This menu provides facilities for the analysis of a row-column design fitting a spatial trend with a two-dimensional spline surface using the V2DSPLINE procedure. The spline surface is made up of groups of basis functions modelling row, column and row × columns trends. Variance components for each group are estimated using the method of residual maximum likelihood (REML). The data should be supplied in a single variate with factors or variates to specify the rows and columns of the grid of plots. The grid does not need to be a complete rectangular array. An optional factor can be provided to specify the replicates in the trial. In a variety trial, the entries (cultivars, varieties, genotypes) can be set as either a fixed model term if BLUEs are required or as a random term if BLUPs are required. Other covariates and model terms can be added to the Fixed model or Additional random terms fields. A choice of the basis functions making up the spline surface is provided in the Spline model section. The default model (Spline basis=p-spline, Smoothness=1 and Degree=1) is the **LVIS** model recommended by Piepho et *al.* (2022).

## Available data

This lists data structures appropriate to the current input field. The contents will change as you move from one field to the next. Double-click on a name to copy it to the current input field or type the name.

## Y-variate

Specifies a variate containing the data values.

## Replicate factor

An optional factor specifying the blocks containing replicates in the trial.

## Rows

A factor or variate specifying the rows in the design.

## Columns

A factor or variate specifying the columns in the design.

## Fixed model

The fixed model terms describe imposed treatment factors and covariates for which the effect of specified levels or values are of interest. The fixed model is described using a formula, which can combine main effects and interactions of factors and also covariates.

## Additional random terms

The random model is generally used to describe those factors for which the values present in an experiment can be considered drawn from some large homogeneous population. This specifies terms extra to the spatial layout defined by the replicates, rows and columns. The additional terms are described using a formula, which can combine main effects and interactions of factors and also covariates.

## Spline Model

These options allow you to select the basis functions used to fit the spline surface. Using a Smoothness of 1 and Degree of 1 gives the **LVIS** model of Piepho et *al.* (2022). Using a Smoothness of 2 and Degree of 3 gives the **SpATS** model of Rodríguez-Álvarez et *al.* (2018).

## Spline Basis

This selects the form of basis functions used in the splines.

P-spline | P-spline generated using the TENSORSPLINE procedure. |

Thin-plate spline | Thin-plate spline generated using the THINPLATE procedure. |

The P-spline has a choice of Smoothness, Degree and Penalty that can be used in constructing the basis functions.

## Smoothness (p-spline only)

This controls the order of differencing used in constructing the p-spline. High orders create smoother basis functions.

1 – First differences | First differences between plots will be modelled (X_{i} – X_{i-1}). |

2 – Second differences | Second differences between plots will be modelled (X_{i} – 2*X_{i-1} + X_{i-2}). |

3 – Third differences | Third differences between plots will be modelled (X_{i} – 3*X_{i-1} + 3*X_{i-2} – X_{i-3}). |

## Degree of basis (p-spline only)

1 – Linear | The basis functions are piecewise linear functions |

2 – Quadratic | The basis functions are piecewise quadratic functions |

3 – Cubic | The basis functions are piecewise cubic functions |

## Penalty (p-spline only)

The model fitted to the row and column margins and the row-column interaction can be reduced to simplify the model, which makes it faster to fit.

Unconstrained | The full marginal model is fitted |

Semiconstrained | The marginal model for rows and row by linear column interaction are combined to have just one variance component, and likewise for the marginal column model |

Isotropic | The model is reduced to a single group of basis functions with a common variance component |

The Isotropic penalty is not available for the **LVIS** model.

## Include linear plane (p-spline only)

The **LVIS** model (p-spline basis, smoothness 1, degree 1), does not allow for a linear trend over the grid. Selecting this option will add a linear plane to the fixed effects in the model.

## Operators

This provides a quick way of entering operators in the fixed and random model formulas. Double-click on the required symbol to copy it to the current input field. You can also type in operators directly. See model formula for a description of each.

## Action buttons

Run | Run the analysis. |

Cancel | Close the menu without further changes. |

Options | Opens a dialog where additional options and settings can be specified for the analysis. |

Defaults | Set the menu settings back to the default settings. Clicking the right mouse on this button produces a shortcut menu where you can choose to set the options using the currently stored defaults or the Genstat default settings. |

Save | Opens a dialog where you can save results from the analysis. |

Predict | Allows you form predictions based on the current model. |

Further output | Opens a dialog for specifying further output from the analysis and displaying residual and means plots. |

## References

- Durbán, M., Hackett, C.A., NcNicol, J.W., Thomas, T.B. & Currie, I.D. (2003). The practical use of semiparametric models in field trials.
*JABES*, 8, 48–66. https://doi.org/10.1198/1085711031265 - Piepho, H.P., Boer, M.P. & Williams, E.R. (2022). Two-dimensional P-spline smoothing for spatial analysis of plant breeding trials.
*Biometrical Journal*, 1–23. https://doi.org/10.1002/bimj.202100212 - Rodríguez-Álvarez, M.X., Boer, M.P., van Eeuwijk, F.A. & Eilers, P.H. (2018). Correcting for spatial heterogeneity in plant breeding experiments with P-splines.
*Spatial Statistics*, 23, 52–71. https://doi.org/10.1016/j.spasta.2017.10.003

## See also

- Options for specifying output options.
- Further Output for additional output after fitting a model.
- Save for saving the results from the analysis.
- Predict for forming and plotting predictions.
- Automatic Analysis of Row-Column Design menu.
- Automatic Analysis of Incomplete-Block Design menu.
- Automatic Analysis of Series of Trials menu.
- Generate a Doubly Resolvable Row-Column Designs menu.
- V2DSPLINE procedure fitting the 2-dimensional spline model.
- TENSORSPLINE procedure.
- THINPLATE procedure.
- VAROWCOLUMNDESIGN procedure for automatically fitting an autocorrelation type spatial model.
- AFRCRESOLVABLE procedure for generating designs approximately optimal under this analysis.
- CDNROWCOLUMNDESIGN procedure for generating efficient row-column designs.
- VCHECK procedure to check the residuals.
- VKEEP directive to save results from an REML analysis.
- VAIC procedure to calculate information coefficients.
- VLSD procedure to calculate least significant differences.