Parameter estimation

Parameters were inferred using a Markov chain Monte Carlo (MCMC) method. The prior distribution of the additive genetic effects was a multivariate normal distribution MVN(0, G ⊗ K), where G is the genetic variance-covariance matrix and K is the genomic relationship matrix calculated from the genome-wide markers using the A.mat function of the R package rrBLUP [32, 33]. The prior distribution of the residuals was MVN(0, R ⊗ I), where R is the residual variance-covariance matrix and I is an identity matrix with the size of the number of cultivars. The prior distributions of G and R were inverse Wishart distributions with four degrees of freedom and with scale matrices that were half of the phenotypic variance-covariance matrix, which resulted in the expectations for these distributions being the same as the scale matrices. We assigned non-informative prior distributions to the overall means and major gene effects. The prior distributions of the weights of the basis functions were as follows

where N denotes the normal distribution and σp2 is the variance. The prior distribution of P_m for m ≥2 is equivalent to assuming (Pm−Pm−1)−(Pm−1−Pm−2)~N(0,σp2). The prior distribution of σp2 is a non-informative scaled inverse-chi-square distribution. The weights were inferred using a Metropolis update procedure, which is illustrated in the Appendix. The other parameters were inferred using Gibbs sampling, treating the phenotypic values of CL adjusted for the influence of DH (i.e., y_{i, CL}−L(y_{i, DH})y_{i, DH}) as the dependent variable [34]. The number of iterations was 1.1 × 10⁶ with the first 0.1 × 10⁶ iterations discarded as burnin, and the sampling interval was 100. The prior distributions of the OLM were the same as those of the corresponding parameters of the NSE. The parameters of the OLM were inferred using Gibbs sampling with the MCMC conditions the same as those for the NSE. Major gene effects were judged to be non-significant if 0 lay within the 0.025 and 0.975 quantiles of the MCMC samples. The calculation was performed using a program written in C language.