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The Interpretation of Interaction in Contingency Tables
Abstract
Summary The definition of second order interaction in a (2 × 2 × 2) table given by Bartlett is accepted, but it is shown by an example that the vanishing of this second order interaction does not necessarily justify the mechanical procedure of forming the three component 2 × 2 tables and testing each of these for significance by standard methods.
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“Indeed, the relationship between body composition and BMI represents a clear example of Simpson’s paradox (Simpson, 1951).”
From The degree to which the cultural ideal is internalized predicts judgments of male and female physical attractiveness (2022) · cites this paper as Simpson, 1951
In the citing paper
“The classical example demonstrating the obstacle of causal vocabulary is Simpson's paradox (Simpson, 1951) - a reversal phenomenon that earns its claim to fame only through a causal interpretation of the data (Pearl, 2000a, Chapter 6).”
From Causal inference in statistics: An overview (2009) · cites this paper as Simpson, 1951
In the citing paper
“Our work extends this to the causal domain, showing that models will endorse logical traps (like Simpson's Paradox (Simpson, 1951)) if pressed.”
From Diagnosing and Mitigating Sycophancy and Skepticism in LLM Causal Judgment (2026) · cites this paper as Simpson, 1951
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“For instance, upon observing that more men than women are scientists, one should eliminate differences in socialization before concluding genetic differences as the cause (see also Simpson’s paradox; Simpson, 1951).”
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“We elected not to include BMI at age 7 yr in our primary model, because of what is commonly known in the statistical literature as the “reversal paradox” (20).”
MethodAn Introduction to Causal InferencePreprint
Dablander, F. (2020) · doi:10.31234/osf.io/b3fkw · Cited by 35
“While Karl Pearson has been aware of similar effects already in 1899 (Aldrich, 1995), it was the article by Simpson (1951) which drew renewed attention to this fact; Blyth (1972) was the first to call it a “paradox”.”
BackgroundInconvenient truths about logistic regression and the remedy of marginal effects
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“14 Result 3a is a special case of the Simpson (1951) Paradox: because of the improved shooting ability of the HH player, he takes all the difficult shots (while sharing with the AH player the easy shots).”
BackgroundEstimating the causal effects of smoking
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“But, do we really have to work this hard? If smokers on average have more expenditures than never-smokers, does this not suggest that smokers would expend more on average even after adjustment for background characteristics? Although intuitively plausible, this claim is incorrect, and extreme counterexamples are somet…”
ResultCondon, D. E., Tran, P. V., Lien, Y., Schug, J., Georgieff, M., Simmons, R. A., & Won, K. (2018) · BMC Bioinformatics · doi:10.1186/s12859-018-2037-1 · Cited by 36
“The coverage-weighted means used in WWE allow Defiant to avoid artifacts such as Simpson’s paradox [49].”
MethodThe rising share of nonmarital births: Fertility choice or marriage behavior?
Gray, J. A., Stockard, J., & Stone, J. A. (2006) · Demography · doi:10.1353/dem.2006.0012 · Cited by 34
“The birth rate behavior described here is an example of Simpson’s Paradox (Simpson 1951).”
BackgroundCombined Linkage and Association Tests in Mx
Posthuma, D., de Geus, E. D., Boomsma, D., & Neale, M. (2004) · Behavior Genetics · doi:10.1023/b:bege.0000013732.19486.74 · Cited by 32
“This statistical effect is known as Simpson’s paradox (Simpson, 1951; Yule, 1900) and refers to the reversal of the direction of an association when data from several groups are combined to form a single group.”
BackgroundCommentary: Individual, ecological and multilevel fallacies
Oakes, J. M. (2009) · International Journal of Epidemiology · doi:10.1093/ije/dyn356 · Cited by 34
Lee, K., Ng, S. F., & Bull, R. (2018) · Developmental Psychology · doi:10.1037/dev0000561 · Cited by 33
Comment: Understanding Simpson’s Paradox
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Fradin, D., & Bougneres, P. F. (2011) · Journal of Nutrition and Metabolism · doi:10.1155/2011/647514 · Cited by 34
“Introducing epigenetic disease markers in epidemiological studies will face the general Simpson’s paradox [207], a statistical phenomenon in which marginal effects, for example, effects associated with a given factor, such as genetic or epigenetic variants can be masked, enhanced, or even reversed in the presence of i…”
BackgroundOrnaghi, V., Pepe, A., & Grazzani, I. (2016) · Frontiers in Psychology · doi:10.3389/fpsyg.2016.01534 · Cited by 33
The relationship between latitude, migration and the evolution of bird song complexity
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“This could be a telling symptom of an ecological fallacy, where this apparent ‘effect of scale’ could be due to one or several causal variables going unmeasured (Simpson 1951, Selvin 1958).”
