The interpretation of interaction in contingency tables
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
Registry
Cites
Statistics
Claims the paper makes
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
The relationship between body composition and BMI represents a clear example of Simpson's paradox.
“Indeed, the relationship between body composition and BMI represents a clear example of Simpson’s paradox (Simpson, 1951).”
From The degree to which the cultural ideal is internalized predicts judgments of male and female physical attractiveness (2022) · cites this paper as Simpson, 1951
Simpson's paradox earns its claim to fame only through a causal interpretation of the data.
“The classical example demonstrating the obstacle of causal vocabulary is Simpson's paradox (Simpson, 1951) - a reversal phenomenon that earns its claim to fame only through a causal interpretation of the data (Pearl, 2000a, Chapter 6).”
From Causal inference in statistics: An overview (2009) · cites this paper as Simpson, 1951
Models will endorse logical traps, such as Simpson's Paradox, if pressed.
“Our work extends this to the causal domain, showing that models will endorse logical traps (like Simpson's Paradox (Simpson, 1951)) if pressed.”
From Diagnosing and Mitigating Sycophancy and Skepticism in LLM Causal Judgment (2026) · cites this paper as Simpson, 1951
Cites
The Logic of Inductive Inference
Fisher, R. A. (1935) · Journal Of The Royal Statistical Society · doi:10.2307/2342435 · Cited by 1,292 · see citing works
124: The Logic of Inductive Inference.
Fisher, R. A. (1935) · Adelaide Research & Scholarship (AR&S) (University of Adelaide) · Cited by 872 · see citing works
The Advanced Theory of Statistics.
P., E. S., & Kendall, M. G. (1944) · Journal Of The Royal Statistical Society · doi:10.2307/2981279 · Cited by 671 · see citing works
Contingency Table Interactions
Bartlett, M. S. (1935) · Journal of the Royal Statistical Society Series B (Statistical Methodology) · doi:10.2307/2983639 · Cited by 313 · see citing works
Calculation of Chi-Square for Complex Contingency Tables
Norton, H. W. (1945) · Journal of the American Statistical Association · doi:10.1080/01621459.1945.10501855 · Cited by 66 · see citing works
Chance in Medicine and Research
Norton, H. W. (1939) · BMJ · doi:10.1136/bmj.2.4103.467-a · Cited by 5 · see citing works
Flagged first, then by citations
Cited by
Sorted by citations
(2005) · Springer eBooks · doi:10.1007/0-387-27255-0_9 · Cited by 20,333 · see citing works
Trapnell, C., Cacchiarelli, D., Grimsby, J., Pokharel, P., Li, S., Morse, M., Lennon, N. J., Livak, K. J., Mikkelsen, T. S., & Rinn, J. L. (2014) · Nature Biotechnology · doi:10.1038/nbt.2859 · Cited by 7,314 · see citing works
“Drawing conclusions from a group of individuals based on the properties of their average is a hazardous practice because the average can mask important trends among the individuals, resulting in phenomena such as Simpson's paradox 14 .”
BackgroundCausal inference in statistics: An overview
Pearl, J. (2009) · Statistics Surveys · doi:10.1214/09-ss057 · Cited by 2,336 · see citing works
“The phenomenon was discovered by statisticians a century ago (Pearson et al., 1899; Yule, 1903) analyzed by statisticians for half a century (Simpson, 1951; Blyth, 1972; Cox and Wermuth, 2003) lamented by statisticians (Good and Mittal, 1987; Bishop et al., 1975) and wrestled with by statisticians…”
BackgroundInfluentialMultimodal population brain imaging in the UK Biobank prospective epidemiological study
Miller, K. L., Alfaro-Almagro, F., Bangerter, N. K., Thomas, D. L., Yacoub, E., Xu, J., Bartsch, A. J., Jbabdi, S., Sotiropoulos, S. N., Andersson, J. L. R., et al. (2016) · Nature Neuroscience · doi:10.1038/nn.4393 · Cited by 2,020 · see citing works
“This may be due to key differences in the HCP and Biobank datasets, including different non-imaging measures, the use of only rfMRI in the HCP analysis, the different cohort profiles (e.g., age range) and the ability to separate more modes from the larger Biobank cohort.”
