The Egger intercept (EI) test is a widely used tool to detect horizontal pleiotropy in two-sample summary-data Mendelian randomization. A significant test result suggests that either the average pleiotropic effect differs from zero (i.e., directional pleiotropy) or the InSIDE (Instrument Strength Independent of Direct Effect) assumption is violated (i.e., correlated pleiotropy) or both. As such, the EI test provides a diagnostic tool for assessing violations of the instrumental variable assumptions, as a nonzero intercept may indicate bias in the commonly used inverse-variance weighted (IVW) estimator. However, the EI test may exhibit inflated type I error rates due to estimation bias in Egger regression arising from measurement error and winner's curse. In this article, we propose a modified EI (MEI) test based on a bias-corrected EI estimator under the null hypothesis of no directional or correlated pleiotropy, leveraging the recently developed rerandomized IVW estimator. We then establish the asymptotic properties of the proposed test under appropriate conditions. Like the EI test, we find that the power of the MEI test is also affected by SNP orientation. To improve the robustness of power, we further develop strategies for selecting appropriate allele coding schemes. Both simulation and real-data studies demonstrate that the MEI tests outperform the EI test in terms of type I error control and power.

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