BackgroundCurrent review and next steps for artificial intelligence in multiple sclerosis risk research
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“The most dramatic effect of this can be seen when the ‘correct’ results are completely reversed due to overlooking the confounder; this is called Simpson’s Paradox [127].”
BackgroundWismans, A., Letina, S., Wennberg, K. J., Thurik, R., Baptista, R., Burke, A., Dejardin, M., Janssen, F., Santarelli, E., Torrès, O., et al. (2021) · Personality and Individual Differences · doi:10.1016/j.paid.2021.110925 · Cited by 33
“…sample did not provide a strong indication of the existence of opposite relationships within groups, something which is known as Simpson’s paradox (Simpson, 1951), there could be other unobserved factors that affect the relationship between compliance and discount rate in different subgroups in…”
Bodendorf, F., Sauter, M., & Franke, J. E. (2022) · International Journal of Production Economics · doi:10.1016/j.ijpe.2022.108708 · Cited by 40
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Cross‐lags and the unbiased estimation of life‐history and demographic parameters
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Snyder, J. J., Musgrove, D. R., Zaun, D. A., Wey, A. R., Salkowski, N., Rosendale, J. D., Israni, A. K., Hirose, R., & Kasiske, B. L. (2020) · American Journal of Transplantation · doi:10.1111/ajt.15842 · Cited by 32
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Capitalizing on Variation: How Conflicting Research Findings Can Be Helpful for Policy
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“This phenomenon is known as Simpson’s paradox (Simpson 1951) and is due to the fact that the distribution of self assessed health status is very di ff erent for Web and phone respondents.”
BackgroundInvited Commentary: The Perils of Birth Weight--A Lesson from Directed Acyclic Graphs
Wilcox, A. J. (2006) · American Journal of Epidemiology · doi:10.1093/aje/kwj276 · Cited by 32
“(Indeed, the birth weight paradox has generated just such discussions (7).) Alternatively, we can see the figure as an illustration of Simpson’s Paradox ( 8 ), in which a factor associated with risk within subsets of a population has the opposite association for the population as a whole.”
BackgroundKaufmann, E., Reips, U.-D., & Maag Merki, K. (2016) · Zeitschrift für Psychologie · doi:10.1027/2151-2604/a000251 · Cited by 31
“The Prevention of Simpson’s Paradox The IPD approach avoids a second type of ecological fallacy aggregation bias, namely, Simpson’s paradox (Simpson, 1951), which occurs when the heterogeneity in the population is underestimated.”
MethodMigration from developing countries: Selection, income elasticity, and Simpson’s paradox
Clemens, M. A., & Mendola, M. (2024) · Journal of Development Economics · doi:10.1016/j.jdeveco.2024.103359 · Cited by 31
Dhillon, M. K., & Sharma, H. C. (2013) · PubMed · Cited by 30
Why multilevel selection matters
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Robust Text Classification in the Presence of Confounding Bias
Landeiro, V., & Culotta, A. (2016) · Proceedings of the AAAI Conference on Artificial Intelligence · doi:10.1609/aaai.v30i1.9997 · Cited by 34
“As another way of considering the effect of BA, recall the notion of Simpson’s paradox (Simpson 1951).”
BackgroundMaximizing Growth Yield and Dispersal via Quorum Sensing Promotes Cooperation in Vibrio Bacteria
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“This is known as Simpson (1951)’s paradox.”
Do Places Have Value?: Quantifying the Intrinsic Value of Housing Neighborhoods Using MGWR
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Iterating on a single model is a viable alternative to multimodel inference
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Verification: Assessment of Calibration and Accuracy
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“While it is common and advisable to look at a model’s performance in certain weather situations or for certain periods of time, it is important to be aware of Simpson’s paradox Verification: assessment of calibration and accuracy 26 (Simpson, 1951).”
