Latent variable models (LVMs), including the special case of factor analysis when the responses are conditionally normally distributed, are gaining traction in many scientific fields owing to both their statistical and computational advantages as a means of dimension reduction, and their attractive interpretation with the latent variables representing unmeasured predictors and factor loadings corresponding to the coefficients. Model selection for LVMs has an additional striking twist: as the name suggests, the latent variables are unobserved and have to be estimated from the data. Therefore, we need to select both the order and the structure of the factor loadings, where the former involves choosing the number of latent variables. I introduce a method for order selection in LVMs known as the Ordered FActor Lasso (OFAL, Hui et al., 2018), which utilises penalised likelihood methods to encourage both element-wise and group-wise sparsity in the loadings. Specifically, we show how the OFAL penalty exploits both the grouped and hierarchical nature of the loadings, thus providing a natural approach to order selection, while also circumventing the issue of identifiability without the use of an arbitrary constraint and offering the potential for easier interpretability of the factor loadings. Additionally, I will discuss a computational algorithm for calculating the OFAL estimates based on a convenient reparameterisation of the penalty.
Invited Session “Dimension Reduction” organised by Kei Hirose.