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DIVISION S-8—NUTRIENT MANAGEMENT & SOIL & PLANT ANALYSIS

Yield Models Implied by Traditional Fertilizer Recommendations and a Framework for Including Nontraditional Information

^a 304F Waters Hall, Dep. of Agricultural Economics, Kansas State University, Manhattan, KS 66506
^b 2004 Throckmorton Plant Sciences Center, Dep. of Agronomy, Kansas State University, Manhattan, KS 66506
^c 306B Waters Hall, Dep. of Agricultural Economics, Kansas State University, Manhattan, KS 66506

* Corresponding author (tkastens{at}agecon.ksu.edu)

	ABSTRACT

TOP NOTES ABSTRACT INTRODUCTION Yield Models MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B APPENDIX C REFERENCES

 
Recently emerged technologies permitting precision farming provide opportunities to collect and evaluate vast amounts of data; yet, a formal method for incorporating additional yield-affecting information into fertilizer-yield response models is still largely unavailable to site-specific managers. This research develops an integrated yield model that combines laboratory-based fertilizer recommendations with data from multiple independent variables. Kansas wheat (Triticum aestivum L. sp. aestivum) was the target crop, and simulated soil test and yield data were characteristic of data from the literature and this geographic area. Independent variables in the plateau-type yield model included fertilizer N and P, soil test N (STN) and soil test P (STP), and Z (representing all other causal factors). Consideration of the potential benefits of managing STP is an explicit outcome of this research. To expand the modeling procedures to include actual farm information, data from a northwest Kansas farm were used to replace the simulated Z variable. Additionally, maximum entropy (ME) was examined as a statistical technique for using the farm's N and P data to modify the predicted N- and P-response of laboratory-based models. Based on out-of-sample yield prediction accuracy, the ME models were generally the most accurate, improving root mean squared prediction error by as much as 14% relative to a model estimated with only farm data. Because the ME models were the most accurate and provided fertilizer recommendations similar to those from soil testing laboratories, this approach should provide a reliable framework for developing site-specific fertilizer recommendations that depend on factors other than STN and STP.

Abbreviations: CSU, Colorado State University • CV, coefficient of variation • H, high • KSU, Kansas State University • L, low • M, medium • MAPE, mean absolute percentage error • ME, maximizing entropy • OAL, Olsen's Agricultural Laboratory • om, soil organic matter content • RMSE, root mean squared error • SSE, sum of squared errors • STN, soil test N • STP, soil test P • UNL, University of Nebraska-Lincoln • VL, very low • VH, very high • VRA, variable rate application

	INTRODUCTION

TOP NOTES ABSTRACT INTRODUCTION Yield Models MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B APPENDIX C REFERENCES

 
LOW-COST DATA COLLECTION methods wrought by new sensing technologies have vastly increased the amount of farm-level information available—information on more and new variables, and information at a finer spatial scale. The preponderance of such data has caused crop input decision makers to demand ever more complex crop input recommendations. For example, software guiding variable rate fertilizer applications (VRA) both allow for and demand an explicit fertilizer rate for each portion of a field. So, farmers are asking questions such as, How much fertilizer should I apply here, conditional upon factors x, y, and z observed or assumed present here? Meanwhile, universities, industry, and farmers, themselves, are struggling to provide the research base and analytical methods to help answer those questions.

A number of issues likely are implicit within the fertilizer recommendation questions being asked by precision agriculture practitioners. First, is an acknowledgment that crop yield is a function of many factors, each interacting with the others. Second, is a belief that the yield function is continuous, or at least can be treated as such (Berck and Helfand, 1990). In other words, to accommodate small changes in soil fertility across a field, VRA providers routinely transform a traditional step-type function (one that assigns the same rate to a broad range of soil fertility) to a continuous fertilizer rate algorithm—typically by interpolating between the steps. Third, is a belief that the yield function, or at least its relevant features, can be made quantitatively explicit, making it possible to determine appropriate fertilizer rates. In short, if the profitability associated with different fertilizer decisions is to be successfully evaluated, an explicit, continuous, and reliable yield response function is necessary. However, such functions are rarely provided to decision makers.

Besides the problem of yield response functions not being available to the decision maker, current fertilizer recommendations typically depend on crop response experiments in which spatial variability has been minimized for every independent variable affecting crop yield except for the nutrient in question. This has been the experimental approach by agronomists for many years. Although agronomists realize that many nonfertility variables (e.g., soil texture, soil bulk density, available water content) and other fertility variables significantly impact crop yield, capturing the effects and interactions of all these variables in a controlled experimental framework generally is infeasible. Thus, single-nutrient fertilizer recommendations have typically been the norm, with occasional adjustments for one or two other factors. Given the constraints associated with traditional fertilizer experiments, development of multi-factor yield response functions for VRA practitioners ultimately will depend on farm data. Presumably, farm information across years might be used to statistically estimate a continuous, reliable, multifactor mathematical yield model.

Although farm information may be necessary to estimate the relationships between crop yield and nonfertility factors, there are at least two reasons why the relationships between yield and fertility or fertilizer derived from university or industry fertilizer trials may be more appropriate than those deduced from farm data. First, if farmers do not include very low or zero fertilizer rates (incurring reduced profitability in the current year), farm data may not have sufficient range in fertilizer rates to reliably estimate yield response. Second, many producers have used university- or industry-derived fertilizer recommendations (from multiyear multilocation fertilizer trials) for years, implying these recommendations must be reasonably reliable as a guide to profit maximization; otherwise, the recommendation guides would fall to disuse.