BackgroundApplications for single-cell and spatial transcriptomics in plant research
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Homophily and the Speed of Social Mobilization: The Effect of Acquired and Ascribed Traits
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“These interactions of recruiter and recruit age are an instance of the Yule-Simpson paradox [33,34], in which two variables viewed in isolation appear to have one set of behaviors, but their interaction reveals that they in fact have an opposite set of behaviors.”
BackgroundSimpson's paradox: how performance measurement can fail even with perfect risk adjustment
de Mheen, P. J. M., & Shojania, K. G. (2014) · BMJ Quality & Safety · doi:10.1136/bmjqs-2014-003358 · Cited by 30
On the importance of thinking locally for statistics and society
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On Educating about Medical Data Management
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“This example is based on a similar one in [6] and refers to effects already described in the 1950s by Edward Hugh Simpson ([21], see also [22-24]).”
MethodZakko, S. F., Barton, G., Weber, E., Dunger‐Baldauf, C., & Rühl, A. (2011) · Alimentary Pharmacology & Therapeutics · doi:10.1111/j.1365-2036.2011.04656.x · Cited by 30
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Causal regression models I: Individual and average causal effects
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The logic of Simpson’s paradox
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Milligan, G. W. (1980) · Psychological Bulletin · doi:10.1037//0033-2909.87.2.238 · Cited by 30
Limits to Prediction of Phenotypes from Knowledge of Genotypes
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DECAF: Generating Fair Synthetic Data Using Causally-Aware Generative NetworksPreprint
van Breugel, B., Kyono, T., Berrevoets, J., & van der Schaar, M. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2110.12884 · Cited by 30
“For example, in Simpson’s paradox [33] there seems to be a bias between gender and college admissions, but this is only due to women applying to more competitive courses.”
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Tsai, C., & Camargo, C. A. (2009) · Journal of Clinical Epidemiology · doi:10.1016/j.jclinepi.2008.10.005 · Cited by 29
Disentangling Intrinsic and Extrinsic Gene Expression Noise in Growing Cells
Lin, J., & Amir, A. (2021) · Physical Review Letters · doi:10.1103/physrevlett.126.078101 · Cited by 29
Simpson’s Paradox in Natural Resource Evaluation
Zee, Y. (2008) · Mathematical Geosciences · doi:10.1007/s11004-008-9187-z · Cited by 29
“Blyth (1972) described the boundary conditions for Simpson’s reversal using conditional probabilities and Pearl (2000) discussed them from a causal perspective.”
BackgroundSmall-Area Racial Disparity in Stroke Mortality
Tassone, E. C., Waller, L. A., & Casper, M. L. (2009) · Epidemiology · doi:10.1097/ede.0b013e3181935aee · Cited by 29
Statistical Considerations in Meta-analysis
Barza, M. J., Trikalinos, T. A., & Lau, J. (2009) · Infectious Disease Clinics of North America · doi:10.1016/j.idc.2009.01.003 · Cited by 29
DeYoung, C. G., & Blain, S. D. (2020) · Cambridge University Press eBooks · doi:10.1017/9781108264822.026 · Cited by 34
Qualitative or Quantitative Differences?
Yang, X., Shaftel, J., Glasnapp, D. R., & Poggio, J. P. (2005) · The Journal of Special Education · doi:10.1177/00224669050380040101 · Cited by 29
Hook, E. B., & Regal, R. R. (1991) · Teratology · doi:10.1002/tera.1420430107 · Cited by 29
Sample size, aggregation, and statistical reasoning in social inference
Schaller, M. (1992) · Journal of Experimental Social Psychology · doi:10.1016/0022-1031(92)90032-f · Cited by 29
Single-Cell Sequencing in Precision Medicine
Wiedmeier, J. E., Noel, P., Lin, W., Hoff, D. D. V., & Han, H. (2019) · Cancer treatment and research · doi:10.1007/978-3-030-16391-4_9 · Cited by 31
Nieuwenhuis, R., Lancker, W. V., Collado, D., & Cantillon, B. (2019) · Italian Economic Journal · doi:10.1007/s40797-019-00115-x · Cited by 28
“The paradox between women’s employment growth and trends in poverty, as introduced above, is a typical example of an aggregation paradox (Yule 1903; Simpson 1951).”
