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a Iowa Soybean Assoc., 4554 114th Street, Urbandale, IA 50322
b Dep. of Agronomy, Iowa State Univ., Ames, IA 50010
c Dep. of Plant Science, Univ. of Connecticut, Storrs, CT 06269
* Corresponding author (pkyveryga{at}iasoybeans.com)
Received for publication November 26, 2006.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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Abbreviations: EOR, economic optimum rate • EXP, exponential model • LRP, linear response and plateau model • QRP, quadratic response and plateau model • QUAD, quadratic model • SR, square root model
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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Uncertainty is always involved when observations made in the past are used to predict what rates will be optimal in the future. Uncertainty is unavoidable when predicting optimal rates because fertilizer is often applied before crops are grown and because weather and other factors that occur after fertilization influence the magnitude of yield response to added N. Researchers in economics (Myrdal and Wicksell, 1939; Rima, 2001; Stonier and Hague, 1980) and other disciplines (Babu and Rhoe, 2003; Jack, 2002; Ulph, 1982) have addressed this problem by using the concepts of ex post and ex ante analyses to specify the time calculations are made. Ex post calculations are after-the-fact analyses in which the only problem is to interpret what happened in the past. Ex ante calculations are made in advance of future applications and require the use of ex post calculations as well as assumptions concerning the likelihood of future events. The distinction between ex post and ex ante analysis has been made when estimating N fertilizer requirements (Anselin et al., 2004; Babcock and Blackmer, 1994; Bullock and Bullock, 2000), but there is a need for more discussion of how these concepts should be used when estimating N fertilizer requirements.
Evidence of systematic errors associated with selection of a model to describe relationships between N rates and yields is another important problem when calculating EORs (Cerrato and Blackmer, 1990; Colwell, 1994; Waugh et al., 1973). Systematic bias imposed by the model selected is indicated by disagreements among calculated EORs when different models are used to describe the same set of data. Although these disagreements often are large enough to be of economic and (or) environmental importance, methods have not been developed to ensure that errors imparted by models are not a serious problem when calculating EORs. The problem of model bias interacts with the problem of variability in yield responses to N in such a way that have made both problems difficult to solve.
The task of disaggregating the separate problems of model bias and variability in yield responses undoubtedly has been hampered by limited numbers of observations, which can be attributed to the relatively high costs of measuring yield responses in field trials. Recent advances in precision farming technologies, however, have made it practical and relatively inexpensive to measure yield responses in large numbers of trials where fertilizer treatments are applied in replicated strips (Bermudez and Mallarino, 2002; Blackmer and White, 1998; Scharf et al., 2005). These new technologies have the potential to collect the unprecedented amounts of data needed to calculate EORs with little uncertainty. These advances have generated a need for discussion of new methodology that is appropriate for calculation of EORs for conditions where the cost of collecting data is small compared with the cost only a few years ago. Calculating EORs for large datasets by using multiple regression analysis and site-specific crop response functions has solved some of the problems about how such analyses should be done (Anselin et al., 2004; Hurley et al., 2004; Lambert et al., 2006; Lark and Wheeler, 2003), but there is still a lack of practical methods for reducing the effects of systematic errors of models and variability in yield response when estimating EORs.
Here we illustrate how the concepts of ex post and ex ante analyses can be used in a multistep procedure to disaggregate the problems of model bias and variability in crop responses to N when calculating ex post EORs in situations where essentially unlimited numbers of yield response observations are available for analysis. Our studies of the problems encountered when calculating EORs were prompted by interest in exploiting data collected using precision farming technologies, but the data we analyzed were collected in conventional small-plot trials to circumvent the need for discussions concerning the details of how precision farming technologies should be used to collect yield-response data (Arslan and Colvin, 2002; Pringle et al., 2004). This report is accompanied by a separate report (Kyveryga et al., 2007) that illustrates how discrete marginal analysis and alternative economic benchmarks can be used to calculate and interpret ex post optimal rates of N fertilization after problems associated with model bias and variability in yield response have been minimized by using the procedures described here.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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Data from the trials were analyzed individually (yields of replications at each site) and as a composite sample comprised of many trials (mean yields across all sites). In this analysis the NONLIN procedure of SAS software (SAS Institute, 2002) was used to fit five yield response models: quadratic (QUAD), exponential (EXP), square root (SR), linear response and plateau (LRP), and quadratic response and plateau (QRP). Ex post EORs were calculated by equating first derivatives of the fitted yield response models to the fertilizer–corn price ratio and solving equations for rates of N fertilization. The corn price was set at $86.50 Mg–1 and the fertilizer N price was set at $0.44 kg–1 for all calculations. Ex post EORs calculated from nonstatistically significant (at P < 0.05) models were assigned a value of 0 kg N ha–1.
