Comment: Understanding Simpson’s Paradox

I thank the editor, Ronald Christensen, for the opportunity to discuss this important topic and to comment on the article by Armistead. Simpson’s paradox is often presented as a compelling demonstration of why we need statistics education in our schools. It is a reminder of how easy it is to fall into a web of paradoxical conclusions when relying solely on intuition, unaided by rigorous statistical methods. In recent years, ironically, the paradox assumed an added dimension when educators began using it to demonstrate the limits of statistical methods, and why causal, rather than statistical considerations are necessary to avoid those paradoxical conclusions (Wasserman 2004; Arah 2008; Pearl 2009, pp. 173–182). My comments are divided into three parts. First, I will give a brief summary of the history of Simpson’s paradox and how it has been treated in the statistical literature in the past century. Next, I will ask what is required to declare the paradox “resolved,” and argue that modern understanding of causal inference has met those requirements. Finally, I will answer specific questions raised in Armistead’s article and show how the resolution of Simpson’s paradox can be taught for fun and progress.