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a Dep. of Appl. Econ., Univ. of Minnesota, 1994 Buford Ave., St. Paul, MN 55108-6040
b Dep. of Soil, Water, and Climate, Univ. of Minnesota, 1991 Upper Buford Circle, St. Paul, MN 55108-6040
c Instituto Centroamericano de Administración de Empresas (INCAE)–Latin American Center for Competitiveness and Sustainable Development, Alejuela, Costa Rica
* Corresponding author (thurley{at}apec.umn.edu)
Received for publication July 14, 2003.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONCONCEPTUAL FRAMEWORKMATERIALS AND METHODSRESULTSSUMMARY AND CONCLUSIONSAPPENDIX: REFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONCONCEPTUAL FRAMEWORKMATERIALS AND METHODSRESULTSSUMMARY AND CONCLUSIONSAPPENDIX: REFERENCES |
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One approach that has emerged to test the PA hypothesis for VRA is the estimation and comparison of SSCRFs using multiple regression analysis (e.g., Davis et al., 1996; Malzer et al., 1996; Bongiovanni and Lowenberg-DeBoer, 2001a, 2001b; Lambert et al., 2003; Hurley et al., 2002, 2003; Mamo et al., 2003). Early applications relied on ordinary least squares (OLS), which does not account for heteroscedastic or correlated errors. While OLS estimates may remain unbiased even with heteroscedasticity and correlation, they are typically not efficient and can convey a false sense of precision (Schabenberger and Pierce, 2002). Having confirmed the presence of spatial correlation, recent applications have used more sophisticated statistical models to address this problem. Still, the conceptual foundations used to justify these models are seldom explicit, making it difficult to judge the merit of the method.
The purpose of this paper is to provide a conceptual framework to illuminate how SSCRFs can be used to test the PA hypothesis. The framework is useful because it identifies an appropriate hypothesis and explains recent evidence of site- and treatment-dependent heteroscedasticity and spatial correlation (Hernandez and Mulla, 2003; Hurley et al., 2002; Lambert et al., 2003). The framework is used to guide the development of a heteroscedastic, fixed and random effects, geostatistical model for estimating SSCRFs and testing the PA hypothesis using field data from a common experimental design. The novelty of the model is the inclusion of site-, spatial-, treatment-, and treatment strip–dependent heteroscedasticity and correlation. The model is applied to 1995 field data to demonstrate the importance of the conceptual results, test the PA hypothesis, and estimate the potential value of PA.
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CONCEPTUAL FRAMEWORK |
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TOPABSTRACTINTRODUCTIONCONCEPTUAL FRAMEWORKMATERIALS AND METHODSRESULTSSUMMARY AND CONCLUSIONSAPPENDIX: REFERENCES |
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If a farmer's objective is to manage the variable input to optimize the net return, the classic rule from economic theory says to choose x* such that py
= px where py and px are the price per unit of crop yield and variable input. In economic parlance, the rule states that an input's value of marginal product should equal its marginal cost. The optimal amount of variable input depends on the crop price, variable input price, and most importantly for PA, amount of fixed input. How the optimal amount of variable input depends on the amount of fixed input is found using the implicit function theorem: 
= –
. Note that the optimal amount of variable input does not change with the amount of fixed input if = 0, which means there is no interaction between the variable and fixed input. For example, if soil organic matter does not influence crop response to N, there is no value to varying N applications in response to variation in soil organic matter.
Testing the Precision Agriculture Hypothesis with an Observable Fixed Input
Observational and experimental field data provide an opportunity to test the PA hypothesis, but the development of appropriate statistical models has proven challenging. To understand why, consider the Taylor series expansions,
 | [1] |
 | [2] |
for all kx and kz are real constants that indicate how variable and fixed inputs combine to influence yield. Equation [1] is a general decomposition of yield into the familiar constant, main, and interaction effects. Equation [2] suggests the null hypothesis ßkxkz = 0 for all kx > 0 and kz > 0, which implies PA cannot be used to the benefit of a farmer or the environment because there is no interaction effect. Consider a set of data collected from a controlled field experiment: (yi, xi, zi) for i = 1, 2, ... , N. An individual data point consists of yi, an observed yield; xi, an observed variable input; and zi, an observed fixed input. To test the PA hypothesis with this data, the constant ß coefficients in Eq. [1] must be estimated, a task that is generally not feasible.