This research has two objectives. Because academics repeatedly challenge mathematical yield models (e.g., Griffin et al., 1987; Berck and Helfand, 1990; Paris, 1992), often for being ad hoc (Frank et al., 1990), objective one is to develop a theoretical and empirical yield model that is more acceptable to academics. To accomplish this, we use simulated data to develop yield models that are consistent with traditional fertilizer N and P recommendations provided by soil testing laboratories. Besides ensuring model acceptability, the results of this process illustrate interesting beliefs likely held by mainstream agronomists and soil scientists, specifically regarding the management of STP over time. Objective two is to develop and evaluate a formal process for expanding laboratory recommendation-based yield models to include less-traditional information available at the farm level, resulting in a yield model suitable for site-specific fertilizer recommendations. To achieve this, we develop and test yield models that integrate data from a northwest Kansas farm with fertilizer recommendation data from soil-testing laboratories in the area.

	Yield Models

TOP NOTES ABSTRACT INTRODUCTION Yield Models MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B APPENDIX C REFERENCES

 
Four key features underlie generally accepted descriptions of yield response to an input: (i) diminishing returns, (ii) limiting factor response, (iii) plateau yields, and (iv) fertilizer and fertility are substitutes. One basic approach to yield model construction that accommodates the first three features is represented in the general multiplicative expression:

[1]

where Y_i denotes crop yield at some point in space or time i, A is some theoretical plateau yield, and Z_k,i denotes some numerical expression relating to causal factor k at point i and is assumed to range in value from 0 to 1. If all Z values equal 1, then model-predicted yield would equal the plateau, A.

Because identifying all of the independent variables is impossible, a finite form of Eq. [1] is considered that contains K factors and an error term, E_i (E can alternatively be thought of as denoting unobserved factors), so that the equation can be true at each observed point i:

[2]

Where N and P are considered important causal factors, along with some other observed factor Z, Eq. [2] can be modified to Eq. [3]:

[3]

where N and P denote the general expressions for N fertilizer and STN, and P fertilizer and STP, respectively, and Z denotes the general expression for any other observed factor of interest (though it also could represent the combined effect of all other observed factors of interest). A more explicit representation of the N and P terms, accommodating all four of the key features noted above, can be presented as Eq. [4]:

[4]

In Eq. [4], Y_i is crop (here, wheat) yield (Mg ha^-1), fertN_i is fertilizer N (kg N ha^-1), STN_i is soil test N (kg NO₃–N ha^-1), fertP_i is fertilizer P (kg P ha^-1), STP_i is soil test P (mg P kg^-1), and i is an index for each observation in time or space. Numerical constants (parameters) that must be estimated include A, B₁, B₂, B₃, B₄, G₁, and G₂. The following restrictions must be applied to correctly describe a yield response function: A > 0, 0 < G₁ 1, 0 < G₂ 1, and B₁, B₂, B₃, B₄ > 0. Once these parameters are estimated, expected yield, Y_i, can be calculated for specific values of Z_i, fertN_i, STN_i, fertP_i, and STP_i, because the expected value of the error, E[E_i], is assumed to be 0. The reader is referred to Melsted and Peck (1977) and Griffin et al. (1987) for the theoretical reasons why an exponential function is preferred to other functional forms, especially the quadratic. Despite its asymptotic plateau characteristic, Eq. [4] allows for interactions of factors. A calculation of the second-order cross-partial derivatives, for example ²Y/fertNSTP 0, illustrates this point, as does the fact that optimal fertilizer rates (shown later as Eq. [6] and [7]) depend on other factors. In fact, the interactions are defined so that fertN and STN behave as substitutes (negative cross-partials), as do fertP and STP; whereas, other factors behave as complements (positive cross-partials).

Using Eq. [4], the expected profit (crop revenue less fertilizer costs) associated with point i can be described as in Eq. [5]:

[5]

where WPRICE, NPRICE, and PPRICE are expected wheat price ($ Mg^-1), fertilizer N price ($ kg^-1), and fertilizer P price ($ kg^-1), respectively, PROF_i is expected profit at point i ($ ha^-1), and Y_i is expected yield (as described by Eq. [4]).

If Eq. [4] is substituted for Y_i in Eq. [5], profit-maximizing levels of fertilizer N and fertilizer P (conditional on specific values of Z_i, STN_i, STP_i, and prices) for point i can be determined by taking the first derivative of PROF_i with respect to fertN_i and fertP_i, setting those derivatives equal to 0, and solving for fertN_i and fertP_i (Eq. [6] and [7]):

[6]

and

[7]

where fertN^*_i and fertP^*_i denote the model-recommended optimal levels of fertilizer N and P, respectively, for point i.

The framework for developing a wheat yield model and associated fertilizer recommendations that are consistent with agronomic and economic theory is represented in Eq. [1] through [7]. Model parameters could be estimated based on farm data using an algorithm that minimizes in-sample yield prediction error (usually the sum of squared errors, SSE) for Eq. [4]. However, as noted earlier, depending solely on farm data may be unreliable. In that case, using university- or industry-derived fertilizer recommendations to constrain the selection of parameters in Eq. [4] may be an appropriate step in improving site-specific fertilizer recommendations based on farm data.

	MATERIALS AND METHODS

TOP NOTES ABSTRACT INTRODUCTION Yield Models MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B APPENDIX C REFERENCES

 
To address the first objective, this research relies on primarily assumption-based simulations of soil tests, yields, fertilizer decisions, and expected profitability. The production system used to empirically test the framework (objective two) is western Kansas nonirrigated wheat. Data from a farm located in Rawlins County, KS are used for model validation. Summary statistics for that farm's data are provided in Table 1. Soil sample procedures are described in Appendix A. Assumptions used are believed reasonable for this farm and production system and, the simulation-based framework provided here could be applied to other geographic areas and other crops.

