Solution Path Clustering for Fixed-Effects Models in a Latent Variable Context

The main drawback of estimating latent variable models with fixed effects is the direct dependence between the number of free parameters and the number of observations. We propose to apply a well suited penalization technique in order to regularize the parameter estimates. In particular, we promote sparsity based on the pairwise differences of subject-specific parameters, inducing the latter to shrink on each other. This method allows to group statistical units into clusters that are homogeneous with respect to a latent attribute, without the need to specify any distributional assumption, and without adopting random effects. In practice, applying the proposed penalization, the number of free parameters is reduced and the adopted model becomes more parsimonious. The estimation of the fixed effects is based on an algorithm that builds a solution path, in the form of a hierarchical aggregation tree, whose outcome depends on a single tuning parameter. The method is intended to be general, and in principle it can be applied on the likelihood of any latent variable model with fixed effects. We describe in detail its application to the Rasch model, for which we provide a real data example and a simulation study. We then extend the method to the case of a latent variable model for continuous data, where the number of fixed effects to be estimated is higher.