Hierarchical Spatial Process Models for Multiple Traits in Large Genetic Trials

Abstract

This article expands upon recent interest in Bayesian hierarchical models in quantitative genetics by developing spatial process models for inference on additive and dominance genetic variance within the context of large spatially referenced trial datasets of multiple traits of interest. Direct application of such multivariate models to large spatial datasets is often computationally infeasible because of cubic order matrix algorithms involved in estimation. The situation is even worse in Markov chain Monte Carlo (MCMC) contexts where such computations are performed for several thousand iterations. Here, we discuss approaches that help obviate these hurdles without sacrificing the richness in modeling. For genetic effects, we demonstrate how an initial spectral decomposition of the relationship matrices negates the expensive matrix inversions required in previously proposed MCMC methods. For spatial effects we discuss a multivariate predictive process that reduces the computational burden by projecting the original process onto a subspace generated by realizations of the original process at a specified set of locations (or knots). We illustrate the proposed methods using a synthetic dataset with multivariate additive and dominant genetic effects and anisotropic spatial residuals, and a large dataset from a scots pine (Pinus sylvestris L.) progeny study conducted in northern Sweden. Our approaches enable us to provide a comprehensive analysis of this large trial which amply demonstrates that, in addition to violating basic assumptions of the linear model, ignoring spatial effects can result in downwardly biased measures of heritability.

Figure 1

The top row shows estimates of univariate model’s spatial variance σ² (left) and residual variance τ² (right) given increasing number of knots. The second row provides corresponding estimates from the bias-adjusted predictive process model.

Figure 2

The left column shows the interpolated surfaces of the three response variables over the 850 synthetic data points (overlaid on top left surface). The right column shows the associated estimates of the random spatial effects based on 80 knots (overlaid on the top right surface).

Figure 3

First row: Each point represents a scots pine tree location. The left column shows interpolated surface of nonspatial model residuals for stem height (H), diameter (D), and branch angle (A), respectively from top to bottom. The middle and right columns show associated estimated surfaces of spatial random effects for the 78 and 160 knot intensities modeled with the stationary isotropic Matérn correlation function. Note that to reduce clutter, the knots are overlaid on only the 78 knot H surface.

Figure 4

Comparative ranking of the additive genetic effects for scots pine height (left), diameter (middle), and branch angle (right) from the nonspatial and 160 knot predictive process models. The x-axis plots the nonspatial model estimated position of the 100 highest ranked individuals and the y-axis plots the corresponding ranking from the spatial model. The straight line indicates no discrepancy in rank between the two models.