm RS trials conducted in Canadian County in 2008. To create these data, two pairs of RSs were applied in each field as topdress ureaammonium nitrate solution 14 Table I1. Number of Observations, Mean Ramped Strip Nitrogen Recommendation, Mean Nitrogen Recommendations, and Mean Predicted Plateau Yield by County for Dataset Two County Trials RS Rate (kg ha1) NFOA Rate (kg ha1) NFOA Intercept (kg ha1) NFOA Plateau (kg ha1) Total Rainfall (cm) Blaine 10 24.53***a (6.01)b c c 3448.70*** (125.87) 132.23 Canadian 44 66.18*** (7.19) 21.51*** (2.23) 2781.62*** (71.76) 3395.95*** (91.94) 135.94 Ellis 5 22.62* (8.81) 5.38** (1.64) 1710.91*** (154.12) 1837.25*** (150.71) 58.29 Grant 20 47.77*** (6.76) 22.68*** (3.80) 2893.63*** (190.74) 3648.15*** (209.50) 103.73 Greer 3 45.17** (7.71) 15.68* (3.88) 1870.40** (266.96) 2273.60** (358.34) 77.13***d (6.42) Jackson 6 64.49** (21.51) 22.40** (6.16) 2619.68*** (130.69) 3178.00*** (246.80) 55.35 Kingfisher 2 83.44 (22.96) 23.52 (5.60) 2701.44 (739.20) 3944.64*** (60.48) 146.46 Muskogee 83 60.75*** (4.77) 26.21*** (2.50) 2925.95*** (70.85) 3599.05*** (92.96) 121.87 Noble 19 67.61*** (11.18) 24.93*** (3.04) 2639.90*** (140.87) 3355.76*** (152.51) 150.80 Nowata 15 58.54*** (6.54) 29.27*** (3.59) 3084.93*** (102.25) 4271.23*** (133.21) 108.43 Okmulgee 5 60.70*** (7.12) 17.47** (4.05) 2870.52*** (162.74) 3316.45*** (241.04) 112.70***e (27.84) Ottowa 33 63.57*** (4.78) 26.57*** (2.43) 2643.20*** (59.50) 3322.53*** (74.31) 121.92 Pawnee 10 77.62*** (20.43) 35.39*** (5.54) 2461.54*** (136.95) 3423.84*** (202.78) 135.08 Payne 5 88.48*** (7.10) 40.77* (16.61) 3240.38*** (352.27) 4359.94*** (262.83) 137.03***f (4.89) Wagoner 8 66.36*** (15.49) 25.06*** (3.77) 2872.80*** (112.59) 3492.30*** (175.43) 111.89 a One, two or three asterisks indicate statistical significance at the 0.10, 0.05 or 0.01 levels, respectively. The null hypothesis is that the means are zero. b Numbers in parentheses are standard errors. c This variable is not available for observations in Blaine County. d This is the average measure from the three closest Mesonet stations. e This is the average measure from the two Mesonet stations in Okmulgee County. f This is the average measure from the three Mesonet stations in Payne County. 15 (UAN) after plant emergence. Paired strips were made by making two adjacent passes over the field with the RS applicator, so that the rates in the paired strips increase in opposite directions. Each of the four strips was analyzed with a handheld Greenseeker optical sensor three times during the growing season, so three RS recommendations, three NFOA recommendations, and three yield plateaus and intercepts predicted by the NFOA are available from each strip. It should be noted that in this dataset (but not in the countylevel data) the predicted yield plateaus from the NFOA are right censored at 6048 kg ha1 (90 bu ac1) even when the predicted intercept is above this level. Such censoring may mean that the NFOA predicts no N response even when the raw NDVI data clearly show N response. Table I2 lists the planting dates and sensing dates for each field. The amount of N applied by producers prior to sensing was not recorded. These data are used to determine how repeatable NFOA and RS recommendations are over space and through time within fields as a measure of how much noise is present in the predictions. Procedures The important question of whether the NFOA and RS recommended N application rates vary by county within a single year is addressed using the countylevel data. If different counties have significantly different N requirements, and if these can be predicted by the RS or NFOA, a regional N requirement prediction system based on NDVI may have predictive value. To test for countylevel effects, the following Tobit model is estimated: (5) = + + + ≤ + + = + + + > = Σ Σ Σ − = − = − = 0 if 0 if 0 1 1 * 1 1 * 1 1 jk K k jk jk k k jk K k jk jk k k K k jk k k jk r N D N D r N D r α β δ μ α β δ α β δ μ 16 Table I2. Planting Date and Sensing Dates for Each Field in Dataset Three Field Planting Date Sensing Dates AC 11/6/2007 01/31/2008 AM 10/10/2007 02/01/2008 02/19/2008 03/11/2008 DE 10/14/2007 01/31/2008 02/19/2008 03/11/2008 JL 10/12/2007 01/31/2008 02/20/2008 03/11/2008 KM 10/5/2007 01/31/2008 02/19/2008 03/11/2008 LZ 10/9/2007 01/23/2008 01/31/2008 02/19/2008 RZ 10/12/2007 02/04/2008 02/19/2008 03/11/2008 SN 10/12/2007 02/04/2008 02/20/2008 03/11/2008 TZ 10/10/2007 01/31/2008 02/19/2008 03/11/2008 where jk r is the RS recommendation from strip at site j in county k; α is the intercept recommendation; β is the effect of preplant N application on the RS recommendation; jk N is the amount of preplant nitrogen applied at site j in county k; k δ is a fixed effect affecting the mean N recommendation for county k; k D is an indicator variable equal to one when county is k, and zero otherwise; K is the number of counties; * jk r is an index of the crop’s predicted “need” for N at site j in county k; jk μ is a normally distributed random deviation in predicted N requirements at site j in county k, with mean zero and 17 variance 2 μ σ . Based on this model, a likelihood ratio test is used to test the null hypothesis that county level variation in RS recommendations does not exist (i.e., δ k = 0, ∀ k ). A ttest is used to determine whether preplant application of N has any impact on RS recommendations (whether β = 0 ). The estimation is done using PROC QLIM in SAS. The above estimation in equation (5) is repeated using the NFOA recommendations as the dependent variable, and perform the hypothesis tests again to determine whether NFOA recommendations vary by county. The important questions of repeatability of RS and NFOA recommendations across time and space are addressed using the fieldlevel data. Poor repeatability of these recommendations at the same strip over time, or low correlation between recommendations from two adjacent strips would indicate that the RS or NFOA recommendations are too noisy to be useful in predicting N requirements at the singlefield level. Such noise could stem from either measurement error or high spatial variability within the field. To determine whether RS detects significant withinfield variability of N requirements, the following nointercept Tobit model is estimated: (6) = + ≤ = + > = Σ Σ Σ = = = 0 if 0 if 0 1 * 1 * 1 ijt J j ijt j j ijt J j ijt j j J j j j ijt r D D r D r δ ε δ δ ε where ijt r is the predicted optimal N application rate on strip i in pair j on sensing date t; j δ is a fixed effect for pair j; j D is an indicator variable equal to one for pair j, and zero otherwise; * ijt r is a latent variable representing the level of N (including residual and applied N) the plants in strip i in pair j on sensing date t need to reach the predicted 18 plateau yield; ijt ε is a random error term distributed with mean zero and variance 2 ε σ ; and J is the number of strip pairs. The first hypothesis tested is that N requirement predictions from the RS do not vary between pairs located within the same field—i.e., δ = δ δ = δ δJ = δJ 1 2 3 4 −1 , , Κ , . Rejection of this hypothesis would indicate that predicted N requirements from the RS vary consistently by pair within each field. Failure to reject the hypothesis would indicate either 1) that there is little variability of N requirements between locations within a field or 2) that the RS is not precise enough to detect this variability. Next, the model is restricted so that predicted N requirements do not vary by field—i.e., j y j y δ =δ , ∀ , —to determine whether the RS detects significant variability of N requirements between fields. Equation (6) is then reestimated using the NFOA predictions as the dependent variable ( ijt r ) to determine whether the NFOA recommendations vary consistently within and between fields. Additionally, graphical analyses and correlation coefficients are used to determine the strength and significance of the relationships between both RS and NFOA recommendations from 1) strips in the same pair at the same sensing date, 2) different pairs (mean recommendation) in the same field at the same sensing date, and 3) the same strip at the second and third sensing dates. The second and third sensing dates were chosen because the second date is (usually) closest to Feekes 5—the growth stage at which topdress N is normally applied—and because the third sensing date (usually in March) is closest to harvest, and may therefore be the most accurate. The correlation and plot of the relationship between RS and NFOA recommendations at the same strip for the same sensing date are also provided. 19 Results Based on the countylevel data, results from equation (5) are presented in table I3. Here, the predicted optimal topdress application rate (either from the RS or the NFOA) is modeled as a function of 1) the preplant N application rate for the field and 2) the county in which the field is located. Notably, the mean RS recommendation (64.01 kg N ha1) is more than twice the mean recommendation from the NFOA (31.14 kg N ha1). The signs of the β coefficients for the RS models are negative, which is expected because higher preplant N applications reduce the need for topdress N. Student’s ttests, however, indicate that preplant N application has no statistically significant effect on predicted topdress N requirements from the RS method—i.e., the null hypothesis β = 0 cannot be rejected. On the other hand, the β coefficients for the NFOA models are not only negative but are also statistically significant. Assuming NUE of 32% and 50% for preplant and topdress N, respectively, one kg ha1 of preplant N should reduce the need for topdress N by 0.46 kg ha1, but the coefficients are much smaller: estimated reductions of topdress needs range from 0.12 to 0.22 kg ha1 per additional kg ha1 of preplant N, depending on the model. The likelihood ratio statistic to determine whether RS method recommendations vary by county is LR = −2(991.16 − 998.61) = 14.90 , and is distributed chisquare with 13 degrees of freedom. The chisquare critical statistic at the 0.10 level is 19.81, so the test provides no evidence that RS recommendations vary by county. Similarly, no evidence is found to indicate that NFOA recommendations vary by county. The likelihood ratio statistic for this test is LR = −2(845.85 − 