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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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The Logic of Inductive Inference
Fisher, R. A. (1935) · Journal Of The Royal Statistical Society · doi:10.2307/2342435 · Cited by 1,308
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Snedecor, 1946, Statistical Methods.

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“well-known Simpson’s paradox [19].”
Pooling data for Number Needed to Treat: no problems for apples
Moore, R. A., Gavaghan, D. J., Edwards, J. E., Wiffen, P. J., & McQuay, H. J. (2002) · BMC Medical Research Methodology · doi:10.1186/1471-2288-2-2 · Cited by 51
“Simpson 50 years ago [2], and is now a staple of any undergraduate statistics course.”
Background
Nonparametric Analysis of Covariance and Its Role in Noninferiority Clinical Trials
Koch, G. G., & Tangen, C. M. (1999) · Drug Information Journal · doi:10.1177/009286159903300419 · Cited by 51
“Conditioning on covariables leads to a different interpretation of the treatment parameter relative to the model with no covariables (unconditional) (4,6-9).”
Background
Impact of 12 step mutual help groups on drug use disorder patients across six clinical trials
Humphreys, K. N., Barreto, N. B., Alessi, S. M., Carroll, K. M., Crits-Christoph, P. F., Donovan, D. M., Kelly, J. F., Schottenfeld, R. S., Timko, C., & Wagner, T. H. (2020) · Drug and Alcohol Dependence · doi:10.1016/j.drugalcdep.2020.108213 · Cited by 50
“Controlling for study level parameters helps remove this noise (Simpson, 1951).”
Background
Investigating geographic variation in use of psychotropic medications among youth in child welfare
Leslie, L. K., Raghavan, R., Hurley, M., Zhang, J., Landsverk, J. A., & Aarons, G. A. (2011) · Child Abuse & Neglect · doi:10.1016/j.chiabu.2011.01.012 · Cited by 54
“Also, heterogeneity across the catchment areasmay be obscuring findings related to catchment area variation, as can be seenwith Simpson’s or amalgamation paradox (Simpson, 1951).”
Background
Directed acyclic graphs (DAGs): an aid to assess confounding in dental research
Merchant, A. T., & Pitiphat, W. (2002) · Community Dentistry And Oral Epidemiology · doi:10.1034/j.1600-0528.2002.00008.x · Cited by 50
Functional assessment of animal interactions with shrub‐facilitation complexes: a formal synthesis and conceptual framework
Lortie, C. J., Filazzola, A., & Sotomayor, D. A. (2015) · Functional Ecology · doi:10.1111/1365-2435.12530 · Cited by 46
C2C Versus B2C: A Comparison of the Winner's Curse in Two Types of Electronic Auctions
Oh, W. (2002) · International Journal of Electronic Commerce · doi:10.1080/10864415.2002.11044249 · Cited by 50
“These mixed results suggest an interesting phenomenon referred to as the “Simpson Paradox,” meaning that the aggregate statistics are the reverse of the disaggregated data [16, 36].”
Background
The effectiveness of compulsory wearing of seat-belts in casualty reduction (with an appendix on chi-square partitioning-tests of complex contingency tables)
Foldvary, L. A., & Lane, J. C. (1974) · Accident Analysis & Prevention · doi:10.1016/0001-4575(74)90016-5 · Cited by 50
How Pooling Failure Data May Reverse Increasing Failure Rates
Gurland, J., & Sethuraman, J. (1994) · doi:10.21236/ada281909 · Cited by 50
“This phenomenon is somewhat reminiscent of Simpson's paradox (Blyth 1972; Simpson 1951), wherein a positive partial association between two variables may exist at each level of a third variable, yet a negative overall unconditional association holds between the two original variables.”
