Estimating Effect Size for Mood’s Median Test

Effect-size estimation for Mood’s median test has received relatively little methodological attention despite the test’s widespread use in robust and nonparametric analysis. This study evaluates four candidate effect-size estimators: the median absolute deviation-based estimator (Delta–MAD), the probability of superiority (PS), Cramér’s V, and a newly proposed bootstrap-standardized median difference (Delta-Boot) across simulation settings involving normal data with equal variances, log-normal skewness, and heteroscedasticity with a twofold variance difference. Under equal variances, PS achieved the highest classification accuracy for moderate and large effects, with Delta–MAD and Delta–Boot close behind and Cramér’s V performing worst. Performance under log-normal skewness was nearly unchanged, demonstrating the robustness of median- and rank-based methods to heavy right-tailed distributions. Notably, Delta–Boot began to show improved performance for moderate effect sizes in the log-normal setting. Under heteroscedasticity, estimator behaviour diverged sharply: PS remained highly effective for distinguishing no and large effects but showed reduced accuracy for moderate effects due to its sensitivity to spread differences; Cramér’s V degraded substantially across all effect sizes; and the two median-standardized estimators—especially Delta–Boot—were more resilient, stabilizing more rapidly with increasing sample size and achieving the highest accuracy for moderate and large shifts at larger n. These patterns indicate that PS (or Delta–MAD) is most appropriate when variances are equal or nearly so, whereas Delta–Boot provides the most reliable performance in settings where variance imbalance is likely. Finally, a real-world application to fasting glucose data from the 2024 WHO STEPS survey in Trinidad and Tobago illustrates the practical utility of these approaches.