BackgroundCausal Collaborative Filtering
Xu, S., Ge, Y., Li, Y., Fu, Z., Chen, X., & Zhang, Y. (2023) · ACM SIGIR International Conference on Theory of Information Retrieval (ICTIR) · doi:10.1145/3578337.3605122 · Cited by 43
“This phenomenon was originally observed in 1951 [47] and was later named as the “Simpson’s Paradox”[6].”
MethodRosenthal, L. M., Simler‐Williamson, A. B., & Rizzo, D. M. (2021) · Ecology Letters · doi:10.1111/ele.13871 · Cited by 29
Estimation of Local Employment Growth: Do Sectoral Aggregation and Industry Definition Matter?
Mameli, F., Faggian, A., & McCann, P. (2013) · Regional Studies · doi:10.1080/00343404.2012.756578 · Cited by 28
Robust discovery of local patterns
Hopstadius, J., & Norén, G. N. (2012) · International Health Informatics Symposium · doi:10.1145/2110363.2110395 · Cited by 28
Much Pain, Little Gain? Paradigm-Specific Models and Methods in Experimental Psychology
Meiser, T. (2011) · Perspectives on Psychological Science · doi:10.1177/1745691611400241 · Cited by 28
“Figure 1B therefore forms an instance of Simpson’s paradox (Simpson, 1951).”
BackgroundEvolution of Ego-networks in Social Media with Link Recommendations
Aiello, L. M., & Barbieri, N. (2017) · ACM International Conference on Web Search and Data Mining (WSDM) · doi:10.1145/3018661.3018733 · Cited by 29
“This problem is known as the Simpson’s Paradox [46] and it is usually addressed by fixing the sample set [12].”
BackgroundBennett, P. R., & Xie, Y. (2003) · American Sociological Review · doi:10.1177/000312240306800404 · Cited by 28
“The supposition that the net black advantage is due to blacks' enrollment in HBCUs is essentially a hypothesis about an aggregation error (also called Simpson's paradox), in which the direction of a relationship between two variables reverses when data are aggregated over a third dimension (Simpson 1951).”
BackgroundGroup selection and the evolution of altruism
Cooper, B. (2004) · Oxford Economic Papers · doi:10.1093/oep/gpf043 · Cited by 25
“Naturally, although each group sees a fall in the proportion of altruists, the population as a whole may see a rise (see Simpson, 1951).”
BackgroundCollapsibility for Directed Acyclic Graphs
Xie, X., & Geng, Z. (2009) · Scandinavian Journal of Statistics · doi:10.1111/j.1467-9469.2008.00627.x · Cited by 27
“This phenomenon is known as the YuleSimpson paradox (Yule, 1903; Simpson, 1951).”
BackgroundJudicious Judgment Meets Unsettling Updating: Dilation, Sure Loss and Simpson’s Paradox
Gong, R., & Meng, X. (2021) · Statistical Science · doi:10.1214/19-sts765 · Cited by 27
A revisit of global wind-sea and swell climate and variability using multiplatform altimeters
Jiang, H., & Zheng, Y. (2022) · Remote Sensing of Environment · doi:10.1016/j.rse.2022.112922 · Cited by 27
Calanchini, J., Hehman, E., Ebert, T., Wilson, L., Simon, D., & Wilson, L. (2022) · Advances in experimental social psychology · doi:10.1016/bs.aesp.2022.04.003 · Cited by 46
Massa, M. S., Wang, N., Bickerton, W.-L., Demeyere, N., Riddoch, M. J., & Humphreys, G. W. (2015) · Cortex · doi:10.1016/j.cortex.2015.06.006 · Cited by 27
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Repetition effects in grasping.
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“Under these circumstances, simple calculations of grip proportion can lead to averaging artifacts such as Simpson’s paradox (Simpson, 1951) in which overall proportions do not reflect the patterns observed for individual subjects.”
BackgroundEnvironmental and genetic perturbations reveal different networks of metabolic regulation
Greenberg, A. J., Hackett, S. R., Harshman, L. G., & Clark, A. G. (2011) · Molecular Systems Biology · doi:10.1038/msb.2011.96 · Cited by 27
“Pooling groups is also problematic because it risks creating spurious relationships (an instance of Simpson’s paradox; Simpson, 1951).”
