Regularization and Model Selection with Categorial Predictors and Effect Modifiers in Generalized Linear Models

Varying-coefficient models with categorical effect modifiers are considered within the framework of generalized linear models. We distinguish between nominal and ordinal effect modifiers, and propose adequate Lasso-type regularization techniques that allow for (1) selection of relevant covariates, and (2) identification of coefficient functions that are actually varying with the level of a potentially effect modifying factor. We investigate large sample properties, and show in simulation studies that the proposed approaches perform very well for finite samples, too. In addition, the presented methods are compared with alternative procedures, and applied to real-world medical data. 1. Introduction. In regression modeling, categorical predictors, also called factors, are a standard case. Nevertheless, variable selection for discrete covariates and the connected problem which categories within one factor are to be distinguished has been somewhat neglected. More concrete, in our application, we model the effects of pregnancy related covariates on the type of delivery, that is, if birth was given vaginally or by means of a Cesarean. Cases were observed over a period of several years. As medical standards typically change over time, modeling the type of delivery requires to consider discrete time-effects, and more importantly, to consider how effects change over years. In general, we are going to address model selection with discrete covariates in a slightly extended version of generalized linear models (GLMs), namely GLMs with varying coefficients. Varying-coefficient models (Hastie and Tibshirani, 1993) are a quite flexible tool to capture complex model structures and interactions. In the setting of GLMs, regression coefficients β j are allowed to vary with the value of other variables u j. Hence the linear predictor has the form