Varietal selection for perennial crops where data relate to multiple harvests from a series of field trials

Abstract

Varietal selection for yield from a series of multi-environment trials can be regarded as a multi-trait selection problem in which the yields in different environments are synonymous with traits. As such an analysis of the data combined across environments should be conducted in order to form an index for selection. Analytical methods that include appropriate models for both the genetic variance structure (that is, the variances and covariances of genotype effects from different environments) and the residual variance structure (which typically comprises spatial covariance models for each trial) have been published previously. In the case of perennial crops, yields are often obtained from multiple harvests which implies that the data comprise short sequences of repeated measurements. Varietal performance in individual harvests is important for selection so that a combined analysis across both trials and harvests is required. The repeated measures nature of the data provides additional modelling challenges. In this paper we propose an approach for the analysis of multi-environment, multi-harvest data that accommodates the major sources of variation and correlation (including temporal). The approach is illustrated using two examples from sugarcane breeding programmes. The proposed models were found to provide a superior fit to the data and thence more accurate selection decisions than the common practice of conducting separate analyses of individual trials and harvests.

Appendix Kronecker products

Definition

Let ${\user2{A} = \{a_{ij}\}}$ be an m  ×  n matrix and ${\user2{B} = \{b_{kl}\}}$ be a p  ×  q matrix. Then the Kronecker product of ${\user2{A}}$ and ${\user2{B}}$, denoted ${\user2{A} \otimes \user2{B}}$, is given by the mp  ×  nq matrix

$$ \left[ {\begin{array}{llll} a_{11} \user2{B} & a_{12} \user2{B} & \ldots & a_{1n} \user2{B} \\ a_{21}\user2{B} & a_{22}\user2{B} & \ldots & a_{2n}\user2{B} \\ \cdot & \cdot & & \cdot \\ \cdot & \cdot & & \cdot \\ \cdot & \cdot & & \cdot \\ a_{m1} \user2{B} & a_{m2}\user2{B} & \ldots & a_{mn}\user2{B} \end{array}}\right] $$

Worked example

We consider the form for the genetic variance model given in Eq. 5 and assume there are h  =  2 harvests and t  =  3 trials with the component matrices ${\user2{G}_{\user2 h}}$ and ${\user2{G}_{\user2 t}}$ given by

$$ \user2{G}_{\user2 h} = \left[{ \begin{array}{ll} a_{11} &a_{12} \\ a_{21} & a_{22}\end{array}}\right] \hbox{and}\, \user2{G}_{\user2 t} = \left[{\begin{array}{lll} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} &b_{23} \\ b_{31} & b_{32} & b_{33} \end{array}}\right] $$

Then ${\user2{G}_{\user2 h} \otimes \user2{G}_{\user2 t}}$ is the 6  ×  6 matrix given by

$$ \left[ {\begin{array}{llllll} a_{11}b_{11} & a_{11}b_{12} & a_{11}b_{13} & a_{12}b_{11} & a_{12}b_{12} & a_{12}b_{13} \\ a_{11}b_{21} & a_{11}b_{22} & a_{11}b_{23} & a_{12}b_{21} & a_{12}b_{22} & a_{12}b_{23} \\ a_{11}b_{31} & a_{11}b_{32} & a_{11}b_{33} & a_{12}b_{31} & a_{12}b_{32} & a_{12}b_{33} \\ a_{21}b_{11} & a_{21}b_{12} & a_{21}b_{13} & a_{22}b_{11} & a_{22}b_{12} & a_{22}b_{13} \\ a_{21}b_{21} & a_{21}b_{22} & a_{21}b_{23} & a_{22}b_{21} & a_{22}b_{22} & a_{22}b_{23} \\ a_{21}b_{31} & a_{21}b_{32} & a_{21}b_{33} & a_{22}b_{31} & a_{22}b_{32} & a_{22}b_{33} \end{array}}\right] $$