852.50) = 13.30 , which is also 20 Table I3. Ramped Strip and Nitrogen Fertilizer Optimization Algorithm Recommendations as Functions of FarmerPractice Preplant Nitrogen Rate and County Ramped Strip Recommendations Nitrogen Fertilizer Optimization Algorithm Recommendations Parameter Definition Unrestricted Restricted Unrestricted Restricted α Intercept 71.29***a (6.61)b 64.01*** (3.69) 27.18*** (3.34) 31.14*** (1.86) β Effect of Preplant Nitrogen 0.13 (0.11) 0.12 (0.10) 0.18*** (0.06) 0.22*** (0.05) 1 δ Effect for County 2 37.88** (18.79)  12.67 (9.48)  2 δ Effect for County 3 17.59* (9.80)  3.91 (4.95)  3 δ Effect for County 4 25.03 (21.61)  9.58 (10.90)  4 δ Effect for County 5 6.00 (16.97)  1.04 (8.69)  5 δ Effect for County 6 19.45 (26.08)  6.68 (13.14)  6 δ Effect for County 7 14.82* (8.21)  6.19 (4.15)  7 δ Effect for County 8 6.88 (10.80)  4.95 (5.45)  8 δ Effect for County 9 10.40 (10.90)  5.42 (5.50)  9 δ Effect for County 10 8.40 (17.06)  6.61 (8.61)  10 δ Effect for County 11 6.58 (9.32)  1.71 (4.71)  11 δ Effect for County 12 9.74 (12.63)  13.05** (6.38)  12 δ Effect for County 13 17.19 (17.36)  13.58 (8.76)  13 δ Effect for County 14 2.30 (13.90)  1.60 (7.02)  2 μ σ Error Variance 35.89*** (1.81) 37.28*** (1.88) 18.10*** (0.92) 18.73*** (0.95) Log Likelihood 991.16 998.61 845.85 852.50 Notes: The unrestricted models allow the mean N recommendation to vary by county, while the restricted models estimate a single mean for all counties. Units are kg ha1. a One, two or three asterisks represent statistical significance at the 0.10, 0.05 or 0.01 confidence levels, respectively. b Numbers in parentheses are standard errors. c No standard error is estimated because the parameter is restricted. 21 distributed chisquare with 13 degrees of freedom. Thus, neither the NFOA nor the RS predicts any statistically significant variability of N requirements by county. This does not mean, however, that actual N requirements do not vary by county, nor does it mean that this variability cannot be predicted using NDVI data—only that it was not predicted by the RS and NFOA methods used in the countylevel data from 2007. The issues of within and betweenfield variability of N requirements are addressed using the fieldlevel data, which includes data from nine fields in Canadian county in 2008. These data are used to estimate equation (6), which models the predicted optimal N application rate (from the RS or NFOA) as a function of the set of paired adjacent strips in which the strip is located. The estimated parameters of this equation for the RS are contained in table I4. The model with pair effects allows the mean predicted N requirement to be unique for each pair of adjacent strips, while the model with field effects is restricted such that pairs in the same field must have the same mean prediction, and the pooled model assumes the same mean N requirement for all strips in the dataset. To determine whether field affects the N recommendation from the RS, the field effects model is tested against the pooled model using a likelihood ratio test. The test statistic (chisquare with 8 degrees of freedom) is LR = −2(−479.26 + 473.43) = 11.66 , but the chisquare critical statistic at the 0.10 level is 13.36, so the test provides no evidence of variation in N requirements predicted by the RS among fields. Because variation in N requirements among fields is well documented (see Lobell et al., 2005; Mamo et al., 2003; Washmon et al., 2002), this result likely indicates that the RS technology is not precise enough to detect this variability. The test to determine whether mean N recommendations vary among pairs of adjacent strips compares the model with 22 Table I4. Mean Ramped Strip Recommendation, with and without Fixed Effects for Strip Pair and Field Model Parameter Definition Pair Effects Field Effectsa Pooledb 1 δ Fixed effect for pair 1 10.08 (21.18) 19.04 (16.07) 35.19*** (3.42) 2 δ Fixed effect for pair 2 28.00 (21.18) 19.04 (16.07) 35.19*** (3.42) 3 δ Fixed effect for pair 3 65.15*** (12.23) 48.91*** (9.28) 35.19*** (3.42) 4 δ Fixed effect for pair 4 32.67 (12.23) *** 48.91*** (9.28) 35.19*** (3.42) 5 δ Fixed effect for pair 5 59.36*** (12.23) 49.75*** (9.28) 35.19*** (3.42) 6 δ Fixed effect for pair 6 40.13*** (12.23) 49.75*** (9.28) 35.19*** (3.42) 7 δ Fixed effect for pair 7 35.47*** (12.23) 23.07** (9.37) 35.19*** (3.42) 8 δ Fixed effect for pair 8 10.40 (12.54) 23.07** (9.37) 35.19*** (3.42) 9 δ Fixed effect for pair 9 26.88** (12.23) 31.08*** (9.28) 35.19*** (3.42) 10 δ Fixed effect for pair 10 35.28*** (12.23) 31.08*** (9.28) 35.19*** (3.42) 11 δ Fixed effect for pair 11 35.47*** (12.23) 34.91*** (9.28) 35.19*** (3.42) 12 δ Fixed effect for pair 12 34.35*** (12.23) 34.91*** (9.28) 35.19*** (3.42) 13 δ Fixed effect for pair 13 16.07 (12.51) 18.61** (9.49) 35.19*** (3.42) 14 δ Fixed effect for pair 14 21.72* (12.48) 18.61*** (9.49) 35.19*** (3.42) 15 δ Fixed effect for pair 15 24.64** (12.23) 33.13*** (9.28) 35.19*** (3.42) 16 δ Fixed effect for pair 16 41.63*** (12.23) 33.13*** (9.28) 35.19*** (3.42) 23 Table I4. Mean Ramped Strip Recommendation, with and without Fixed Effects for Strip Pair and Field Model Parameter Definition Pair Effects Field Effectsa Pooledb 17 δ Fixed effect for pair 17 68.48*** (12.33) 47.25*** (9.34) 35.19*** (3.42) 18 δ Fixed effect for pair 18 26.69** (12.23) 47.25*** (9.34) 35.19*** (3.42) 2 ε σ Variance of error 29.95*** (2.18) 32.15*** (2.34) 34.05*** (2.48) Log Likelihood 466.79 473.43 479.26 Note: Units are kg ha1. a This model is restricted such that 1 2 3 4 5 6 17 18 δ =δ , δ =δ , δ =δ , Κ , δ =δ . b This model is restricted such that 1 2 3 18 δ =δ =δ =, Κ ,= δ . c One, two or three asterisks (*) indicate statistical significance at the 0.10, 0.05 or 0.01 confidence level, respectively. d Numbers in parentheses are standard errors. pair effects to the pooled mean model. The likelihood ratio statistic, which is distributed chisquare with 17 degrees of freedom, is LR = −2(−479.26 + 466.79) = 24.94 . Since the likelihood ratio statistic is slightly greater than the critical value—24.77 at the 0.10 confidence level—the test provides some evidence that mean N recommendations vary among pairs of strips in a consistent way. However, because yield data are not provided, nothing can be said about the economic significance of this finding. What is surprising, though, is that the statistical significance is not stronger. The inference is that recommendations from two adjacent strips in a pair selected at random are only slightly more homogeneous than readings from two randomly selected strips from different pairs—perhaps on opposite sides of Canadian county. The fact that RS predictions of N requirements do not show strong spatial correlation within pairs perhaps indicates that the predictions are imprecise. The lack of precision could be caused by measurement error, 24 such as would occur if the person reading the strip walked at an uneven pace while using the handheld sensor. It should also be noted that the pair effects model does not have a significantly better fit than the field effects model. The likelihood ratio statistic is LR = −2(−473.43 + 466.79) = 13.28 , and is less than 14.68—i.e., the chisquare critical statistic with 9 degrees of freedom at the 0.10 confidence level. This means that the RS detects no within field variability of N requirements. Table I5 shows the mean N application rate recommended by the NFOA with and without fixed effects for strip pair and field. The likelihood ratio test for field effects compares the model with field effects to the pooled model. The likelihood ratio statistic is LR = −2(−358.39 + 301.65) = 113.48 with 8 degrees of freedom, which exceeds the chisquare critical value of 20.09 at the 0.01 level. The likelihood ratio statistic to determine whether pair effects improve the fit of the model relative to field effects alone is LR = −2(−301.65 + 292.12) = 19.06, and is distributed chisquare with 9 degrees of freedom, and is greater than the critical statistic at the 0.10 level (16.92). Thus, the test finds (marginal) evidence that different sets of paired strips within the same field can have significantly different N recommendations—or that recommendations from adjacent strips in the same pair are more homogeneous than two randomly selected strips from different pairs but within the same field. However, the economic significance of this finding is unknown because yield data are unavailable to verify prediction accuracy. Figures I4 and I5 show plots and correlations of the recommendations from strips in the same pair at the same sensing date for the RS and NFOA, respectively. Note that the correlation between RS recommendations from adjacent strips in figure I4 is 25 Table I5. Mean Nitrogen Fertilizer Optimization Algorithm Recommendation, with and without Fixed Effects for Strip Pair and Field Model Parameter Definition Pair Effects Field Effectsa Pooledb 1 δ Fixed effect for pair 1 7.28 (19.12) 10.64 (15.47) 16.11*** (6.24) 2 δ Fixed effect for pair 2 14.00 (19.12) 10.64 (15.47) 16.11*** (6.24) 3 δ Fixed effect for pair 3 156.84 (0.00) 46.00*** (17.01) 16.11*** (6.24) 4 δ Fixed effect for pair 4 30.53* (16.83) 46.00*** (17.01) 16.11*** (6.24) 5 δ Fixed effect for pair 5 63.65*** (11.04) 63.00*** (8.93) 16.11*** (6.24) 6 δ Fixed effect for pair 6 62.35*** (11.04) 63.00*** (8.93) 16.11*** (6.24) 7 δ Fixed effect for pair 7 156.84 (0.00) 186.66 (0.00) 16.11*** (6.24) 8 δ Fixed effect for pair 8 156.84 (0.00) 186.66 (0.00) 16.11*** (6.24) 9 δ Fixed effect for pair 9 107.71*** (11.04) 81.48*** (8.93) 16.11*** (6.24) 10 δ Fixed effect for pair 10 55.25*** (11.04) 81.48*** (8.93) 16.11*** (6.24) 11 δ Fixed effect for pair 11 43.12*** (11.04) 26.81*** (9.20) 16.11*** (6.24) 12 δ Fixed effect for pair 12 9.86 (12.00) 26.81*** (9.20) 16.11*** (6.24) 13 δ Fixed effect for pair 13 26.74** (11.23) 39.94*** (9.06) 16.11*** (6.24) 14 δ Fixed effect for pair 14 53.94*** (11.15) 39.94*** (9.06) 16.11*** (6.24) 15 δ Fixed effect for pair 15 29.28* (16.63) 45.06*** (16.81) 16.11*** (6.24) 16 δ Fixed effect for pair 16 156.84 (0.00) 45.06*** (16.81) 16.11*** (6.24) 26 Table I5. Mean Nitrogen Fertilizer Optimization Algorithm Recommendation, with and without Fixed Effects for Strip Pair and Field Model Parameter Definition Pair Effects Field