Conditional Independence in Statistical Theory
Dawid, A. P. (1979) · Journal of the Royal Statistical Society Series B (Statistical Methodology) · doi:10.1111/j.2517-6161.1979.tb01052.x · Cited by 1,507 · see citing works
What to add to nothing? Use and avoidance of continuity corrections in meta‐analysis of sparse data
J. Sweeting, M., J. Sutton, A., & C. Lambert, P. (2004) · Statistics in Medicine · doi:10.1002/sim.1761 · Cited by 1,734 · see citing works
Lack of group-to-individual generalizability is a threat to human subjects research
Fisher, A. J., Medaglia, J. D., & Jeronimus, B. F. (2018) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1711978115 · Cited by 1,099 · see citing works
“Simpson’s paradox (12) is a statistical effect in which trends in subgroups differ from (or are even inverse to) the aggregate trend when the groups are combined.”
BackgroundTesting for Factorial Invariance in the Context of Construct Validation
Dimitrov, D. M. (2010) · Measurement and Evaluation in Counseling and Development · doi:10.1177/0748175610373459 · Cited by 1,177 · see citing works
“They explained the paradoxical nature of this finding as a continuous version of Simpson’s paradox (Simpson, 1951), which describes how effects that are observed in subpopulations can differ dramatically from effects observed in the aggregates of the subpopulations.”
BackgroundThe Gaussian Graphical Model in Cross-Sectional and Time-Series Data
Epskamp, S., Waldorp, L., Mõttus, R., & Borsboom, D. (2018) · Multivariate Behavioral Research · doi:10.1080/00273171.2018.1454823 · Cited by 1,183 · see citing works
“Such a different effect depending on the level of analysis is well known in the statistical literature as Simpson’s paradox (Simpson, 1951).”
BackgroundHand, D. J. (2007) · Drug Safety · doi:10.2165/00002018-200730070-00010 · Cited by 1,057 · see citing works
Confounding and Collapsibility in Causal Inference
Greenland, S., Pearl, J., & Robins, J. M. (1999) · Statistical Science · doi:10.1214/ss/1009211805 · Cited by 933 · see citing works
“Noncollapsibility (violation of collapsibility) is sometimes referred to as Simpson’s paradox, after a celebrated article by Simpson (1951).”
BackgroundForecasting: theory and practice
Petropoulos, F., Apiletti, D., Assimakopoulos, V., Babai, M. Z., Barrow, D. K., Taieb, S. B., Bergmeir, C., Bessa, R., Bijak, J., Boylan, J., et al. (2020) · International Journal of Forecasting · doi:10.1016/j.ijforecast.2021.11.001 · Cited by 630 · see citing works
“This is closely related to the well-known Simpson’s Paradox (Simpson, 1951).”
Cicero predicts cis-regulatory DNA interactions from single-cell chromatin accessibility data
Pliner, H. A., Packer, J. S., McFaline-Figueroa, J. L., Cusanovich, D. A., Daza, R. M., Aghamirzaie, D., Srivatsan, S., Qiu, X., Jackson, D., Minkina, A., et al. (2018) · Molecular Cell · doi:10.1016/j.molcel.2018.06.044 · Cited by 789 · see citing works
Evolutionary Personality Psychology
Buss, D. M. (1991) · Annual Review of Psychology · doi:10.1146/annurev.ps.42.020191.002331 · Cited by 930 · see citing works
DuMouchel, W. (1999) · The American Statistician · doi:10.1080/00031305.1999.10474456 · Cited by 883 · see citing works
“jk = C N i j k of these variables’ mutual association with C; that is, one wants to avoid being misled by the well-known Simpson’s (1951) paradox.”
