Model diagnostics of discrete data regression: a unifying framework using functional residuals
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离散数据回归的模型诊断：基于函数残差的统一框架
- 研究问题
- 二元、序数与计数结局的离散性限制了传统残差诊断，如何保留残差随机性的信息并识别不同类型的模型设定错误？
- 方法与思路
- 作者将残差定义为函数，建立其理论性质，并提出函数残差与协变量图及函数对函数图，用于统一诊断离散回归模型。
- 材料陈述的贡献
- 摘要称这些工具在数值研究中能揭示遗漏变量、交互项、离散参数和零膨胀等问题，并将代理残差纳入更一般框架。
- 阅读关注点
- 应核对函数残差的定义、理论性质的适用条件、诊断图的判读规则，以及不同错设类型能否被可靠区分。
- 阅读边界
- 这是 arXiv 预印本，同行评审及期刊接收未核实；摘要未提供具体诊断性能，离散结局本身也不足以认定属于经济离散选择。
原文摘要
Model diagnostics is an indispensable component of regression analysis, yet it is not well addressed in standard textbooks on generalized linear models. The lack of exposition is attributed to the fact that when outcome data are discrete, classical methods (e.g., Pearson/deviance residual analysis and goodness-of-fit tests) have limited utility in model diagnostics and treatment. This paper establishes a novel framework for model diagnostics of discrete data regression. Unlike the literature defining a single-valued quantity as the residual, we propose to use a function as a vehicle to retain the residual information. In the presence of discreteness, we show that such a functional residual is appropriate for summarizing the residual randomness that cannot be captured by the structural part of the model. We establish its theoretical properties, which leads to the innovation of new diagnostic tools including the functional-residual-vs covariate plot and Function-to-Function (Fn-Fn) plot. Our numerical studies demonstrate that the use of these tools can reveal a variety of model misspecifications, such as not properly including a higher-order term, an explanatory variable, an interaction effect, a dispersion parameter, or a zero-inflation component. The functional residual yields, as a byproduct, Liu-Zhang's surrogate residual mainly developed for cumulative link models for ordinal data (Liu and Zhang, 2018, JASA). As a general notion, it considerably broadens the diagnostic scope as it applies to virtually all parametric models for binary, ordinal and count data, all in a unified diagnostic scheme.