Walking through the statistical black boxes of plant breeding

Abstract

Key message

The main statistical procedures in plant breeding are based on Gaussian process and can be computed through mixed linear models.

Abstract

Intelligent decision making relies on our ability to extract useful information from data to help us achieve our goals more efficiently. Many plant breeders and geneticists perform statistical analyses without understanding the underlying assumptions of the methods or their strengths and pitfalls. In other words, they treat these statistical methods (software and programs) like black boxes. Black boxes represent complex pieces of machinery with contents that are not fully understood by the user. The user sees the inputs and outputs without knowing how the outputs are generated. By providing a general background on statistical methodologies, this review aims (1) to introduce basic concepts of machine learning and its applications to plant breeding; (2) to link classical selection theory to current statistical approaches; (3) to show how to solve mixed models and extend their application to pedigree-based and genomic-based prediction; and (4) to clarify how the algorithms of genome-wide association studies work, including their assumptions and limitations.

Appendix: Numerical example of design matrices

Suppose that a breeding program is conducting a test with a three-way hybrid (${\text{A}} \times {\text{B}} \times {\text{C}}$) to find out the narrow-sense heritability of the trait of interest. The only genetic information available is a short pedigree that describes the three-way cross, as follows:

This evaluation was conducted in a single environment, growing two replicates of each parent (${\text{A}},{\text{B}},{\text{C}}$) and the final hybrid (${\text{E}}$). Considering that a plot with genotype C was lost during the growing season, the design matrices are given by:

The example above was run using the Gibbs sampling algorithm shown in the manuscript, with the prior suggested here ($\upnu^{*} = 5 \;{\text{and }}{\rm S}^{*} = 0.5 \times {\text{var}}\left( {\mathbf{y}} \right) = 5.17$). The outcome was:

$$\textbf{b} = [23.812]\quad \textbf{u} = \left[ {\begin{array}{*{20}c} {1.191} \\ {0.172} \\ { - 1.291} \\ {0.799} \\ { - 0.060} \\ \end{array} } \right]\quad \sigma _{a}^{2} = 4.004\quad \sigma _{e}^{2} = 6.987$$

which yields a narrow-sense heritability of 0.364, and breeding values (u) computed for all genotypes, including the parental line D not grown in the field.