The occurrence of Simpson's paradox (SP) in $2\times 2$ contingency tables has been well studied. The present work comprehensively revisits this problem using a combination of philosophical reflections, causal considerations, and probability theory. The first contribution is to provide a schematic analysis of SP in $2\times 2$ contingency tables and present new results, detailed proofs of previous results and a unifying view of the important examples of SP that have been reported in the literature. The second contribution of the paper suggests a new perspective on the surprise element of SP, raises some critical questions regarding the influential causal analyses of SP and provides a broad perspective on logic, probability, and statistics with SP at its center. The upshot of this research is that we need both causal concepts and statistical tools coupled with philosophical analyses to sort out issues regarding SP.
翻译:对辛普森在2美元应急表格中的悖论(SP)的出现进行了认真的研究,目前的工作利用哲学思考、因果考虑和概率理论的结合,全面重新审视了这一问题。第一个贡献是在2美元应急表格中提供对SP的示意图分析,并提出新的结果、以往结果的详细证明以及对文献中报道的SP重要例子的统一看法。文件的第二稿对SP的突袭要素提出了新的观点,提出了关于SP具有影响力的因果分析的一些关键问题,并对SP的逻辑、概率和统计数字提供了广泛的视角。这项研究的结果是,我们需要有因果概念和统计工具,同时进行哲学分析,以理清与SP有关的问题。