Resolving the ambiguity of random‐effects models with singular precision matrix

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.

Linear Variance, P-splines and Neighbour Differences for Spatial Adjustment in Field Trials: How are they Related?

Nearest-neighbour methods based on first differences are an approach to spatial analysis of field trials with a long history, going back to the early work by Papadakis first published in 1937. These methods are closely related to a geostatistical model that assumes spatial covariance to be a linear function of distance. Recently, P-splines have been proposed as a flexible alternative to spatial analysis of field trials. On the surface, P-splines may appear like a completely new type of method, but closer scrutiny reveals intimate ties with earlier proposals based on first differences and the linear variance model. This paper studies these relations in detail, first focussing on one-dimensional spatial models and then extending to the two-dimensional case. Two yield trial datasets serve to illustrate the methods and their equivalence relations. Parsimonious linear variance and random walk models are suggested as a good point of departure for exploring possible improvements of model fit via the flexible P-spline framework.