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：纵向研究中经常遇到二值或有序响应数据，这类分析的核心在于刻画各响应类别概率随时间的动态演变。传统方法多基于广义混合效应模型框架，即便在固定或随机效应上施加非参数先验，仍会因对边际期望和协方差结构的隐含假设而存在局限性。本文从函数型数据分析视角处理该问题，将每个受试者的观测视为其在测量时间点上受试者专属随机过程的实现。我们首先针对二值响应建立方法论，假设其随机过程服从二项边际分布。利用对数几率表示法，我们将离散空间过程建模为连续空间过程的序列。通过引入层次框架，我们分别利用高斯过程先验和逆Wishart过程先验，对连续空间过程的均值核与协方差核进行非参数联合建模。该先验结构既能灵活推断二值响应的演变规律与相关性，又可实现跨受试者的信息共享。该建模方法可自然扩展至有序响应，其中多项分布的续存比对数几率分解是实现高效建模与推断的关键——包括处理非平衡纵向数据的实用方案。通过合成数据示例及大学生心理健康状况数据分析，验证了该方法的有效性。