The N recommendation from all four laboratories is a function of yield goal and STN (Fig. 1) . Yield goal was selected as 10% greater than the 10-yr (1991–2000) average of 2.56 Mg ha^-1 (38 bu acre^-1) reported for the northwest Kansas Crop Reporting District (CRD) (Crops, Kansas Agricultural Statistics, Crops). This approach was used based on related publications (e.g., Devlin et al., 1996; Franzen and Goos, 1997). Because the OAL and KSU recommended N levels are the same, they cannot be distinguished in Fig. 1. The most prominent distinction among laboratories is that the CSU N recommendation is less for lower STN and more for higher STN.

View larger version (22K): [in this window] [in a new window]   Fig. 1. Fertilizer N recommendation from four laboratories as a function of soil test N. The yield goal was 2.83 Mg ha-1 (42 bu acre-1).

View larger version (24K): [in this window] [in a new window]   Fig. 2. Fertilizer P recommendation from four laboratories as a function of soil test P. The yield goal was 2.83 Mg ha-1 (42 bu acre-1).

Because Z represents all non-N and non-P independent variables, explicit quantification of Z is not possible; however, inferences about Z can be obtained through information about Y. Spatial and temporal crop yield CVs can vary greatly. For example, Jaynes and Colvin (1997) reported spatial CVs for corn (Zea mays L. sp. mays) and soybean (Glycine max [L.] Merr.) yield ranging from 0.12 to 0.33 and noted that others have reported spatial CVs for wheat and soybean yield ranging from 0.15 to 0.43. Eghball and Varvel (1997) reported temporal CVs for corn, soybean, and grain sorghum (Sorghum bicolor [L.] Moench) yield ranging from 0.26 to 0.53. The spatio-temporal wheat yield CV for the data from the Rawlins County farm was approximately 0.25, and this was considered realistic for our simulations.

In the simulations (described later), the plateau parameter, A, can be combined with the Z variable to form a single independent variable A x Z, henceforth referred to as AZ. The parenthetical terms for N and P in Eq. [4] are represented as N and P, respectively. To denote statistical expectations, E[.] is used. The terms AZ, N, and P are assumed to be statistically independent. Given that the expected error, E[E], in Eq. [4] is equal to 0, then E[Y] = E[AZ] x E[N] x E[P], or E[AZ] = E[Y] x (E[N x P])^-1. Since we assumed that E[Y] = 2.56 Mg ha^-1, E[AZ] can be derived by knowing E[N x P].

Based on the yield model presented as Eq. [4], fertN can substitute for STN and fertP for STP. If fertilizer were free, then the optimal quantities would be an infinite amount of each, so that N and P each equal 1. In that case, E[N x P] would equal 1. However, because fertilizer is not free, E[N x P] would be less than 1 (for example, D, where 0 < D < 1) for some profit maximizing levels of fertN and fertP. Then the simulation mean for AZ would be E[AZ] = E[Y] x D^-1 = (2.56)D^-1. In practice, D is a parameter determined within the model estimation process (described later).

Although model-based optimal fertilizer rates from Eq. [6] and [7] do not depend on yield goal (YG), lab-based fertilizer recommendations do depend on yield goal. As noted earlier, yield goals are assumed to be 10% greater than expected yields. Here, the expected yield for the ith simulated observation is considered to be the model-predicted yield (^*_i) given optimally selected fertilizer rates:

[8]

where ₁,..., ₄; ₁, and ₂ denote the estimated model parameters and other terms have already been defined. Thus, YG_i = 1.10i*.

Although crop and fertilizer prices would have random components, prices were assumed fixed. The wheat price was assumed to equal $118.31 Mg^-1 ($3.22 bu^-1), which is the 1991 through 2000 monthly average price reported for the northwest Kansas CRD (Kansas Agricultural Statistics, Prices). Fertilizer N price was assumed to equal $0.42 kg^-1 ($0.19 lb^-1) and fertilizer P price was $1.16 kg^-1 ($0.53 lb^-1). The fixed cost of fertilizer application was ignored because fertilizer N and P are often simultaneously applied in northwest Kansas wheat production, and often in conjunction with a tillage operation. Moreover, if this simulation were used to represent a within-field VRA program, parts of a field with zero to low fertilizer requirements would probably still require the underlying application field operation.

The simulations that were required to estimate a yield response model from laboratory fertilizer recommendations relied on two key assumptions. First, the model-generated optimal fertilizer rates (fertN* and fertP*) should be as close as possible to the laboratory's recommendations for N and P (Fig. 1 and 2), conditional on all prices and on simulated YG, AZ, STN, and STP values. Close is defined as selecting the model parameters (B₁, B₂, B₃, B₄, G₁, and G₂) to minimize the SSE between the laboratory's N and P fertilizer recommendations and fertN* and fertP*, respectively. Second, when simulated variable values and model-determined optimal fertilizer rates are applied to Eq. [8], the predicted yield, averaged across many observations, should approximately equal 2.56 Mg ha^-1 (38 bu acre^-1). Implied in the second assumption is that, on average, producers have applied the correct (profit-maximizing) amount of fertilizer, so model-predicted yield equals observed yield.

Three series are simulated, yield (Y), STN, and STP. The simulated populations of Y (thus AZ), STN, and STP were assumed to be log-normally distributed. Kravchenko and Bullock (1999) and Schmidt et al. (2002) provided evidence supporting the assumption that STP data are log-normally distributed. Additionally, Parkin et al. (1988) illustrated that frequency distributions for many physical, chemical, and microbiological soil properties are skewed to the right and are better approximated by the log-normal distribution than by the normal (Gaussian) distribution. To summarize, a brief description of the simulation modeling process follows.