The first obstacle is the dimension of the problem. Since the true relationship between yield and inputs is seldom (if ever) known, some approximation is necessary. Additionally, there is the potential for measurement error. Both problems are universal, and the common solution (explicit or implicit) is to truncate the expansion in Eq. [1] and add an error:
 | [3] |
Testing the Precision Agriculture Hypothesis with an Unobservable Fixed Input
Another obstacle more specific to PA is that zi is often unobserved. A researcher or farmer may suspect some fixed input interacts with the variable input but not know which fixed input is important. Confirming the PA hypothesis without knowledge of important fixed inputs is useful because it indicates whether searching for such inputs is worth an effort. If the PA hypothesis cannot be confirmed generally or the value of discovering which fixed inputs are important is small, it makes sense to devote research effort elsewhere.
When zi is unobserved, it can be treated as another source of error. Equation [3] becomes
 | [4] |
kx = ßkx0 + 
ßkxkz Zkz for kx = 0, ..., Kx, and Zk are real constants and 
i = 
i
+ ei is the regression error. Under the traditional assumption that the expected value of the regression error is zero, Zkz is the expectation of zkzi. Two important implications emerge from Eq. [4]. First, the parameters associated with the constant and main effect of the variable input depend on the interaction between the variable and fixed inputs. Second, there is another source of error attributable to the unobserved fixed input that is dependent on the variable input and interactions between the variable and fixed input. Note that in a perfectly controlled experiment, the value of the fixed input is constant for all observations: zkzi – Zkz = 0 for all i and kz. Therefore, the only source of error is related to approximation and measurement. Unfortunately, most field experiments are not perfectly controlled, so error attributable to variation in the unobserved fixed input can be important.
Testing the null hypothesis for PA using Eq. [4] is complicated by the fact that the interaction parameters of interest are inextricably embedded in the parameters for the main effect of the variable input and in the error. This complication highlights the utility of estimating SSCRFs to test the PA hypothesis. Suppose the data is partitioned by dividing the field into R distinct sites such that ri 
{1, ..., R} denotes the ith observation's assigned site. Separate parameters can be estimated for each site by rewriting Eq. [4] as
 | [5] |
kxri = ßkx0 +ßkxkzZkzri for kx = 0, ..., Kx, and Zkzri are real constants; and i = 
+ ei is the regression error. Under the assumption that the expected value of the regression error is zero, Zkzri is the expectation of zkzi for all observations falling in site ri.
Equation [5] shows that the parameters for the main effect of the variable input at a site can be decomposed into a main effect of the variable input 
that does not vary by site and an interaction 
that does vary by site. Therefore, if the PA hypothesis is true and Zkzr varies by site, the parameters for the main effect of the variable input will vary by site. This implies that if the null hypothesis (ßkxkz = 0 for kx = 1,...Kx and kz = 1, ... Kz) is correct, kxri = kxrj for kx = 1, ... Kx and all ri and rj.
Site-specific crop response functions allow the PA hypothesis to be tested by comparing parameter estimates for the main effect of the variable input in Eq. [5] across sites—parameters for which efficient and unbiased estimates can usually be obtained even in the presence of heteroscedastic and correlated errors. It is important to note that this test does not imply the equality of site constants 
. When there is no interaction between the variable and fixed input, check plot yields (yields with no variable input) can vary across sites even though crop response to the variable input does not. Equation [5] shows this is possible because the main effect of the fixed input is absorbed into the site constants.
