# Assessment of the Use of Geographically Weighted Regression for Analysis of Large On-Farm Experiments and Implications for Practical Application

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## Abstract

**:**

## 1. Introduction

- is influenced by spatial auto-autocorrelation;
- is not due to the treatment being tested [34]; and
- is often much larger than the variation due to treatment effects.

## 2. Materials and Methods

#### 2.1. Simulated Data

- Global linear response to fertiliser
- Linear response to fertiliser that varies within spatial zones
- Locally varying linear response to fertiliser

^{−}

^{1}, using an exponential variogram with sill equal to 2500 × 10

^{2}and range equal to 30 pixels. A fertiliser strip trial is simulated using a randomised complete block design with four rates (0, 30, 60, and 100) and two repetitions. Each strip is 3 pixels (30 m) wide. The response to fertiliser is added to the minimum yield.

#### 2.1.1. Global Linear Response to Fertiliser

^{−1}. The coefficient 6 corresponds to a global response of 6 kg of yield per kilogram of fertiliser applied. The simulated experiment, comprised of $yield.min$, $rate$, and $yield$, is shown in Figure 1.

#### 2.1.2. Linear Response to Fertiliser that Varies within Spatial Zones

#### 2.1.3. Locally Varying Linear Response to Fertiliser

#### 2.2. Potassium Trials

#### 2.2.1. Cunderdin Trial

- Extreme high and low yields.
- Extreme high and low harvester speeds.

#### 2.2.2. Narambeen Trial

#### 2.3. Geographically Weighted Regression (GWR)

#### 2.3.1. Basic GWR

#### 2.3.2. Bandwidth Selection for GWR

#### 2.3.3. Mixed GWR and Model Selection

- For each column of ${X}_{a}$,
- i.
- Regress the column against ${X}_{b}$ using basic GWR.
- ii.
- Compute the residual from the above regression.

- Regress y against ${X}_{b}$ using basic GWR.
- Compute the residual from the above regression.
- Regress the y residuals against the ${X}_{a}$ residuals using ordinary least squares regression to get $\widehat{a}=\left\{{\widehat{\beta}}_{1},\dots ,{\widehat{\beta}}_{q}\right\}$.
- Regress $y-{X}_{a}\widehat{a}$ against ${X}_{b}$ using basic GWR to get the spatially varying coefficients.

#### 2.3.4. GWmodel R Package

## 3. Results

#### 3.1. Simulated Data

#### 3.1.1. Bandwidth Selection

#### Summary of Bandwidth Selection Results and Recommended Approach to Bandwidth Selection for OFE

#### 3.1.2. Model Selection

#### Summary of Model Selection Results and Recommended Approach to Model Selection for OFE

- Calculate the bandwidth that minimises AICc.
- Fit a mixed GWR that allows only the intercept to vary spatially.
- Fit a basic GWR that allows both the intercept and yield response to treatment to vary spatially.
- Select the model from 2 and 3 that has the lowest AICc.

#### 3.2. Potassium Trials

#### 3.2.1. Cunderdin Trial

^{−1}. The global response (rate of change in yield due to applied MOP) was 1.7 kg ha

^{−1}of barley yield per kg ha

^{−1}of MOP applied. This response is extremely low and would not justify the use of MOP fertilisation: the expected yield with no fertiliser applied is 4605 kg ha

^{−1}across the trial. The addition of 40 kg MOP ha

^{−1}would result in a total yield increase of only 61 kg ha

^{−1}, and 80 kg MOP ha

^{−1}would result in an increase of only 121 kg ha

^{−1}.

^{−1}, and approximately 50% of the absolute errors are less than 250 kg ha

^{−1}. The correlation between observed and predicted yields is 0.63.

#### 3.2.2. Narambeen Trial

^{−1}, with three quarters of the paddock yielding less than 2000 kg ha

^{−1}.

^{−1}per kg of MOP applied, and the maximum response is 5 kg ha

^{−1}yield per kg of MOP applied. A recent analysis of potassium cost to wheat grain prices [69] showed that the that the break-even ratio for the fertilisation of barley with MOP was 7.4; thus, this analysis shows that MOP fertilisation is not profitable for this paddock. This begs the question: Is statistical significance as important as practical or economic significance? Since farmers are ultimately interested in profit, statistical significance is not necessarily a useful piece of information for OFE.

^{−1}of MOP had been applied. The predicted average improvement in yield achieved across the trial by MOP fertiliser application was 106 kg ha

^{−1}at a rate of 66 kg MOP ha

^{−1}, 215 kg ha

^{−1}at a rate of 134 kg MOP ha

^{−1}, and 321 kg ha

^{−1}at 200 kg MOP ha

^{−1}.

^{−1}with 90% of absolute errors less than 275 ha

^{−1}. The correlation between observed and predicted yields is 0.90.

## 4. Discussion

^{2}was 0.63 for the Cunderdin trial and 0.9 for the Narambeen trial, which compares favourably with another published potassium-yield model derived from on-farm experiments, which had an R

^{2}of 0.48 [71]. Unfortunately, both of our potassium trials showed small yield responses to MOP fertiliser; and moreover, a response that was below the break-even ratio, indicating that fertilisation with MOP was not profitable at either location. This may because there was sufficient plant-available potassium in the soil, and it would be useful to compare the results of soil tests taken prior to experimentation with locally varying response in future trials. We intend to follow up this study by applying our recommended approach in situations where larger yield responses might be expected—for example, nitrogen trials in known deficient conditions. We note that while our applications are in broadacre cereal cropping, OFE can also be used for any horticultural or viticultural cropping system.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Cunderdin (

**a**) trial design (kg Muriate of Potash (MOP) ha

^{−1}) and (

**b**) cleaned yield data (kg ha

^{−1}).