Conditional on an estimate for D, the AZ series is calculated as AZ_i = Y_iD^-1. Then, conditional on the other parameter estimates (B₁, B₂, B₃, B₄, G₁, and G₂), Eq. [6] and [7] are used to determine optimal fertilizer rates, which are used to determine model-predicted optimal yields through Eq. [8]. Model-predicted optimal yields are used to determine yield goals as described earlier, which, along with STN and STP, determine a laboratory's recommended fertilizer rates. Then, model- and laboratory-recommended fertilizer rates together are used to determine the SSE to be minimized. Finally, the whole process is numerically constrained so that average model-predicted optimal yield equals average simulated yield. A step-by-step description of the simulation process used to generate the yield models (referred to as lab-based models) from the laboratory-based fertilizer recommendations is presented in Appendix B.

To evaluate the success of the response function generating process, two prediction accuracy statistics were used to measure how accurately the model-generated optimal fertilizer rates (fertN* and fertP*) predicted the laboratory fertilizer recommendations. The first measure was relative root mean square error (RMSE) and was calculated as the square root of the average squared error associated with the fertilizer recommendations, divided by the standard deviation of the laboratory's recommendations. Because the standard deviation is the RMSE associated with using the mean as a predictor for each data point, it is a natural benchmark. An advantage of relative RMSE is that it is a unitless measure that allows series with different scales to be compared, such as comparing the accuracy of N predictions and P predictions. The second measure, r², is the squared-linear correlation coefficient between the laboratory's fertilizer recommendation and the model-generated optimal fertilizer recommendation. Accuracy measures were calculated for N and P separately, and for each of the four laboratories. Each accuracy measure was calculated across the 10 000 pairs (model vs. laboratory) of fertilizer recommendations associated with 10 000 simulated observations of Y, STN, and STP.

Including Farm Information with Laboratory-Recommendation Information
We propose that a yield model that incorporates both laboratory-recommendation information and farm information could be used to generate practical and reliable site-specific fertilizer recommendations. However, for such a model-generating framework to have merit, it should result in fertilizer N and P recommendations that are an improvement over those generated directly from a yield model estimated from only farm data. Although it is impossible to be conclusive with results presented here, we examine some validation of our procedures by way of example, by incorporating the N and P yield-response information from lab-based models into a yield model estimated using data from the example northwest Kansas wheat farm.

As considered here, the validation process begins by using historical data from the example farm to estimate a multifactor yield model of the general form shown in Eq. [4]. Model parameters are selected that minimize the sum of squared in-sample yield-prediction errors (SSE), subject only to certain parametric constraints such as those described immediately following Eq. [4] (this optimization process is often referred to as constrained nonlinear regression). The yield model estimated, referred to as the Farm-Only Model, is

Information from the lab-based models is incorporated into the farm information by estimating a second group of yield models of the form shown in Eq. [9]. However, in these models, during the SSE-minimizing process, N- and P-related parameters (the G₁, B₁, B₂, G₂, B₃, and B₄ values) are constrained to exactly equal those of the lab-based models depicted as Eq. [4]. This forces the yield response to fertN, STN, fertP, and STP to equal (in a proportion-of-plateau-yield sense) that of the lab-based models. In other words, farm data on N and P are essentially ignored in model estimation. These yield models are referred to as Lab-Constrained models.

To improve yield models based on farm data, complete disregard for farm data on N and P during the model estimation process may be inappropriate. For example, a farm manager may be gathering ever better information on N and P response over time by conducting on-farm experiments with substantial fertilizer rate variability. Thus, another approach to combining information from soil testing laboratories with farm information is considered here. This approach takes the soil testing laboratory's N and P fertilizer recommendations for the farm as what the recommendations are expected to be. Then, farm information on N, P, and other factors is used to estimate a yield model conditional on the prior fertilizer recommendation information. Because it requires less restrictive distributional assumptions than more traditional information-updating statistical processes such as Bayesian inference, the formal process used here is ME and the models are referred to as Lab-ME models. For a procedural discussion of ME see Golan et al. (1996); for a comparison with ordinary least-squares estimators see Preckel (2001); for more detail regarding our use of ME see Appendix C. Within an estimation sample, increasing the constraints on model parameters will always reduce the yield prediction accuracy relative to less constrained optimization. That means the Farm-Only Model will always predict yield more accurately in sample than will the Lab-Constrained models (but not necessarily better than the Lab-ME models since those models are not estimated with the minimizing yield error criterion). The hope is that imposing greater parametric constraints during model estimation will lead to a model that performs better (predicts yield more accurately) out-of-sample. This way a manager would: (i) have at least as much confidence in his yield model that incorporated soil laboratory information as he might have in a model estimated using only his farm's information, (ii) especially have confidence in the model-generated yield relationships with managed variables such as N and P (because they are consistent with the selected laboratory's recommendations), and (iii) have a model that allows optimal fertilizer rates to vary with other measured variables of interest, allowing VRA N and P to be based on other variables besides only STN and STP.

To test the out-of-sample prediction accuracy of the Farm-Only, Lab-Constrained, and Lab-ME Models discussed above, a jackknife approach is used. The jackknife approach predicts a group of one or more yield observations using a model estimated with only the other observations. After each yield observation for the example farm is predicted, the entire predicted series is compared with the actual yield series using two measures of prediction accuracy: (i) RMSE and (ii) mean absolute percentage error (MAPE), which is the average absolute percentage error. Mean absolute percentage error is added as a test of accuracy because it is conducive to statistically comparing accuracy of models (using a paired-t test).

