Inflated Discrete Beta Regression (IDBR)

• Inflated Discrete Beta Regression (IDBR) is a statistical framework for bounded ordinal responses that integrates discretized beta regression with an explicit inflation component.
• The model jointly regresses location, dispersion, and inflation probabilities, distinguishing systematic invariant responses from latent variability.
• IDBR's application in survey research, marketing, and policy analysis enhances prediction accuracy on Likert scales and deepens understanding of respondent heterogeneity.

Inflated Discrete Beta Regression (IDBR) is a statistical modeling framework specifically designed to analyze ordinal discrete outcomes, such as Likert and rating scale data, which are bounded, potentially skewed, and frequently exhibit disproportionate response frequencies—“inflation”—at a specific scale point. IDBR simultaneously accounts for key data attributes: discreteness, boundedness, potential skewness, and inflation. By coupling a discretized latent beta regression with a mixture component for the inflated point, the model enables joint regression on the location and dispersion of the latent variable as well as on the propensity to select the inflated response, providing nuanced insights beyond conventional methods (Taverne et al., 2014).

1. Model Architecture and Likelihood Formulation

The core of IDBR is an extension of discrete beta regression (DBR). Suppose the observed outcome $y^*$ is an ordinal variable with $K$ equally spaced levels $\{a, a+h^*, \ldots, b-h^*, b\}$. It is rescaled to the unit interval via

$y = \frac{y^* - a + h^*}{b - a + h^*}$

yielding discrete support at $\{h, 2h, \ldots, 1-h, 1\}$ with $h = 1/K$.

The non-inflated component assumes an underlying continuous latent variable $U \sim \mathrm{Beta}(p, q)$, observed through discretization:

$$P(Y = y) = P(y - h < U \leq y) = \frac{\int_{y - h}^y u^{p-1}(1-u)^{q-1} \, du}{B(p, q)}$$

with $B(p, q)$ the beta function.

Parameters are reparameterized via mean $\mu$ and dispersion $\phi$, i.e., $E(U) = \mu = p/(p+q)$, $\mathrm{Var}(U) = \mu(1-\mu)\phi$, with $\mu$ and $\phi$ both linked to covariates through respective link functions ($g_1$, $g_2$), typically employing a logit transformation to constrain them to $(0,1)$.

The inflated component acknowledges excess mass at a specific category (e.g., midpoint for Likert scales, $y=kh$), introducing a mixture:

$$P(Y = y) = I(y=kh)\cdot\pi + [1-\pi]P(y-h < U \leq y)$$

where $\pi$ (the inflation probability) is regressed on covariates ($W$) via a link function $g_0(\pi) = f_0(W, \gamma)$.

The resulting likelihood incorporates three regression submodels: the probability of inflation ($\pi$), location ($\mu$), and dispersion ($\phi$), each with its own (potentially distinct) set of covariates.

2. Key Model Features

• Discrete, Bounded Ordinal Support: IDBR natively accommodates Likert or rating scales, preserving the discrete bounded support through integration over discretization intervals, avoiding mis-specification incurred by continuous or unbounded models.
• Jointly Modeled Location and Dispersion: The model flexibly links both mean (location) and dispersion parameters to covariates, facilitating differentiation not only of central tendency but also heterogeneity/precision across respondent groups.
• Inflation Mechanism: By allocating an explicit mixture mass at any selected level, IDBR distinguishes between “invariant choosers” (systematically choosing the level) and respondents whose choices reflect latent variation. This granularity addresses the commonly confounded mixture of certainty and proximity responses in scale data.
• Generalizability: The architecture allows for extensions to multiple inflated levels or hierarchical modeling, enabling adaptation to complex survey and panel data structures.

3. Statistical Properties and Simulation Performance

Simulation studies demonstrate the following IDBR properties (Taverne et al., 2014):

• Consistency and Efficiency: As the sample size increases, bias and root mean squared error (RMSE) in parameter estimates decrease, confirming consistency and efficiency.
• Sharp Predictive Performance: IDBR attains a higher proportion of correct predictions for discrete responses than standard alternatives. Predictive intervals, computed via sampling from the posterior parameter distribution, are typically narrower yet maintain nominal coverage rates.
• Accurate Credible Intervals: Highest posterior density (HPD) intervals for regression parameters are well-calibrated, with interval lengths reducing as sample size grows.
• Model-Based Covariate Insight: Simulation under varying data-generating mechanisms confirms that the model accurately recovers known covariate effects for exhaustion, inflation, and dispersion submodels.