Using Eq. [5] to test the PA hypothesis is still not trivial because of the covariance
 | [6] |
Equation [6] provides an explanation for three phenomena reported in the literature. The first and most common is spatial correlation where regression errors tend to be more correlated for observations that are closer in distance to each other. If fixed inputs are spatially correlated, zkzi – Zkzri and zk'zj – Zk'zrj will be spatially correlated. Hernandez and Mulla (2003) also reports semivariogram estimates that vary by treatment, a result explained by the dependence of Eq. [6] on the variable input, xkxi and xkxj, when the PA hypothesis is true. Hurley et al. (2002) and Lambert et al. (2003) report site-specific heteroscedasticity, a result consistent with the dependence of Eq. [6] on Zkxri and Zkzrj—constants that vary by site. Each of these phenomena implies estimates from Eq. [5] using OLS will be inefficient.
A variety of methods have been proposed to deal with the estimation problems posed by these phenomena. Spatial econometric and geostatistical models have been estimated to address problems arising from spatial correlation. Hernandez and Mulla (2003) estimate treatment-specific semivariograms to deal with treatment-dependent spatial correlation. Hurley et al. (2002) and Lambert et al. (2003) incorporate site-dependent heteroscedasticity using OLS, spatial econometric, and geostatistical models. None of these models or others we are aware of address site-, spatial-, and treatment-dependent heteroscedasticity and correlation jointly.
The practical relevance of these problems is now explored using data from a common experimental design. The experiment was constructed to test within-field variation in corn response to N. After discussing the design details, a new statistical model is proposed using insights gleaned from the experimental design and conceptual framework.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONCONCEPTUAL FRAMEWORKMATERIALS AND METHODSRESULTSSUMMARY AND CONCLUSIONSAPPENDIX: REFERENCES |
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Each 4.5-ha experimental field plot was 164 by 274 m. Within this area, six 27.3- by 274-m replications of six 4.6- by 274-m treatment strips were established in a randomized complete block design (for an illustration of an analogous design see Mamo et al., 2003). The strips in each replication included N rates of 0, 67, 101, 134, 168, and 202 kg ha–1 applied as anhydrous ammonia. Treatments were applied on 4 Nov. 1994 using a radar-controlled variable-rate applicator to ensure a constant application rate within each strip.
Corn (cv. Pioneer 3531) was planted during the first week of May in 0.76-m rows at approximately 76500 seeds ha–1. Grain yield was determined by harvesting the center two rows (six row strips) with a Massey Ferguson plot combine equipped with a ground distance monitor and a computerized Harvestmaster weigh cell. Each of the 36 strips was divided into seventeen 4.6- by 15.2-m harvest segments. Approximately 8 m was discarded from the end of each strip to eliminate border effects. No headlands were harvested. The experiment produced 612 yield observations at each location. Subsamples of grain were collected to adjust yields to reflect 15.5% moisture. Dikici (2000) reports more details and a descriptive summary of the data.
Empirical
Estimating Eq. [5] with these data provides an opportunity to test the PA hypothesis for VRA. One feature of these data is that they provide observations for each of the six treatments in 102 15.2- by 27.6-m sites at each location. Therefore, Eq. [5] can be used to estimate up to 102 SSCRFs with a full complement of treatments. Another feature is that treatments were randomly assigned across, but not within, strips. This lack of randomization within strips may introduce additional correlation.
The conceptual framework and experimental design suggest that estimation of Eq. [5] using OLS is not efficient. Ordinary least squares estimates of the standard errors for the parameters can be either upward or downward biased. The conceptual framework shows the covariance of regression errors will exhibit site and treatment spatial dependencies. Lack of randomization within strips suggests the covariance of regression errors may also exhibit strip dependencies. Therefore, estimates of Eq. [5] should incorporate an error structure that permits strip as well as site and treatment spatial dependencies.
