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.

Integrity signals

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

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BackgroundResult
An internally replicated quasi-experimental comparison of checklist and perspective based reading of code documents
Laitenberger, O., Emam, K. E., & Harbich, T. (2001) · IEEE Transactions on Software Engineering · doi:10.1109/32.922713 · Cited by 117
“However, it is known that pooling data from different studies can mask even strong effects [79][90], making it much preferable to combine the results through a meta-analysis [50].”
Background
Beyond social capital: The norms, belief systems, and agency embedded in social networks shape resilience to climatic and geophysical hazards
MacGillivray, B. H. (2018) · Environmental Science & Policy · doi:10.1016/j.envsci.2018.07.014 · Cited by 135
Performance of the international physical activity questionnaire (short form) in subgroups of the Hong Kong chinese population
Lee, P. H., Yu, Y., McDowell, I., Leung, G. M., Lam, T., & Stewart, S. M. (2011) · International Journal of Behavioral Nutrition and Physical Activity · doi:10.1186/1479-5868-8-81 · Cited by 120
“To rule out Simpson’s paradox [28] (i.”
Two New Approaches to Assessing Differential Item Functioning: Standardization and the Mantel--Haenszel Method
Dorans, N. J. (1989) · Applied Measurement in Education · doi:10.1207/s15324818ame0203_3 · Cited by 116
The Hot (Invisible?) Hand: Can Time Sequence Patterns of Success/Failure in Sports Be Modeled as Repeated Random Independent Trials?
Yaari, G., & Eisenmann, S. (2011) · PLoS ONE · doi:10.1371/journal.pone.0024532 · Cited by 116
“But, as we learned from the Simpson’s Paradox, this alone is not enough to declare that there is a tendency for a player to improve with consecutive free throw attempts (NS) nor that there is a better chance to score after a success in the previous attempt (CP).”
BackgroundInfluential
The individualistic fallacy, ecological studies and instrumental variables: a causal interpretation
Loney, T., & Nagelkerke, N. J. D. (2014) · Emerging Themes in Epidemiology · doi:10.1186/1742-7622-11-18 · Cited by 116
“Although originally formulated with nuance, the Simpson’s “paradox” which has also become part of the canon of epidemiology, has strengthened this attitude [7].”
Background
Plant Systems Biology at the Single-Cell Level
Libault, M., Pingault, L., Zogli, P., & Schiefelbein, J. (2017) · Trends in Plant Science · doi:10.1016/j.tplants.2017.08.006 · Cited by 114
Single-cell RNA-sequencing of peripheral blood mononuclear cells reveals widespread, context-specific gene expression regulation upon pathogenic exposure
Oelen, R., de Vries, D. H., Brugge, H., Gordon, M. G., Vochteloo, M., Ye, C., Westra, H.-J., Franke, L., & van der Wijst, M. G. P. (2022) · Nature Communications · doi:10.1038/s41467-022-30893-5 · Cited by 120
“As a consequence, the co-expression between two genes as calculated from bulk RNA-seq data may be different from the true individual-speci ﬁ c co-expression relationship as extracted from scRNA-seq data (due to Simpson ’ s paradox 42 ).”
Method
Dental transposition as a disorder of genetic origin
Ely, N. J., Sherriff, M., & Cobourne, M. T. (2005) · European Journal of Orthodontics · doi:10.1093/ejo/cji092 · Cited by 114
Frankly, We Do Give a Damn
Feldman, G., Lian, H., Kosiński, M., & Stillwell, D. J. (2017) · Social Psychological and Personality Science · doi:10.1177/1948550616681055 · Cited by 119
“Finally, Simpson’s Paradox (Simpson, 1951) points to conceptual and empirical differences in testing a relationship on different levels of analysis, and therefore the state-level findings of Study 3 are conceptually broader than the findings in Studies 1 and 2.”
Result
Simpson's paradox visualized: The example of the Rosiglitazone meta-analysis
Rücker, G., & Schumacher, M. C. (2008) · BMC Medical Research Methodology · doi:10.1186/1471-2288-8-34 · Cited by 117
Roles of alcohol consumption in fatty liver: A longitudinal study
Moriya, A., Iwasaki, Y., Ohguchi, S., Kayashima, E., Mitsumune, T., Taniguchi, H., Ando, M., & Yamamoto, K. (2014) · Journal of Hepatology · doi:10.1016/j.jhep.2014.11.025 · Cited by 112
“There seems to be a discrepancy between their results and ours, which however may be explained by Simpson’s paradox [42].”
Result
Standardizing fishery-dependent catch and effort data in complex fisheries with technology change
Bishop, J. (2006) · Reviews in Fish Biology and Fisheries · doi:10.1007/s11160-006-0004-9 · Cited by 112
“Such examples of confounding have been known as Simpson’s Paradox (Simpson 1951); ‘‘the aggregation paradox’’ (MacKinnon and Glick 1999); ‘‘the aggregation problem’’ by geographers and demographers (Duncan et al.”
Background
Interactions in Multi-Factor Contingency Tables
Darroch, J. N. (1962) · Journal of the Royal Statistical Society Series B (Statistical Methodology) · doi:10.1111/j.2517-6161.1962.tb00457.x · Cited by 112
An integrative approach for building personalized gene regulatory networks for precision medicine
van der Wijst, M. G. P., de Vries, D. H., Brugge, H., Westra, H.-J., & Franke, L. (2018) · Genome Medicine · doi:10.1186/s13073-018-0608-4 · Cited by 112
“Another advantage of scRNA-seq data is that true correlations between genes can be identified that would otherwise be masked by the effects of averaging in bulk RNA-seq data due to Simpson’s paradox [76] (Box 1).”
Background
The determinants of CO 2 emissions in MENA countries: a responsiveness scores approach
Magazzino, C., & Cerulli, G. (2019) · International Journal of Sustainable Development & World Ecology · doi:10.1080/13504509.2019.1606863 · Cited by 110

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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