Effectsa Pooledb 17 δ Fixed effect for pair 17 45.36*** (11.17) 40.42*** (8.99) 16.11*** (6.24) 18 δ Fixed effect for pair 18 36.03*** (11.04) 40.42*** (8.99) 16.11*** (6.24) 2 ε σ Variance of error 27.04*** (2.53) 30.93*** (2.89) 55.42*** (5.56) Log Likelihood 292.12 301.65 358.39 Note: Units are kg ha1. a This model is restricted such that 1 2 3 4 5 6 17 18 δ =δ , δ =δ , δ =δ , Κ , δ =δ . b This model is restricted such that 1 2 3 18 δ =δ =δ =, Κ ,= δ . c One, two or three asterisks (*) indicate statistical significance at the 0.10, 0.05 or 0.01 confidence level, respectively. d Numbers in parentheses are standard errors. slightly negative, though not significant (p = 0.61). This result indicates that the RS is a noisy predictor of N requirements. On the other hand, the correlation between NFOA recommendations from adjacent strips in figure I5 is 0.56, and is statistically significant (p < 0.01). Figure I6 shows the mean RS recommendation from one pair of strips plotted against the mean RS recommendation from the other pair of strips in the same field at the same sensing date, while figure I7 plots the NFOA recommendations in the same manner. The mean RS recommendations from pairs in the same field have low correlation (0.01) that it is not statistically significant (p = 0.98). However, the mean NFOA recommendations from the different pairs are highly (0.74) and significantly correlated (p < 0.01). Figures I8 and I9 show plots of recommendations at the same strip at the second sensing date (usually February) and the third sensing date (usually March) for the RS and 27 Correlation = 0.08 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 Ramped Strip Recommendation (kg N ha1) Paired Ramped Strip Recommendation (kg N ha1 ) Figure I4. Ramped strip recommendation at one strip vs. ramped strip recommendation from the other strip in the same pair at the same sensing date. NFOA, respectively. For the RS measures, the correlation is only 0.10, and is not statistically significant (p = 0.57). The correlation for the NFOA recommendations is 0.56, and is significant at the 0.01 confidence level. The plots and correlations in figures I4 through I9 indicate that the RS recommendations are not stable over time and space within the same growing season. This result likely indicates that RS recommendations in the fieldlevel data do not very accurately represent actual N requirements. However, the relative spatial and temporal stability of the NFOA recommendations does not necessarily mean that NFOA recommendations are any more accurate than the RS predictions. To explicitly determine whether NFOA predictions are accurate, production 28 Correlation = 0.56 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 Nitrogen Fertilizer Optimization Algorithm Recommendation (kg N ha1) Paired Nitrogen Fertilizer Optimization Algorithm Recommendation (kg N ha1 ) Figure I5. Nitrogen fertilizer optimization algorithm recommendation at one strip vs. nitrogen fertilizer algorithm recommendation from the other strip in the same pair at the same sensing date. functions would have to be estimated using yield response data (which were not recorded) from the fields in the fieldlevel dataset. One reason why the NFOA recommendations show higher spatial relatedness may be the NFOA’s propensity to predict optimal rates of zero kg ha1. The NFOA, as used in the fieldlevel dataset, restricts the predicted plateau yield for each strip to be no greater that 6048 kg ha1. Thus, in cases where the NFOA predicts a yield intercept greater than 6048 kg ha1 the predicted plateau yield is still no greater than 6048 kg ha1, without regard to NDVI response to N. However, if NDVI is a noisy predictor of yield—i.e., if the relationship between NDVI data and yields varies among fields or by wheat variety— 29 Correlation = 0.01 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 Mean Ramped Strip Recommendation (kg N ha1) Mean Ramped Strip Recommendation (kg N ha1 ) Figure I6. Mean ramped strip recommendation from one pair of strips vs. mean ramped strip recommendation from the other pair in the same field at the same sensing date. then imposing this restriction on the plateau yield could bias the NFOA to predict that no N should be applied when, in fact, it would be optimal to apply N in some quantity. Figure I10 shows a plot of NFOA recommendations against RS recommendations from the same strip at the same sensing date. Note that the NFOA often recommends no application while the RS recommends some positive application rate (36 of 100 observations). This means that even when NDVI data indicate an N response— i.e., the average NDVI reading at one end of the strip is different from the average NDVI reading at the other end—the NFOA still assumes no N response by assuming that the relationship between NDVI and yields is estimated without error. However, the error 30 Correlation = 0.74 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 Mean Nitrogen Fertilizer Optimization Algorithm Recommendation (kg N ha1) Mean Nitrogen Fertilizer Optimization Algorithm Recommendation (kg N ha1) Figure I7. Mean nitrogen fertilizer optimization algorithm recommendation from one pair of strips vs. mean nitrogen fertilizer optimization algorithm recommendation from the other pair in the same field at the same sensing date. variance may be large, or may be heteroskedastic such that it increases for higher NDVI readings, or may be unique to each field. Thus, imposing this type of restriction on a plateau predicted with error may bias the NFOA predictions toward zero. Perhaps this problem could be solved by explicitly introducing this error variance into the NFOA. Conclusions First and foremost, the results indicate that the RS technique for N requirements prediction in growing winter wheat is likely too noisy to be useful in terms of accurately 31 Correlation = 0.10 0 20 40 60 80 100 120 140 0 50 100 150 200 250 Ramped Strip Recommendation, MidFebruary (kg N ha1) Ramped Strip Recommendation, MidMarch (kg N ha1 ) Figure I8. Ramped strip recommendation from MiMarch vs. ramped strip recommendation from the same strip in MidFebruary. and consistently predicting optimal N application levels. For example, the RS does not detect any significant, consistent variability of N requirements between counties, between fields, or within fields (tables I3 and I4, respectively). Furthermore, RS recommendations are neither 1) significantly correlated with RS recommendations from nearby strips (figure I4) nor steady across sensing dates (figure I8). These facts together indicate that the RS technology requires continuing development to address the sources of noise that adversely affect the consistency of its predictions. 32 Correlation = 0.56 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 Nitrogen Fertilizer Optimization Algorithm Recommendation, MidFebruary (kg N ha1) Nitrogen Fertilizer Optimization Algorithm Recommendation, MidMarch (kg N ha1 ) Figure I9. Nitrogen fertilizer optimization algorithm recommendation from Mid March vs. nitrogen fertilizer optimization algorithm recommendation from the same strip in MidFebruary. The NFOA recommendations (as opposed to the RS recommendations) seem more consistent with expectations about variability of N requirements between and within fields (table I5 and associated hypothesis tests). NFOA recommendations are also significantly correlated within pairs (figure I5), within fields (figure I7) and across time within the growing season (figure I9). However, the reason for this high correlation may be the restriction on the plateau yield predicted by the NFOA. Because the plateau and intercept are predicted based on the estimated (with error) relationship between NDVI data and yields, the predictions are uncertain. Because of this estimation error, the NFOA 33 Correlation = 0.11 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 Nitrogen Fertilizer Optimization Algorithm Recommendation (kg N ha1) Ramped Strip Recommendation (kg N ha1) Figure I10. Ramped strip recommendation vs. nitrogen fertilizer optimization algorithm recommendation from the same strip at the same sensing date. occasionally predicts crop yields will be unresponsive to N (by capping the predicted yield plateau) even when NDVI data are N responsive. Ultimately, both the NFOA and RS methods used to create these data are too noisy to accurately predict crop N requirements. However, these techniques have been— and continue to be—used by producers (Raun et al., 2008). Producers using the RS and NFOA technologies do so because they believe it is profitable. Perhaps these producers are not using the technology precisely as intended. For example, they may be integrating farmer intuition into the process of choosing an N application rate—using the NFOA or RS in addition to rules of thumb they have always used. It may be optimal to use a combined information set that includes the old (farmer practice) and new (NFOA or RS) 34 decision tools in the choice of N application rates. The results in this paper suggest several potential avenues of related research, including: 1) the creation of a formal Bayesian framework that will allow producers to input a set of fieldspecific rules of thumb, say, into the NFOA and RS methodologies, 2) the development of a framework for including uncertainty (such as the error variance of the relationship between NDVI and yield) in the NFOA or RS methodologies, 3) use of improved estimation methods for the ramped strip linear responseplateau functions and 4) development of more accurate measurement techniques for collecting NDVI data (as opposed to walking with a handheld sensor. Any of these pursuits (or several jointly) might improve the accuracy of midseason N requirements predictions based on the RS and NFOA. SubPaper 2: Prediction Uncertainty and the Value of Increasingly Spatially Precise Sampling of Optical Reflectance Data Data The dataset used in this subpaper consists of experiments conducted at ten sites throughout the state of Oklahoma between 1998 and 2008. The ten sites are located at the Efaw, Haskell, Hennessey, Lahoma, Lake Carl Blackwell, Perkins, Stillwater, and Tipton agricultural experiment stations. Table I6 contains the specifics about N treatment levels, replications, soil types, and dates for each