The proposed model is based on the geostatistical framework. First, let Kx = 2, so Eq. [5] becomes
 | [7] |
i and j is assumed to be
 | [8] |
2risi > 0 and 2rjsj > 0 are the site- and strip-specific variances for observations i and j; dij is the distance in meters between observations i and j; xij is an indicator variable equal to 1.0 if observations i and j had the same treatment applied and 0.0 otherwise; sij is an indicator variable equal to 1.0 if observations i and j came from the same strip and 0.0 otherwise; C1 
0, Cs 0, and Cx 0 are spatial, strip, and treatment correlation parameters that assume positive correlation; 1 g(dij, a) 0 is a permissible semivariogram distance function (e.g., see McBratney and Webster, 1986); and a is a range or shape parameter for the semivariogram distance function.
Dividing Eq. [8] by risi and rjsj results in the correlation coefficient. When i 
j, this correlation coefficient is comprised of three elements: spatial correlation [C1(1 – g(dij, a)], strip correlation (Cssij), and treatment correlation (Cxxij). Since the correlation coefficient must always lie between 1.0 and –1.0, 1.0 C1 + Cs + Cx 0.0, assuming spatial, strip, and treatment correlation are positive to ensure the covariance matrix satisfies the necessary regularity conditions (i.e., is positive definite).
The classical geostatistical approach decomposes variation in the dependent variable into a trend, local variance (nugget), and distance effect. Equations [7] and [8] accomplish a similar decomposition but add heteroscedasticity, strip effects, and treatment effects. The trend is captured by 0ri + 1rixi + 2rix2i, which is site specific. The semivariogram is
 | [9] |
risirjsj C0 = risirjsj 
can be interpreted as the nugget and risirjsj as the sill. Equation [9] shows precisely how the standard geostatistical model is modified by heteroscedasticity and strip and treatment correlation.
Estimation
Equations [7] and [8] can be estimated using a variety of methods after choosing a division of the field into various sites and a distance function for spatial correlation (Schabenberger and Pierce, 2002). The method employed uses FGLS for the parameters. Estimates of the covariance parameters [C1, Cs, Cx, a, and 2rs for all r R and s (1, ..., 36)] are obtained using ML after substituting the FGLS estimator in for the parameters. The parameters are substituted or profiled in this manner because the FGLS estimator for is the ML estimator given the covariance parameters. The procedure also substantially speeds computation.
The data can be divided into 102 sites with a full complement of treatments, but with only a single observation per strip in each of these sites, it is not possible to identify strip correlation. Therefore, fewer sites are necessary given these data. To illustrate the benefit of estimating Eq. [7] and [8] for smaller management units, two site partitions are explored. The first divides each location into six contiguous sites of about 0.75 ha: four sites with 108 observations and two sites with 90 (Fig. 1) . The second divides each location into 48 contiguous sites of about 0.094 ha: six sites with 18 observations and 42 sites with 12.
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With these two partitions, the variance parameters for every possible site and strip combination [2rs for all r (1, ..., R) and s (1, ..., 36)] cannot be identified without additional simplifying assumptions. The identification problem is analogous to trying to use an independent variable in a multiple regression analysis that is a linear combination of other independent variables. To identify the model, the site and strip variances were assumed to be multiplicatively separable (i.e., 2rs = 2r2s), and 2s is set to 1 for s (1, 19) for six sites and s (1, 7, 13, 19, 25, 31) for 48 sites. Additively separable variances (i.e., 2rs = 2r + 2s) were also explored but did not fit the data as well.
There are a variety of possible distance functions. However, the computational intensity of the model restricts the practicality of comparing lots of functions. Since the primary purpose of the paper is to explore the value of incorporating site, treatment, and strip dependencies into a model with spatially correlated errors, attention is focused on a single distance function. Comparing the fit of a standard geostatistical model at both locations based on the maximized log-likelihood using the exponential, Gaussian, and spherical distance functions suggested the Gaussian model fit best. Therefore, the full model with site, treatment, and strip spatial dependencies was estimated with the Gaussian function: g
= 1 – e–
2.