BackgroundDefining cell types and states with single-cell genomics
Trapnell, C. (2015) · Genome Research · doi:10.1101/gr.190595.115 · Cited by 813 · see citing works
“The misleading effects of Simpson’s Paradox are likely to be pervasive in modern experiments using bulk assays.”
BackgroundInappropriate use of bivariable analysis to screen risk factors for use in multivariable analysis
Sun, G.-W., Shook, T. L., & Kay, G. L. (1996) · Journal of Clinical Epidemiology · doi:10.1016/0895-4356(96)00025-x · Cited by 933 · see citing works
Converging Technologies for Improving Human Performance
Roco, M. C., & Bainbridge, W. S. (2003) · Springer eBooks · doi:10.1007/978-94-017-0359-8 · Cited by 792 · see citing works
An Introduction to Causal Inference
Pearl, J. (2010) · The International Journal of Biostatistics · doi:10.2202/1557-4679.1203 · Cited by 755 · see citing works
Out of One, Many: Using Language Models to Simulate Human Samples
Argyle, L. P., Busby, E. C., Fulda, N., Gubler, J. R., Rytting, C., & Wingate, D. (2023) · Political Analysis · doi:10.1017/pan.2023.2 · Cited by 783 · see citing works
“The conditional nature of GPT-3’s text completions creates a situation analogous to Simpson’s Paradox [28], in which the trends evident in a combined group do not reflect the trends of its composite distributions.”
BackgroundDetecting circular RNAs: bioinformatic and experimental challenges
Szabo, L. A., & Salzman, J. (2016) · Nature Reviews Genetics · doi:10.1038/nrg.2016.114 · Cited by 684 · see citing works
“Tests of genome-wide conservation are always subject to confounding by Simpson’s paradox 68 : it could be the case that genes that host circRNAs expressed in the brain are statistically conserved between humans and mice, whereas, after collapsing over all organs, conservation of gene sets is lost;…”
BackgroundTSCAN: Pseudo-time reconstruction and evaluation in single-cell RNA-seq analysis
Ji, Z., & Ji, H. (2016) · Nucleic Acids Research · doi:10.1093/nar/gkw430 · Cited by 624 · see citing works
“Sometimes, using the population average to study cell type specific behavior can also be misleading due to Simpson’s paradox (7,8).”
BackgroundThe Risk of Stomach Cancer in Patients with Gastric or Duodenal Ulcer Disease
Hansson, L., Nyrén, O., Hsing, A. W., Bergström, R., Josefsson, S., Chow, W., Fraumeni, J. F., & Adami, H. (1996) · New England Journal of Medicine · doi:10.1056/nejm199607253350404 · Cited by 637 · see citing works
On Simpson's Paradox and the Sure-Thing Principle
Blyth, C. R. (1972) · Journal of the American Statistical Association · doi:10.1080/01621459.1972.10482387 · Cited by 630 · see citing works
What mediation analysis can (not) do
Fiedler, K., Schott, M., & Meiser, T. (2011) · Journal of Experimental Social Psychology · doi:10.1016/j.jesp.2011.05.007 · Cited by 609 · see citing works
Theory of Decision under Uncertainty
Gilboa, I. (2009) · Cambridge University Press eBooks · doi:10.1017/cbo9780511840203 · Cited by 598 · see citing works
Extending the Linear Model with R
Faraway, J. J. (2016) · Chapman & Hall/CRC eBooks · doi:10.1201/9781315382722 · Cited by 832 · see citing works
Lawlor, N. T., George, J., Bolisetty, M. T., Kursawe, R., Sun, L., Vijayakumar, S., Kycia, I., Robson, P., & Stitzel, M. L. (2016) · Genome Research · doi:10.1101/gr.212720.116 · Cited by 540 · see citing works
“…’ s paradox, approximately half of these genes in each major islet cell type (64% beta, 45% alpha) and ∼ 90% of these in the less abundant delta cells were not detected in whole islet or single-cell islet transcriptomes when they were not stratified by cell type (Simpson 1951; Trapnell 2015).”
