Breeding analytics: Using quantitative genetics to drive product development

Alencar Xavier

Quantitative Geneticist, Corteva Agrisciences
Adjunct Professor at Purdue University
http://alenxav.wixsite.com/home/

Outline

Part 1: Overview

Introduction
Breeding analytics, impact, history
General framework

Part 2: Operations

Routine analytics
Data visualization
Exploratory analysis

Part 1 - Overview

Introduction

Analytics help us to work smart and to achieve our goals more efficiently.
Empower breeders to understand the data nuances, strengths and pitfalls.

Breeding analytics

Data visualization: Quality control of locations, experiments, traits
Phenotypic analysis: BLUPs, repeatability, spatial, product placement
Genomic analysis: Predictions, crosses-combinations, diversity, QTLs
Phenomic analysis: Automation, multi-trait analysis, quality control
Envirotyping analysis: Environmental characterization, GxE predictions
Optimization analysis: Exp. Designs, resource allocation (Reps vs Locs)

Impact of breeding analytics

Where is the main impact of breeding analytics?

ADD VALUE: Optimize the breeding pipeline
INTUITUVE DECISIONS: Descriptive + Predictive + Prescriptive
EMPOWERMENT: Enhance understanding and free breeder’s time

Analytics must be (1) simple, (2) useful and (3) interpretable

History of breeding analytics

Francis Galton (1886): Regression and heritability
Ronald Fisher (1918): Infinitesimal model, P = G + E
Sewall Wright - (1921,1922): Inheritance and genetic relationship
Charles Henderson (1968): Modeling genetics with relationship

General framework

Why are mixed models so popular?

The primary approach for breeding analytics involves MIXED MODELS:

Simple to compute, robust, and well-known statistical properties
Heritability and repeatability computed from variance components
BLUPs are utilized for selection and advancement
Easily accommodate known sources of variation (spatial variation)
Enable the prediction of unobserved individuals and GxE patterns
Covariances used for genetic correlations and selection indexes
Whole selection theory is based on truncating normal distributions
Statistical inference (detect QTLs)

Common model setup

Mixed model notation

Linear model: \(y=Xb+Zu+e\)
Genetic variance: \(V(u)=G=A\sigma^2_a\)
Residual variance: \(V(e)=R=I\sigma^2_e\)
Henderson’s equation (\(Cg=r\))

\[ \left[\begin{array}{rr}X'R^{-1} X & Z'R^{-1}X \\X'R^{-1}Z & Z'R^{-1}Z+G^{-1}\end{array}\right] \left[\begin{array}{r} b \\ u \end{array}\right] =\left[\begin{array}{r} X'R^{-1}y \\ Z'R^{-1}y \end{array}\right] \]

We know (data): \(x=\{y,X,Z,A\}\)
We want (parameters): \(\theta=\{b,u,\sigma^{2}_{a},\sigma^{2}_{e}\}\)
Parameter estimation based on Gaussian likelihood: \(L(x|\theta)\)

Mixed model solutions

Variance components (\(\partial L / \partial \sigma^2_i\))

\(\sigma^2_a = \frac{\hat{u}' A^{-1}\hat{u} + C^{22}}{q} \approx \frac{y'S'Z\hat{u}}{tr(SZ'AZ)}\)
\(\sigma^2_e = \frac{y'\hat{e}}{n-rank(X)}\)

Key summary statistics

\(h^2 = \frac{V(g)}{V(y)} = V^{-1}G\)
\(sd(\hat{g}) = \sqrt{Diag(G-C^{22})}\)
\(Acc = Cor(\hat{g},g) = \frac{Cov(\hat{g},g)}{\sqrt{V(\hat{g})V(g)} } = \sqrt{\frac{GV^{-1}G}{G}}\)

Multivariate mixed models

Mixed models also enable us to evaluate multiple traits

\[ y=\{y_1,y_2,...,y_k\} \]

With multiple traits, the relation among traits is modeled

\[ V(u) = A \otimes \Sigma_a = \left[\begin{array}{rr} A \sigma^2_{a_1} & A \sigma_{a_{12}} \\ A \sigma_{a_{21}} & A \sigma^2_{a_2} \end{array}\right] \] \[ V(e) = I \otimes \Sigma_e = \left[\begin{array}{rr} I\sigma^2_{e_1} & I\sigma_{e_1e_2} \\ I\sigma_{e_2e_1} & I\sigma^2_{e_2} \end{array}\right] \] Why does it matter? Covariances (\(\sigma_{a_{12}}\), \(\sigma_{e_{12}}\)) are extra information!!