Longitudinal studies with binary or ordinal responses are widely encountered in various disciplines, where the primary focus is on the temporal evolution of the probability of each response category. Traditional approaches build from the generalized mixed effects modeling framework. Even amplified with nonparametric priors placed on the fixed or random effects, such models are restrictive due to the implied assumptions on the marginal expectation and covariance structure of the responses. We tackle the problem from a functional data analysis perspective, treating the observations for each subject as realizations from subject-specific stochastic processes at the measured times. We develop the methodology focusing initially on binary responses, for which we assume the stochastic processes have Binomial marginal distributions. Leveraging the logits representation, we model the discrete space processes through sequences of continuous space processes. We utilize a hierarchical framework to model the mean and covariance kernel of the continuous space processes nonparametrically and simultaneously through a Gaussian process prior and an Inverse-Wishart process prior, respectively. The prior structure results in flexible inference for the evolution and correlation of binary responses, while allowing for borrowing of strength across all subjects. The modeling approach can be naturally extended to ordinal responses. Here, the continuation-ratio logits factorization of the multinomial distribution is key for efficient modeling and inference, including a practical way of dealing with unbalanced longitudinal data. The methodology is illustrated with synthetic data examples and an analysis of college students' mental health status data.

翻译：在众多学科领域中广泛存在以二值或有序响应为观测指标的纵向研究，其核心关注点在于各响应类别概率随时间演变的规律。传统方法基于广义混合效应建模框架构建。即使对固定效应或随机效应施加非参数先验，此类模型仍受限于其对响应变量边际期望与协方差结构的隐含假设。本文从函数型数据分析视角切入，将每个受试者的观测数据视为特定受试者随机过程在测量时间点上的实现。我们首先针对二值响应发展方法论框架，假设相应随机过程具有二项边际分布。借助对数比表示法，我们通过连续空间过程序列对离散空间过程进行建模。我们采用分层框架，分别通过高斯过程先验和逆威沙特过程先验对连续空间过程的均值函数与协方差核进行非参数化同步建模。该先验结构能够灵活推断二值响应的演变规律与相关性，同时实现所有受试者间的信息共享。本建模方法可自然拓展至有序响应场景。其中，多项分布的连续比对数比分解是实现高效建模与推断的关键，这为处理非平衡纵向数据提供了实用途径。我们通过合成数据示例与大学生心理健康状态数据分析来展示该方法的应用。