Bouman, T., Verschoor, M., Albers, C. J., Böhm, G., Fisher, S. D., Poortinga, W., Whitmarsh, L. E., & Steg, L. (2020) · Global Environmental Change · doi:10.1016/j.gloenvcha.2020.102061 · Cited by 560 · see citing works
“Before running the regression and mediation analyses, for which we used the R-package lavaan (Rosseel, 2012), we standardized all variables at the country level to prevent the occurrence of Yule's (Yule, 1903) and Simpson's (Simpson, 1951) paradoxes, and to”
MethodTowards a standard for identifying and managing bias in artificial intelligence
Schwartz, R., Vassilev, A., Greene, K., Perine, L., Burt, A., & Hall, P. (2022) · doi:10.6028/nist.sp.1270 · Cited by 548 · see citing works
“Simple but effective techniques, such as class imbalance measures or label imbalance measures, or analysis using statistical phenomena such as SIMPSON’S PARADOX,[136] can be used to detect bias in datasets, and sometimes help mitigate it [86, 137–140].”
MethodBurgess, S., Daniel, R. M., Butterworth, A. S., Thompson, S. G., & the EPIC-InterAct Consortium (2015) · International Journal of Epidemiology · doi:10.1093/ije/dyu176 · Cited by 543 · see citing works
“This is an analogous problem to Simpson’s paradox, whereby the average effect in the population can be in the opposite direction to the average effects in each of the substrata of the population.(36,37) Hence the use of IVs for both the exposure and the mediator formally requires the assumption of homogeneity across i…”
Simpson's paradox in psychological science: a practical guide
Kievit, R. A., Frankenhuis, W. E., Waldorp, L. J., & Borsboom, D. (2013) · Frontiers in Psychology · doi:10.3389/fpsyg.2013.00513 · Cited by 542 · see citing works
Causal Laws and Effective Strategies
Cartwright, N. D. (1979) · Noûs · doi:10.2307/2215337 · Cited by 538 · see citing works
“This example depends on a fact about probabilities known as Simpson's paradox (Simpson [11]), or sometimes as the Cohen-Nagel-Simpson paradox, for it is presented as an exercise in Morris Cohen's and Ernest Nagel's text, An Introduction to Logic and Scientific Method [4].”
MethodForm and function: Metabolic footprints of nematodes in the soil food web
Ferris, H. (2010) · European Journal of Soil Biology · doi:10.1016/j.ejsobi.2010.01.003 · Cited by 531 · see citing works
“The differences between the three samples are not revealed by Shannon's or Simpson's Diversity Indices [34,35].”
BackgroundStatistically small effects of the Implicit Association Test can have societally large effects.
Greenwald, A. G., Banaji, M. R., & Nosek, B. A. (2014) · Journal of Personality and Social Psychology · doi:10.1037/pspa0000016 · Cited by 508 · see citing works
How meta-analysis increases statistical power.
Cohn, L. D., & Becker, B. J. (2003) · Psychological Methods · doi:10.1037/1082-989x.8.3.243 · Cited by 500 · see citing works
“Indeed the latter strategy is not recommended (Bravata & Olkin, 2002) and can result in outcomes characterized by Simpson’s paradox (Simpson, 1951).”
BackgroundCounterfactual reasoning and learning systems: the example of computational advertising
Bottou, L., Peters, J., Quiñonero-Candela, J., Charles, D. X., Chickering, D. M., Portugaly, E., Ray, D., Simard, P., & Snelson, E. (2013) · Cited by 497 · see citing works
Wrzus, C., & Neubauer, A. B. (2023) · Assessment · doi:10.1177/10731911211067538 · Cited by 500 · see citing works
“It needs to be acknowledged although that inferring such recommendations for individual studies still entails uncertainty because the current findings refer to the level of a population of studies (e.g., see the Simpson paradox, Kievit et al., 2013; Simpson, 1951).”
