Component-wise Z-residual Diagnosis for Bayesian Hurdle Models

In many applied research areas, count data often exhibit more zeros than expected under standard distributions like the Poisson or negative binomial. These “excess zeros” are common in fields such as health economics, criminology, and ecology, and can distort inference if not properly accounted for. Traditional models assume specific data-generating mechanisms and often perform poorly when zero counts dominate. Hurdle models provide a flexible solution by separating the modeling process into two parts: (1) a binary model that predicts whether the count is zero or positive, and (2) a zero-truncated model that describes the distribution of positive counts. This two-part structure captures and interprets the distinct processes generating zeros and positive counts. In frequentist statistics, residuals are widely used to assess model fit and detect problems such as misspecification. In Bayesian analysis, however, residual diagnostics have been less common. A recent development is the use of Z-residuals, which are especially useful because they are approximately standard normal when the model is correctly specified. While Z-residuals have shown effectiveness in frequentist settings, their application in Bayesian hurdle models remains limited, and existing approaches typically apply Z-residuals to the model as a whole, without distinguishing between its individual components. This is a limitation in the context of hurdle models, where the two components represent fundamentally different data-generating processes. Aggregated residuals can hide which part of the model is responsible for the poor fit, making it difficult to diagnose and address specific sources of misspecification. To address this gap, this study proposes a component-wise approach to Z-residual diagnostics for Bayesian hurdle models. By constructing and analyzing Z-residuals separately for the logistic and count components, we aim to develop diagnostic tools that are more localized to model misspecified components and more informative for guiding model refinement. This approach enables practitioners to identify whether deficiencies lie in the logistic side or the count distribution, allowing for targeted adjustments and avoiding unnecessary changes to well-specified components. Ultimately, component-wise Z-residuals offer a more precise and interpretable diagnostic strategy, enhancing the reliability and transparency of Bayesian hurdle modeling. As part of this work, we developed a collection of generic R functions that implement both aggregated and component-wise Z-residuals, together with associated statistical tests and diagnostic plots, forming a central contribution to an R package currently in development. Simulation studies show that Z-residuals follow an approximate standard normal distribution when the model is correctly specified, while clearly identifying the component responsible for lack of fit under misspecification. The simulations further demonstrate that Type I error rates are well controlled and that the method has high power to detect deviations from the true model. Finally, the proposed component-wise approach was applied to a real-world dataset, illustrating how component-wise Z-residuals can provide deeper insights into checking model adequacy and enhance interpretation in practice.

Name thesis:degree_name

Master of Science (M.Sc.)

Level thesis:degree_level

Masters

Discipline thesis:degree_discipline

Statistics

Grantor

University of Saskatchewan

Year dc:date.issued

2025

Author dc:creator

• Hettiarachchige Alias Egodage, Dananji

Advisors dc:contributor.advisor

• Li, Longhai
• Feng, Cindy

Committee members dc:contributor.committeemember

• Xing, Li
• Liu, Juxin
• Soteros, Chris

• Z-residual
• Component-wise
• Bayesian hurdle models

Language dc:language.iso

en

Handle dc:identifier.uri

https://hdl.handle.net/10388/17250

OAI identifier oai:identifier

oai:harvest.usask.ca:10388/17250