Hypotheses
Eight models based on Eq. [7] and [8] were estimated for each location to test a variety of hypotheses. Table 1 summarizes while detailing the applicable model restrictions. Model 1 used six sites in an ML analogy to OLS. Model 2 used six sites in a standard geostatistical model. Model 3 used six sites while adding site and treatment heteroscedasticity and correlation to Model 2. Model 4 used six sites while adding strip heteroscedasticity and correlation to Model 3. Model 5 is similar to Model 4 except it assumed no interaction between N and fixed inputs. Models 6, 7, and 8 used 48 sites but were otherwise identical to Models 1, 4, and 5.
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2 with the degrees of freedom equal to the number of parameter restrictions. The benefit of incorporating site-, spatial-, treatment-, and strip-dependent heteroscedasticity and correlation was evaluated by comparing Models 1 and 2, 2 and 3, 3 and 4, and 6 and 7. The comparison of 1 and 2 evaluates the importance of spatial correlation. The comparison of 2 and 3 evaluates the importance of conditioning the variance and spatial correlation on the site and treatment. The comparison of 3 and 4 evaluates the importance of also conditioning on strips. These three comparisons are all based on six sites. The comparison of 6 and 7 evaluates the importance of incorporating site, spatial, treatment, and strip effects with smaller management units (0.094 vs. 0.75 ha).
Comparing Models 4 and 5 and 7 and 8 tests the PA hypothesis. The comparison between 4 and 5 evaluates whether there were significant differences in crop response to N between the six sites in the first partition. The comparison between 7 and 8 evaluates whether there were significant differences in crop response to N between the 48 sites in the second partition. If there is a significant difference in crop response to N within a field, VRA can potentially improve N returns.
Finally, comparing 4 and 7 evaluates variation in crop response functions within the six sites in the first partition. The test determines if dividing a field into smaller management units significantly improves explanatory power.
Potential Value of Variable-Rate Nitrogen Applications
The potential value of the increased N return from VRA was calculated using coefficient estimates for the parameters in Eq. [7] and [8]. The estimated N return above fertilizer costs was defined as 
= 
. The optimal VRA was calculated by choosing xi for i = 1, ..., 612 to maximize . Alternatively, an optimal uniform rate (URA) was calculated by choosing x = xi for i = 1, ..., 612 to maximize . These optimal rates were constrained between 0 and 202 kg ha–1 to avoid predicting yields outside the range of available data. Nitrogen returns for the optimal VRA and URA were compared with the University of Minnesota (UMN) recommendation (140 kg ha–1 for both Hanska and Morgan) to determine the potential value of VRA within and between fields assuming the price of corn and N was $98.21 t–1 ($2.50 bu–1) and $0.374 kg–1 ($0.17 lbs–1).
Let VRA, URA, and UMN be the estimated N return for the optimal VRA, optimal URA, and UMN rate. The potential return to switching to the optimal VRA from the UMN rate was calculated as VRA – UMN, which represents the potential value of varying N applications within a field using VRA. It is important to note that this potential value is exclusive of the cost of implementing a VRA strategy (e.g., the cost of information acquisition and variable-rate application equipment or services). The standard deviation and 90% confidence interval were calculated using a Taylor series expansion (see Casella and Berger, 1990, p. 328–331) and assuming normality.
The potential value of switching to the optimal VRA from the UMN rate was decomposed as VRA – UMN = VRA – URA + URA – UMN. The potential value of VRA due to switching to the optimal URA from the UMN rate or of getting the right average rate for a field is URA – UMN. The potential value of VRA due to switching to the optimal VRA from the optimal URA or to varying the right average rate optimally within a field is VRA – URA.
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RESULTS |
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TOPABSTRACTINTRODUCTIONCONCEPTUAL FRAMEWORKMATERIALS AND METHODSRESULTSSUMMARY AND CONCLUSIONSAPPENDIX: REFERENCES |
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Error Structure
Table 3 reports the correlation parameters along with the shape parameter (a) and the average standard deviation for selected models.