location, while the map in figure I11 shows the locations of the sites. Each siteyear had at least three different levels of N treatment, which differed across sites, and occasionally between years at the same location. The number of replications at each N application rate varies by siteyear. NDVI measures for 35 Table I6. Locations, Years, Soil Types, and Nitrogen Levels, and Replications for Experiments in Dataset One Experiment Station Years Soil Type Nitrogen Treatment Levels (kg ha1) Efaw 1 19992006 Easpur loam 0 (3) 45 (3) 90 (3) 179 (3) 269 (3) 538a (3)b Efaw 2 19992003 Easpur loam 0 (3) 56 (6) 90 (6) 123 (6) Haskell 19992002 Taloka silt loam 0 (8) 112 (16) 168 (4) Hennessey 20002003 Shellabarger sandy loam 0 (3) 56 (5) 90 (6) 123 (6) Lahoma 19992008 Grant silt loam 0 (8) 22 (4) 45 (4) 67 (4) 90 (4) 112 (4) Lake C.B. 2004, 2006 Port silt loam 0 (4) 50 (4) 100 (4) Perkins 1c 19982006 Teller sandy loam 0 (3) 56 (3) 112 (3) 168 (3) Perkins 2 1998 Teller sandy loam 0 (9) 56 (9) 112 (9) 168 (9) Stillwater 19992006, 2008 Norge silt loam 0 (8) 45 (4) 90 (4) 134d (4) Tipton 1998 Tipton silt loam 0 (12) 56 (12) 112 (12) 168 (12) a Rate not available in 2000. b Numbers in parentheses are the number of replications at each rate each year. c Numbers of replications are the same in 1998 as at Perkins 2. d Rate not available in 2004, 2005, 2008. each observation were collected around Feekes growth stage 5, and yield was measured at harvest. These data are used to 1) determine whether year has a significant impact on N requirements across locations throughout the state of Oklahoma, 2) determine the relationship between NDVI information and the parameters of the LRP functions yield response to N, 3) create a framework for introducing the uncertainty about this relationship into a RStype N requirements prediction techniques, and 4) estimate the relative profitability of the different N requirement prediction systems described in the introduction. 36 Figure I11. Map of experimental locations. Based on local cooperative prices on February 14, 2009, assumed prices of N from UAN and NH3 are $1.10 kg1 and $0.57 kg1, respectively. Custom application costs for UAN are assumed to be $9.71 ha1 and custom application of NH3 is $20.49 ha1 (Doye, Sahs and Kletke, 2007). The wheat price is $0.24 kg1 RS application is assumed to take place early after planting as topdress UAN. Producers are advised to apply as many as 3 strips per field, each measuring 3 m by 55 m (0.0165 ha), starting at an application rate of 0 kg ha1, and increasing the application rate in increments of 14.56 kg ha1, until reaching the maximum rate that could possibly be used by the plants (assumed to be 134 kg ha1). Thus, the average N application rate in the three RSs is 67 kg ha1. It is also assumed that because the RSs are applied separately 37 Table I7. Partial Budget for Creation and Use of Three Ramped Strips in a 63 ha field Operating Input Units Price Quantity Cost UAN kg 1.10 3.31 3.64 Road Time km 4.12 8 32.96 Coop Labor hr 17.50 2.50 43.75 Sensor ha 1.08 63.00 68.04 Producer Labor hr 17.50 2.50 43.75 Total Cost of RS Field 192.14 Total Cost of RS ha 3.04 from preplant N, the producer pays road time totaling eight km per field at $4.12 km1 for delivery of the RS applicator. It is assumed that the custom application of the strips takes 2.5 hours of custom labor, and that the producer later spends 2.5 hours reading the three strips with his own Greenseeker® sensor. Thus, the total cost of creating and using three RSs is $192.14 per field, or $3.04 ha1 for a 63 ha field on a 1000 ha farm, where the cost of the sensor is spread over the entire farm. Table I7 is a partial budget for the creation and use of the strips. Procedures Variability of Nitrogen Needs by Year and Location One objective of this subpaper is to quantify variation in N requirements by year and experimental site. This is of interest because, for a regional N requirement prediction system to be of value, annual effects on N needs within the region must exist and be predictable with some accuracy. Thus, tests for fieldspecific and year specific effects on N needs are conducted based on the following model: (7) t t pit N i i i N i t i i N i pit pit i i y = + N +Σv D + P +Σv D +Σ D + + + u − = − = − = min( , ) 1 1 1 1 1 1 0 1 β β ε ω ε υ , 38 pit y is yield on plot p of field i in year t; 0 β is the intercept yield; 1 β is the crop N response rate; P is the expected yield plateau; i v and i ω are fixed effects for field i, shifting the intercept and plateau, respectively; i D is an indicator variable equal to one for field i and zero otherwise; t ε and t υ are random effects for year t, also shifting the intercept and plateau, respectively; pit u is a random disturbance; and t ε , t υ and pit u are from independent normal distributions with means zero and variances 2 ε σ , 2 υ σ , and 2 u σ , respectively. The determination of whether plateau yield shifts randomly (and independently of intercept yield) by year is made using a likelihood ratio test with one degree of freedom to test the restriction 2 = 0 υ σ . Rejection of this restriction would be evidence that accurate predictions of annual effects could be valuable information to producers making a choice of N application rate in the region for which the prediction was made. The restriction , , 0 1 2 1 = = = = N− ω ω Κ ω is also tested, where N −1 is the number of estimated fieldspecific plateau fixed effects in the model, to determine whether the mean yield plateau varies by site (at the fieldlevel, say). If this type of variation of plateau yield is found, it will indicate that N requirements also vary by field, which is expected on the basis of the literature (Lobell et al., 2005; Mamo et al., 2003; Washmon et al., 2002). Defining a Predictive Relationship This paper develops an N requirements prediction system based on the RS methodology that accounts for two types of uncertainty: 1) estimation uncertainty—or uncertainty about the value of the parameters of NDVI response to N—and 2) prediction 39 uncertainty—or uncertainty in the predictive relationship between NDVI response and yield response to N. To this end, the following equations are estimated for each siteyear: (8) it t t it t it y = min( + N , P ) + u 0 1 β β and (9) it t t it t it insey = min(α +α N , φ ) +η 0 1 , where it y is the measured yield on plot i in fieldyear t; 0t β is the intercept yield for fieldyear t; 1t β is the yield response to N in fieldyear t; t P is the plateau yield in fieldyear t; it u is a normally distributed disturbance with mean zero and variance u t σ 2 for plot i in fieldyear t; it insey is the measured NDVI on plot i in fieldyear t; 0t α is the NDVI intercept for fieldyear t; 1t α is the NDVI response to N for fieldyear t; t φ is the NDVI plateau for fieldyear t; and it η is a normally distributed disturbance with mean zero and variance 2 t η σ for plot i in fieldyear t. Of paramount interest is the accuracy with which the parameters of equation (8) can be predicted by the parameters of equation (9). In other words, how do the LRP functions of NDVI compare with the LRP functions of actual yields? To answer this question, seemingly unrelated regression is used in SAS PROC MODEL to estimate the following: (10) t t t β = λ +λ α +ε 0 0 1 0 ˆ ˆ (11) t t t = + + r 1 0 1 1 βˆ γ γ αˆ , and (12) t t t Pˆ = ρ + ρ φˆ + e 0 1 , where 0t ˆβ , 1t ˆβ and t P ˆ are the estimated parameters of the LRP response of yield to N application from equation (8); 0t αˆ , 1t αˆ and t φˆ are the estimated parameters of the LRP 40 response of NDVI to N application from equation (9); 1 λ and 2 λ are the intercept and slope, respectively, of the relationship between the NDVI intercepts and the yield intercepts; 1 γ and 2 γ are the respective intercept and slope of the relationship between the responses of yield and NDVI to N application; 0 ρ and 1 ρ are the intercept and slope of the relationship between the NDVI plateau and yield plateau; and t ε , t r and t e are random, correlated error terms with means zero and variancecovariance matrix: (13) I Σ I σ σ σ σ σ σ σ σ σ Σ ⊗ = ⊗ = c e er ee r rr er r e ε ε εε ε ε , where σεε is the n by n variancecovariance matrix for equation (10); where σrr is the n by n variancecovariance matrix for equation (11); where σee is the n by n variancecovariance matrix for equation (12); the offdiagonal elements are nonzero crossmodel correlation matrices of the contemporaneous error terms, and I is an n by n identity matrix. The parameters estimated in equations (8) through (13) are used to determine the optimal N application rates for 1) the fieldlevel perfect predictor, 2) the fieldlevel NDVIbased predictor, and 3) the regional NDVIbased predictor, as well as to calculate the net returns above Nrelated costs1 (hereafter simply called “net returns”) for each prediction system. 1 Nrelated costs as defined here include 1) the cost of purchasing N, 2) the cost of custom application of N and 3) the cost of any technology and/or experimental strip required any given system for predicting the optimal N application rate. 41 The Perfect Prediction Nitrogen Rate The perfect prediction N application rate for each siteyear is determined based on the yield data. This rate produces the maximum possible expected profit for a topdressonly N application system. However, because the true parameters of the LRP functions estimated in equation (8) are unknown, and because the LRP functional form is nonlinear in parameters, the true optimal N application rate cannot be calculated deterministically (Babcock, 1992). Babcock (1992) and Tembo et al. (2008), however, do not consider uncertainty in all parameters of the LRP functional form—only in the yield plateau. The solution derived here is different, in that it accounts for uncertainty in the intercept, slope and plateau parameters. This work also differs from that of the preceding authors by considering parameter estimation uncertainty, rather than uncertainty caused by annual variability of the plateau. To account for estimation uncertainty, ten thousand Monte Carlo observations are used to determine the expected profit maximizing N application rate for each siteyear. These simulated observations are obtained by the process: (14) jt t t j βˆ = βˆ +Q 'z and QtQt '= t , where jt β ˆ is the jth simulated 4 by 1 vector of LRP parameters for siteyear t based on the estimation of equation (8)—i.e., 0 jt ˆβ , 1 jt ˆβ , jt P ˆ , and u jt σˆ 2 ; t β ˆ is the 3 by 1 vector of LRP parameter estimates for siteyear t from equation (8); ' t Q is the 3 by 3 lower triangular Cholesky decomposition matrix of t , which is the 3 by 3 variancecovariance matrix of parameter estimates for siteyear t; j z is the jth 3 by 1 vector of random deviates from a standard normal distribution; j = 1, Κ , J ; and J is ten thousand. 