2D Spline Analysis of Row-Column Design

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 – 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

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