4. Empirical Application: Political Self-Placement

IDBR has been applied to Belgian respondents in the 2012 European Social Survey to model self-placement on an 11-point left–right political scale (Taverne et al., 2014):

• Data Characteristics: Substantial inflation at the central value (“5”) was observed (≈35% of responses).
• Model Specification: Inflation ($\pi$) was regressed on gender, education, and self-placement in society; location ($\mu$) on area of residence, gender, income, and social placement; dispersion ($\phi$) on age and economic comfort.
• Findings: Women and respondents with lower educational attainment exhibited stronger tendencies toward invariant mid-scale choices; social self-placement influenced both inflation and location. The dispersion submodel highlighted varying ideological variance by age and economic circumstance.
• Interpretive Richness: IDBR facilitated nuanced disaggregation: separating respondents systematically at the midpoint from those exhibiting latently centrist but non-invariant preferences.

5. Comparative Evaluation with Alternative Models

IDBR exhibits several advantages over standard approaches:

Method	Discreteness	Boundedness	Inflation Handling	Covariate Links
Linear Regression (LM)	No	No	No	Mean only
Continuous Beta Regression	No	Yes	No	Mean/dispersion
Ordered Logit/Probit	Yes	Implicit	Limited	Location only
Multinomial Models	Yes	Implicit	Limited	Location only
IDBR	Yes	Yes	Flexible	All three

• Standard Linear Models disregard both bounds and discreteness, leading to potentially biased predictions.
• Continuous Beta Regression (e.g., Simas et al.) assumes continuous outcomes, inapplicable to truly discrete data and unable to resolve inflation.
• Ordered Logit/Probit and multinomial models recognize ordinality but do not directly accommodate skewness or empirical inflation and may lack interpretability due to overparameterization.
• IDBR alone models all four dimensions—discreteness, boundedness, dispersion, and inflation—jointly and (unlike composite likelihoods in augmented beta regression) does so within a unified likelihood.

Potential limitations include the challenge of correctly specifying which response level(s) should be subject to inflation and the interpretational complexity arising from maintaining and explaining multiple linked submodels.

6. Practical Implications and Methodological Extensions

The practical scope of the IDBR model comprises:

• Survey Research: Well-suited for analysis of Likert and rating scales where invariant responses (e.g., persistent neutral or extreme choices) are frequent.
• Marketing and Social Science: Enables isolation of latent “non-choosers” or “invariant” clusters (e.g., non-buyers or respondents with fixed stances), facilitating tailored interventions.
• Richer Inference: The model structure promotes the separation and interpretation of central tendency, variability, and pronounced respondent behavior within a single coherent framework.
• Potential for Extension: The structure is amenable to modeling multiple inflations, hierarchical (multi-level) designs, or adaption for discrete outcomes within broader [0,1] models incorporating recent advances in endpoint modeling (Hahn, 2023). A plausible implication is that recent unified beta modeling for the full $[0,1]$ interval may suggest analogous mixture-based strategies for extending the IDBR to situations with endpoint inflations, mitigating the necessity for discrete/continuous composite likelihoods and simplifying expectation derivations.

7. Conclusion

Inflated Discrete Beta Regression encompasses a flexible likelihood-based approach for discrete ordinal data, uniquely accommodating inflation at a specified response level, joint covariate effects on location and dispersion, and possessing favorable statistical performance in estimation and inference. The separation of systematic invariant choice from beta-driven variation offers interpretive depth often unattainable with classic ordinal or multinomial models. Empirical studies confirm superior predictive sharpness and inferential precision. The model’s conceptual architecture coheres with recent developments in unified [0,1]-interval regression modeling, portending potential methodological synergies especially for modeling endpoint or multiple-category inflation.

IDBR is thus positioned as a principal analytic framework for modern discrete ordinal response modeling in survey, marketing, and policy research contexts.