42 The true (or perfect prediction) application rate that maximizes expected profit for siteyear t is then calculated based on the Monte Carlo observations generated in equation (14) using the following maximization problem: (15) n t a t J j c jt jt t jt jt t N p N p J p N P E N t δ β β β π − − + =Σ= 1 0 1 0 max(min( ˆ ˆ , ˆ ), ˆ ) max ( ( )) , where π is profit; t N is the uniform N application rate for siteyear t; c p is the wheat price; 0 jt ˆβ is the jth simulated intercept coefficient for siteyear t; 1 jt ˆβ is the jth simulated slope coefficient for siteyear t; jt P ˆ is the jth simulated plateau coefficient for siteyear t; n p is the price of N from UAN solution; a p is the custom application cost for UAN solution; t δ is an indicator variable equal to one if > 0 t N ; J is ten thousand; and the max function ensures that yield is always greater than or equal to the intercept yield. Nitrogen Needs Predictions by SiteYear Next, the predicted economically optimal N application rate must be predicted for each siteyear based on the available NDVI data. The methods use to predict these application rates differ from those of Raun et al. (2008) by accounting for estimation uncertainty about the estimated parameters in equation (9) and of the parameters estimated in equations (10) through (12). To begin the prediction process, ten thousand sets of Monte Carlo simulated parameters are generated for each siteyear based on the parameter estimates from equation (9). The process for generating these Monte Carlo simulations is the same as that described in equation (14), and is used (as before) to account for parameter estimation uncertainty. However, these simulated parameters cannot be 43 directly used to predict the expected profit maximizing N application rate because they represent the response of expected NDVI measures (rather than yields) to N application. To predict the economically optimal N application rate based on these LRP functions of NDVI, the Monte Carlo simulated parameters based on equations (9) and (14) must be converted to expected yield parameters. This transformation is made using the seemingly unrelated regression parameters estimated in equations (10), (11) and (12), where the parameters of the expected yield functions depend on the parameters of the NDVI functions. However, the parameters describing the relationships between the LRP functions of NDVI and yield data are also estimated with error. Thus, Monte Carlo simulation is again used to generate ten thousand vectors of simulated parameters based on the joint normal distributions of the parameters estimated in equations (10), (11), and (12). These vectors are generated as follows: (16) j j λˆ = λˆ +Q'z , and QQ'= (X'Σ−1X)−1 where j λ ˆ is the jth simulated 6 by 1 vector of parameter estimates based on the estimated system in equations (10), (11) and (12)—i.e., 0 j ˆλ , 1 j ˆλ , 0 j γ , 1 j γ , 0 j ρ , and j 1 ρ ; λ ˆ is the 6 by 1 vector of estimated parameters from equations (10), (11) and (12); Q' is the lower triangular Cholesky decomposition of (X'Σ−1X)−1 , which is the 6 by 6 variancecovariance matrix of the parameters in λ ˆ , where: (17) = 3 2 1 0 0 X 0 X 0 X 0 0 X and Σ−1 = Σ−1 ⊗I c , such that 1 X , 2 X and 3 X are the n by 2 matrices with n 1s and n N recommendations from equations (10), (11) and (12); and Σc and I are defined in equation (13). 44 Then, using the Monte Carlo simulated parameters from equation (16), the simulated parameters of the LRP functions of NDVI for each siteyear—see equations (9) and (14)—are transformed from NDVI parameters to expected yield LRP parameters as follows: (18) 0 jt 0 j 1 j 0 jt ˆ ˆ ˆ ~ β = λ +λ α , (19) 1 jt 0 j 1 j 1 jt ˆ ˆ ˆ ~ β = γ +γ α , and (20) jt j j jt P ρˆ ρˆ φˆ ~ 0 1 = + where 0 jt ~β , 1 jt ~β and jt P ~ are, respectively, the jth simulated intercept, slope and plateau coefficients of the predicted expected yield LRP function for siteyear t; 0 jt αˆ , 1 jt αˆ and jt φˆ are the jth simulated intercept, slope and plateau coefficients, respectively, of the LRP function of NDVI measures for siteyear t; 0 j ˆλ , 1 j ˆλ , 0 j γˆ , 1 j γˆ , 0 j ρˆ and 1 j ρˆ comprise the jth simulated set of parameters relating LRP functions of yield and NDVI. 0 jt ~β , 1 jt ~β and jt P ~ in place of 0 jt ˆβ , 1 jt ˆβ and jt P ˆ in equation (15) to calculate the predicted expected profit maximizing N application rate. Nitrogen Needs Predictions by RegionYear The process for making regionyear predictions of the economically optimal N application rate is similar to the process for obtaining siteyear predictions. To begin, data from all sites in a given year are pooled to estimate: (21) iy y y iy y iy insey = min(α +α N , φ ) +ε 0 1 , 45 where iy insey is the NDVI measure on plot i in year y; 0 y α , y 1 α , y φ are, respectively, the intercept, slope, and plateau of the LRP response of NDVI measures to N application in year y; iy N is the N application rate on plot i in year y; and iy ε is a stochastic error term with mean zero and variance 2 ε σ . The parameter estimates from equation (21) are then used with their estimated variancecovariance matrix to simulate ten thousand 4 by 1 vectors of parameters. These simulated parameters are transformed to parameters of expected yield LRP functions of N using the process described in equations (18), (19) and (20). Finally, these simulated parameters are used to predict the optimal topdress N application rate for the statewide region in year y using the maximization problem in equation (15). Calculation of Expected Yield and Expected Profit Next, because one of the major objectives of this paper is to estimate the differences in relative profitability between the perfect predictor, the siteyearspecific predictor, the regionyear predictor and the historically recommended extension rate, the expected yield and expected profit are calculated for each system in each siteyear as follows: (22) Σ= + = J j jt jt kt jt kt J N P E y N 1 0 1 min( ˆ ˆ , ˆ ) [ ( )] β β , (23) n kt a kt k J j c jt jt kt jt kt p N p p J p N P E N − − − + =Σ= δ β β π 1 0 1 min( ˆ ˆ , ˆ ) [ ( )] , where y is yield; kt N is the N application rate prescribed by system k for siteyear t; 0 jt ˆβ is the jth simulated intercept coefficient of the yield response function for siteyear t; 1 jt ˆβ 46 is the jth simulated slope coefficient of the production function in siteyear t; jt P ˆ is the jth simulated yield plateau for siteyear t; kt δ is an indicator variable equal to one if > 0 kt N and zero otherwise; k p is the cost of acquiring and using the information set for system k; k is either the regionyear, siteyear, historical extension rate, or perfect prediction system; and all other symbols are previously defined. Testing for Differences in Expected Profit, Expected Yield, and Nitrogen Application Rates Based on the calculations of expected yields and profits in equations (23) and (24), and the predicted economically optimal N application rates for each system and siteyear, paired differences tests are used to determine whether any statistically significant differences exists between three systems in terms of yields, profitability and N use. These paired differences are calculated as: (24) D E y N E y N q k qt kt y qkt = [ ( )] − [ ( )], ≠ (25) D E N E N q k qkt qt kt π = [π ( )] − [π ( )], ≠ , and (26) D N N q k qt kt N qkt = ( ) − ( ), ≠ , where y qkt D is the difference between the expected yield for methods q and k in siteyear t; qt N is the amount of N prescribed by system q in siteyear t; kt N is the N application rate prescribed by method k in siteyear t; π qkt D is the difference of expected profit from methods q and k for siteyear t; N qkt D is the difference of the N application rates prescribed by methods q and k for siteyear t; methods q and k are two N application 47 recommendation systems selected from the siteyear, regionyear, historical extension, and perfect predictor systems; and all other symbols are previously defined. Because the student’s t test relies on normality of the data, nonparametric bootstrapping of these differences is performed to test the null hypothesis that the mean paired differences of profits, yields and N application rates are zero. This is done by random sampling with replacement from the original sample of observations on the 52 siteyears2 to create ten thousand random samples of 52 siteyears each. Using the means of the sample means and simulated standard errors (i.e., standard deviations of the sample means), t tests are conducted to determine whether the siteyear, regionyear, or historical extension recommendation system should be recommended for expected profit maximization. Sensitivity analysis is performed to determine whether the results are sensitive to assumptions about NUE from topdress N application, as compared with preplant N application. NUE levels assumed for the purpose of sensitivity analysis are 32%, 45% and 50%—with 32% being the average NUE for preplant N applications (Roberts, 2009; Raun et al., 1999), 50% being the NUE for topdress applications assumed by Raun et al. (2005) and 45% being an intermediate level of NUE. These assumed levels of NUE correspond to multiplying Monte Carlo simulated slope parameters for topdress systems by 1, 1.41 and 1.56, respectively, before solving for the optimal N application rates and proceeding with the calculation of expected profits and yields. 2 The original sample contains 53 siteyears; however, experiments were only conducted at one location in 2007. As a result, a regionyear N application could not be calculated for the Lahoma site in 2007. 