**Figure 6.**Results for data simulated with global linear response using bandwidth selection by (

**a**) minimal corrected Akaike Information Criterion (AICc) and (

**b**) based on experimental design.

**Figure 7.**Results for data simulated with local linear response by zones using bandwidth selection by (

**a**) minimal AICc and (

**b**) based on experimental design.

**Figure 8.**Results for data simulated with locally varying linear response using bandwidth selection by (

**a**) minimal AICc and (

**b**) based on experimental design.

**Figure 9.**Distribution of AICc and error components calculated for 500 simulations of locally varying linear response of yield to fertiliser application with varying bandwidths.

**Figure 10.**(

**a**) Yield intercept (kg ha

^{−1}) from geographically weighted regression (GWR) applied to the Cunderdin trial and (

**b**) model error for the Cunderdin trial.

**Figure 11.**Results of GWR applied to the Narambeen trial: (

**a**) yield intercept (kg ha

^{−1}); (

**b**) response (kg yield kg

^{−1}MOP); (

**c**) predicted change in yield when 134 kg ha

^{−1}MOP is applied; and (

**d**) adjusted p-values of the estimated response coloured such that green areas are significant at the 0.05 probability level.

Model | GW Model Function and Formula |
---|---|

Spatial regression | $\mathrm{gwr}.\mathrm{mixed}\left(\mathrm{yield}\text{}~\text{}\mathrm{rate},\mathrm{fixed}.\mathrm{vars}=\u201c\mathrm{rate}\right)$ |

Local linear regression (SVC) | $\mathrm{gwr}.\mathrm{basic}\left(\mathrm{yield}\text{}~\text{}\mathrm{rate}\right)$ |

**Table 2.**Distance weighting kernels implemented in the GWmodel R package where ${w}_{ij}$ is the j-th element of the diagonal of the weight matrix W used to estimate the local regression at location i, ${d}_{ij}$ is the distance between the i-th and j-th observations, and b is the bandwidth.

Kernel Name | Formula |
---|---|

Gaussian | ${w}_{ij}\text{}=\mathit{exp}\left(-\frac{1}{2}{\left(\frac{{d}_{ij}}{b}\right)}^{2}\right)$ |

Exponential | ${w}_{ij}\text{}=\mathit{exp}\left(-\frac{\left|{d}_{ij}\right|}{b}\right)$ |

Bi-Square | ${w}_{ij}=\left\{\begin{array}{c}{\left(1-{\left({d}_{ij}\right)}^{2}\right)}^{2},if\left|{d}_{ij}\right|b,\\ 0,otherwise.\end{array}\right.$ |

Tri-Cube | ${w}_{ij}=\left\{\begin{array}{c}{\left(1-{\left({d}_{ij}\right)}^{3}\right)}^{3},if\left|{d}_{ij}\right|b,\\ 0,otherwise.\end{array}\right.$ |

Box-car | ${w}_{ij}=\left\{\begin{array}{c}1if\left|{d}_{ij}\right|b,\\ 0,otherwise.\end{array}\right.$ |

**Table 3.**AICc for models of increasing order of complexity fitted to data with simulated global and locally varying linear response (minimum values of AICc identify optimal models; these are marked with asterisks, *).

Model | Global Linear (gwr.mixed) | Local Linear (gwr.basic) | Global Linear (gwr.mixed) | Local Linear (gwr.basic) |
---|---|---|---|---|

Simulated data with global linear response | ||||

Bandwidth (pixels) | 0.87 | 0.87 | 4.5 | 4.5 |

AICc | 28,468 * | 28,795 | 31,397 | 31,227 * |

Simulated data with local linear response | ||||

Bandwidth (pixels) | 0.88 | 0.88 | 4.5 | 4.5 |

AICc | 30,032 | 29,303 * | 33,184 | 31,973 * |

**Table 4.**Percentage of 500 simulations in which the correct model was identified by model selection.

Bandwidth | Global Response | Local Response |
---|---|---|

AICc-minimising | 100 | 78 |

45 m | <1 | 100 |

**Table 5.**Percentage of 500 simulations in which the correct model was identified by a Monte Carlo test for spatial variability.

Bandwidth | Global Response | Local Response |
---|---|---|

AICc-minimising | 70 | 69 |

45 m | 18 | 99 |

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**MDPI and ACS Style**

Evans, F.H.; Recalde Salas, A.; Rakshit, S.; Scanlan, C.A.; Cook, S.E. Assessment of the Use of Geographically Weighted Regression for Analysis of Large On-Farm Experiments and Implications for Practical Application. *Agronomy* **2020**, *10*, 1720.
https://doi.org/10.3390/agronomy10111720

**AMA Style**

Evans FH, Recalde Salas A, Rakshit S, Scanlan CA, Cook SE. Assessment of the Use of Geographically Weighted Regression for Analysis of Large On-Farm Experiments and Implications for Practical Application. *Agronomy*. 2020; 10(11):1720.
https://doi.org/10.3390/agronomy10111720

**Chicago/Turabian Style**

Evans, Fiona H., Angela Recalde Salas, Suman Rakshit, Craig A. Scanlan, and Simon E. Cook. 2020. "Assessment of the Use of Geographically Weighted Regression for Analysis of Large On-Farm Experiments and Implications for Practical Application" *Agronomy* 10, no. 11: 1720.
https://doi.org/10.3390/agronomy10111720