Simpson's Paradox is suppression, but Lord's Paradox is neither: clarification of and correction to Tu, Gunnell, and...

Tu et al. (Emerg Themes Epidemiol 5:2, 2008. https://doi.org/10.1186/1742-7622-5-2) asserted that suppression, Simpson's Paradox, and Lord's Paradox are all the same phenomenon-the reversal paradox. In the reversal paradox, the association between an outcome variable and an explanatory (predictor) variable is reversed when another explanatory variable is added to the analysis. More specifically, Tu et al. (2008) purported to demonstrate that these three paradoxes are different manifestations of the same phenomenon, differently named depending on the scaling of the outcome variable, the explanatory variable, and the third variable. According to Tu et al. (2008), when all three variables are continuous, the phenomenon is called suppression; when all three variables are categorical, the phenomenon is called Simpson's Paradox; and when the outcome variable and the third variable are continuous but the explanatory variable is categorical, the phenomenon is called Lord's Paradox. We show that (a) the strong form of Simpson's Paradox is equivalent to negative suppression for a $$ contingency table, (b) the weak form of Simpson's Paradox is equivalent to classical suppression for a $$ contingency table, and (c) Lord's Paradox is not the same phenomenon as suppression or Simpson's Paradox.