The complement of the relative variance (CRV) method as described by Webster and Oliver (1990) was used to calculate the percentage of variability in ex post EORs explained by classifying EORs (i.e., dividing into mutually exclusive categories) based on the previous crop. Efficiency of classification was calculated as
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w is pooled within class variance, and total is total pooled variance. |
RESULTS AND DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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Testing model residuals (observed yields minus predicted yields) for deviation from normality revealed that most of the model residuals followed a standard normal distribution. A Shapiro-Wilk test indicated that the residuals obtained from all five models for the corn after soybean category were normally distributed (Fig. 3 ), yet these models substantially disagreed on calculated EOR values. Difficulty of choosing one model over another is also shown by visual examination of the residuals obtained from the EXP model (Fig. 3B). The EXP model fit data with the smallest residual values among the models for this crop category; however, this model tended to systematically underestimate yields in the range of N fertilization from 168 to 280 kg N ha–1.
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Considering Only Important Variability in Yields
Analyses showed that ex post EORs calculated from models fitted to the treatment means of yields for the individual trials (Fig. 5
) were exactly the same as the ex post EORs calculated from models fitted to the treatment means across all trials (Fig. 2). This result should be expected because an ex post EOR for a sample of trials explicitly denotes the single rate that would have maximized mean net returns to fertilization if the one rate were applied across all sites under conditions assumed in the analyses. This rate is determined solely by the mean yield response to fertilizer, it is not influenced by the amounts of variability in yield responses included in that mean. Although variability in yield responses is of great importance when calculating ex post EORs, it is not important in this particular step of the calculation.
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When the models were fitted to the datasets showing the yield variability within treatments (Fig. 5), the magnitude of the r2 values were greatly influenced by the mean yield response. Most models had an r2 value of 0.20 when fitted to data from all sites (Fig. 5A), all models had an r2 value of 0.35 when fitted to data for corn after corn (Fig. 5B), and all models had an r2 value of 0.10 when fitted to data for corn after soybean (Fig. 5C). The highest r2 values occurred in the most responsive category and the lowest r2 occurred in the least responsive category. The effects of responsiveness on r2 values was also shown in the analyses showing that the r2 values at near-optimal rates of N fertilization in Fig. 4 were reduced to 0.01 when variability among trials was included in the analysis (Fig. 6 ). The r2 values were reduced because deleting rates outside the near-optimal range decreased responsiveness. The fact that the variability in yield response among sites was large compared with the mean response to fertilizer N within the near-optimal range does not minimize the importance of the fact that the mean yield responses are the sole factor determining ex post EORs in this step of the analysis.
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Analyses summarized in Table 1 illustrate how the use of r2 values for models to assess uncertainty in calculated ex post EORs can lead to inaccurate estimates of uncertainty. For example, the mean r2 value for the EXP model is 0.58 for all trials and the mean r2 value for the QUAD model is 0.61, but the QUAD model has a much wider range of EORs for the individual trials. In addition, high r2 values do not describe the uncertainty in the calculation of EORs when r2 values are compared for models fit to individual trials. For example, Trial 1 had r2 values for the calculation of EORs ranging from 0.90 to 0.92 for the five models, but there was much uncertainty in the calculation of EORs varying from 133 to 308 kg N ha–1, whereas Trial 53 had r2 values ranging from 0.10 to 0.17 for the five models, but each model uniformly predicted an EOR of 0 kg N ha–1 for Trial 53 (data not shown). These data demonstrate that high r2 values can occur in situations where there is a high degree of uncertainty about the EOR, and that low r2 values can occur in situations where there is a high degree of certainty that no fertilizer was needed. Regression analyses (not shown) suggested that the range in EORs, calculated for individual trials by using different models, significantly increased with increasing yield responsiveness.
