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Modeling Zero-Inflated Longitudinal Circular Data Using Bayesian Methods: Application to Ophthalmology

Prajamitra Bhuyan, Soutik Halder, Jayant Jha

TL;DR

This work addresses zero-inflated longitudinal circular data in ophthalmology by developing a Bayesian two-stage mixed-effects model based on the projected normal distribution. It combines latent censoring for zeros with instrumental variables to model a circular response and a circular covariate, and it employs Gibbs sampling for efficient Bayesian inference under identifiability constraints. The method is validated through extensive simulations, showing robust performance and clear advantages over non–zero-inflation alternatives, and is demonstrated on post-operative astigmatism data to yield clinically actionable insights into treatment effects and recovery trajectories. The approach offers a flexible, tractable framework for zero-inflated circular data in longitudinal settings with practical impact for treatment planning and follow-up decision-making.

Abstract

This paper introduces the modeling of circular data with excess zeros under a longitudinal framework, where the response is a circular variable and the covariates can be both linear and circular in nature. In the literature, various circular-circular and circular-linear regression models have been studied and applied to different real-world problems. However, there are no models for addressing zero-inflated circular observations in the context of longitudinal studies. Motivated by a real case study, a mixed-effects two-stage model based on the projected normal distribution is proposed to handle such issues. The interpretation of the model parameters is discussed and identifiability conditions are derived. A Bayesian methodology based on Gibbs sampling technique is developed for estimating the associated model parameters. Simulation results show that the proposed method outperforms its competitors in various situations. A real dataset on post-operative astigmatism is analyzed to demonstrate the practical implementation of the proposed methodology. The use of the proposed method facilitates effective decision-making for treatment choices and in the follow-up phases.

Modeling Zero-Inflated Longitudinal Circular Data Using Bayesian Methods: Application to Ophthalmology

TL;DR

This work addresses zero-inflated longitudinal circular data in ophthalmology by developing a Bayesian two-stage mixed-effects model based on the projected normal distribution. It combines latent censoring for zeros with instrumental variables to model a circular response and a circular covariate, and it employs Gibbs sampling for efficient Bayesian inference under identifiability constraints. The method is validated through extensive simulations, showing robust performance and clear advantages over non–zero-inflation alternatives, and is demonstrated on post-operative astigmatism data to yield clinically actionable insights into treatment effects and recovery trajectories. The approach offers a flexible, tractable framework for zero-inflated circular data in longitudinal settings with practical impact for treatment planning and follow-up decision-making.

Abstract

This paper introduces the modeling of circular data with excess zeros under a longitudinal framework, where the response is a circular variable and the covariates can be both linear and circular in nature. In the literature, various circular-circular and circular-linear regression models have been studied and applied to different real-world problems. However, there are no models for addressing zero-inflated circular observations in the context of longitudinal studies. Motivated by a real case study, a mixed-effects two-stage model based on the projected normal distribution is proposed to handle such issues. The interpretation of the model parameters is discussed and identifiability conditions are derived. A Bayesian methodology based on Gibbs sampling technique is developed for estimating the associated model parameters. Simulation results show that the proposed method outperforms its competitors in various situations. A real dataset on post-operative astigmatism is analyzed to demonstrate the practical implementation of the proposed methodology. The use of the proposed method facilitates effective decision-making for treatment choices and in the follow-up phases.
Paper Structure (18 sections, 1 theorem, 17 equations, 8 figures, 9 tables, 2 algorithms)

This paper contains 18 sections, 1 theorem, 17 equations, 8 figures, 9 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Challenges in analyzing astigmatism data
  3. Preliminaries
  4. Projected normal regression model
  5. Proposed model for zero-inflated longitudinal circular data
  6. Identifiability issues
  7. Inferential methodology
  8. Bayesian estimation
  9. Sampling procedures
  10. Generalization
  11. Simulations
  12. Model comparison
  13. Sensitivity analysis
  14. Sensitivity to model misspecification
  15. Sensitivity to prior specifications
  16. ...and 3 more sections

Key Result

Theorem 1

If $\theta_{Y_{ij}} \sim \mathcal{PN}_2 (\boldsymbol{\mu}_{Y_{ij}}, \boldsymbol{\Gamma}_b)$, where $\boldsymbol{\mu}_{Y_{ij}} = \textbf{B}^{\top} \widetilde{\boldsymbol{x}}_{ij}$ and $\boldsymbol{\Gamma}_b = \boldsymbol{I}_2 + \boldsymbol{\Sigma}_b$, then the model, given by (EQN:LCRM:Stage-I), is i

Figures (8)

  • Figure 1: Focus of light rays on the retina in a normal eye (left) and an astigmatic eye (right).
  • Figure 2: Visual distortions under WTR, ATR and Oblique astigmatism.
  • Figure 3: Circular density plots of $\mathcal{PN}_2((\beta_{10} + \beta_{11}x, \beta_{20} + \beta_{21}x)^{\top}, \boldsymbol{I}_2)$ for $x = 0$ (dark orchid), $x = 1$ (dark green), $x = 2$ (blue), $x = 5$ (red) and $x = 100$ (black) with coefficients $\beta_{10} = 1, \beta_{11} = 3, \beta_{20} = -2$ and $\beta_{21} = 2$.
  • Figure 4: Confidence ellipses at 95% level depicting the significance of (A) intercept, (B) Age, (C) Gender, (D) Surgery, (E) $t_1$, (F) $t_2$, and (G) the intensity of astigmatism on the 1st day post-surgery at Stage-I, where '$\color{blue}{\bullet}$' indicates (0,0) position.
  • Figure 5: Confidence ellipses at 95% level depicting the significance of (H) intercept and (I) the intensity of astigmatism before surgery at Stage-II, where '$\color{blue}{\bullet}$' indicates (0,0) position.
  • ...and 3 more figures

Theorems & Definitions (2)

  • Theorem 1
  • proof