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Spatial Example: Uniformity Trial

Consider the Uniformity Trial.jmp sample data table. An agronomic uniformity trial was conducted on an 8x8 grid of plots. In a uniformity trial, a test crop is grown on a field with no experimental treatments applied. The response variable, often yield, is measured. The idea is to narrate variability in the field every bit background for planning a designed experiment to be conducted on that field. (See Littell et al. 2006 , p. 447.)

Your objective is to use the data from these data to design a yield trial with 16 treatments. Specifically, you desire to decide whether to bear future experiments on the field:

• a complete block design with 4 blocks (denoted Quarter in the data)

• an incomplete block design with 16 blocks (denoted Subquarter in the data)

• a completely randomized design with spatially correlated errors

With this objective, spatial data can exist treated equally repeated measures with 2 or more dimensions every bit repeated effects. And then, you tin compare and choose an appropriate model using the values in the Fit Statistics report. You lot commencement by determining if in that location is significant spatial variability, so you determine whether there is a asset effect.

Once yous have established whether there is a nugget effect, you determine the best fitting spatial covariance structure. Finally, you fit the blocking models and compare these to the best spatial structure. In this case, both AICc and BIC are used to select a best model. Spatial Correlation Structure provides more information about nugget effects and other spatial terminology.

Tip: This section walks you through many aspects of fitting spatial data (from fitting the model to deciding on the best covariance structure). Exit the Fit Model launch window open equally you work through each case.

Fit a Spatial Construction Model

To decide whether at that place is significant spatial variability, yous can fit a model that accounts for spatial variability. Then you tin compare the likelihood for this spatial model to the likelihood for a model that does non account for spatial variability. You tin can exercise this because the contained errors model is nested within the spatial model family: The independent errors model is a spatial model with spatial correlation, ρ, equal to 0. This means that you can perform a formal likelihood ratio exam of the ii models.

Start, fit the model that accounts for spatial construction.

i. Select Help > Sample Data Library and open Uniformity Trial.jmp.

ii. Select Analyze > Fit Model.

iii. Select Keep dialog open up and then that yous tin return to the launch window in the adjacent example.

iv. Select Yield and click Y.

five. Select Mixed Model from the Personality list.

6. Select the Repeated Construction tab.

7. Choose Spatial from the list next to Structure.

eight. Cull Spherical from the list next to Blazon.

nine. Select Row and Column and click Repeated.

Effigy 8.32 Completed Fit Model Launch Window Showing Repeated Structure Tab

x. Click Run.

Figure viii.33 Fit Mixed Report - Spatial Spherical

In the Fit Mixed report, the Actual by Predicted Plot shows that the predicted yield is a single value. This is because but spatial covariance was fit. The Fit Statistics report shows that -2 Log Likelihood is 227.68, and the AICc is 234.08.

Because an isotropic spatial construction was fit, a Variogram plot is shown. Because the trials are laid out in an 8 past viii grid, there are more pairs of points at pocket-size distances than at very large distances. Run into Effigy 8.37 for the layout. The Variogram shows that a spherical spatial structure is an first-class fit for distances up to nearly eight.iv. The altitude class for the final distance consists of just the 2 diagonal pairs of points.

The Repeated Furnishings Covariance Parameter Estimates report gives estimates of the range (Spatial Spherical = 2.71) and the sill (Residual = 3.26). See Variogram.

Fit the Independent Errors Model

Next, fit the contained errors model.

1. Return to the Fit Model Launch Window.

two. Select Repeated Construction tab.

3. Select Residual from the Structure listing.

4. Remove Row and Cavalcade from the Repeated furnishings list.

Otherwise, a warning appears: "Repeated columns and bailiwick columns are ignored when the Residuum covariance structure is selected."

v. Click Run.

The fit statistics for the contained errors model are: -ii Log Likelihood = 254.22, and AICc = 258.41. Each of these exceeds the corresponding value for the spatial correlation model, where -2 Log Likelihood is 227.68 and the AICc is 234.08. Because smaller values of these statistics betoken a better fit, the spatial model might provide a amend fit.

Bear a Likelihood Ratio Examination (Optional)

A formal likelihood ratio test shows whether the spatial correlation model explains pregnant variation. One model must exist nested in some other model to create valid likelihood ratio tests.

Typically, spatial models are compared using AICc or BIC rather than through formal likelihood ratio testing. Evaluating the AICc or BIC is faster, and many spatial models are non nested.

You tin conduct a likelihood ratio test in this example, because the contained errors model is nested within the spatial model family. The contained errors model is a spatial model with spatial correlation, ρ, equal to 0. This means that you tin perform a formal likelihood ratio examination of the two models.

In this example, the likelihood ratio exam statistic is 254.22-227.68=26.54. Comparing this to a Chi-square distribution on 1 caste of freedom, the nil hypothesis of no spatial correlation is rejected with a p-value < 0.0001. You tin conclude that these data contain significant spatial variability.