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The fact that it is not appropriate to calculate the ex post EOR for a group of trials by calculating the means of EORs determined by fitting models to individual trials was recognized by Bullock and Bullock (1994). They proposed a formula to address this problem by calculating the ex post EOR for different trials and solving for the rate at which the mean of the slopes of regression lines indicates that marginal returns are equal to marginal costs. The importance of using EORs calculated for a group of trials rather than using mean EORs of individual trials also was shown by Heckman et al. (1996).
Controlling Variability in Economic Optimum Rates
Analysis of variance showed that classification of all sites into two categories based on the previous crop explained 30% of the total variability in EORs when the EXP model was used (data not shown). The corresponding percentages were 18, 17, 7, and 7% when the SR, LRP, QRP, and QUAD models were used. The finding that the models disagreed concerning the amounts of variability explained is less important than the finding that all models agreed that some variability was explained by classification.
The observed difference in EORs due to preceding crop is consistent with existing knowledge and can be explained by processes known to occur in the soil (Blackmer, 2000; Green and Blackmer, 1995). Producers usually know the previous crop at the time of N fertilization, so this information is appropriate for use in ex post calculations to refine estimates of EORs for individual sites. This information must be used twice, once when dividing all sites into categories to calculate ex post EORs and once when deciding that a specific area of land to be fertilized falls within this category. Distinguishing between sources of variability that can be, and cannot be known at the time of fertilization is of vital importance in efforts to learn how much of the variability in EORs can be explained by forming appropriate categories.
Many different factors are known at the time of fertilization (e.g., preceding weather, corn hybrid, tillage system, soil texture, soil nitrate concentrations, time of fertilization, placement of fertilizer, form of N, etc.), and each of these factors may serve as a potential basis for category formation to help reduce amounts of unexplained variability in yield response. We have not considered factors other than previous crop because we recognize that our sample of yield responses is too small and was not collected to represent the distribution of yield responses within a specified geographic area. Our sample size is adequate; however, to demonstrate that the problem of model bias is different from the problem of variability in EORs and how these problems can be disaggregated by performing multistep calculations.
Multistep Method for Calculating Ex Post Economic Optimum Rates
Our observations suggest that calculation of ex post EORs should be considered a multistep process where each step is critical for a different reason. Step 1 is to collect a sample of yield responses that represents the population of yield responses within a given geographic area, specified range of conditions within this area, and specified period. Because it is self-evident that the final estimate of ex post EOR can be no better than the sample of yield responses measured, there is a need for more emphasis on the methods for selecting the locations of trials. Care should be taken to ensure that bias is not introduced by collecting a nonrandom sample, and independent samples must be collected to provide meaningful assessments of the uncertainty in calculated values for EORs.
Step 2 is to calculate an ex post EOR for the range of conditions represented by the sample of yield responses. This step involves fitting models to the mean yields observed for each N treatment across all trials. Excluding treatments outside the near-optimal range can reduce disagreements among models. It should be noted that the near-optimal range can be defined only because the sample analyzed was selected to represent a population of responses expected across space and time.
Step 3 is to reduce unexplained variability in ex post EORs by using information that can be available at the time of fertilization to classify sites into categories. This step involves calculating EORs for individual sites and doing analyses to establish the amount of variability that can be explained by forming categories. Knowledge about how the site conditions differ (e.g., different sand content of soils) and about processes (e.g., different leaching losses of N fertilizer applied in fall compared with sidedress application) that may influence yield response can be used to identify factors that are likely to explain as much of the variability in EORs as possible. This step requires a large number of experimental sites so that each factor can be assessed by analysis of the reduction in variability caused by forming new categories. Unlike when calculated from data collected at a single trial and year, ex post EORs should vary little among samples collected to represent the large population of yield responses expected across space, time, and different categories.
