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

Abstract

The Wald test remains ubiquitous in statistical practice despite shortcomings such as its inaccuracy in small samples and lack of invariance under reparameterization. This article 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 p-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.