Factor analytic (FA) models are widely used across many disciplines, owing to both the statistical and computational advantages from dimension reduction and the possible ability to interpret the factors. In plant breeding specifically, FA models provide a natural framework for parsimoniously modelling genotype x environment interactions, as often the number of environments can be relatively large compared to the number of genotypes and the elements of the FA model, e.g. specific factors, may have meaningful interpretation for the aim of the experiment, e.g. environmental specific genotype effects are not replicable in other environments and thus may not be of interest in the aim for selection.
An FA model is dictated to a large extent by the number of factors (the order of the model). A higher order leads to more parameters in the model, which necessitates that order selection be used to achieve parsimony while also ensuring that a sufficient number of factors is selected to flexibly model between-environment interactions. We introduce a new method for order selection in FA models known as the Ordered FActor Lasso (OFAL), which induces both element-wise and group-wise sparsity in the loadings. By exploiting both the grouped and hierarchical nature of the loadings, the OFAL penalty provides a natural approach to order selection, while also circumventing the issue of identifiability without use of an arbitrary constraint and offering the potential for easier interpretability of the factor loadings. We compare its performance based on a simulation motivated from a real wheat yield multi-environmental trial.