Spline Model Options

Automatic REML 2D Spline Options

This menu provides facilities for adding two-dimensional spline surfaces to the models by fitted the automatic REML menus: Automatic Analysis of Row-Column Design, Automatic Analysis of Incomplete-Block Design and Automatic Analysis of Series of Trials. Selecting the Try 2d-spline surface across rows and columns in their Options dialogs and clicking the Specify spline models button will open this dialog. The automatic REML modelling will then try all the specified models and will select one of these if it is the optimal model.

The spline surfaces are 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). Three commonly used models are initialized by default. These are the LVIS model recommended by Piepho et al. (2022), the SpATS model of Rodríguez-Álvarez et al. (2018) and a thin-plate spline. Further models can be added to the list of splines to try by entering a new model specification and Description, and then clicking the Add button. (At most 20 models can be defined.) The order of fitting can be changed using the Up and Down buttons. The settings of the currently selected model can be changed with the Update button. Models can be removed with the Remove button, but at least one model must be retained. The Knots to be used in a basis function apply to all models. For a large grid of plots, you may want to reduce the number of knots by specifying the number, or supplying a variate of knot positions for even more control.

Spline Model

This gives a list of the spline models currently defined. Clicking on a model in the list will put its settings in the items below. You can change the settings and modify the model by clicking Update button, or click Add to add it as a new model.

The 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 – 3X_i-1 + 3X_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

Include linear plane (p-spline with first differences 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.

Constrain spline variance components to be positive

If this is selected, the variance components associated with the spline models will be constrained to be positive.

Description

A description of the model to be used in the REML output. This must be unique for each model.

Generate description

If this clicked, a unique default description for the currently selected spline settings will be put into the Description field.

Knots to be used in a basis function

This allows the knots to be used in the spline models to be specified. The Default setting uses all the unique row and column values for knots. If the grid is large, you may wish to reduce the number of knots by selecting the Number of knots setting and specifying the number of knots in the Rows (Y) and Columns (X) directions. You can gain even more control by selecting the Specify knot positions setting, and providing two variates to define the positions. These variates can be entered by double-clicking variates in the Available data list. Alternatively, you can enter a numerical list of values in the position fields.

Available data

This lists variates that can be entered in the Knot position fields. Double-click on a name to copy it to the current input field, or type the name.

Action buttons

OK	Save the specified models.
Cancel	Close the menu without further changes.
User defaults	Resets the models to those set up when the menu was first opened.
Genstat defaults	Resets the models to the three common spline models.
Remove	Remove the selected model.
Up	Move the selected model up in the list.
Down	Move the selected model down in the list.
Update	Update the selected model.
Add	Add the currently selected options as a new model.

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