Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Modeling Ordinal Survey Data with Unfolding Models

Rayleigh Lei, Abel Rodriguez

TL;DR

This work addresses the limitation of monotone response functions in standard ordinal factor models by introducing OPUM, an ordinal probit unfolding model grounded in random utilities that accommodates monotonic and non-monotonic item responses. OPUM relies on a latent-variable representation with auxiliary vectors $\boldsymbol{z}_{i,j}$ and a symmetric mixture to learn the correct item directionality, enabling more flexible and interpretable scaling of Likert-type data. A tunning-free MCMC algorithm with data augmentation enables Bayesian inference, and an empirical application to immigration attitudes demonstrates that OPUM achieves better complexity-adjusted fit (WAIC) than both GRMs and GGUMs, with informative, sometimes multimodal, posterior distributions for latent traits. The approach offers practical benefits for survey design and scaling, and opens avenues for extensions to multivariate traits, dependent item parameters, and nonparametric response curves, potentially improving measurement in political science and marketing contexts.

Abstract

Surveys that rely on ordinal polychotomous (Likert-like) items are widely employed to capture individual preferences because they allow respondents to express both the direction and strength of their preferences. Latent factor models traditionally used in this context implicitly assume that the response functions (the cumulative distribution of the ordinal outcome) are monotonic on the latent trait. This assumption can be too restrictive in several application areas, including in political science and marketing. In this work, we propose a novel ordinal probit unfolding model that can accommodate both monotonic and non-monotonic response functions. The advantages of the model are illustrated by analyzing an immigration attitude survey conducted in the United States.

Modeling Ordinal Survey Data with Unfolding Models

TL;DR

This work addresses the limitation of monotone response functions in standard ordinal factor models by introducing OPUM, an ordinal probit unfolding model grounded in random utilities that accommodates monotonic and non-monotonic item responses. OPUM relies on a latent-variable representation with auxiliary vectors and a symmetric mixture to learn the correct item directionality, enabling more flexible and interpretable scaling of Likert-type data. A tunning-free MCMC algorithm with data augmentation enables Bayesian inference, and an empirical application to immigration attitudes demonstrates that OPUM achieves better complexity-adjusted fit (WAIC) than both GRMs and GGUMs, with informative, sometimes multimodal, posterior distributions for latent traits. The approach offers practical benefits for survey design and scaling, and opens avenues for extensions to multivariate traits, dependent item parameters, and nonparametric response curves, potentially improving measurement in political science and marketing contexts.

Abstract

Surveys that rely on ordinal polychotomous (Likert-like) items are widely employed to capture individual preferences because they allow respondents to express both the direction and strength of their preferences. Latent factor models traditionally used in this context implicitly assume that the response functions (the cumulative distribution of the ordinal outcome) are monotonic on the latent trait. This assumption can be too restrictive in several application areas, including in political science and marketing. In this work, we propose a novel ordinal probit unfolding model that can accommodate both monotonic and non-monotonic response functions. The advantages of the model are illustrated by analyzing an immigration attitude survey conducted in the United States.
Paper Structure (14 sections, 25 equations, 9 figures)

This paper contains 14 sections, 25 equations, 9 figures.

Table of Contents

  1. Introduction
  2. A brief review of GRMs and GGUMs
  3. An ordinal probit unfolding model based on random utilities
  4. Symmetrizing the model
  5. Prior distributions
  6. Computation
  7. Illustration
  8. Discussion
  9. Properties of the GGUM response functions
  10. Minimum
  11. Symmetry
  12. Limit as $| \beta_i | \to \infty$
  13. OPUM's latent variable formulation
  14. Gibbs sampler

Figures (9)

  • Figure 1: Response function for a BGGUM where $K_j=3$, $\alpha_j = 1$, $\delta_j = 1.1$, $\bm{\tau} = (0, -1.2, -1.0, -0.7)$.
  • Figure 2: Examples of response functions for OPUM where $K_j=3$. Panel (a) shows an example where the response functions are monotonic over the range of interest, which corresponds to $\boldsymbol{\alpha}_j = (-3.25, -2.50, -2.25, 0.50, 0.75, 2.00)$ and $\boldsymbol{\mu}_j = (-4.00, -10.50, -4.00, 1.50, 1.50, 1.50)$. Panel (b) shows an example of non-monotonic response functions, which corresponds to $\boldsymbol{\alpha}_j = (-3.0, -2.5, -2.5, 0.5, 1.0, 2.0)$ and $\boldsymbol{\mu}_j = (-2.0, -5.25, -2.0, 1.5, 1.5, 1.5)$.
  • Figure 3: Prior mean response functions for $K_j=4$. Panel (a) corresponds to a BGRM with $\alpha_j \sim N(0, 0.5)$ and a $\boldsymbol{\tau}_j \sim N((-4, -2, 2, 4)', 9 I_{4}) \mathbbm{I}\left(\tau_{j, 0} < \tau_{j, 1} < \tau_{j, 2} < \tau_{j, 3}\right)$. Panel (b) corresponds to a BGGUM with $\alpha_j \sim \textrm{Beta}_{(0.25, 4)}(1.5, 1.5)$, $\mu_j \sim \textrm{Beta}_{(-5, 5)}(2, 2)$ and $\tau_{j, l} \sim \textrm{Beta}_{(-6, 6)}(2, 2)$. Panel (c) corresponds to an OPUM with $\kappa^2_j = 10$, $\omega^2_j = 25$, $\upsilon_{j, -k} = -4 - 1.5 k$ and $\upsilon_{j, k} = 4 + 1.5 k$.
  • Figure 4: Comparison of the posterior median ranks of individual preferences. Panel (a) compares the ranks from BGRM from those from OPUM, while panel (b) compares the ranks from BGGUM to those of OPUM. Symbols are used to signal self-reported party affiliation (triangles for Republicans and circles for Democrats), while color saturation indicates the strength of the affiliation.
  • Figure 5: Posterior mean of the response functions for Question 3 estimated by each of BGRM (panel A), GGUM (panel B) and OPUM (panel C). The horizontal axis of each panel captures the range of of values of the latent traits estimated by each model.
  • ...and 4 more figures