Optimal Scaling transformations to model non-linear relations in GLMs with ordered and unordered predictors

TL;DR

This paper introduces an optimal scaling approach for GLMs that models nonlinear predictor-outcome relations, allowing for direct quantification of categorical levels, monotonicity constraints, and mixed scaling levels, demonstrated on logistic regression datasets.

Contribution

It presents a novel optimal scaling methodology integrated with GLMs, enabling flexible, interpretable modeling of nonlinear effects and categorical predictor quantification.

Findings

Enhanced interpretability of predictor effects.

Ability to include mixed scaling levels.

Demonstrated effectiveness on logistic regression datasets.

Abstract

In Generalized Linear Models (GLMs) it is assumed that there is a linear effect of the predictor variables on the outcome. However, this assumption is often too strict, because in many applications predictors have a nonlinear relation with the outcome. Optimal Scaling (OS) transformations combined with GLMs can deal with this type of relations. Transformations of the predictors have been integrated in GLMs before, e.g. in Generalized Additive Models. However, the OS methodology has several benefits. For example, the levels of categorical predictors are quantified directly, such that they can be included in the model without defining dummy variables. This approach enhances the interpretation and visualization of the effect of different levels on the outcome. Furthermore, monotonicity restrictions can be applied to the OS transformations such that the original ordering of the category values is preserved. This improves the interpretation of the effect and may prevent overfitting. The scaling level can be chosen for each individual predictor such that models can include mixed scaling levels. In this way, a suitable transformation can be found for each predictor in the model. The implementation of OS in logistic regression is demonstrated using three datasets that contain a binary outcome variable and a set of categorical and/or continuous predictor variables.