Step 4 is to calculate the ex post EORs for the range of conditions represented by each new category formed. The methods are the same as in Step 2.
Step 5 (not illustrated) would be to calculate the economic and (or) environmental benefits of forming new categories. A complete discussion of the methodology for estimating the value of forming new categories is beyond the scope of this discussion, but it should be noted that economic benefits could be easily calculated from information generated during the analyses if simplifying assumptions are made. It could be assumed that a new category should be formed if the direct costs of changing how fertilizer is applied, for example, side-dressing N compared with fall application of N, are less than the expected change in value of the grain produced.
Step 6 (not illustrated) would be to repeat the first five steps starting with the new categories formed. Important considerations include taking steps to ensure that a representative sample of yield responses is selected for each new category that will be used in the future. As important sources of variability in EORs are removed by classification, it should be easier to identify factors that explain more of the remaining variability within each new category. Step 6, therefore, may have to be repeated each time the data show a useful category should be established. The underlying reason is that judgments concerning the appropriateness of any calculated estimate of ex post EORs should be expected to vary with amount of knowledge available.
Iteration of these steps to reduce unexplained variability in crop responses before calculating EORs is of great importance because one of the assumptions of models used to calculate EORs is the condition of technological efficiency (Binger and Hoffman, 1998; Perloff, 1999; Varian, 1992). Technologically efficient production excludes obtaining the same level of output with fewer levels of input: yields should be maximized at each fertilizer rate. Technological efficiency when calculating EORs for N means that it would be inappropriate to calculate an EOR from data collected in trials where methods of application were known to result in much less N available for crop use than is available where the application methods used were more efficient. We think the concept of technological efficiency also makes it inappropriate to calculate an EOR for a wide range of conditions when information is available to divide the data into categories that make the calculations more relevant to the specific conditions of interest. Although technological efficiency has received relatively little attention in discussions of EORs for N in the past, methods that are technologically efficient and reduce N losses, and hence increase profitability, are of great importance because N is highly mobile in soils and substantial amounts of N often move from fields to water supplies before plants begin transpiring large amounts of water (Balkcom et al., 2003).
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CONCLUSIONS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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Unlike EORs calculated from individual trials or a small group of trials, category-specific ex post EORs do not vary much from year to year due to normal variation in weather and other factors. These EORs, however, should be expected to change with the amount of knowledge used in the analyses. Because knowledge is acquired during analyses, the multistep procedure should be viewed as a systematic search for the rate most likely to maximize profits for specific crop rotations, N practices, and management systems. Searches are needed because many factors can affect yield responses, because these factors are usually difficult to quantify, and because the importance of each factor should be expected to vary with each new category formed. These searches also should be expected to greatly simplify the task of enumerating clear steps for calculating ex ante EORs because of substantial reduction in the amounts of uncertainty. Although the methods described here probably are not often practical when data are collected in conventional small-plot trials, use of these methods with data collected in such trials reveals some problems in commonly used methods for calculating EORs and illustrates how these problems can be avoided when larger amounts of data are collected by using new technologies.
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ACKNOWLEDGMENTS |
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REFERENCES |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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This article has been cited by other articles:
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  P. M. Kyveryga, A. M. Blackmer, and J. Zhang Characterizing and Classifying Variability in Corn Yield Response to Nitrogen Fertilization on Subfield and Field Scales Agron. J., February 4, 2009; 101(2): 269 - 277. [Abstract] [Full Text] [PDF]
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J. A. Hernandez and D. J. Mulla Estimating Uncertainty of Economically Optimum Fertilizer Rates Agron. J., September 1, 2008; 100(5): 1221 - 1229. [Abstract] [Full Text] [PDF]
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P. M. Kyveryga, A. M. Blackmer, and T. F. Morris Alternative Benchmarks for Economically Optimal Rates of Nitrogen Fertilization for Corn Agron. J., June 5, 2007; 99(4): 1057 - 1065. [Abstract] [Full Text] [PDF]
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