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Potential Value of Variable-Rate Nitrogen Applications
Figure 2
reports estimates of the potential value of VRA and the decomposition of this value into the effect of switching to the optimal URA from the UMN rate and to the optimal VRA from the optimal URA. While the results of Table 1 show that Model 7 is the best-fitting model, results for other models are also reported to demonstrate the practical importance of using a model that incorporates site, treatment, and strip as well as spatial effects.
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The estimated potential value of VRA increases as the size of the management unit decreases, but the precision of the estimate (width of the confidence interval) may increase or decrease. Estimating SSCRFs with 48 instead of 6 sites (Model 7 vs. 4) increased the estimated potential value of VRA (EXCLUSIVE OF IMPLEMENTATION COSTS) by 133 and 88% for Hanska and Morgan. For Hanska, smaller management units increased the width of the confidence interval for the estimate by about 2% while for Morgan, it decreased it by about 30%.
More of the spatial variability in corn yields and corn response to N was captured by estimating more SSCRFs for smaller sites within the field. This allows N application rates to be better tailored to within-field variability and increases the potential N return. It also reduced the error in the estimated SSCRFs, which tended to reduce the width of the confidence intervals, making the estimate more precise. However, estimating more SSCRFs increased the number of estimated parameters, reducing the model's degrees of freedom, which tended to increase the width of the confidence intervals, making the estimate less precise. This result reflects the classic tradeoff between degrees of freedom and error reduction that comes from increasing the number of estimated parameters. For Hanska, the loss of degrees of freedom dominates, so the confidence interval got wider, and the estimate became less precise with smaller management units. For Morgan, the reduction in error dominated, so the confidence interval shrank and the estimate became more precise with smaller management units.
Comparing Model 4 with 1 through 3 and 7 with 6 provides insight into the practical importance of using a model with site, treatment, and strip as well as spatial effects. Two features of this comparison are of particular interest.
First, for Hanska, Models 1 through 4 produced similar estimates of the potential value of VRA. Models 6 and 7 also produced similar estimates. These results are consistent with the findings of Lambert et al. (2003). However, for Morgan, the estimate for Model 4 is notably lower than the estimates for 1 through 3, and the estimate for Model 7 is notably lower than for 6. These results are contrary to the findings of Lambert et al. (2003).
The notable reduction in the value of VRA for Morgan using Models 4 and 7 can be explained by the increased precision of the estimates of the quadratic parameters in Eq. [7]. Figures 3 and 4 explain why by reporting and illustrating the estimated SSCRFs for Models 1 and 4. All the parameter estimates are significant (p < 0.05) for Model 1 and 4 at Hanska, and both models produced similar crop response functions for each site. Both models indicated the response functions were concave (a positive linear and negative quadratic parameter), implying limited N returns at Hanska. For Morgan, both models produced significant estimates for the constant and linear parameters but not for the quadratic parameters. Only Model 4 produced significant estimates for all quadratic parameters. For Sites 1 through 4, Model 1 produced larger linear estimates but smaller insignificant quadratic estimates, implying linear response functions or unlimited N returns. Model 4 produced smaller linear estimates but larger significant quadratic estimates, implying concave response functions or limited N returns. The unlimited N returns implied by Model 1 for Sites 1 through 4 result in larger predicted yield increases and a higher estimated value for the optimal VRA.
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Second, the confidence intervals for Model 4 were wider than for 1, 2, and 3 as were the confidence intervals for Model 7 when compared with 6. While these results seem to suggest OLS produced more precise estimates for the potential value of VRA, this is an erroneous conclusion. Ordinary least squares confidence intervals are reliable only if it is reasonable to assume errors are homoscedastic and uncorrelated. Table 1 rejected these assumptions, so the OLS confidence intervals are unreliable and, even worse, convey a false sense of precision. For example, with 48 sites, OLS could lead to the false conclusion that there was greater than a 95% chance that the potential value of VRA exceeds $15 ha–1 for Hanska (Model 6 vs. 7 in Fig. 2).
