Cited by 2,301Probability and Statistical ResearchStatistical Methods and ApplicationsStatistical Methods and Bayesian Inference

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The Interpretation of Interaction in Contingency Tables

Simpson, E. H.·1951·Journal of the Royal Statistical Society Series B: Statistical Methodology

DOI 10.1111/j.2517-6161.1951.tb00088.x

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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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Crossref · 10.1111/j.2517-6161.1951.tb00088.x

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“Situations in which weakly consistent results appear reversed overall have been named the Yule-Simpson paradox (Yule 1900; Simpson 1951).”
Background
How to remove or control confounds in predictive models, with applications to brain biomarkers
Chyzhyk, D., Varoquaux, G., Milham, M., & Thirion, B. (2022) · GigaScience · doi:10.1093/gigascience/giac014 · Cited by 65
“Conditioning on the third variable z can then reverse the correlation between 2 variables X and y , a phenomenon known as the Berkson or Simpson statistical paradox [66,67].”
CAUSALITY, MEASUREMENT ERROR AND MULTICOLLINEARITY IN EPIDEMIOLOGY
Zidek, J. V., Wong, H., Le, N. D., & Burnett, R. T. (1996) · Environmetrics · doi:10.1002/(sici)1099-095x(199607)7:4 · Cited by 74
Taking the aggravation out of data aggregation: A conceptual guide to dealing with statistical issues related to the pooling of individual‐level observational data
Pollet, T. V., Stulp, G., Henzi, S. P., & Barrett, L. (2015) · American Journal of Primatology · doi:10.1002/ajp.22405 · Cited by 75
“A special case of the fallacy is known as Simpson’s paradox (SP) [Simpson, 1951], although Pearson et al. [1899] and most notably Yule [1900, 1903] pointed to this phenomenon earlier [Good & Mittal, 1987], hence it also known as the Yule– Simpson effect, the reversal paradox and the amalgamation…”
Background
Habitat Fragmentation Effects on Birds in Grasslands and Wetlands: A Critique of our Knowledge
Johnson, D. H. (2001) · Lincoln (University of Nebraska) · Cited by 74
Causal Reasoning for Algorithmic FairnessPreprint
Loftus, J. R., Russell, C., Kusner, M. J., & Silva, R. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1805.05859 · Cited by 163
“The Berkeley admissions example [3] is often used to explain Simpson’s paradox [38] and highlight the importance of adjusting for covariates.”
Method
Measuring relatedness between technological fields
Joo, S. H., & Kim, Y. (2009) · Scientometrics · doi:10.1007/s11192-009-0108-9 · Cited by 73
“Simpson’s paradox ( Simpson 1951 ) shows that the relatedness of i and j with the rest of the TFs, despite the assumption to the contrary, may affect results.”
Background
Causal Inference with Observational Data
Nichols, A. (2007) · The Stata Journal Promoting communications on statistics and Stata · doi:10.1177/1536867x0700700403 · Cited by 73
Criterion validity and relationships between alternative hierarchical dimensional models of general and specific psychopathology.
Moore, T. M., Kaczkurkin, A. N., Durham, E. L., Jeong, H. J., McDowell, M. G., Dupont, R. M., Applegate, B., Tackett, J. L., Cardenas-Iniguez, C., Kardan, O., et al. (2020) · Journal of Abnormal Psychology · doi:10.1037/abn0000601 · Cited by 72
On the Applicability of Machine Learning Fairness Notions
Makhlouf, K., Zhioua, S., & Catuscia Palamidessi, A. (2021) · ACM SIGKDD Explorations Newsletter · doi:10.1145/3468507.3468511 · Cited by 96
“Such scenario can lead to statistical anomalies such as Simpson’s paradox [87, 76] where the statistical conclusions drawn from the sub-populations differ from that from the whole population.”
Background
Bench-to-bedside review: Avoiding pitfalls in critical care meta-analysis – funnel plots, risk estimates, types of heterogeneity, baseline risk and the ecologic fallacy
Reade, M. C., Delaney, A. P., Bailey, M., & Angus, D. C. (2008) · Critical Care · doi:10.1186/cc6941 · Cited by 76
“[19], where (for example) a beneficial effect in each study can”
Background
A reversal paradox.
Messick, D. M., & de Geer, J. P. V. (1981) · Psychological Bulletin · doi:10.1037/0033-2909.90.3.582 · Cited by 71
The Cognitive-Ecological Approach to Rationality in Social Psychology
