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.

翻译：本文介绍了在纵向框架下对具有过量零值的圆形数据进行建模的方法，其中响应变量为圆形变量，协变量可同时包含线性和圆形类型。现有文献已对各类圆形-圆形及圆形-线性回归模型进行了研究，并将其应用于不同的实际问题。然而，目前尚缺乏针对纵向研究中零膨胀圆形观测数据的建模方法。基于真实案例研究的驱动，本文提出了一种基于投影正态分布的混合效应两阶段模型来处理此类问题。文中讨论了模型参数的解释性，并推导了可识别性条件。针对模型参数估计，开发了基于吉布斯采样技术的贝叶斯方法。仿真结果表明，所提方法在多种情境下均优于现有方法。通过分析术后散光真实数据集，展示了所提方法在实际应用中的实施过程。该方法的使用有助于在治疗选择及随访阶段进行有效决策。