Yule-Simpson's paradox in research

Yule (1903) and Simpson (1951) described a statistical paradox that occurs when data is aggregated. In such situations, aggregated data may reveal a trend that directly contrasts those of sub-groups trends. In fact, the aggregate data trends may even be opposite in direction of sub-group trends. To reveal Yule-Simpson's paradox (YSP)-type occurrences, researchers must simultaneously consider the effect of an intervention at specific levels and on the overall model to ensure datasets are accurately analyzed and research findings are appropriately interpreted. The primary objectives of this manuscript are to: (1) examine the history of YSP; (2) describe necessary and sufficient causes for YSP occurrences; (3) provide examples of YSP in research and explain YSP's relationship to multi-level modeling including Hierarchical Linear Modeling (HLM); and (4) discuss YSP's implications for researchers.