Bayesian Additive Main Effects and Multiplicative Interaction Models using Tensor Regression for Multi-environmental Trials

TL;DR

This work introduces BAMMIT, a Bayesian Additive Main effects and Multiplicative Interaction Tensor model that generalizes AMMI to multiple categorical factors via tensor regression. It combines a hierarchical Bayesian framework with identifiability-enforcing transformations and a spike-and-slab prior to select relevant interactions, enabling robust prediction in complex multi-environment trials. Across simulated scenarios and a real wheat-yield case study (Ireland, 2010–2019), BAMMIT outperforms competitive methods (including RF, XGB, AMMI, AMBARTI, and Bayesian factorial models) and provides interpretable, uncertainty-aware visuals of interactions through heatmaps and VSUP plots. The approach supports imputing missing combinations and offers practical insights for selecting high-performing genotype-environment-year combinations, with potential extensions to continuous variables and more sophisticated rank selection to further enhance scalability and applicability.

Abstract

We propose a Bayesian tensor regression model to accommodate the effect of multiple factors on phenotype prediction. We adopt a set of prior distributions that resolve identifiability issues that may arise between the parameters in the model. Further, we incorporate a spike-and-slab structure that identifies which interactions are relevant for inclusion in the linear predictor, even when they form a subset of the available variables. Simulation experiments show that our method outperforms previous related models and machine learning algorithms under different sample sizes and degrees of complexity. We further explore the applicability of our model by analysing real-world data related to wheat production across Ireland from 2010 to 2019. Our model performs competitively and overcomes key limitations found in other analogous approaches. Finally, we adapt a set of visualisations for the posterior distribution of the tensor effects that facilitate the identification of optimal interactions between the tensor variables, whilst accounting for the uncertainty in the posterior distribution.

Figures (9)

• Figure 1: In-sample scatterplots of true versus estimated interaction term for simulation scenario (ii) part (a), setting $Q_{\text{sim}} = 1$ and $\lambda = 10$. The left panel shows only the estimated interaction effects whilst the right panel shows the full estimated fitted values. The models were fitted with $Q = 1$. The blue points represent the posterior median and the grey bars represent the $95\%$ credible intervals.
• Figure 2: Scatterplots of true versus estimated interaction term for simulation scenario (ii) and Equation (\ref{['int_3_way']}), setting $Q_{1, \text{sim}} = Q_{2, \text{sim}} = 1$ and $\lambda = 10$. The model was fitted with $Q = \{1,2,3,4\}$. The blue points represent the posterior median and the grey bars represent the $95\%$ credible intervals.
• Figure 3: Scatterplots of true versus estimated additive terms for simulation scenario (iii), setting $Q_{\text{sim}} = 2$, $\lambda = \{8,10\}$. The blue points represent the posterior median and the grey bars represent the $95\%$ credible intervals.
• Figure 4: Scatterplots of true versus estimated interaction terms for simulations scenarios (i), (ii) and (iii) setting $Q_{\text{sim}} = 1$ and $\lambda = \{10\}$. The interactions generated were 2-way ($V = 2$), 3-way ( $V = 3$) and 4-way ($V = 4$). The blue points represent the posterior median and the grey bars represent the $95\%$ credible intervals.
• Figure 5: Estimated values of $\hat{p}^{(v)}_q$ and the 95% credible intervals. Values closer to zero indicate an increasing probability that the variable was included in the interaction term. The general low values indicate a high degree of interaction, with environment being particularly important. We note that the uncertainty ranges in these values are far smaller than that of the $Be(1, 10)$ prior.