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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“…on the first appearance of (real) dependence between two variables—or sets of variables, bears on cases collectively included under the name ofSi ps n’s paradox (Yule, 1903; Simpson, 1951; Eells, 1991, Ch. 2; Edwards, 2000, pp. 8–11; Spirteset al., 2000, pp. 38–42; Frosini, 2001, pp. 263, 290–292).”
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“3 Simpsons paradox (Simpson 1951) indicates that correlations between variates observed in the whole population can change when subsets of samples are examined separately.”
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Method
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“Data presented by Soini et al. (2020) shows that the thermal conductivities of ordinary and carbonaceous chondrites are in agreement when the sample porosities are taken into account.”
Background
Taking Corporate Culture Seriously: Group Effects in the Trust Game
Arce, D. G. (2006) · Southern Economic Journal · doi:10.1002/j.2325-8012.2006.tb00755.x · Cited by 21
Make your data fair: A survey of data preprocessing techniques that address biases in data towards fair AI
Tawakuli, A., & Engel, T. (2024) · Journal of Engineering Research · doi:10.1016/j.jer.2024.06.016 · Cited by 21
Simpson's Paradox … and How to Avoid it
Norton, H. J., & Divine, G. (2015) · Significance · doi:10.1111/j.1740-9713.2015.00844.x · Cited by 21
Can Simpson's paradox explain co-operation inPseudomonas aeruginosabiofilms?
Penn, A. S., Conibear, T. C. R., Watson, R. A. R., Kraaijeveld, A. R., & Webb, J. S. (2012) · FEMS Immunology & Medical Microbiology · doi:10.1111/j.1574-695x.2012.00970.x · Cited by 21
“Under these conditions, a ‘Simpson’s paradox’ (Simpson, 1951) may arise, in which the global proportion of co-operators can increase even whilst it decreases within any one group.”
Background
Ventral Striatum and the Evaluation of Memory Retrieval Strategies
Badre, D., Lebrecht, S., Pagliaccio, D., Long, N. M., & Scimeca, J. M. (2014) · Journal of Cognitive Neuroscience · doi:10.1162/jocn_a_00596 · Cited by 21
“However, we recognize that correlations observed across the group, although common in neuroimaging, may not represent the within-subject patterns of correlation (Simpson, 1951).”
The redundancy in cumulative information and how it biases impressions.
Alves, H., & Mata, A. (2019) · Journal of Personality and Social Psychology · doi:10.1037/pspa0000169 · Cited by 21
Latitudinal gradients in population growth do not reflect demographic responses to climate
DeMarche, M. L., Bailes, G., Hendricks, L. B., Pfeifer‐Meister, L. E., Reed, P. B., Bridgham, S. D., Johnson, B. R., Shriver, R. K., Waddle, E., Wroton, H., et al. (2020) · Ecological Applications · doi:10.1002/eap.2242 · Cited by 21
Can Bayes' Rule be Justified by Cognitive Rationality Principles?
Walliser, B., & Zwirn, D. (2002) · Theory and Decision · doi:10.1023/a:1021227106744 · Cited by 21
“Remark 2 : The fact that Bayes' rule does not satisfy Homomorphism can be considered as an explanation of the well-known Simpson paradox (Simpson, 1951), illustrated by the following example : Consider a town T with 12000 inhabitants, i.e. 8000 men among which 5000 are sick and 4000 women among…”
Background
New estimates of age‐ and sex‐specific earnings and the male–female earnings gap in the British cotton industry, 1833–19061
Boot, H. M., & Maindonald, J. H. (2007) · The Economic History Review · doi:10.1111/j.1468-0289.2007.00398.x · Cited by 21
Resolution of the Happiness–Income Paradox
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Factors in choosing atypical antipsychotics: Toward understanding the bases of physicians’ prescribing decisions☆
Hoblyn, J. C., Noda, A. M., Yesavage, J. A., BROOKSIII, J., Sheikh, J. I., Lee, T. T., Tinklenberg, J. R., Schneider, B., O'Hara, R. M., & LESLIE, D. L. (2006) · Journal of Psychiatric Research · doi:10.1016/j.jpsychires.2005.06.004 · Cited by 21
Data Information in Contingency Tables: A Fallacy of Hierarchical Loglinear Models
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Think outside the grids: An objective approach to define spatial strata for catch and effort analysis
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The measurement and cost of removing unexplained gender differences in faculty salaries
Becker, W. E., & Toutkoushian, R. K. (1995) · Economics of Education Review · doi:10.1016/0272-7757(95)00006-6 · Cited by 21
Dominance and rarity in tree communities across the globe: Patterns, predictors and threats
Hordijk, I., Bialic‐Murphy, L., Lauber, T., Routh, D., Poorter, L., Rivers, M., ter Steege, H., Liang, J., Reich, P. B., de‐Miguel, S., et al. (2024) · Global Ecology and Biogeography · doi:10.1111/geb.13889 · Cited by 20
“At the global scale, patterns of community dominance and rarity are predicted by species richness, whereas at a regional scale there is more direct competition between dominant and rare species for space and resources (Markham, 2015).”