Large‐scale phenotyping links adult hippocampal neurogenesis to the reaction to novelty
van Dijk, R. M., Lazic, S. E., Slomianka, L., Wolfer, D. P., & Amrein, I. (2015) · Hippocampus · doi:10.1002/hipo.22548 · Cited by 26
“In our own data, between-group differences can show a reversed relationship compared to the within-group analyses (Simpson’s effect: Simpson, 1951), i.e., C57BL/6 mice with overall more neurogenesis have a higher frequency of pokes compared to DBA mice (Fig.”
ResultAsaduzzaman, M., Luckett, D. J., Cowley, R. B., An, M., Pratley, J. E., & Lemerle, D. (2014) · Biocontrol Science and Technology · doi:10.1080/09583157.2014.942596 · Cited by 27
“Weed diversity and frequency were summarised using Simpson’s Diversity Index (SDI; Simpson, 1951).”
MethodExplainable and Non-explainable Discrimination in Classification
Kamiran, F., & Žliobaitė, I. (2012) · Studies in applied philosophy, epistemology and rational ethics · doi:10.1007/978-3-642-30487-3_8 · Cited by 27
“The light grey area illustrates the Simpson’s paradox (Simpson, 1951), in which a relation exists in different groups is reversed when the groups are combined.”
BackgroundMethods to Quantify Regional Differences in Land Cover Change
Comber, A., Balzter, H., Cole, B., FISHER, P. F., Johnson, S., & Ogutu, B. (2016) · Remote Sensing · doi:10.3390/rs8030176 · Cited by 27
“As a result, the odds ratios may present an example of Simpson’s paradox [26], where different rates (and directions) of per class changes may be masked by the aggregate gross changes.”
BackgroundInférence statistique, échangeabilité et approche mutiniveau
Courgeau, D. (2007) · Mathématiques et sciences humaines · doi:10.4000/msh.6793 · Cited by 27
Tangen, C. M., & Koch, G. G. (1999) · Journal of Biopharmaceutical Statistics · doi:10.1081/bip-100100999 · Cited by 27
Contingency matrix theory: Statistical dependence in a contingency table
Tsumoto, S. (2008) · Information Sciences · doi:10.1016/j.ins.2008.11.023 · Cited by 27
“The second important observation is that if the rank of a matrix is not full, merging columns or rows will generate a new table with statistical independence, which exactly corresponds to Simpson’s paradox.”
BackgroundResultInfluentialA Product-Quality View of the Linder Hypothesis
Hallak, J. C. (2006) · National Bureau of Economic Research · doi:10.3386/w12712 · Cited by 27
Brunsdon, C., & Comber, A. (2012) · GeoInformatica · doi:10.1007/s10707-012-0159-6 · Cited by 26
“This is an example of Simpson’s Paradox [33].”
MethodCoexistence and cooperation in structured habitats
Geyrhofer, L., & Brenner, N. (2020) · BMC Ecology · doi:10.1186/s12898-020-00281-y · Cited by 24
“(4) allows to encode effects that have been termed Simpson’s paradox (Blyth, 1972; Simpson, 1951).”
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Metadata
- Authors
- Simpson, E. H.
- Corresponding author
- Simpson, E. H.
- Venue
- Journal of the Royal Statistical Society Series B: Statistical Methodology
- Publisher
- Oxford University Press (OUP)
- Published
- 1951-07-01
- Volume
- 13
- Issue
- 2
- Pages
- 238-241
- ISSN
- 1369-7412 (ISSN-L)
- 1467-9868
- OpenAlex
- W2905472553
- Cited by
- 2,301
- References
- 6
- Citations by year
- 2026: 92 · 2025: 136 · 2024: 122 · 2023: 113 · 2022: 116 · 2021: 121 · 2020: 165 · 2019: 101 · 2018: 94 · 2017: 78 · 2016: 87 · 2015: 84 · 2014: 75 · 2013: 57 · 2012: 109
- FWCI
- 0.00
- Type
- journal article
- Language
- en
- Open access
- closed
- Licence
- academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model (vor)
- Locations
- Journal of the Royal Statistical Society Series B (Statistical Methodology) (publishedVersion)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Probability and Statistical Research
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Applications
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Bayesian Inference
- Keywords
- second-order interactions, three-way contingency tables, marginal homogeneity, interaction testing, categorical data analysis
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
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