	RESULTS AND DISCUSSION

TOP NOTES ABSTRACT INTRODUCTION Yield Models MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B APPENDIX C REFERENCES

 
Parameter estimates for each lab-based model (the yield models represented by Eq. [4]) and corresponding measures of accuracy for the model-generated N and P recommendations (Eq. [6] and [7]) are provided in Table 2. A genetic algorithm was coded in Matlab Numeric Computation Software (Math Works Inc., Natick, MA) to implement the minimizing-SSE parameter selection process in these highly nonlinear circularly recursive (optimal N depends on optimal P and vice versa) models. Overall, the model-generated N and P recommendations are quite similar to the laboratory fertilizer recommendations, with r² degree of fit values ranging between 0.74 and 0.97, and 0.66 and 0.93 for N and P, respectively (Table 2). For N, the OAL- and KSU-based models generated recommendations closest to those of the underlying laboratory. For P, the KSU-based model clearly resulted in the best fit with the underlying laboratory's recommendations (because KSU's P recommendations were divided into smaller increments). Figure 3 depicts the fertilizer N recommendation as a function of STN for the KSU-based model and for the underlying laboratory. Figure 4 depicts similar information for P. Note that if the model-based fertilizer recommendations had not been constrained to be nonnegative, the model-based line in Fig. 3 or 4 would have been identical to a straight line drawn to best fit the laboratory fertilizer recommendation data (using a linear least-squares fit). This can be seen in the fact that Eq. [6] and [7] are of the general form y = a + bx. However, ignoring the nonnegative fertilizer issue, the parameters in Eq. [6] and [7] (ultimately needed for the yield model) could not have been recovered from the simple equation behind the laboratory recommendations least-squares trend line; hence, the reason for the simulation and modeling process used here.

View this table: [in this window] [in a new window]   Table 2. Laboratory-based yield model parameter estimates and fertilizer recommendation accuracy measures. Each lab-based yield model was estimated from the same simulated series of 10 000 data points.

View larger version (15K): [in this window] [in a new window]   Fig. 3. Fertilizer N recommendation as a function of soil test N for Kansas State University (KSU) Laboratory and KSU-based Model. Soil test P was fixed at 16 mg kg-1; fertilizer P was optimized; yield goal was fixed at 2.83 Mg ha-1 (42 bu acre-1); AZ simulation term was fixed at its mean.

View larger version (14K): [in this window] [in a new window]   Fig. 4. Fertilizer P recommendation as a function of soil test P for Kansas State University (KSU) laboratory and KSU-based Model. Soil test N was fixed at 45 kg ha-1 (40 lb acre-1); fertilizer N was optimized;yield goal was fixed at 2.83 Mg ha-1 (42 bu acre-1); AZ simulation term was fixed at its mean.

View larger version (22K): [in this window] [in a new window]   Fig. 5. Predicted wheat yield response to fertilizer P at different levels of soil test P (mg kg-1) for the Kansas State University-based Yield Model. Soil test N was fixed at 45 kg ha-1 (40 lb acre-1); fertilizer N was optimized; AZ simulation term was fixed at its mean.

Long Run Steady-State Soil Test Phosphorus Level
A particularly interesting question that can be addressed with this type of simulation exercise is: Given that a producer adheres to a lab-based fertilizer recommendation, what will be the long run steady-state STP level? More to the point, despite not being explicitly addressed by a laboratory's fertP recommendations, might STP be treated as a managed variable, and not just fertP? To address these questions, a transformation rate between fertilizer P applied in excess of crop removal (EfertP) and change in STP must be estimated. Removal rate for wheat is assumed to be 4.4 kg P Mg^-1 (0.3 lb P bu^-1). Constant transformation rates of 4 to 15 kg EfertP ha^-1 for each 1 mg kg^-1 change in STP have been used in the literature, representing soils in the Midwest, USA (Peck et al., 1971; Barber, 1979; Leikam, 1992; Randall et al., 1997; Kastens et al., 2000). The transformation rate for most soils is probably not constant with increasing STP, accounting for mechanisms responsible for adsorption isotherms (Bohn et al., 1979); so more EfertP is probably required to change STP at lower STP than required at higher STP levels. The formula provided in Eq. [10] accommodates this change in transformation rate:

[10]

where TR is the transformation rate and H and W are positive parameters to be estimated. Given Eq. [10], the amount of EfertP required to change STP from some initial level (STP_initial) to a final level (STP_final), or Efert P_{STPinitial,STPfinal}, can be expressed as

[11]

with the corresponding solution,

[12]

Equation [12] can be solved for STPfinal, given STPinitial and EfertP_{STPinitial,STPfinal}, giving Eq. 13:

[13]

The STP build-up recommendations from the University of Missouri (Buchholz et al., 1993) were used to estimate parameters H and W in Eq. [10]. These authors provided 20 build-up values along with corresponding initial and final STP levels (EfertP_{STPinitial,STPfinal}, STPinitial, and STPfinal in Eq. [12]). Soil test P levels provided by Buchholz et al. (1993) in units of pounds per acre (lb acre^-1) were divided by 2 to achieve units of milligrams per kilogram (mg kg^-1). The parameters H and W were estimated by minimizing the SSE for predictions of EfertP_{STPinitial,STPfinal}. The estimated model fit the data very well (r² = 0.99). Estimated parameters were H = 53.78 and W = 0.6707.

Given any initial STP level, for example 5 mg kg^-1, the optimal fertP rates from each of the four lab-based models can now be determined as a function of time. The underlying time-dependent simulation depended on lab-based optimal fertN and fertP each year, EfertP each year as determined from model-predicted yield (Eq. [4]), and year-end STP level each year (Eq. [13]). Fertilizer P rates for successive years are depicted in Fig. 6 . The amount of P applied is initially greatest when based on the OAL Model (29 kg P ha^-1), whereas the P recommendation from the other three lab-based models initially ranged between 17 and 21 kg P ha^-1. As soil test P increased, the amount of fertP applied decreased, with recommendations from all lab-based models decreasing to about 12 kg P ha^-1 in 30 to 40 yr.