48 Table I8. Wheat Yield as a Function of Nitrogen Application with Site and Year Specific Effects on the Intercept and Plateau Yields Model Parameter Definition Unrestricted No Plateau Random Effects No Plateau Fixed Effects 0 β Expected intercept yield 4144.99***a (196.52)b 3699.72*** (182.26) 4327.57*** (172.44) 1 β Crop response to nitrogen 19.41*** (1.70) 19.30*** (1.79) 65.45*** (15.98) P Expected yield plateau 5522.14*** (188.54) 5535.95*** (175.22) 5243.81*** (154.43) 2 u σ Variance of error term 631321.00*** (26599.00) 677913.00*** (28651.00) 729510.00*** (30954.00) 2 ε σ Variance of intercept random effects for year 167307.00*** (23791.00) 188344.00*** (20907.00) 144457.00*** (31221.00) 2 υ σ Variance of plateau random effects for year 170663.00*** (33424.00)  208312.00*** (40004.00) 2 v σ Variance of intercept fixed effects for site 51143301.00*** (9613634.00) 38989411.00*** (7422990.00) 61587700.00*** (8685127.00) 2 ω σ Variance of plateau fixed effects for site 13124989.00*** (1610810.00) 14357724.00*** (4374240.00)  Log Likelihood 9187.50 9203.50 9251.50 a Three asterisks (*) indicate statistical significance at the 0.01 confidence level. b Numbers in parentheses are standard errors. Results Table I8 contains estimates of the parameters from equation (5), where wheat yield is a function of site, year and the preplant N application rate. The unrestricted model allows plateau and intercept yields to vary by site and year—i.e., 2 , 2 , 2 , 2 > 0 ε υ ω σ σ σ σ v . The model with no plateau random effects is restricted such that 2 = 0 υ σ , limiting the model so that the average plateau yield across all locations does not vary by year. The likelihood ratio statistic—LR = −2(−9203.50 + 9187.5) = 32.00—is distributed chisquare with one 49 degree of freedom, and exceeds the critical value at the 0.01 confidence level (6.64). This result indicates that the average plateau yield for the entire state of Oklahoma varies consistently by year. The implication is that if these annual effects on the plateau yield are predictable over a large region, NDVI data from locations (experiment stations, for example) dispersed throughout the region would provide valuable information to all producers therein. However, at the statewide level, the variability represented in the annual plateau effects is only about 10% of the total plateau variability—i.e., /( + ) = 0.10 υ υ ω σ σ σ . At the state level, annual effects have relatively little (albeit statistically significant) influence on N requirements; however, this may not be trivial. For example, if a perfect predictor—accounting for both field and annual effects— improves profit above the current practice by $7.01 ha1(Roberts, 2009), a system that perfectly predicts annual effects would improve profits by about $0.70 ha1. It is also possible that at smaller spatial resolutions, such as at the county level, annual effects might play a relatively larger role in variation of N requirements. The model with no plateau fixed effects is restricted such that , i i ω = 0 ∀ — meaning that there is no individual effect on the average plateau yield for site i within the state of Oklahoma. Given = 0 i ω , the restriction may also be expressed as 2 = 0 ω σ ; however, because the model is estimated with fixed effects for site, the likelihood ratio statistic has nine degrees of freedom—i.e., the number of fixed effects estimated. The likelihood ratio statistic for this test is LR = −2(−9251.50 + 9187.50) = 128.00 , which exceeds the chisquare critical value of 21.67, indicating that the average plateau yield over all years varies from site to site. Farmers who have fieldspecific experience and expectations could then adjust their expectations (and topdress N applications) annually 50 based on midseason regional NDVI data collected at agricultural experiment stations and disseminated by the Cooperative Extension Service. Tables I9 and I10 contain the estimated LRP parameters for yield and NDVI data as functions of preplant N for each siteyear from equations (8) and (9), respectively. Table I11 displays the results from the annual, statewide estimation of the LRP function of NDVI as a function of N from equation (21). As noted in these tables, some of the estimated parameters have no standard errors. This occurs because the data for some siteyears do not reach a plateau. In these cases, PROC NLMIXED estimated a linear model, but generated a plateau equal to the expected yield at the maximum rate applied in the data for these siteyears. These estimates without standard errors are biased downward, because they tell us only that the plateau is expected to be greater than or equal to the estimate. This is also the case for estimates of the slope given without standard errors. At the Lahoma site in 2007, for instance, it appears “likely” that no data points are found on the slope of the production function. Figure I12 illustrates this type of data limitation. In such instances, the estimate is a lower bound on the expected value of the slope parameter. The dashed lines show how the true production function might deviate from the estimated function, but exactly how the true slope deviates from the parameter estimated in PROC NLMIXED is uncertain. Additionally, for the Perkins 1 site in 2001 there are no standard errors for the intercept or plateau parameters. In this case, PROC NLMIXED estimated the mean yield for the siteyear, but failed to provide standard errors because of data constraints. The fact that all points occur on the plateau means Monte Carlo simulation to account for estimation uncertainty is unnecessary because the 51 Table I9. Estimated Wheat Yield Response to Nitrogen by SiteYear Site Year Intercept Slope Plateau Perkins 1 1998 1134.16***a (132.79)b 8.30*** (1.80) 2102.70*** (131.46) Perkins 2 1998 1316.98*** (94.25) 1.22 (1.30) 1487.41*** (107.84) Tipton 1998 2942.65*** (93.34) 12.46*** (0.43) 5037.68*** (21.57) Efaw 1 1999 1040.52*** (226.84) 5.46*** (1.50) 3068.36*** (323.60) Efaw 2 1999 2169.07*** (192.95) 19.27*** (4.22) 3514.67*** (96.48) Haskell 1999 1767.41*** (288.21) 7.71c 2072.13*** (182.28) Lahoma 1999 1515.22*** (116.66) 26.28*** (2.28) 4443.08*** (181.36) Perkins 1 1999 1077.20*** (177.94) 12.71** (4.49) 2431.26*** (125.83) Stillwater 1999 856.12*** (103.51) 10.90** (4.00) 1712.27*** (110.65) Efaw 1 2000 911.11** (380.28) 26.84*** (6.57) 3384.06*** (294.56) Efaw 2 2000 2246.40*** (579.52) 1.53 (6.18) 2160.87*** (415.54) Haskell 2000 4262.17*** (212.53) 13.77*** (1.20) 2719.13*** (212.53) Hennessey 2000 3833.55*** (453.84) 0.29 (4.84) 3817.26*** (324.55) Lahoma 2000 1944.08*** (152.73) 25.03*** (6.09) 3515.75*** (130.79) Perkins 1 2000 2599.85*** (714.43) 6.55 (14.72) 3333.56*** (319.59) Stillwater 2000 1120.71*** (83.13) 17.05*** (1.34) 3414.03*** (96.79) Efaw 1 2001 921.82*** (215.47) 15.52** (6.80) 2024.16*** (112.53) Efaw 2 2001 2693.37*** (285.19) 8.80 (6.23) 3301.97*** (142.60) Haskell 2001 3669.98** (1368.34) 6.77 (10.92) 3121.59*** (387.02) 52 Table I9. Estimated Wheat Yield Response to Nitrogen by SiteYear Site Year Intercept Slope Plateau Hennessey 2001 1951.38*** (184.75) 7.01*** (0.76) 2815.16*** (91.34) Lahoma 2001 1495.54*** (201.16) 3.48 (17.18) 1651.35*** (142.25) Perkins 1 2001 2602.15d 1.35 (1.09) 2602.15d Stillwater 2001 1054.21*** (142.89) 12.70** (5.52) 1636.39*** (142.89) Efaw 1 2002 732.37** (325.25) 30.95*** (10.26) 2705.91*** (178.15) Efaw 2 2002 1811.65*** (305.03) 19.95*** (6.67) 3575.11*** (152.52) Haskell 2002 3500.96*** (938.17) 13.98* (1.45) 3112.43*** (262.23) Hennessey 2002 3898.07*** (28.52) 10.17*** (2.44) 2986.17*** (189.00) Lahoma 2002 2711.28*** (194.42) 16.54c 3075.88*** (122.96) Perkins 1 2002 2711.83*** (192.26) 1.55*** (0.18) 2971.97*** (161.91) Stillwater 2002 961.60*** (77.43) 16.03*** (1.54) 2987.25*** (114.85) Efaw 1 2003 1077.11** (477.42) 24.02*** (8.25) 3996.63*** (320.26) Efaw 2 2003 2792.10*** (403.20) 20.31*** (6.03) 4950.90*** (312.61) Hennessey 2003 2337.13*** (256.09) 14.67*** (3.65) 3760.42*** (166.31) Lahoma 2003 2760.86*** (209.35) 46.43*** (8.30) 5716.37*** (177.55) Perkins 1 2003 2796.69*** (190.99) 12.81** (4.82) 3779.32*** (135.05) Stillwater 2003 1136.43*** (176.83) 19.88*** (6.86) 2473.30*** (144.36) Efaw 1 2004 2079.37*** (570.45) 22.90 (18.01) 4132.65*** (285.13) Lahoma 2004 1871.40*** (313.47) 29.23c 2526.56*** (198.26) 53 Table I9. Estimated Wheat Yield Response to Nitrogen by SiteYear Site Year Intercept Slope Plateau Lake C.B. 2004 2227.34*** (248.19) 18.21*** (2.14) 4063.86*** (32.38) Perkins 1 2004 1936.34*** (393.48) 19.77* (9.93) 3399.90*** (278.24) Stillwater 2004 2080.99 (2250.37) 2.77 (28.29) 1895.02*** (220.59) Efaw 1 2005 1164.41*** (210.37) 4.56*** (1.39) 2845.72*** (300.10) Lahoma 2005 1754.09*** (188.07) 18.44** (7.27) 2683.34*** (151.63) Perkins 1 2005 3494.44*** (267.04) 9.84c 4021.48*** (178.03) Stillwater 2005 1764.35*** (145.62) 15.36 2223.53*** (118.90) Efaw 1 2006 1081.14*** (275.92) 8.05 (4.77) 2291.79*** (174.51) Lahoma 2006 2229.78*** (199.48) 4.03 (3.18) 2680.96c Lake C.B. 2006 1277.42*** (291.04) 37.69*** (8.16) 4377.41*** (291.04) Perkins 1 2006 917.24*** (113.69) 12.33*** (2.87) 2053.63*** (80.39) Stillwater 2006 1333.57*** (0.17) 5.64*** (0.68) 772.77*** (40.72) Lahoma 2007 2540.65*** (177.01) 28.81c 3162.98*** (129.27) Lahoma 2008 2761.46*** (294.09) 59.55*** (11.73) 5525.64*** (251.85) Stillwater 2008 1381.12 (174.25) 15.99*** (4.31) 2697.59*** (251.12) Mean for all siteyears 2004.70*** (124.73) 13.19*** (1.92) 3071.95*** (139.93) Note: Units are kg ha1. a One, two, or three asterisks (*) indicate statistical significance at the 0.10, 0.05 or 0.01 level, respectively. b Numbers in parentheses are standard errors. c Standard error cannot be estimated due to lack of data points on the slope or plateau. The estimated parameter is biased downward. d Standard errors for the intercept and plateau are not estimated because all available data are on the plateau. 