Background
North American red squirrels mitigate costs of territory defence through social plasticity
Siracusa, E. R., Wilson, D. R., Studd, E. K., Boutin, S. A., Humphries, M. M., Dantzer, B., Lane, J. E., & McAdam, A. G. (2019) · Animal Behaviour · doi:10.1016/j.anbehav.2019.02.014 · Cited by 49
Exploring relationships between climate change beliefs and energy preferences: A network analysis of the European Social Survey
Verschoor, M., Albers, C. J., Poortinga, W., Böhm, G., & Steg, L. (2020) · Journal of Environmental Psychology · doi:10.1016/j.jenvp.2020.101435 · Cited by 49
“To prevent the possibility of country 287 differences in means driving the overall network model and distorting the correlations 288 (i.e., Simpson’s paradox; Simpson, 1951), we standardized the data by rescaling all 289 variables such that for each country every variable had a mean of 0 and a standard 290 deviation o…”
Tracheotomy in COVID‐19 Patients: A Systematic Review and Meta‐analysis of Weaning, Decannulation, and Survival
Benito, D. A., Bestourous, D. E., Tong, J. Y., Pasick, L. J., & Sataloff, R. T. (2021) · Otolaryngology · doi:10.1177/0194599820984780 · Cited by 51
Visual Causality Analysis Made Practical
Wang, J., & Mueller, K. D. (2017) · IEEE Conference on Visual Analytics Science and Technology (VAST) · doi:10.1109/vast.2017.8585647 · Cited by 49
“We then further subdivide the first sub-dataset based on students’ major at T1 and the second based on major at T2, so that students selecting different fields are studied separately, avoiding possible disturbances by Simpson’s Paradox.”
BackgroundInfluential
The Design and Analysis of the Observational Study--A Review
McKinlay, S. M. (1975) · Journal of the American Statistical Association · doi:10.1080/01621459.1975.10482464 · Cited by 50
Pooling, meta-analysis, and the evaluation of drug safety
Lièvre, M. M., Cucherat, M., & Leizorovicz, A. (2002) · Current Controlled Trials in Cardiovascular Medicine · doi:10.1186/1468-6708-3-6 · Cited by 49
“Because pooling focuses on treatment groups rather than on studies, this approach does not consider the validity of the comparisons and, therefore, is subject to a bias termed "Simpson's paradox in probability" (also known in epidemiology as "confusion bias" [1,2].”
Psychology Is a Property of Persons, Not Averages or Distributions: Confronting the Group-to-Person Generalizability Problem in Experimental Psychology
McManus, R. M., Young, L., & Sweetman, J. (2023) · Advances in Methods and Practices in Psychological Science · doi:10.1177/25152459231186615 · Cited by 50
“paradox—when group-level patterns poorly represent lower-level units constituting the group (Kievit et al., 2013; Simpson, 1951; for an illustrative example on the relation between typing speed and mistake frequency, also see Hamaker, 2012).”
Background
On the selection and classification of independent features
Bressan, M., & Vitria, J. M. (2003) · IEEE Transactions on Pattern Analysis and Machine Intelligence · doi:10.1109/tpami.2003.1233904 · Cited by 48
“The consequence of this marginalization on the Bayesian decision scheme is well-known, resulting in the naive Bayes classifier.”
Background
Marginal or conditional regression models for correlated non‐normal data?
Muff, S., Held, L., & Keller, L. F. (2016) · Methods in Ecology and Evolution · doi:10.1111/2041-210x.12623 · Cited by 48
“Already Lindsey & Lambert (1998) illustrated how marginal model parameters may have a misleading interpretation that is equivalent to that of Simpson’s paradox (Simpson 1951).”
Background
Nonsensical and biased correlation due to pooling heterogeneous samples
Hassler, U., & Thadewald, T. (2003) · Journal of the Royal Statistical Society Series D (The Statistician) · doi:10.1111/1467-9884.00365 · Cited by 48
Marital Instability After Midlife
Wu, Z. L., & Penning, M. J. (1997) · Journal of Family Issues · doi:10.1177/019251397018005001 · Cited by 48
Comparison of Nebulized Epinephrine to Albuterol in Bronchiolitis
Walsh, P. F., Caldwell, J. W., McQuillan, K. K., Friese, S., Robbins, D., & Rothenberg, S. J. (2008) · Academic Emergency Medicine · doi:10.1111/j.1553-2712.2008.00064.x · Cited by 48
“The Yule-Simpson ‘‘paradox’’ describes just this situation, where the success of several subgroups may be reduced or even reversed when these subgroups are combined.(32) The conditions required for the ‘‘paradox,’’ namely, the combination of an imbalance in the proportion of each subgroup receiving each intervention a…”
Background
Good for Women, Good for Men, Bad for People: Simpson's Paradox and the Importance of Sex-Specific Analysis in Observational Studies
Baker, S. G., & Kramer, B. S. (2001) · Journal of Women s Health & Gender-Based Medicine · doi:10.1089/152460901753285769 · Cited by 48
“[1], for the combined study population: Survival rate for treatment A 5 (1 2 0.67) 3 60% 1 0.67 3 95% 5 83% Survival rate for treatment B 5 (1 2 0.67) 3 50% 1 0.67 3 85% 5 73% indicating no bias in the difference in treatment rates.”
Influential

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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