View larger version (24K): [in this window] [in a new window]   Fig. 6. Optimal fertilizer P as a function of time for the four lab-based models. Soil test N was fixed at 45 kg ha-1 (40 lb acre-1); fertilizer N and fertilizer P were optimized annually; initial soil test P was 5 mg kg-1; AZ simulation term was fixed at its mean.

View larger version (26K): [in this window] [in a new window]   Fig. 7. Soil test P as a function of time for the four lab-based models. Soil test N was fixed at 45 kg ha-1 (40 lb acre-1); fertilizer N and P were optimized annually; initial soil test P was 5 mg kg-1; AZ simulation term was fixed at its mean.

Profit and Amortized Profit
An important aspect of this simulation exercise, given a model that incorporates and adequately represents reliable N and P recommendations from soil testing laboratories, is the ability to evaluate annual profit (crop revenue less fertilizer expense) for each year into the future. Each model, representing the optimal fertilizer recommendation from each laboratory, shows an increase in profitability with time (Fig. 8) as STP increases to the steady-state level.

View larger version (27K): [in this window] [in a new window]   Fig. 8. Profit (crop revenue less fertilizer cost) as a function of time for the four lab-based models. Soil test N was fixed at 45 kg ha-1 (40 lb acre-1); fertilizer N and P were optimized annually; initial soil test P was 5 mg kg-1; AZ simulation term was fixed at its mean.

View larger version (27K): [in this window] [in a new window]   Fig. 9. Annually amortized profit (crop revenue less fertilizer cost) as a function of time for the four lab-based models if following the program for the number of years shown on the x-axis. Soil test N was fixed at 45 kg ha-1 (40 lb acre-1); fertilizer N and P were optimized annually; initial soil test P was 5 mg kg-1; AZ simulation term was fixed at its mean.

Because the fertilizer P recommendations from the four laboratories considered here are conditional on STP, they are essentially 1-yr horizon recommendations. They do not explicitly consider the potential benefit associated with increasing STP with time (or allowing STP to decrease when it is sufficiently high for obtaining optimum yield). If producers expect to control land more than 1 yr, these fertilizer P recommendations should be considered minimum levels (conditional on STP) and not optimal. Other P recommendations explicitly address the issue of increasing STP. For example, the University of Missouri recommends increasing STP over a period of 8 yr (Buchholz et al., 1993); however, whether its recommendation includes an economic analysis of time horizons is not evident.

A benefit of the simulation process developed here is that resultant yield models can help explicitly answer the question: How much fertilizer P should be applied each year to maximize profits over some finite time horizon? As one example, Fig. 10 shows the P recommendations for the OAL-based model over 10 yr, assuming either a 1-yr horizon (same as the OAL line of Fig. 6) or a 10-yr horizon. Relative to the 1-yr horizon, maximizing profits over a 10-yr horizon resulted in fertilizer P rates that were 23% higher in Year 1, 16% lower in Year 10, and 10% higher overall. Although not shown in Fig. 10, despite applying more fertilizer P, model-predicted profits were $0.87 ha^-1 yr^-1 greater with the 10-yr horizon given the prices assumed. Model-predicted STP after Year 10 was 15.4 mg kg^-1 following the 1-yr horizon recommendation each year and 18.1 mg kg^-1 following the 10-yr horizon recommendations.

View larger version (14K): [in this window] [in a new window]   Fig. 10. Optimal fertilizer P as a function of time for the Olsen's Agriculture Laboratory-based Model, based on a 1-yr and a 10-yr time horizon. Soil test N was fixed at 45 kg ha-1 (40 lb acre-1); fertilizer N was optimized annually; initial soil test P was 5 mg kg-1; AZ simulation term was fixed at its mean.

The jackknife out-of-sample validation procedure used here involved predicting wheat yields from 1 yr using a model estimated with farm data from all other years. For example, 1998 yields were predicted by inserting independent variable data from 1998 into a model estimated using only data from 1994 through 1997 and 1999 through 2001. Table 3 shows the jackknife accuracy measures associated with using the example farm's data for estimating the various wheat yield models described. A Naive model was included as a reference. The Naive model used the mean yield as a predictor, where the mean was calculated over the same range as the estimated model. Given the reported accuracy measures in Table 3, none of the models is especially accurate relative to simply using the mean yield as a predictor (the Naive model). This is probably because random weather is still a large causal factor. However, this does not preclude using the model to improve fertilizer decisions, which are almost always made without knowing an upcoming growing season's weather.

In Table 3, generally the ME models stand out as being the best models for predicting out-of-sample wheat yields. By the MAPE criterion, the KSU–ME model was more accurate than the Naive model (the OAL–ME and CSU–ME models also were more accurate at a 0.10 level of significance). Three of four ME models were statistically more accurate than the Farm-Only model, and all four ME models were statistically more accurate than their constrained counterparts. These results suggest that maximizing entropy, which is a systematic and theoretically well-supported modeling approach, may be a plausible way to incorporate ever better farm-level N and P information over time. As with the Lab-Constrained models, the farm manager might consider the ME models appropriate for VRA of N or P.