BackgroundA Class of Bivariate Distributions
Plackett, R. L. (1965) · Journal of the American Statistical Association · doi:10.1080/01621459.1965.10480807 · Cited by 492 · see citing works
Methodology Review: Statistical Approaches for Assessing Measurement Bias
Millsap, R. E., & Everson, H. T. (1993) · Applied Psychological Measurement · doi:10.1177/014662169301700401 · Cited by 492 · see citing works
“of the linking. The available research does not clearly support the superiority of either type of method in all situations. Stocking & Lord (1983) found that their characteristic curve method was more accurate than the moment method developed by Linn et al.”
BackgroundMethodInfluentialDIF DETECTION AND DESCRIPTION: MANTEL‐HAENSZEL AND STANDARDIZATION1,2
Dorans, N. J., & Holland, P. W. (1992) · ETS Research Report Series · doi:10.1002/j.2333-8504.1992.tb01440.x · Cited by 452 · see citing works
Esserman, L. J., Berry, D. A., DeMichele, A. M., Carey, L., Davis, S., Buxton, M., Hudis, C., Gray, J. W., Perou, C. M., Yau, C. S. F., et al. (2012) · Journal of Clinical Oncology · doi:10.1200/jco.2011.39.2779 · Cited by 439 · see citing works
“This is an instance of Simpson’s paradox.(18) However, the parameters of which these are estimates are not precisely known and so they may not have the same relationships.”
BackgroundData Cleaning: Overview and Emerging Challenges
Chu, X., Ilyas, I. F., Krishnan, S., & Wang, J. (2016) · Proceedings of the 2016 International Conference on Management of Data · doi:10.1145/2882903.2912574 · Cited by 488 · see citing works
“Aggregates over mixtures of different populations of data can result in spurious relationships due to the well-known phenomenon called Simpson’s paradox [68].”
BackgroundIntuitions about combining opinions: misappreciation of the averaging principle
Larrick, R. P., & Soll, J. B. (2006) · Management Science · doi:10.1287/mnsc.1050.0459 · Cited by 410 · see citing works
“We measured extensional reasoning ability by testing performance on two reasoning problems that are formally unrelated to combining opinions: The Will Rogers effect (Messick & Asuncion, 1993) and Simpson's paradox (Simpson, 1951).”
Single-cell RNA sequencing identifies celltype-specific cis-eQTLs and co-expression QTLs
Study, L. C., van der Wijst, M. G. P., BIOS Consortium, Brugge, H., de Vries, D. H., Deelen, P., Swertz, M. A., & Franke, L. (2018) · Nature Genetics · doi:10.1038/s41588-018-0089-9 · Cited by 399 · see citing works
Correlations Genuine and Spurious in Pearson and Yule
Aldrich, J. (1995) · Statistical Science · doi:10.1214/ss/1177009870 · Cited by 401 · see citing works
Rapid Naming, Phonological Awareness, and Reading: A Meta-Analysis of the Correlation Evidence
Swanson, H. L., Trainin, G., Necoechea, D. M., & Hammill, D. D. (2003) · Review of Educational Research · doi:10.3102/00346543073004407 · Cited by 404 · see citing works
“This aggregation may yield an error referred to as “Simpson’s paradox” (Simpson, 1951; see Nunnally & Bernstein, 1995, for further discussion).”
BackgroundFive types of OECD healthcare systems: Empirical results of a deductive classification
Böhm, K., Schmid, A., Götze, R., Landwehr, C., & Rothgang, H. (2013) · Health Policy · doi:10.1016/j.healthpol.2013.09.003 · Cited by 398 · see citing works
Confounding in Health Research
Greenland, S., & Morgenstern, H. (2001) · Annual Review of Public Health · doi:10.1146/annurev.publhealth.22.1.189 · Cited by 390 · see citing works
Simpson's Paradox in Real Life
Wagner, C. H. (1982) · The American Statistician · doi:10.1080/00031305.1982.10482778 · Cited by 389 · see citing works
“SIMPSON, E. H. (1951), "The Interpretation of Interaction in Contingency Tables," Journal of the Royal Statistical Society, Ser.”