Select the Type of Spatial Covariance

Next, y'all determine which spatial covariance construction best fits the data:

• with or without a nugget consequence (variation over relatively modest distances)

• isotropic (spatial correlation is equal in all directions) or anisotropic (spatial correlation differs in the two directions)

• blazon of structure, spherical, Gaussian, exponential, or ability.

1. Render to the Fit Model launch window.

2. Select the Repeated Structure tab.

iii. Select Row and Cavalcade and click Repeated.

4. Select Spatial with Nugget from the Structure listing.

5. Select Spherical from the Type list.

6. Click Run.

The Fit Mixed report is shown in Figure 8.34. Detect that the log-likelihoods are substantially equal to those of the spherical with no nugget model, and the AICc is slightly higher (236.36 compared to 234.08). The Repeated Effects Covariance Parameter Estimates study shows that the Nugget covariance parameter has an estimate of zero. In that location does not announced to exist any evidence for a nugget effect.

Figure 8.34 Fit Mixed Report - Spatial Spherical with Nugget

vii. Click the Variogram ruby triangle and select Spatial > Spherical.

Figure 8.35 Fit Mixed Report - Variogram

The two variograms are virtually identical. This also suggests that there is no evidence of a nugget effect.

8. Return to the Fit Model launch window.

9. Select Repeated Structure tab.

10. To test anisotropicity, select Spatial Anisotropic from the Structure list.

11. Select Spherical from the Type list.

12. Click Run.

The Fit Mixed report is shown in Figure eight.36. The fit statistics indicate not equally good a fit every bit the isotropic (spatial construction) spherical model (AICc 240.54 compared to 234.08). The Repeated Effects Covariance Parameter Estimates report shows that the estimates for the Row (Spatial Spherical Row) and Column (Spatial Spherical Column) covariances are very close. There is no evidence to suggest that spatial correlations within rows and columns of the grid differ.

Figure 8.36 Fit Mixed Written report - Spatial Anisotropic Spherical

Decide the Type of the Spatial Construction

An isotropic spatial structure with no nugget is appropriate. To determine the type of the spatial structure, you can compare the available types.

ane. Return to the Fit Model launch window.

2. Select the Repeated Construction tab.

3. Select Row and Column and click Repeated.

4. Select Spatial from the Construction list.

5. Select Power from the Type listing.

half dozen. Click Run.

Note the AICc and BIC values for the model fit.

seven. Return to the Fit Model launch window.

viii. Select Exponential from the Type list.

9. Click Run.

Notation the AICc and BIC values for the model fit.

10. Return to the Fit Model launch window.

11. Select Gaussian from the Blazon list.

12. Click Run.

Annotation the AICc and BIC values for the model fit.

The observed AICc values for these types and the other fits that yous performed are summarized in Table 8.two.

Tabular array viii.2 Fit Statistics for Spatial Models Fit

Structure	Type	AICc	BIC
Spatial	Spherical	234.08	240.16
Residue		258.41	262.53
Spatial with Nugget	Spherical	236.36	244.31
Spatial Anisotropic	Spherical	240.54	248.50
Spatial	Power	240.24	246.32
Spatial	Exponential	240.24	246.32
Spatial	Gaussian	238.37	244.44

The best fitting model (using the smallest AICc value) is the Spatial construction model with Spherical covariance construction. Now y'all will compare this model to the consummate and incomplete block models to complete the objectives of the uniformity trial.

Compare the Model to Block Designs

i. Return to the Uniformity Trial data table.

two. In the Tables panel, run the Graph Builder script.

Figure 8.37 Graph Builder Plot of Proposed Complete and Incomplete Block Designs

This plot shows the proposed complete and incomplete block designs for the field. The color indicates the quarter fields that would serve as consummate blocks. The numbered points correspond the sub-quarter fields that would serve every bit incomplete blocks.

To fit the complete block model, follow these steps:

1. Return to the Fit Model launch window.

2. Select Repeated Structure tab.

3. Select Residuum from the Construction list.

4. Remove Row and Cavalcade from the effect.

Otherwise, a pop-upward window appears, stating, "Repeated columns and subject columns are ignored when the Residue covariance construction is selected."

5. Select the Random Effects tab.

six. Select Quarter and click Add.

7. Click Run.

To fit the incomplete block model, follow these steps:

i. Return to the Fit Model launch window.

2. Select the Random Effects tab.

3. Select Quarter and click Remove.

4. Select Subquarter and click Add together.

v. Click Run.

The post-obit list shows both AICc and BIC values for the competing models. The spherical covariance structure results in the best model fit. This indicates that, for future studies using this field, a completely randomized blueprint with spatially correlated errors is preferred.

• Spherical model (see Determine the Type of the Spatial Structure)

– AICc: 234.08

– BICc: 240.sixteen

• Complete cake (RCBD) model

– AICc: 259.90

– BICc: 265.97

• Incomplete block model

– AICc: 248.77

– BICc: 254.85

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