Generally, the fertilizer recommendations from the ME models did not depart substantially from the soil laboratory recommendations implicit in the constrained models. For example, varying STN over the range of the data (approximately 20–70 kg ha^-1), while holding other variables at their means, generated recommended fertN rates from Lab-ME models similar to those of corresponding Lab-Constrained models. At the low end of the range (20 kg ha^-1 STN), the largest difference was for the UNL models, where UNL-Constrained recommended 69 kg ha^-1 fertN and UNL–ME recommended 60 kg ha^-1 fertN. At the high end of the range (70 kg ha^-1 STN), the largest difference was for the CSU models, where CSU-Constrained recommended 66 kg ha^-1 fertN and CSU–ME recommended 75 kg ha^-1 fertN. Similarly, varying STP over the range of the data (approximately 7–35 mg kg^-1), while holding other variables at their means, generated recommended fertP rates from Lab-ME models similar to those of corresponding Lab-Constrained models. At the low end of the range (7 mg kg^-1 STP), the largest difference was for the CSU models, where CSU-Constrained recommended 21.4 kg ha^-1 fertP and CSU–ME recommended 20.1 kg ha^-1 fertP. At the high end of the range (35 mg kg^-1), the largest difference was again for the CSU models, where CSU-Constrained recommended 3.5 kg ha^-1 fertP and CSU–ME recommended 0 kg ha^-1 fertP.

Using Models to make Variable Rate Application Decisions
In general, using one of the Lab-Constrained models for making N or P VRA decisions for the example northwest Kansas farm would result in yield responses to N and P very similar to those of the lab-based models discussed earlier (e.g., like those shown in Fig. 5 regarding P). Generally, using one of the Lab-ME models for making N or P VRA decisions for the example farm would result in yield responses to N and P somewhere between those of the Farm-Only model and the Lab-Constrained model, but typically closer to the Lab-Constrained model (presumably, at least partly because of inadequate range of fertilizer rates in the farm data). Similar to using soil laboratory fertilizer recommendation formulas directly, STN would be the dominant factor in determining optimal fertN rates and STP in determining optimal fertP rates for both Lab-Constrained and Lab-ME models.

Interestingly, based on the models, optimal fertN rates do not depend on STP (nor fertP on STN) as long as both fertN and fertP are optimally selected. This is because optimally selected fertN will fully compensate for yield differences that result from differences in STN and optimally selected fertP will fully compensate for yield differences in STP. In other words, regardless of STP, fertP would be applied to make P nonlimiting, which implies that optimal fertN rates will not depend on STP. Mathematically, at optimal fertilizer rates, the parenthetical N term in Eq. [9] will always be valued the same regardless of STN, and the parenthetical P term will always be valued the same regardless of STP. This is not to say that STN and STP levels do not greatly impact profitability. Nor is it to say that these conditions hold when one fertilizer rate is not chosen optimally. For example, consider the OAL–ME Model and a constant fertP rate of 10.3 kg ha^-1 (the average rate actually observed in the farm data). Now, optimally selecting fertN while holding STN at the data mean of 43.3 kg ha^-1, but varying STP over its data range of 7 to 35 mg kg^-1, results in fertN rates ranging from 52 to 70 kg ha^-1. Thus, when the manager is presumed to have both STN and STP information and has the ability to perform VRA on only N, optimal fertN does indeed depend on STP. Current soil laboratory fertilizer recommendations do not have this functionality.

Of special interest for VRA of fertilizer is how much other variables are expected to impact the choice of optimal fertilizer rates. Depending on the model used and the nonfertility variable examined, several substantial impacts emerged (only Lab-ME models are considered here because they are considered the most plausible). For example, holding other variables (including STN and STP) at their means, varying pH across its observed data range (5.8 to 7.9) while optimally selecting fertN and fertP resulted in fertN rates varying from 68 to 52 kg N ha^-1 and fertP rates from 9 to 8 kg P ha^-1 for the KSU–ME Model. In this data set, higher pH results in reduced yields and lower fertP rates. As another example, using the CSU–ME Model, optimal fertN and fertP rates varied by 3.4 kg N ha^-1 and 0.2 kg P ha^-1, respectively, as clay was varied over its data range (2.4–3.5 g kg^-1). Though not associated with particularly large impacts, these examples show how the example farm's manager might use nonfertility information to help improve VRA N and P decisions that otherwise would be based on only STN and STP.

For the example farm, based on the results of this section, VRA N and P fertilizer rates would depend principally on measures of STN and STP, respectively, but also on other information. Increased profits as a result of VRA might be expected from non-N non-P information if it were sufficiently inexpensive to obtain. Regardless, the Lab-Constrained and Lab-ME modeling approaches would provide this farm's manager with reasonable ways to incorporate other information into his fertilizer management program—methods that would not allow his fertilizer rates to stray too far from tried and true soil laboratory recommendations. Additionally, these approaches could easily accommodate new and better farm information as it became available over time.

	CONCLUSIONS

TOP NOTES ABSTRACT INTRODUCTION Yield Models MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B APPENDIX C REFERENCES

 
Fertilizer N and P recommendations explicitly or implicitly depend on yield models that mathematically characterize yield response to fertilizer and soil fertility. Site-specific fertilizer recommendations depend on models that include additional factors besides N and P. Yet, because there are numerous ways to construct yield models, they are often criticized for being ad hoc. To ensure more believable yield models, this research, in the context of northwest Kansas wheat production, first developed a simulation process for generating yield models from fertilizer recommendations provided by soil testing laboratories in the area. Besides making well-accepted fertilizer recommendations, these yield models were mathematically designed to be consistent with agronomic and economic theory, further ensuring their acceptability to both researchers and fertilizer decision makers.