BackgroundPeto's Paradox: evolution's prescription for cancer prevention
Caulin, A. F., & Maley, C. C. (2011) · Trends in Ecology & Evolution · doi:10.1016/j.tree.2011.01.002 · Cited by 384 · see citing works
“This divergent trend within versus between species is an example of Simpson’s paradox [24].”
BackgroundMethods for dealing with time‐dependent confounding
Daniel, R., Cousens, S. N., Stavola, B. L. D., Kenward, M. G., & Sterne, J. A. C. (2012) · Statistics in Medicine · doi:10.1002/sim.5686 · Cited by 378 · see citing works
“Because of the non-collapsibility of the odds ratio [24,25], it is not possible for P.”
BackgroundSexual and Drug Behavior Patterns and HIV and STD Racial Disparities: The Need for New Directions
Hallfors, D. D., Iritani, B. J., Miller, W. C., & Bauer, D. J. (2006) · American Journal of Public Health · doi:10.2105/ajph.2005.075747 · Cited by 379 · see citing works
Independence, Invariance and the Causal Markov Condition
Hausman, D. M., & Woodward, J. (1999) · The British Journal for the Philosophy of Science · doi:10.1093/bjps/50.4.521 · Cited by 371 · see citing works
“10 Such mixing has the same structure as Simpson’s Paradox (Cartwright [1979], pp.”
Test of Significance for 2 × 2 Contingency Tables
Yates, F. (1984) · Journal of the Royal Statistical Society Series A (General) · doi:10.2307/2981577 · Cited by 364 · see citing works
“If anyone is unfamiliar with the dangers of hidden classifications, I recommend Simpson (1951).”
BackgroundRoussey-Kesler, G., Gadjos, V., Idres, N., Horen, B., Ichay, L., Leclair, M., Raymond, F., Grellier, A., Hazart, I., de Parscau, L., et al. (2007) · The Journal of Urology · doi:10.1016/j.juro.2007.09.090 · Cited by 360 · see citing works
Models of accuracy in repeated-measures designs
Dixon, P. (2008) · Journal of Memory and Language · doi:10.1016/j.jml.2007.11.004 · Cited by 359 · see citing works
“In Simpson’s paradox (Simpson, 1951), for example, the pattern of overall means may not reflect the pattern observed in individual conditions or with individual subjects.”
BackgroundCooper, N. J., Sutton, A. J., Abrams, K. R., Wailoo, A., Turner, D., & Nicholson, K. G. (2003) · BMJ · doi:10.1136/bmj.326.7401.1235 · Cited by 357 · see citing works
The differential impact of risk factors on mortality in hemodialysis and peritoneal dialysis
Vonesh, E. F., Snyder, J. J., Foley, R. N., & Collins, A. J. (2004) · Kidney International · doi:10.1111/j.1523-1755.2004.66028.x · Cited by 351 · see citing works
“Since this older population contributes the majority of deaths, ignoring the age by modality interaction can lead to an estimate of overall risk that is unduly influenced by the RR of death in this group, a result known as Simpson’s paradox [25].”
ResultCooper, H., & Patall, E. A. (2009) · Psychological Methods · doi:10.1037/a0015565 · Cited by 339 · see citing works
“Related to the ecological fallacy is another phenomenon called “Simpson’s paradox” (Simpson, 1951).”
BackgroundCausal inference with observational data
Nichols, A. (2007) · The Stata Journal: Promoting communications on statistics and Stata · doi:10.1177/1536867x0800700403 · Cited by 339 · see citing works
Evolutionary games and population dynamics: maintenance of cooperation in public goods games
Hauert, C., Holmes, M., & Doebeli, M. (2006) · Proceedings of the Royal Society B Biological Sciences · doi:10.1098/rspb.2006.3600 · Cited by 314 · see citing works
“The fact that evolution nevertheless favours cooperation represents an instance of Simpson’s paradox (Simpson 1951; Hauert et al. 2002b).”