54 Table I10. Estimated Wheat Optical Reflectance Response to Nitrogen by SiteYear, Scaled by a Factor of Ten Thousand Site Year Intercept Slope Plateau Perkins 1 1998 595.56***a (24.82)b 1.60*** (0.34) 804.24*** (25.90) Perkins 2 1998 571.95*** (23.12) 0.87 (0.58) 663.34*** (16.35) Tipton 1998 693.77*** (8.83) 1.18*** (2.23) 804.08*** (6.20) Efaw 1 1999 383.81*** (32.08) 3.34*** (1.01) 618.43*** (16.04) Efaw 2 1999 693.32*** (17.56) 1.44*** (0.38) 783.52*** (8.71) Haskell 1999 619.21*** (23.99) 2.29c 669.79*** (15.17) Lahoma 1999 615.96*** (13.21) 2.04*** (0.35) 785.83*** (14.55) Perkins 1 1999 466.16*** (23.44) 1.93*** (0.59) 591.37*** (16.48) Stillwater 1999 553.76*** (32.94) 3.52c 634.99*** (26.90) Efaw 1 2000 702.43*** (93.12) 7.82*** (1.61) 1488.30*** (72.13) Efaw 2 2000 864.23*** (38.39) 7.21c 891.76*** (15.67) Haskell 2000 600.48*** (37.85) 2.10c 625.01*** (23.94) Hennessey 2000 961.20*** (2.36) 0.14 (0.22) 978.53*** (25.06) Lahoma 2000 784.50*** (18.58) 5.10*** (0.74) 1092.05*** (15.92) Perkins 1 2000 652.11*** (53.91) 3.99c 770.82*** (31.12) Stillwater 2000 558.14*** (21.62) 7.19*** (0.84) 935.22*** (21.62) Efaw 1 2001 627.63*** (37.81) 2.46*** (0.65) 876.16*** (25.37) Efaw 2 2001 896.45*** (24.13) 0.21 (0.36) 922.09*** (18.71) Haskell 2001 674.80*** (32.57) 0.36 (0.31) 822.37 55 Table I10. Estimated Wheat Optical Reflectance Response to Nitrogen by SiteYear, Scaled by a Factor of Ten Thousand Site Year Intercept Slope Plateau Hennessey 2001 726.26*** (48.56) 1.29*** (0.56) 912.51c Lahoma 2001 774.70*** (35.75) 0.82 (2.78) 805.71*** (25.28) Perkins 1 2001 834.69d 0c 834.69d Stillwater 2001 677.06*** (42.71) 2.76 (1.65) 824.16*** (42.71) Efaw 1 2002 537.19*** (87.72) 2.39 (2.77) 649.67*** (43.86) Efaw 2 2002 638.31*** (14.87) 1.79*** (0.33) 742.28*** (7.43) Haskell 2002 517.16*** (7.65) 0.92 (1.25) 672.40*** (202.60) Hennessey 2002 652.30*** (0.01) 0.39 (0.41) 616.92*** (29.23) Lahoma 2002 753.81*** (48.93) 4.24c 843.41*** (30.94) Perkins 1 2002 721.90*** (13.34) 0.35** (0.13) 834.69c Stillwater 2002 448.76*** (13.06) 3.71*** (0.50) 692.68*** (13.03) Efaw 1 2003 346.68*** (37.06) 1.55*** (0.36) 670.93*** (33.83) Efaw 2 2003 652.54*** (36.81) 1.38*** (0.55) 816.61*** (28.54) Hennessey 2003 876.40*** (70.44) 1.69 (1.05) 1073.04*** (54.63) Lahoma 2003 570.00*** (1.39) 9.00*** (1.39) 860.00*** (11.94) Perkins 1 2003 496.14*** (19.24) 1.20*** (0.18) 684.12c Stillwater 2003 391.54*** (25.24) 3.71*** (0.62) 648.51*** (25.72) Efaw 1 2004 478.65*** (81.33) 3.40 (2.57) 781.76*** (40.64) Lahoma 2004 598.04 (85.90) 10.32c 757.59*** (54.33) 56 Table I10. Estimated Wheat Optical Reflectance Response to Nitrogen by SiteYear, Scaled by a Factor of Ten Thousand Site Year Intercept Slope Plateau Lake C.B. 2004 418.32*** (46.44) 2.20 (9.16) 639.75 (877.19) Perkins 1 2004 480.82*** (14.34) 1.17*** (0.20) 617.06*** (15.70) Stillwater 2004 727.06* (407.27) 2.45 (5.12) 564.32*** (44.51) Efaw 1 2005 497.75*** (33.02) 2.29*** (0.57) 763.08*** (20.87) Lahoma 2005 543.05*** (15.09) 3.20*** (0.59) 735.75*** (12.11) Perkins 1 2005 471.63*** (21.72) 1.33*** (0.32) 669.34*** (26.93) Stillwater 2005 550.07*** (2.92) 1.66*** (0.46) 699.18*** (38.05) Efaw 1 2006 306.18*** (50.30) 2.05** (0.87) 527.35*** (31.81) Lahoma 2006 484.41*** (30.91) 4.78c 564.29*** (19.55) Lake C.B. 2006 501.91*** (4.79) 1.03 (0.80) 606.19*** (76.37) Perkins 1 2006 268.27*** (24.38) 2.20*** (0.62) 476.83*** (17.25) Stillwater 2006 354.27*** (1.31) 1.00*** (0.29) 488.87*** (37.81) Lahoma 2007 513.11*** (12.08) 3.22*** (0.93) 597.96*** (8.54) Lahoma 2008 508.56*** (19.71) 5.36*** (0.53) 912.38*** (21.66) Stillwater 2008 690.54*** (68.31) 1.82c 771.92*** (55.78) Mean for all siteyears 594.78*** (21.08) 2.56*** (0.33) 756.87*** (23.49) a One, two, or three asterisks (*) indicate statistical significance at the 0.10, 0.05 or 0.01 level, respectively. b Numbers in parentheses are standard errors. c Standard error cannot be estimated due to lack of data points on the slope or plateau. The estimated parameter is biased downward. d Standard errors for the intercept and plateau are not estimated because all available data are on the plateau. 57 Table I11. Estimated Wheat Optical Reflectance Response to Nitrogen by Year, scaled by a Factor of Ten Thousand Yeara Intercept Slope Plateau 1998 618.49***b (16.09)c 1.33*** (0.40) 749.34*** (11.19) 1999 576.33*** (16.65) 2.11*** (0.74) 685.08*** (10.91) 2000 745.87*** (1.95) 1.64*** (0.23) 1187.04*** (60.97) 2001 731.74*** (19.02) 1.81** (0.88) 821.66*** (11.80) 2002 646.57*** (18.50) 0.45 (1.96) 767.48*** (80.44) 2003 537.63*** (33.07) 3.49*** (0.82) 792.95*** (23.71) 2004 574.33*** (30.83) 1.21 (0.82) 677.89*** (28.07) 2005 549.72*** (12.66) 1.57*** (0.21) 739.71*** (19.17) 2006 427.00*** (21.20) 0.81** (0.34) 534.13*** (30.34) 2008 597.96*** (30.92) 3.40*** (0.80) 856.63*** (37.95) Note: Units are kg ha1. a A response function for 2007 is not estimated because only one site is available in this year. b Two or three asterisks (*) indicate statistical significance at the 0.05 or 0.01 level, respectively. c Numbers in parentheses are standard errors. mean is linear in parameters. Thus, the lack of standard errors for the plateau and intercept in this siteyear is not problematic. The estimated relationships between the parameters of NDVI and yield response—estimated in equations (10), (11) and (12)—are presented in table I12. Here, the relationship describes how yield LRP function parameters (table I9) depend upon midseason NDVI parameters (table I10). The signs of the estimated coefficients are as expected—i.e., higher NDVI intercepts predict higher yield intercepts; higher NDVI response (slope) predicts higher yield response; and higher NDVI plateaus predict high 58 NDVI plateaus. Note based on the coefficients of variation for these relationships ( R2 in table I12) that these relationships are very noisy. These parameters (and their variancecovariance matrix) are used to convert LRP parameters of NDVI response into expected yield response through Monte Carlo simulation described in equations (18) to (20). Nonparametrically bootstrapped means of the prescribed N application rates, expected yields, and return above Nrelated costs for each system are displayed in table I13, assuming NUE of 32% for both preplant and topdress applications. Notably, mean net revenue is greatest for the historically recommended rate of 90 kg N ha1 from NH3, at $639.92 ha1, but this is only slightly greater than the $638.46 ha1 earned by the perfect predictor. While N purchase costs are much lower for the historical rate—because it uses NH3 rather than UAN—N application costs are much higher for the historical rate. The increased application cost, along with a slight yield boost for the perfect predictor system, nearly cancels out any saving on N purchase for the historical rate system. Additionally, the mean recommended application rates for the field and regionbased N requirements predictors are 88.92 and 94.11 kg ha1, respectively—apparently not much different from the historically recommended rate. The fieldbased system does appear to have some predictive power (though not statistically significant) because it achieves slightly higher yield than the historical rate while applying slightly less N. However, the total costs of N purchase and application for the two NDVIbased systems are, respectively, $107.52 and $113.23 ha1—relatively high compared to the analogous costs of $71.79 ha1 for the historical rate system. This difference is primarily due to the relative prices of topdress UAN ($1.10 kg1) and preplant NH3 ($0.57 kg1). It is possible that the field and region based systems could save substantially on Nrelated costs by 59 0 500 1000 1500 2000 2500 3000 3500 4000 4500 0 20 40 60 80 100 120 Nitrogen (kg ha1) Wheat Yield (kg ha1 ) Yield Data Estimated Function Figure I12. Plot of yield data and estimated production function for Lahoma 2007. using a split application—i.e., some N applied preplant as NH3 and some as topdress UAN if the RS shows that the crop is responsive. The mean UAN rate applied by the perfect predictor system is 65.41 kg ha1, compared with about 90 kg ha1 for either of the NDVIbased systems, meaning that the NDVIbased systems used in this paper overapply N substantially, as expected. Table I14 shows the nonparametrically bootstrapped means of the paired differences of expected profits, expected N application and expected yield generated in equations (24), (25) and (26). These results confirm that the profitability difference between the perfect predictor and the historical rate is statistically insignificant, despite the historical N application rate being on average 24.60 kg ha1 higher than the perfect predictor rate. Recall from table I3 that the “perfect predictor” fails to provide the 60 Table I12. Response of Yield Intercepts, Slopes and Plateaus to Optical Reflectance Intercepts, Slopes and Plateau, Respectively, Estimated by Seemingly Unrelated Regression Parameter Definition Estimate 0 λ Intercept of intercept response 480.85 (369.50)a 1 λ Slope of intercept response 255830.70***b (59091.90) R2 Coefficient of determination 0.15 0 γ Intercept of slope response 7.65*** (2.06) 1 γ Slope of slope response 217088.40*** (48025.00) R2 Coefficient of determination 0.24 0 ρ Intercept of plateau response 1440.23*** (429.20) 1 ρ Slope of plateau response 215697.40*** (53847.50) R2 Coefficient of determination 0.09 a Numbers in parentheses are standard errors. b Three asterisks (*) represent statistical significance at the 0.01 level. maximum profit primarily because it is a topdress system, using expensive UAN in place of cheaper NH3. Additionally, the perfect predictor system is significantly (p < 0.01) more profitable than either the fieldbased or the regionbased predictors by $35.14 ha1 on average. The historically recommended application of 90 kg N ha1 preplant is significantly more profitable than both the field and regionbased predictors by respective averages of $36.60 and $38.41 ha1. Expected profits from the field and regionbased systems are not statistically different. Table I15 displays the nonparametrically bootstrapped means of the prescribed N application rates, expected yields, and return above Nrelated costs for each system, assuming that NUE is 32% for preplant N applications and 45% for midseason topdress applications. Note that under this assumption, the perfect predictor system maximizes expected profit compared to the other systems (in contrast to the results in table I13). 