Revisiting the Wald Test in Small Case-Control Studies With a Skewed Covariate

The Wald test is routinely used in case-control studies to test for association between a covariate and disease. However, when the evidence for association is high, the Wald test tends to inflate small P values as a result of the Hauck-Donner effect (HDE). Here, we investigate the HDE in the context of genetic burden, both with and without additional covariates. First, we examine the burden-based P values in the absence of association using whole-exome sequence data from 1000 Genomes Project reference samples (n = 54) and selected preterm infants with neonatal complications (n = 74). Our careful analysis of the burden-based P values shows that the HDE is present and that the cause of the HDE in this setting is likely a natural extension of the well-known cause of the HDE in 2 x 2 contingency tables. Second, in a reanalysis of real data, we find that the permutation test provides increased power over the Wald, Firth, and likelihood ratio tests, which agrees with our intuition since the permutation test is valid for any sample size and since it does not suffer from the HDE. Therefore, we propose a powerful and computationally efficient permutation-based approach for the analysis and reanalysis of small case-control association studies.