Bayesian estimation of a surface to account for a spatial trend using penalized splines in an individual-tree mixed model

Unaccounted for spatial variability leads to bias in estimating genetic parameters and predicting breeding values from forest genetic trials. Previous attempts to account for large-scale continuous spatial variation employed spatial coordinates in the direction of the rows (or columns). In this research, we use an individual-tree mixed model and the tensor product of B-spline bases with a proper covariance structure for the random knot effects to account for spatial variability. Dispersion parameters were estimated using Bayesian techniques via Gibbs sampling. The procedure is illustrated with data from a progeny trial of Eucalyptus globulus subsp. globulus Labill. Four different models were used in the sequel. The first model included block effects and the three other models included a surface on a grid of either 8 x 8, 12 x 12, or 18 x IS knots. The three models with B-splines displayed a sizeable lower value of the deviance information criterion than the model with blocks. Also, the mixed models fitting a surface displayed a consistent reduction in the posterior mean of sigma(2)e, an increase in the posterior means of sigma(2)(A) and h(DBM)(2), and an increase of 66% (for parents) or 60% (for offspring) in the accuracy of breeding values.