61 Table I13. Noparametrically Bootstrapped Means of Net Returns, Revenues, NitrogenRelated Costs, Yields and Nitrogen Application Rates for Each Application System, Assuming 32% NitrogenUse Efficiency for Both Topdress and Preplant Nitrogen Applications System Revenue/Cost Perfect Predictor Historical Rate FieldBased Predictor Region Based Predictor Net Revenue ($ ha1) 638.46 (33.82) 639.92 (33.40) 603.32 (33.56) 601.51 (33.17) Yield Revenue ($ ha1) 717.51 (35.19) 711.71 (33.40) 713.88 (33.74) 714.73 (33.41) NH3 Cost ($ ha1)  51.30   Mean UAN Cost ($ ha1) 71.85 (6.96)  97.81 (2.73) 103.52 (2.08) NH3 Application Cost ($ ha1)  20.49   Mean UAN Application Cost ($ ha1) 7.10 (0.60)  9.71 (0.00) 9.71 (0.00) Precision System Cost ($ ha1)   3.04  Average Yield (kg ha1) 2989.64 (146.61) 2965.46 (139.19) 2974.51 (140.60) 2978.05 (139.23) Mean UAN Rate (kg ha1) 65.41 (6.33)  88.92 (2.49) 94.11 (1.89) Note: All estimates are significant at the 0.01 confidence level. This is because assuming topdress NUE of 45% substantially increases the marginal product of topdress N, while leaving the marginal product of preplant N unchanged. Under this assumption, the bootstrapped mean N application for each topdress system is substantially reduced relative to those in table I13. This occurs because an increase in the marginal product of N means that not as much N is required to reach the plateau. Also noteworthy is the result that expected yield for the topdress systems has increased, indicating that this increase in the marginal product of N makes UAN application more 62 Table I14. Nonparametrically Bootstrapped Means of Paired Differences of Expected Profits, Expected Nitrogen Application Rates, and Expected Yields, Assuming 32% NitrogenUse Efficiencies for Both Preplant and Topdress Nitrogen Applications Difference Expected Profit ($ ha1) Expected Nitrogen Rate (kg ha1) Expected Yield (kg ha1) Perfect Predictor  Historical Rate 1.46 (5.31)a 24.60*** (6.33) 24.18 (26.33) Perfect Predictor  FieldBased Predictor 35.14***b (5.51) 23.51*** (5.91) 15.13 (26.78) Perfect Predictor  RegionBased Predictor 36.95*** (5.69) 28.70*** (6.74) 11.58 (27.36) Historical Rate  FieldBased Predictor 36.60*** (2.61) 1.08 (2.49) 9.05 (13.15) Historical Rate  RegionBased Predictor 38.41*** (2.45) 4.11** (1.89) 12.60 (12.03) FieldBased Predictor  RegionBased Predictor 1.81 (4.10) 5.18 (3.28) 3.55 (19.72) a Numbers in parentheses are standard errors. b Two or three asterisks (*) indicate statistical significance at the 0.05 or 0.01 level, respectively. profitable than it otherwise would be, specifically in siteyears where the slope of the response to preplant N is small. Table I16 contains the bootstrapped means of the paired differences of expected profit, expected N application rate and expected yield for each system, assuming 32% and 45% NUE for preplant and topdress applications. These results confirm that the profitability difference of $24.98—favoring the perfect predictor system over the historical recommendation—is statistically significant at the 0.01 confidence level. The perfect predictor system continues to be more profitable than the field and regionbased systems. Notably, though the mean profit paired differences between the historical rate system and the field and regionbased systems continue to be significant in favor of the historical rate—$6.91 and $9.73 ha1, respectively—the differences are smaller in magnitude compared to those in table I15. There still is no statistically significant 63 Table I15. Noparametrically Bootstrapped Means of Net Returns, Revenues, NitrogenRelated Costs, Yields and Nitrogen Application Rates for Each Application System, Assuming 32% and 45% NitrogenUse Efficiency for Preplant and Topdress Nitrogen Applications, Respectively System Revenue/Cost Perfect Predictor Historical Rate Field Based Predictor Region Based Predictor Net Revenue ($ ha1) 664.90 (34.32) 639.12 (33.40) 633.01 (33.48) 630.19 (33.14) Yield Revenue ($ ha1) 728.52 (35.51) 711.71 (33.40) 718.80 (33.60) 721.25 (33.48) NH3 Cost ($ ha1)  51.30   Mean UAN Cost ($ ha1) 56.33 (5.57)  73.05 (2.29) 81.35 (2.48) NH3 Application Cost ($ ha1)  20.49   Mean UAN Application Cost ($ ha1) 7.29 (0.58)  9.71 (0.00) 9.71 (0.00) Precision System Cost ($ ha1)   3.04 (0.00)  Average Yield (kg ha1) 3035.49 (147.96) 2965.46 (139.19) 2995.01 (140.00) 3005.19 (139.52) Mean UAN Rate (kg ha1) 51.21 (5.06)  66.41 (2.09) 73.95 (2.25) Note: All estimates are significant at the 0.01 confidence level. difference between the field and regionbased systems in terms of profitability, though the regionbased system applies more N by an average of 7.55 kg ha1 (p < 0.05). The nonparametrically bootstrapped means of prescribed N application rates, expected yields, and return above Nrelated costs for each system assuming 32% and 50% NUE for preplant and topdress N, respectively, are presented in table I17. Here, the field and regionbased predictors have returns (net of Nrelated costs) very similar to the returns from using the historical rate. The mean of expected net revenue is slightly higher for the fieldbased system and slightly lower for the regionbased system. The costs of N 64 Table I16. Nonparametrically Bootstrapped Means of Paired Differences of Expected Profits, Expected Nitrogen Application Rates, and Expected Yields, Assuming 32% and 45% NitrogenUse Efficiencies for Preplant and Topdress Nitrogen Applications, Respectively Difference Expected Profit ($ ha1) Expected Nitrogen Rate (kg ha1) Expected Yield (kg ha1) Perfect Predictor  Historical Rate 24.98***a (5.43)b 38.79*** (5.06) 70.03** (34.25) Perfect Predictor  FieldBased Predictor 31.90*** (5.72) 15.20*** (4.84) 40.48 (32.38) Perfect Predictor  RegionBased Predictor 34.71*** (5.55) 22.75*** (5.80) 30.30 (30.82) Historical Rate  FieldBased Predictor 6.91** (2.96) 23.59*** (2.09) 29.55* (16.56) Historical Rate  RegionBased Predictor 9.73*** (3.42) 16.05*** (2.25) 39.73** (15.89) FieldBased Predictor  RegionBased Predictor 2.82 (4.89) 7.55** (3.39) 10.18 (24.37) a One, two or three asterisks indicate statistical significance at the 0.10, 0.05 or 0.01 confidence level, respectively. b Numbers in parentheses are standard errors. purchase and application for the historical, fieldbased and region based systems are$71.79, $76.89 and $84.54 ha1, respectively. The field and regionbased systems make up for their increased N expenditures (and the cost of the RS, in the case of the fieldbased system) through increased yields resulting from higher NUE. Table I18 presents the nonparametrically bootstrapped means of the paired differences of expected profits, expected N application rates and expected yields between the four systems. Note that the perfect predictor system is expected to be more profitable than all other systems by at least $31.74 ha1, and that these differences are statistically significant at the 0.01 confidence level. Additionally, the historical rate is higher than the mean of any other system by at least 21.79 kg N ha1. One problem with the field and 65 Table I17. Noparametrically Bootstrapped Means of Net Returns, Revenues, NitrogenRelated Costs, Yields and Nitrogen Application Rates for Each Application System, Assuming 32% and 50% NitrogenUse Efficiency for Preplant and Topdress Nitrogen Applications, Respectively System Revenue/Cost Perfect Predictor Historical Rate FieldBased Predictor Region Based Predictor Net Revenue ($ ha1) 671.84 (34.09) 639.12 (33.40) 640.10 (33.54) 638.01 (33.18) Yield Revenue ($ ha1) 734.11 (34.64) 711.71 (33.40) 720.03 (33.66) 722.55 (33.50) NH3 Cost ($ ha1)  51.30   Mean UAN Cost ($ ha1) 54.79 (5.28)  67.18 (2.12) 74.83 (2.29) NH3 Application Cost ($ ha1)  20.49   Mean UAN Application Cost ($ ha1) 7.48 (0.56)  9.71 (0.00) 9.71 (0.00) Precision System Cost ($ ha1)   3.04 (0.00)  Average Yield (kg ha1) 3058.81 (144.33) 2965.46 (139.19) 3000.14 (140.27) 3010.62 (139.57) Mean UAN Rate (kg ha1) 49.81 (4.80)  61.07 (1.93) 68.03 (2.08) Note: All estimates are significant at the 0.01 confidence level. regionbased systems as developed in this paper is that they always recommend some level of N application. This is evident because mean application costs for these systems, regardless of assumptions about NUE are $9.71 ha1 (see tables I13, I15 and I17). As a result, field and regionbased methods used here apply substantial N in cases where the true expected profit maximizing N rate is actually zero. This results in a substantial increase in N costs relative to the perfect predictor system without a commensurate increase in yield (because yield reaches a plateau at many sites at 65 kg N ha1). 66 Table I18. Nonparametrically Bootstrapped Means of Paired Differences of Expected Profits, Expected Nitrogen Application Rates, and Expected Yields, Assuming 32% and 50% NitrogenUse Efficiencies for Preplant and Topdress Nitrogen Applications, Respectively Difference Expected Profit ($ ha1) Expected Nitrogen Rate (kg ha1) Expected Yield (kg ha1) Perfect Predictor  Historical Rate 31.92*** (5.39) 40.18*** (4.80) 93.35*** (34.14) Perfect Predictor  FieldBased Predictor 31.74*** (5.57) 11.26** (4.52) 58.67* (31.07) Perfect Predictor  RegionBased Predictor 33.83*** (5.21) 18.21*** (5.46) 48.19 (29.41) Historical Rate  FieldBased Predictor 0.18 (3.03) 28.93*** (1.93) 34.68** (16.75) Historical Rate  RegionBased Predictor 1.91 (3.48) 21.97*** (2.08) 45.16*** (16.12) FieldBased Predictor  RegionBased Predictor 2.09 (4.85) 6.96** (3.13) 10.48 (24.24) a One, two or three asterisks indicate statistical significance at the 0.10, 0.05 or 0.01 confidence level, respectively. b Numbers in parentheses are standard errors. Also noteworthy is that the value of a perfect predictor system—i.e., the profit difference between the perfect predictor and the second most profitable system—is highly dependent on NUE. If NUE for topdress applications is the same as for preplant applications (32%), a perfect prediction of topdress N requirements has no value (see table I15). On the other hand tables I17 and I19 indicate that the value of a perfect predictor given 45% and 50% NUE is $24.98 or $31.74 ha1, respectively. Thus, the value of a perfect predictor of topdress N requirements is strongly dependent on the true NUE for topdress applications. 67 Conclusions One important finding of this research is that the historical extension recommendation— i.e., 90 kg N ha1 as NH3—is statistically indistinguishable from the “perfect predictor” topdress application system using UAN, primarily resulting from relative costs of UAN ($1.10 kg1) and NH3 ($0.56 kg1). The value of a perfect predictor, or the mean difference between the perfect predictor and 



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