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.

Keyword:

• Iteratively reweighted least squares algorithm
• p-value
• Regularity conditions
• Significance testing
• Vector generalized linear model

Supplementary Materials

The supplementary materials include specific HDE details needed for the zero-inflated Poisson and cumulative logit model as examples, sandwich estimators, multiple testing, the Cox proportional hazards model, and profile likelihoods. It is shown that it is possible to determine whether the HDE will occur for a binary covariate in a logistic regression. An R script file is also included to run the examples using the VGAM package.

Acknowledgments

Thanks are extended to the Centre for Applied Statistics and School of Mathematics and Statistics at the University of Western Australia for hospitality during a workshop given there and for valuable feedback from participants that helped lead to this work. Constructive comments from the editor, an associate editor, reviewers, George Seber and Elbert Chia are gratefully acknowledged. This article is dedicated to the memory of Prof. Alastair J. Scott FASA FRSNZ (1939–2017), a real statistics professor at the University of Auckland. SDG.