[Paper Review] Non Proportional Odds Models are Widely Dispensable -- Sparser Modeling based on Parametric and Additive Location-Shift Approaches

Gerhard Tutz, Moritz Berger|arXiv (Cornell University)|Jun 6, 2020

Statistical Methods and Bayesian Inference5 citations

TL;DR

This paper proposes location-shift models as a sparse, flexible alternative to non-proportional odds models in ordinal regression, combining global location effects with dispersion effects to capture response tendencies (e.g., extreme or middle category preferences). It demonstrates that these models often outperform both proportional odds and non-proportional odds models in fit and interpretability, with additive extensions allowing smooth effects while avoiding the complexity of full category-specific coefficient models.

ABSTRACT

The potential of location-shift models to find adequate models between the proportional odds model and the non-proportional odds model is investigated. It is demonstrated that these models are very useful in ordinal modeling. While proportional odds models are often too simple, non proportional odds models are typically unnecessary complicated and seem widely dispensable. The class of location-shift models is also extended to allow for smooth effects. The additive location-shift model contains two functions for each explanatory variable, one for the location and one for dispersion. It is much sparser than hard-to-handle additive models with category-specific covariate functions but more flexible than common vector generalized additive models.

Motivation & Objective

• To address the limitations of proportional odds models, which often lack adequate fit in real-world ordinal data.
• To challenge the necessity of non-proportional odds models, which are overly complex and hard to interpret due to numerous category-specific parameters.
• To introduce location-shift models that incorporate both location and dispersion effects, capturing response tendencies without excessive parameterization.
• To extend location-shift models to additive forms that allow smooth, non-linear effects while maintaining sparsity and interpretability.
• To provide a practical R package (ordDisp) for fitting and visualizing these models with standard statistical methods.

Proposed method

• The location-shift model extends the proportional odds model by adding a dispersion term that captures variability in response patterns, such as tendencies to choose extreme or middle categories.
• For parametric models, the location and dispersion effects are modeled via linear predictors: η_loc = Xβ and η_disp = Zγ, with separate link functions.
• For additive models, smooth functions f(x) and f^(z) are estimated using B-spline basis expansions with n_bs basis functions.
• The models are fitted using the VGAM package in R, leveraging its vglm() function for generalized linear modeling with cumulative and adjacent categories link functions.
• The package ordDisp enables model fitting with formula syntax y ~ x1 + ... + xp | z1 + ... + zq, separating location and dispersion covariates.
• The method supports reverse scoring for interpretation consistency and includes tools for plotting effects with 95% confidence intervals.

Experimental results

Research questions

• RQ1 Can location-shift models provide a better balance between model fit and interpretability than proportional or non-proportional odds models?
• RQ2 Are non-proportional odds models frequently unnecessary in ordinal regression due to excessive complexity?
• RQ3 Can smooth location-shift models effectively capture non-linear trends in both location and dispersion without overfitting?
• RQ4 How do location-shift models compare to standard vector generalized additive models in terms of parameter sparsity and flexibility?
• RQ5 What is the empirical performance of location-shift models in real-world ordinal data, such as survey responses on nuclear energy fears?

Key findings

• Non-proportional odds models are often dispensable, as they introduce excessive complexity without significant gains in fit over simpler location-shift models.
• Location-shift models with both location and dispersion effects show significantly better fit than proportional odds models, with likelihood ratio tests yielding p-values < 0.001 for location effects (e.g., 64.62 on 4 df) and dispersion effects (e.g., 21.24 on 1 df) in the nuclear energy example.
• The additive location-shift model successfully captures smooth, non-linear effects of age on fear of nuclear energy, showing increased fear in the sixties and reduced dispersion among older respondents.
• The proposed models maintain interpretability while allowing for flexible, smooth effects, avoiding the need for large numbers of category-specific coefficients.
• The R package ordDisp enables seamless model fitting, visualization, and inference using standard R methods like summary, predict, and plot.
• Adjacent categories models are preferred over cumulative models in additive settings due to the absence of parameter space restrictions, enhancing model stability.