A Functional Coupling of Brain and Behavior During Social Categorization of Faces
Volpert‐Esmond, H. I., & Bartholow, B. D. (2021) · Personality and Social Psychology Bulletin · doi:10.1177/0146167220976688 · Cited by 21
“This could reflect the fact that associations within individuals can vanish or even reverse when individuals’ data are combined (i.e., Simpson’s Paradox; Simpson, 1951; Wagner, 1982).”
Background
The illusion of data validity: Why numbers about people are likely wrong
Jansen, B. J. J., Salminen, J. O., Jung, S., & Almerekhi, H. (2022) · Data and Information Management · doi:10.1016/j.dim.2022.100020 · Cited by 26
“Several articles discuss Simpson's Paradox, along with the varieties and why it occurs (Ameringer et al., 2009; Blyth, 1972; Hern (cid:1) an et al., 2011; Wagner, 1982).”
Resurrecting the Third Variable: A Critique of Pearl's Causal Analysis of Simpson's Paradox
Armistead, T. W. (2014) · The American Statistician · doi:10.1080/00031305.2013.807750 · Cited by 20
“…the paradox was given its name in 1972 by Blyth (Pearson, Lee, and Bramley-Moore 1899; Yule 1903; Niles 1922; Cohen and Nagel 1934; Fisher 1934; Simpson 1951; Blyth 1972; Bickel, Hammel, and O’Connell 1975; Knapp 1985; Aldrich 1995; Lesser 2001; Ooi 2004; Wainer and Brown Timothy W.…”
Background
Robust Text Classification under Confounding Shift
Landeiro, V., & Culotta, A. (2018) · Journal of Artificial Intelligence Research · doi:10.1613/jair.1.11248 · Cited by 21
“As another way of considering the effect of BA, recall the notion of Simpson’s paradox (Simpson, 1951) that describes a situation where the association between two variables X and Y changes direction when we account for a third variable Z (e.g. if X and Y are positively correlated in the general…”
Background
Studying Lexical Dynamics and Language Change via Generalized Entropies: The Problem of Sample Size
Koplenig, A., Wolfer, S., & Müller-Spitzer, C. (2019) · Entropy · doi:10.3390/e21050464 · Cited by 20
“Figure 1 shows a Simpson’s Paradox [18] for the resulting data: an apparent strong positive relationship between H and γ is observed across all datapoints (Spearman ρ = 0.99).”
BackgroundInfluential
The implications of Simpson's paradox for cross-scale inference among lakes
Qian, S. S., Stow, C. A., Farnaz Nojavan, A., Stachelek, J., Cha, Y., Alameddine, I., & Soranno, P. A. (2019) · Water Research · doi:10.1016/j.watres.2019.114855 · Cited by 20
Is time elapsed between cannabis use and sleep start time associated with sleep continuity? An experience sampling method
Sznitman, S. R., Shochat, T., & Greene, T. (2020) · Drug and Alcohol Dependence · doi:10.1016/j.drugalcdep.2020.107846 · Cited by 20
Method for Assessment of Water Supply Diversification
Boryczko, K., & Rak, J. (2020) · Resources · doi:10.3390/resources9070087 · Cited by 20
“Where an approach was taken, the main indices of diversification of water supply used by authors [36–38] are indices from ecology, like Shannon-Weaver [39], Pielou [7,40], Simpson [8,41] and Hurlbert [42].”
Method
HLA Variants and Inhibitor Development in Hemophilia A: A Retrospective Case-Controlled Study Using the ATHNdataset
McGill, J. R., Simhadri, V. L., & Sauna, Z. E. (2021) · Frontiers in Medicine · doi:10.3389/fmed.2021.663396 · Cited by 21
“One possible outcome of this is Simpson’s Paradox in which trends identified in the individual studies cannot be found in the pooled data (16).”
Background

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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