Though site-specific fertilizer recommendations generally must depend on farm data, those data may lack sufficient range in fertilizer rates to reliably estimate yield response to fertilizer. Using an example northwest Kansas wheat farm, this research developed two ways of blending information from university or industry fertilizer recommendations with farm information. One way essentially ignored farm data on N and P, and focused instead on how university or industry recommendations for N and P may change based on other farm information (e.g., soil organic matter, soil texture, soil pH). Recognizing that farm data on N and P may continue to improve over time, this research also considered maximum entropy as a formal process for blending soil lab recommendation information with farm information on all available yield factors, including farm data on N and P. For the example farm, the information-blending techniques resulted in models that predicted yield out of sample as well as or better than a model estimated with only farm data.

Besides helping farm managers make better crop production decisions, this research should help public and private providers of fertilizer recommendations improve those recommendations. For example, one outcome of this research is that producers would benefit from fertilizer P recommendations that address STP management in a way that explicitly depends on length of land tenancy. Regardless, the demand for improved analytical methods likely will continue to increase as farm managers gather more and more farm-level information.

	APPENDIX A

TOP NOTES ABSTRACT INTRODUCTION Yield Models MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS APPENDIX A APPENDIX B APPENDIX C REFERENCES

 
Soil Sample Collection Procedures and Description of N and P Fertilizer
Recommendations for Soil Testing Laboratories
Soil Sample Collection Procedures for Northwest Kansas Example Farm Described in Text.

Each farm soil sample was a field composite (the average field size was 31 ha or 77 acres) made up of 8 to 12 cores that were 2.5 cm (1 inch) in diameter and from the top 20 cm (8 inches) of the soil surface. Samples were analyzed at Olsen's Agricultural Laboratory, McCook, NE.

Nitrogen and Phosphorus Fertilizer Recommendations for Soil Testing Laboratories
Laboratory N and P fertilizer recommendations were generally offered to fertilizer decision makers in English units, as shown here. Before use in SI-unit-based simulations and analyses described in the text, the following conversions were made: pounds per acre (lb acre^-1) was multiplied by 1.1208 to give kilograms per hectare (kg ha^-1); where a bushel (bu) of wheat weighs 60 lb, bushels per acre (bu acre^-1) was multiplied by 0.0673 to give megagrams per hectare (Mg ha^-1); inches of soil depth was multiplied by 2.54 to give centimeters; percentage of organic matter was multiplied by 0.10 to give grams per kilogram (g kg^-1); and units of P₂O₅ were converted to units of P by multiplying the former by 0.436. Some laboratories included manure and previous crop credits in their N recommendation formulas. These credits were ignored in the descriptions below because they were not relevant for the example farm used in this research.

Olsen's Agricultural Laboratory Lab
Nitrogen.

Nrecs are based on the formula, Nrec = (YG x 1.75) - NO3, where YG is the yield goal in bushels per acre (bu acre^-1) and NO3 is the residual soil test NO₃–N in pounds per acre (lb acre^-1) in 0 to 24 inches of soil. Nrecs are rounded to the nearest 10 lb acre^-1. Negative Nrecs are assigned a value of 0.

Phosphorus.

Precs are based on fertility classifications using the Bray-1P test (mg kg^-1). Classifications are very low (VL) with Bray 5; low (L) with 5 < Bray 15; medium (M) with 15 < Bray 25; high (H) with 25 < Bray 30; and very high (VH) with Bray > 30. The associated Precs are: VL 75, L 50, M 25, H 0, and VH 0 lb P₂O₅ acre^-1.

Kansas State University Lab
Nitrogen.

Nrecs are identical to OAL's described above.

Phosphorus.

Precs are in units of pounds of P₂O₅ per acre (lb P₂O₅ acre^-1) and based on the formula, Prec = exp (PX1 + PX2 x 2Bray + PX3 x 4Bray²) x ADJ, where ADJ is the yield goal adjustment for western Kansas, ADJ = (35 x 2 + YG)/(3 x 35), YG is the yield goal in bushels per acre (bu acre^-1), PX1 = 4.088, PX2 = -0.02803, PX3 = -0.0007102, and Bray is the Bray-1P test in milligrams per kilogram (mg kg^-1). Precs are rounded to the nearest 5 lb acre^-1 with this skew: calculated Precs from 0.5 through 5.499 are called 5, from 5.5 through 10.499 are called 10, etc. Calculated Precs above 80 lb acre^-1 are called 80 lb acre^-1.

University of Nebraska–Lincoln Lab
Nitrogen.

University of Nebraska–Lincoln reports Nrecs based on STN as milligrams of NO₃–N per kilogram (mg NO₃–N kg^-1) in 0 to 36 inches of soil. The five classes of fertility, STN 1.1, 1.1 < STN 3.7, 3.7 < STN 7.4, 7.4 < STN 11.1, and STN > 11.1, are assigned Nrecs of 80, 60, 40, 20, and 0 lb acre^-1, respectively. To ensure consistent comparisons across all four soil labs we converted UNL's 0 to 36 inch mg kg^-1 STN tests to 0 to 24 inch lb acre^-1 STN tests. First, because we did not know the typical relationship between 0 to 24 and 0 to 36 inches milligrams of NO₃–N per kilogram, we assumed them to be the same. Then, we assumed 0.3 million pounds of soil per acre inch. Thus, for example, the 1.1 mg kg^-1 reported above in the first classification became 1.1 x 0.3 x 24 = 7.92 lb acre^-1. University of Nebraska–Lincoln's Nrecs do not depend on yield goals.

Phosphorus.

University of Nebraska–Lincoln considers four P fertility classes: VL, Bray < 5, L, 5 Bray < 15, M, 15 Bray < 25, and H, Bray 25; where Bray is STP in milligrams per kilogram (mg kg^-1) using the Bray-1P test. Broadcast fertilizer P rates for the four classes are 80, 60, 40, and 0