Lambert et al. (2003) finds that including spatial correlation improved the precision of the estimated value of VRA. Comparing Models 1 and 2 supports this conclusion. However, also including site, treatment, and strip effects reverses this conclusion. Therefore, accounting for spatial correlation without considering site, treatment, and strip effects resulted in even narrower confidence intervals that exacerbate the false sense of precision obtained from OLS.
Figures 5 and 6 report more detailed spatial results for the best-fitting model (Model 7). The figures highlight the degree of within-field variability at both locations. For Hanska and Morgan, estimated check strip yields ranged from 2.7 to 8.2 t ha–1 and 3.6 to 9.8 t ha–1, with an average of 6.2 and 6.3 t ha–1. The optimal N rates ranged from 97 to 202 kg ha–1 for Hanska, with an average of 154 kg ha–1. These rates correspond to yields ranging from 6.5 to 11.2 t ha–1, with an average of 9.4 t ha–1. For Morgan, the optimal application rates ranged from 109 to 202 kg ha–1, with an average of 184 kg ha–1. Corresponding yields ranged from 8.3 to 12.9 t ha–1, with an average of 10.7 t ha–1. The increase in return when compared with the UMN rate ranged from $0.0 to $176.1 ha–1 for Hanska and $0.0 to $274.2 ha–1 for Morgan. The standard deviation of this increased return ranged from $0.3 to $76.3 ha–1 for Hanska and $0.2 to $78.5 ha–1 for Morgan.
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Holding the price of corn constant at $98.21 t–1 ($2.50 bu–1) and letting the price of N increase from $0.15 kg–1 ($0.07 lbs–1) to $0.59 kg–1 ($0.27 lbs–1), the potential value of VRA compared with the UMN-recommended rate decreases linearly from $32.52 to $24.27 ha–1 for Hanska and from $75.75 to $57.09 ha–1 for Morgan, and the percentage of this value attributable to using the optimal uniform rate decreases from 23 to 1 for Hanska and 78 to 56 for Morgan. Therefore, the importance of varying the right average rate optimally within a field increases with an increase in the price of N.
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SUMMARY AND CONCLUSIONS |
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TOPABSTRACTINTRODUCTIONCONCEPTUAL FRAMEWORKMATERIALS AND METHODSRESULTSSUMMARY AND CONCLUSIONSAPPENDIX: REFERENCES |
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Effort to improve models for testing the PA hypothesis has focused on spatial correlation. Recently however, problems with site-specific and treatment-dependent heteroscedasticity and correlation have been identified. Our conceptual framework shows why this should not be a surprise, and our results show this is not the end of the story for data from a common experimental design. We also find important strip heteroscedasticity and correlation. Failing to account for strip effects resulted in estimates of the potential value of VRA that were too high and confidence intervals that convey a false sense of precision because they were too narrow.
The conceptual and empirical models we developed are most applicable to a single year of data though they could be extended to multiple years. An important consideration for a multiyear extension of the model is the need to differentiate between unmanaged inputs that are temporally stable (e.g., topography and soil type) and those that are not (e.g., rainfall and temperature) (Bullock et al., 2002). Within the context of our conceptual model, one could include two rather than one vector of unmanaged inputs: one that varies with time and one that does not. Empirically, additional parameters would have to be estimated for the fixed effect of time invariant unmanaged inputs and random effect of time variant unmanaged inputs.
The conceptual model points to the importance of site- and treatment-dependent heteroscedasticity and spatial correlation. These results are generally applicable to any field experiment where soil, rainfall, and other important agronomic factors other than the treatment may vary substantially across the experimental plot. The important strip effects found in our analysis are likely specific to complete randomized block design experiments that divide treatment strips within a block into multiple observations. Experimental designs that randomize more completely should eliminate this complication.
