# Insight into Genome-Wide Associations of Growth Trajectories Using a Hierarchical Non-Linear Mixed Model

## TL;DR

This paper introduces a faster and more powerful method to study how genes influence growth patterns over time using a hierarchical non-linear mixed model.

## Contribution

A novel hierarchical non-linear mixed model approach for genome-wide association studies of growth trajectories that improves computational efficiency and detection power.

## Key findings

- The method outperforms traditional approaches in computational efficiency and statistical power for detecting genetic regions influencing growth.
- Applying the model to mouse data identified genetic loci affecting body weight changes over time more effectively than conventional methods.
- Simulations and real data analysis showed advantages over hierarchical random regression models using Legendre polynomials.

## Abstract

Understanding how body weight changes as animals grow helps scientists identify genes that influence health, development, and productivity. Traditional methods analyze body weight at various ages but are slow and often miss important genetic signals. In this study, we developed a faster and more powerful approach that summarizes each animal’s growth pattern using simple biological growth curves instead of many individual measurements. We then linked these growth patterns to genetic markers across the whole genome. Applying this method to a large mouse population, we identified genetic regions that influence how body weight changes over time with much higher computational efficiency and improved detection ability compared with conventional methods. This approach provides a practical way to study genes controlling growth and other time-dependent traits, and can be broadly applied in animal breeding, biomedical research, and studies of development and disease.

In applying a hierarchical mixed model to genome-wide association analysis (GWAS) of longitudinal data, dimensionality reduction through modeling repeated measurements improves both computational efficiency and statistical power. Legendre polynomials can flexibly fit population growth trajectories, but higher orders substantially increase computational complexity. Instead of using Legendre polynomials, we first estimated fewer individual-specific parameters using biologically meaningful non-linear models and then associated these phenotypic regressions with genetic markers using a multivariate linear mixed model (mvLMM). After performing a canonical transformation of the regressions based on the pre-estimated covariance matrices under the null genomic mvLMM, we decomposed the mvLMM into mutually independent univariate models and incorporated EMMAX to enable rapid genome-wide mixed-model associations for each transformed phenotype. Simulations for longitudinal association analysis in maize and GWAS for the growth trajectories of body weights in mice demonstrated the advantages of hierarchical non-linear mixed models in computing efficiency and statistical power for detecting quantitative trait loci (QTL), compared with mvLMM for multiple growth points and the hierarchical random regression model using Legendre polynomials as sub-models.