# Nonlinear genomic selection index accelerates multi-trait crop improvement

## TL;DR

This paper introduces a new method for crop improvement that uses nonlinear relationships between traits to speed up breeding.

## Contribution

The paper introduces the Quadratic Genomic Selection Index (QGSI) to capture nonlinear trait relationships in genomic selection.

## Key findings

- QGSI outperforms linear and quadratic phenotypic and genomic indices in selection response and prediction accuracy.
- QGSI integrates additive, squared, and cross-product terms of genomic estimated breeding values.
- The method was validated using simulated and real maize and wheat datasets.

## Abstract

Linear phenotypic and genomic selection indices assume additivity and linearity, limiting their ability to exploit nonlinear trait relationships. Here, we introduce the Quadratic Genomic Selection Index (QGSI), a genomic extension of the quadratic phenotypic selection index (QPSI) that integrates genomic estimated breeding values (GEBVs) within a unified quadratic framework. QGSI combines additive, squared, and cross-product terms of GEBVs, enabling phenotype-free, rapid-cycle multi-trait selection while capturing genome-wide nonlinear relationships. We evaluate QGSI using two genomic prediction strategies: (i) a maximum-likelihood additive genomic model, and (ii) a nonlinear multi-trait Gaussian kernel model that accommodates epistatic signals. Using 10 simulated maize selection cycles and two real maize and five wheat real datasets, QGSI achieves the highest selection response and the lowest prediction error variance relative to linear and quadratic phenotypic and genomic indices. Thus, combining nonlinear genomic prediction with quadratic selection indices provides a general strategy for accelerating multi-trait crop improvement.

Non-additive effect and trait-trait interactions are often overlooked in current genomic prediction approaches. Here, the authors extend quadratic phenotypic selection index to the genomic domain to enable phenotype-free, rapid-cycle multi-trait selection while capturing genome-wide nonlinear relationships.