Lord’s Paradox Revisited – (Oh Lord! Kumbaya!)

Among the many peculiarities that were dubbed “paradoxes” by well meaning statisticians, the one reported by Frederic M. Lord in 1967 has earned a special status. Although it can be viewed, formally, as a version of Simpson’s paradox (Arah, 2008; Tu et al., 2008; Pearl, 2014b) its reputation has gone much worse. Unlike Simpson’s reversal, Lord’s is easier to state, harder to disentangle (Wainer and Brown, 2007) and, for some reason, it has been lingering for almost four decades, under several interpretations and re-interpretations (Holland and Rubin, 1983), and it keeps coming up in new situations and under new lights (van Breukelen, 2013; Senn, 2006; Eriksson and Häggström, 2014). Most peculiar yet, while some of its variants has received a satisfactory resolution (Glymour, 2006; Hernández-Dı́az et al., 2006), the original version presented by Lord, to the best of my knowledge, has not been given a proper treatment, not to mention a resolution. The purpose of this paper is to trace back Lord’s paradox from its original formulation, resolve it using modern tools of causal analysis, explain why it resisted prior attempts at resolution and, finally, address the general methodological issue of whether adjustments for pre-existing conditions is justified in group comparison applications. 1 Lord’s original dilemma Any attempt to describe Lord’s paradox in words other than those used by Lord himself can only do injustice to the clarity and freshness with which it was first enunciated in 1967. We will begin therefore by listening to Lord’s own words. “A large university is interested in investigating the effects on the students of the diet provided in the university dining halls and any sex difference in these effects. Various types of data are gathered. In particular, the weight of each student at the time of his arrival in September and his weight the following June are recorded. 1 TECHNICAL REPORT R-436 July 2016