Two-dimensional P-spline smoothing for spatial analysis of plant breeding trials

Large agricultural field trials may display irregular spatial trends that cannot be fully captured by a purely randomization-based analysis. For this reason, paralleling the development of analysis-of-variance procedures for randomized field trials, there is a long history of spatial modeling for field trials, starting with the early work of Papadakis on nearest neighbor analysis, which can be cast in terms of first or second differences among neighboring plot values. This kind of spatial modeling is amenable to a natural extension using splines, as has been demonstrated in recent publications in the field. Here, we consider the P-spline framework, focusing on model options that are easy to implement in linear mixed model packages. Two examples serve to illustrate and evaluate the methods. A key conclusion is that first differences are rather competitive with second differences. A further key observation is that second differences require special attention regarding the representation of the null space of the smooth terms for spatial interaction, and that an unstructured variance-covariance structure is required to ensure invariance to translation and rotation of eigenvectors associated with that null space. We develop a strategy that permits fitting this model with ease, but the approach is more demanding than that needed for fitting models using first differences. Hence, even though in other areas, second differences are very commonly used in the application of P-splines, our conclusion is that with field trials, first differences have advantages for routine use.