Bayesian factor analytic model: An approach in multiple environment trials

Abstract

One of the main challenges in plant breeding programs is the efficient quantification of the genotype-by-environment interaction (GEI). The presence of significant GEI may create difficulties for breeders in the selection and recommendation of superior genotypes for a wide environmental network. Among the diverse statistical procedures developed for this purpose, we highlight those based on mixed models and factor analysis that are called factor analytic (FA) models. However, some inferential issues are related to the factor analytic model, such as Heywood cases that make the model non-identifiable. Moreover, the representation of the loads and factors in the conventional biplot does not involve any measurement of uncertainty. In this work, we propose dealing with the FA model using the Bayesian framework with direct sampling of factor loadings via spectral decomposition; this guarantees identifiability in the estimation process and eliminates the need for the rotationality of factor loadings or imposition of any ad hoc constraints. We used simulated and real data to illustrate the method’s application in multi-environment trials (MET) and to compare it with traditional FA mixed models on controlled unbalancing. In general, the Bayesian FA model was robust under different simulated unbalanced levels, presenting the superior predictive ability of missing data when compared to competing models, such as those based on FA mixed models. In addition, for some scenarios, the classical FA mixed model failed in estimating the full FA model, illustrating the parametric problems of convergence in these models. Our results suggest that Bayesian factorial models might be successfully used in plant breeding for MET analysis.

Shrinkage in the Bayesian analysis of the GGE model: A case study with simulation

“…The predictive ability of the models was quantified by the average predicted residual error sum of squares (PRESS) and phenotypic correlation between the predicted ( ) and the observed ( y ij ) values (COR). As described in Nuvunga et al [ 64 ], PRESS and COR are calculated by and where is the mean of the values predicted by the model, is the mean of the values predicted for validation, and n is the number of data removed. As per the PRESS criterion, the lowest value indicates better performance, while in the case of the COR criterion, better performance is indicated by the highest value.…”

Bayesian GGE model for heteroscedastic multienvironmental trials

“…Nuvunga et al. (2019) showed that under the Bayesian approach, full‐dimension models can be considered in the FA analysis without requiring axis rotation. Another advantage is that the estimates are restricted to the parametric space.…”

“…In these models, it is unclear how parametric confidence regions can be constructed for biplot points, and approximate alternatives have been proposed, as highlighted by Crossa et al (2011). Some of these problems were overcome by Nuvunga et al (2019), who applied Bayesian FA.…”

“…Conversely, the Bayesian method has been shown to be promising for estimating parameters related to stability and genotypic adaptability in MET data. Nuvunga et al (2019) showed that under the Bayesian approach, full-dimension models can be considered in the FA analysis without requiring axis rotation. Another advantage is that the estimates are restricted to the parametric space.…”

“…Such an inference is not feasible in fixed effects models and is not usually presented in FA models. The Bayesian version of the FA-SERG (Bayesian Analytical Factor model) by Nuvunga et al (2019), similar to our proposal, is also flexible and allows the incorporation of bivariate credible regions in the biplot. However, the authors themselves recommend caution regarding the interpretation, which must be performed separately for genotypes and environments.…”