The LASSO on latent indices for regression modeling with ordinal categorical predictors

Many applications of regression models involve ordinal categorical predictors. Two common approaches for handling ordinal predictors are to form a set of dummy variables, or employ a two stage approach where dimension reduction is first applied and then the response is regressed against the predicted latent indices. Both approaches have drawbacks, with the former running into a high-dimensional problem especially if interactions are considered, while the latter separates the prediction of the latent indices from the construction of the regression model. To overcome these challenges, a new approach called the LASSO on Latent Indices (LoLI) for handling ordinal predictors in regression is proposed, which involves jointly constructing latent indices for each or for groups of ordinal predictors and modeling the response directly as a function of these. LoLI borrows strength from the response to more accurately predict the latent indices, leading to better estimation of the corresponding effects. Furthermore, LoLI incorporates a LASSO type penalty to perform hierarchical selection, with interaction terms selected only if both parent main effects are included. Simulations show that LoLI can outperform the dummy variable and two stage approaches in selection and prediction performance. Applying LoLI to an Australian household-based panel identified three dimensions of psychosocial workplace quality (job demands, stress, and security) which affect an individual's mental health in an additive and pairwise interactive manner.