On the Hauck-Donner Effect in Wald Tests: Detection, Tipping Points and Parameter Space Characterization

The Wald test remains ubiquitous in statistical practice despite shortcomings such as its inaccuracy in small samples and lack of invariance under reparameterization. This paper develops on another but lesser-known shortcoming called the Hauck–Donner effect (HDE) whereby a Wald test statistic is no longer monotone increasing as a function of increasing distance between the parameter estimate and the null value. Resulting in an upward biased <i>p</i>-value and loss of power, the aberration can lead to very damaging consequences such as in variable selection. The HDE afflicts many types of regression models and corresponds to estimates near the boundary of the parameter space. This article presents several new results, and its main contributions are to (i) propose a very general test for detecting the HDE in the class of vector generalized linear models (VGLMs), regardless of the underlying cause; (ii) fundamentally characterize the HDE by pairwise ratios of Wald and Rao score and likelihood ratio test statistics for 1-parameter distributions with large samples; (iii) show that the parameter space may be partitioned into an interior encased by at least 5 HDE severity measures (faint, weak, moderate, strong, extreme); (iv) prove that a necessary condition for the HDE in a 2 by 2 table is a log odds ratio of at least 2; (v) give some practical guidelines about HDE-free hypothesis testing. Overall, practical post-fit tests can now be conducted potentially to any model estimated by iteratively reweighted least squares, especially the GLM and VGLM classes, the latter which encompasses many popular regression models.

沃尔德检验（Wald test）在统计实践中仍被广泛应用，尽管其存在诸多缺陷，比如小样本下精度不足，以及在重参数化下不具备不变性。本文针对另一项鲜为人知的缺陷展开研究，该缺陷被称为豪克-唐纳效应（Hauck–Donner effect, HDE），即当参数估计值与原假设值之间的距离增大时，沃尔德检验统计量不再单调递增。这种异常会导致*p*值上偏且检验功效下降，进而在变量选择等场景中引发极为严重的后果。豪克-唐纳效应会影响多种回归模型，且多出现在参数空间边界附近的参数估计场景中。

本文提出了多项新的研究结果，主要贡献包括：(i) 针对向量广义线性模型（vector generalized linear models, VGLMs）类，提出了一种通用性极强的检验方法，可用于检测豪克-唐纳效应，且无需考虑其具体成因；(ii) 针对大样本下的单参数分布，通过沃尔德检验、拉奥评分检验（Rao score test）与似然比检验（likelihood ratio test）统计量的两两比值，从根本上刻画了豪克-唐纳效应的特征；(iii) 证明参数空间可被至少5种豪克-唐纳效应严重程度指标（轻微、弱、中等、强、极端）划分出内部区域；(iv) 证实2×2列联表中出现豪克-唐纳效应的必要条件是对数优势比（log odds ratio）至少为2；(v) 给出了若干避免豪克-唐纳效应的实用假设检验指南。

总体而言，如今可针对任意通过迭代加权最小二乘法估计的模型开展实用的拟合后检验，尤其适用于广义线性模型（Generalized Linear Model, GLM）与向量广义线性模型类，后者涵盖了诸多主流回归模型。