Fiedler, K., & Wänke, M. (2009) · Social Cognition · doi:10.1521/soco.2009.27.5.699 · Cited by 72
Plant individuality: a solution to the demographer’s dilemma
Clarke, E. (2012) · Biology & Philosophy · doi:10.1007/s10539-012-9309-3 · Cited by 71
“Now we might be perplexed—how can having a higher metabolic rate end up decreasing a ramet’s fitness on average? This case is supposed to be exactly analogous to a model for the evolution of altruism, and also to Simpson’s paradox (Simpson 1951).”
Temporal Anomaly Detection in Business Processes
Solti, A., & Kasneci, G. (2014) · Lecture notes in computer science · doi:10.1007/978-3-319-10172-9_15 · Cited by 71
Standardization and Control for Confounding in Observational Studies: A Historical Perspective
Keiding, N., & Clayton, D. G. (2014) · Statistical Science · doi:10.1214/13-sts453 · Cited by 68
“A landmark paper was that of Simpson (1951) which dealt with the problem of interpreting associations in three-way contingency tables.”
Background
Evaluating measurement equivalence using the item response theory log-likelihood ratio (IRTLR) method to assess differential item functioning (DIF): applications (with illustrations) to measures of physical functioning ability and general distress
Teresi, J. A., Ocepek‐Welikson, K., Kleinman, M., Cook, K. F., Crane, P. K., Gibbons, L. E., Morales, L. S., Edelen, M. O., & Cella, D. F. (2007) · Quality of Life Research · doi:10.1007/s11136-007-9186-4 · Cited by 70
“The necessity of a conditioning variable has been illustrated by Dorans and Holland [18] and Dorans and Kulick [19], in the context of Simpson’s [ 20 ] paradox.”
Performance analysis of novel methods for detecting epistasis
Shang, J., Zhang, J., Sun, Y., Liu, D., Ye, D., & Yin, Y. (2011) · BMC Bioinformatics · doi:10.1186/1471-2105-12-475 · Cited by 70
“In addition, both methods are based on contingency table, which reflects (direct-) dependence structure between two or more variables, and has been proven to be effective in identifying interactions [51-53].”
Method
Alternative Perspectives on the Biased Foundations of Medical Technology Assessment
Diamond, G. A., & Denton, T. A. (1993) · Annals of Internal Medicine · doi:10.7326/0003-4819-118-6-199303150-00010 · Cited by 70
Recent advances in thyroid hormone regulation: toward a new paradigm for optimal diagnosis and treatment
Hoermann, R., Midgley, J. E. M., Larisch, R., & Dietrich, J. W. (2017) · Frontiers in Endocrinology · doi:10.3389/fendo.2017.00364 · Cited by 69
“The heterogeneity bias is known as Simpson’s amalgamation paradox (75, 76).”
Background
Collaborative diversity in a nanotechnology innovation system: Evidence from the EU Framework Programme
Pandža, K., Wilkins, T., & Alfoldi, E. A. (2011) · Technovation · doi:10.1016/j.technovation.2011.05.003 · Cited by 69
“We adapted the notion of biological diversity (Simpson, 1951) to construct the measurement of institutional diversity.”
Method
When Does Repeat Burglary Victimisation Occur?
Sagovsky, A., & Johnson, S. D. (2007) · Australian & New Zealand Journal of Criminology · doi:10.1375/acri.40.1.1 · Cited by 69
“However, this simple analysis is an example of Simpson’s Paradox (Simpson, 1951) for which ‘the nature of an association can change, and even reverse direction, when data from several groups are combined to form a single group’ (Moore, 1997, p. 290).”
Method
What Is Evolutionary Altruism?
Sober, E. R. (1988) · Canadian Journal of Philosophy Supplementary Volume · doi:10.1080/00455091.1988.10715945 · Cited by 69
Separating Causal Laws from Casual Facts: Pressing the Limits of Statistical Relevance
Cheng, P. W. (1993) · The Psychology of learning and motivation/The psychology of learning and motivation · doi:10.1016/s0079-7421(08)60298-4 · Cited by 69
Academics as Agentic Superheroes: Female academics’ lack of fit with the agentic stereotype of success limits their career advancement
van Veelen, R., & Derks, B. (2021) · British Journal of Social Psychology · doi:10.1111/bjso.12515 · Cited by 91

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
DOI: 10.1111/j.2517-6161.1951.tb00088.x
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

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The Interpretation of Interaction in Contingency Tables

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