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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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“Finally, it avoids the potential of a Yule–Simpson effect (Simpson 1951) due to aggregation across groups of different sizes for the various patient and physician-level characteristics.”
Cross‐Level/Ecological Inference
Cho, W. K. T., & Manski, C. F. (2009) · Oxford University Press eBooks · doi:10.1093/oxfordhb/9780199286546.003.0024 · Cited by 37
“Many statisticians and mathematicians have been captivated by Simpson's Paradox (Simpson 1951), which is the reversal in direction of association between two variables when a third ( " lurking " ) variable is controlled.”
Background
Graphical models illustrated complex associations between variables describing human functioning
Strobl, R., Stucki, G., Grill, E., Müller, M., & Mansmann, U. (2009) · Journal of Clinical Epidemiology · doi:10.1016/j.jclinepi.2009.01.018 · Cited by 37
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“This phenomenon is known as Simpson’s paradox [7], or spatial heterogeneity [30].”
Background
An Excess of Gene Expression Divergence on the X Chromosome in Drosophila Embryos: Implications for the Faster-X Hypothesis
Kayserili, M. A., Gerrard, D. T., Tomančák, P., & Kalinka, A. T. (2012) · PLoS Genetics · doi:10.1371/journal.pgen.1003200 · Cited by 41
“et Bin pairs of species. As it is established that correlation coecients within subsets can vary, sometimes dramatically, from correlation at the level of aggregates (known as the Yule-Simpson eect [91,92]), we believe that it is necessary to account for possible discrepancies when measuring correlation within subsets…”
Background
Unconventional Resources from Exploration to Production
Zee, Y. (2015) · Elsevier eBooks · doi:10.1016/b978-0-12-802238-2.00001-8 · Cited by 36
Inter‐ and intra‐individual coupling between pupillary, electrophysiological, and behavioral responses in a visual oddball task
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A machine-vision-based frailty index for mice
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“Due to the slight bimodality of our data, we tested for the presence of Simpson’s Paradox [34] and did not see any evidence (Figure S4).”
BackgroundMethod
Accident involvement rate and car size
Evans, L. (1984) · Accident Analysis & Prevention · doi:10.1016/0001-4575(84)90052-6 · Cited by 36
Multiple voting location and single voting location on trees
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Repression of competition favours cooperation: experimental evidence from bacteria
Kümmerli, R., van den Berg, P., Griffin, A. S., West, S. A., & Gardner, A. (2010) · Journal of Evolutionary Biology · doi:10.1111/j.1420-9101.2010.01936.x · Cited by 35
“…contrast with the results from an engineered microbial cooperative system (Chuang et al., 2009), in which cooperators increased in frequency under complete dispersal despite being selected against in mixed cultures with cheats (i.e. conforming to the ‘Simpson’s paradox; Simpson, 1951; Blyth, 1972).”
Result
Simpson's paradox and clinical trials: What you find is not necessarily what you prove
Abramson, N. S., Kelsey, S. F., Safar, P. J., & Sutton-Tyrrell, K. (1992) · Annals of Emergency Medicine · doi:10.1016/s0196-0644(05)80066-6 · Cited by 36
No evidence that gender contributes to personal research funding success in The Netherlands: A reaction to van der Lee and Ellemers
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Li, S., Schmid, K., de Vries, D. H., Korshevniuk, M., Losert, C., Oelen, R., van Blokland, I. V., Groot, H. E., Swertz, M. A., van der Harst, P., et al. (2023) · Genome biology · doi:10.1186/s13059-023-02897-x · Cited by 37
“Furthermore, the validity of bulk-based correlations is affected by the possibility of Simpson’s paradox [27] occurring.”
Background
The Danger of Engagement: Behavioral Observations of Online Community Activity and Service Spending in the Online Gaming Context
Kaptein, M. C., Parvinen, P. M. T., & Pöyry, E. (2015) · International Journal of Electronic Commerce · doi:10.1080/10864415.2016.1061414 · Cited by 38
“The above example serves to highlight the fact that different levels of aggregation might paint a different picture—an effect well known in statistics as the Simpson paradox [50, 55].”
Background
Ecological inference in the social sciences
Glynn, A., & Wakefield, J. (2009) · Statistical Methodology · doi:10.1016/j.stamet.2009.09.003 · Cited by 36
Linear trend tests for case–control genetic association that incorporate random phenotype and genotype misclassification error
Gordon, D., Haynes, C. A., Yang, Y., Kramer, P. L., & Finch, S. J. (2007) · Genetic Epidemiology · doi:10.1002/gepi.20246 · Cited by 35
Measures of agency
Grünbaum, T., & Christensen, M. S. (2020) · Neuroscience of Consciousness · doi:10.1093/nc/niaa019 · Cited by 35
“Unfortunately, it cannot be ruled out that Imaizumi and Tanno’s (2019) correlation is an example of Simpson’s paradox (Simpson 1951), where, when combined from multiple action-effect tasks at different delays, a series of non-correlating measurements between implicit and explicit measures of agency…”
Background
Psychological and demographic predictors of plastic bag consumption in transaction data
Lavelle-Hill, R. E., Goulding, J., Smith, G., Clarke, D. D., & Bibby, P. A. (2020) · Journal of Environmental Psychology · doi:10.1016/j.jenvp.2020.101473 · Cited by 35
Detecting Latent Heterogeneity
Pearl, J. (2015) · Sociological Methods & Research · doi:10.1177/0049124115600597 · Cited by 37
Distinguishing different modes of growth using single-cell data
Kar, P., Krishnan, S. T., Männik, J., Männik, J., & Amir, A. (2021) · eLife · doi:10.7554/elife.72565 · Cited by 32
“For example, Willis and Huang, 2017 discusses Simpson’s paradox ( Simpson, 1951 ) where distinct cellular sub-populations might lead to erroneous interpretation of cell cycle mechanisms.”

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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