ArticleThe plant genome2020
On Hadamard and Kronecker products in covariance structures for genotype × environment interaction.
Article in The plant genome, 2020. The graph could read no effect estimate from its abstract, so it casts no vote on the map. Cited by 13 papers.
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Who cites it
13 citing papers in PubMed.
- Genome-wide detection and genomic selection based on genotype-by-environment interaction-associated signals for yak coat color.BMC genomics · 2026Article
- Discovering putative novel stem rust resistance loci in wheat genetic resources.The plant genome · 2026Article
- When does accounting for gene-environment interactions improve complex trait prediction? A case study with Drosophila lifespan.G3 (Bethesda, Md.) · 2026Article
- When does accounting for gene-environment interactions improve complex trait prediction? A case study withbioRxiv : the preprint server for biology · 2025Article
- Article
- Enhancing wheat genomic prediction by a hybrid kernel approach.Frontiers in plant science · 2025Article
- Genomic prediction of synthetic hexaploid wheat upon tetraploid durum and diploid Aegilops parental pools.The plant genome · 2024Article
- Next-Gen GWAS: full 2D epistatic interaction maps retrieve part of missing heritability and improve phenotypic prediction.Genome biology · 2024Article
- A simulation-based assessment of the efficiency of QTL mapping under environment and genotype x environment interaction effects.PloS one · 2023Article
- Estimation of Complex-Trait Prediction Accuracy from the Different Holo-Omics Interaction Models.Genes · 2022Article
- Improving Selection Efficiency of Crop Breeding With Genomic Prediction Aided Sparse Phenotyping.Frontiers in plant science · 2021Article
- Enviromic Assembly Increases Accuracy and Reduces Costs of the Genomic Prediction for Yield Plasticity in Maize.Frontiers in plant science · 2021Article
- On Hadamard and Kronecker products in covariance structures for genotype × environment interaction.The plant genome · 2020Article
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Authors and funding
4 authors.
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Abstract
When including genotype × environment interactions (G × E) in genomic prediction models, Hadamard or Kronecker products have been used to model the covariance structure of interactions. The relation between these two types of modeling has not been made clear in genomic prediction literature. Here, we demonstrate that a certain model based on a Hadamard formulation and another using the Kronecker product lead to exactly the same statistical model. Moreover, we illustrate how a multiplication of entries of covariance matrices is related to modeling locus × environmental-variable interactions explicitly. Finally, we use a wheat and a maize data set to illustrate that the environmental covariance E can be specified easily, also if no information on environmental variables - such as temperature or precipitation - is available. Given that lines have been tested in different environments, the corresponding environmental covariance can simply be estimated from the training set as phenotypic covariance between environments. To achieve a high level of increase in predictive ability, the environmental covariance has to be defined appropriately and records on the performance of the lines of the test set under different environmental conditions have to be included in the training set.
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