Due to the computational intensity of the model and scope of our objectives, we did not systematically explore a wide variety of assumptions regarding the structure of spatial correlation and heteroscedasticity. Specifically, we focused on a multiplicative form of heteroscedasticity and Gaussian spatial correlation. Alternatively, one could explore other forms of heteroscedasticity. With increasing computer power and new experiments with more observations per site, estimating the most general form of heteroscedasticity in our empirical model may soon be practical. There are also a wide variety of both isotropic and anisotropic models of spatial correlation that could be explored in future work.
The range of treatments employed in our experiments was well suited for the Hanska location but not for the Morgan location, which is why we see a greater divergence between the estimated models using the Morgan data. It is also why we had to constrain our estimates of the optimal N rates for many of the sites at Morgan; therefore, the estimated potential value of VRA at this location is likely downward biased.
Finally, our analysis of the potential value of VRA does not include implementation costs. These costs will vary depending on how this potential is tapped (e.g., the information used to guide applications and size of management units). A farmer who uses soil nitrate testing to tap this potential may have lower implementation costs than a farmer who runs controlled field experiments; however, controlled field experiments may provide better information. While demonstrating the potential of VRA under varied field conditions is important, more effort could be devoted to finding better ways to tap this potential.
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APPENDIX: |
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TOPABSTRACTINTRODUCTIONCONCEPTUAL FRAMEWORKMATERIALS AND METHODSRESULTSSUMMARY AND CONCLUSIONSAPPENDIX: REFERENCES |
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LRS, likelihood ratio statistic
ML, maximum likelihood
OLS, ordinary least squares
PA, precision agriculture
SSCRF, site-specific crop response function
UMN, University of Minnesota
URA, uniform-rate nitrogen application
VRA, variable-rate nitrogen application
x, units of variable/managed input
x*, optimal units of variable/managed input
z , units of fixed/unmanaged input
y = f(x, z), units of crop yield as a function of variable and fixed inputs
py and px, price per unit of crop yield and variable input
kx and kz, indexes for units of variable and fixed input
ßkxkz = 
, kxth and kzth order cross partial derivative of crop yield with respect to the variable and fixed input
N, number of observations
i, observation index
ei, approximation and measurement error
Kx and Kz, integer constants
kx, kxth estimable mean parameter
Zkz, kzth unobserved real constant
i, unobserved error
R and ri, number of field sites/partitions and site assigned to the ith observation
kxr, kxth estimable mean parameter for site r
Zkzr, kzth unobserved real constant for site r
E[·], expectation operator
dij, spatial distance between observations i and j
C0, estimable spatial nugget semivariogram parameter
C1, estimable semivariogram distance correlation parameter
a, estimable semivariogram range/shape parameter
g(dij, a), semivariogram distance function
Cx, estimable variable input correlation parameter
xij, indicator variable for observations with the same variable input
Cs, estimable treatment strip correlation parameter
sij, indicator variable for observations from the same strip
(dij), semivariogram
rs, covariance parameter for site r and treatment strip s
r and s, estimable covariance parameters for site r and treatment strip s
, returns to N above the cost of N
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REFERENCES |
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TOPABSTRACTINTRODUCTIONCONCEPTUAL FRAMEWORKMATERIALS AND METHODSRESULTSSUMMARY AND CONCLUSIONSAPPENDIX: REFERENCES |
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 Y. Liu, S. M. Swinton, and N. R. Miller Is Site-Specific Yield Response Consistent over Time? Does It Pay? Am. J. Agr. Econ., May 1, 2006; 88(2): 471 - 483. [Abstract] [Full Text] [PDF]
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Y. Miao, D. J. Mulla, W. D. Batchelor, J. O. Paz, P. C. Robert, and M. Wiebers Evaluating Management Zone Optimal Nitrogen Rates with a Crop Growth Model Agron. J., April 11, 2006; 98(3): 545 - 553. [Abstract] [Full Text] [PDF]
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