BackgroundPatsopoulos, N. A., Barcellos, L. F., Hintzen, R. Q., Schaefer, C., van Duijn, C. M., Noble, J. A., Raj, T., IMSGC, ANZgene, Gourraud, P.-A., et al. (2013) · PLoS Genetics · doi:10.1371/journal.pgen.1003926 · Cited by 312 · see citing works
“This phenomenon can likely be explained by Simpson’s paradox, where two subgroups share the same association but the overall population shows no association (or even a reversed one) [10].”
BackgroundPerson-Oriented and Variable-Oriented Research: Concepts, Results, and Development
von Eye, A., & Bogat, G. A. (2006) · Merrill-palmer Quarterly · doi:10.1353/mpq.2006.0032 · Cited by 310 · see citing works
Zhang, J. L., & Rubin, D. B. (2003) · Journal of Educational and Behavioral Statistics · doi:10.3102/10769986028004353 · Cited by 308 · see citing works
“This same issue of comparing different subgroup of subjects arises with Simpson’s paradox (Simpson, 1951, Wainer, 1999).”
BackgroundTrustworthy Online Controlled Experiments: A Practical Guide to A/B Testing
Kohavi, R., Tang, D., & Xu, Y. (2020) · doi:10.1017/9781108653985 · Cited by 297 · see citing works
Summarizability in OLAP and statistical data bases
Lenz, H. J., & Shoshani, A. (2002) · International Conference on Scientific and Statistical Database Management · doi:10.1109/ssdm.1997.621175 · Cited by 295 · see citing works
“Simpson (1951) proved for 3-way tables with binary value sets as above that such tables are collapsible over an attribute k=3, iff u 13 (3) = 0 or u 23 (3) = 0 or u 13 (3) = u 23 (3) = 0.”
Background<i>Cis</i> -acting expression quantitative trait loci in mice
Doss, S., Schadt, E. E., Drake, T. A., & Lusis, A. J. (2005) · Genome Research · doi:10.1101/gr.3216905 · Cited by 273 · see citing works
Modeling carbon allocation in trees: a search for principles
Franklin, O., Johansson, J., Dewar, R. C., Dieckmann, U., McMurtrie, R. E., Brännström, Å., & Dybzinski, R. (2012) · Tree Physiology · doi:10.1093/treephys/tpr138 · Cited by 280 · see citing works
Nelder, J. A., & Wedderburn, R. W. M. (1992) · Springer series in statistics · doi:10.1007/978-1-4612-4380-9_39 · Cited by 276 · see citing works
Hofer, S. M., & Sliwinski, M. J. (2001) · Gerontology · doi:10.1159/000052825 · Cited by 275 · see citing works
Maximum Likelihood in Three-Way Contingency Tables
Birch, M. W. (1963) · Journal of the Royal Statistical Society Series B (Statistical Methodology) · doi:10.1111/j.2517-6161.1963.tb00504.x · Cited by 274 · see citing works
Twenty years of “hot hand” research: Review and critique
Bar‐Eli, M., Avugos, S., & Raab, M. (2006) · Psychology of sport and exercise · doi:10.1016/j.psychsport.2006.03.001 · Cited by 271 · see citing works
Empirical bayes screening for multi-item associations
DuMouchel, W., & Pregibon, D. (2001) · ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD) · doi:10.1145/502512.502526 · Cited by 268 · see citing works
“The method provides protection against Simpson’s paradox [ 15 ] whereby associations that hold separately within each stratum can disappear or even be reversed when data are pooled across strata.”
MethodActiveClean: interactive data cleaning for statistical modeling
Krishnan, S., Wang, J., Wu, E., Franklin, M. J., & Goldberg, K. (2016) · Proceedings of the VLDB Endowment · doi:10.14778/2994509.2994514 · Cited by 264 · see citing works
“It is known that aggregates over mixtures of different populations of data can result in spurious relationships due to the well-known phenomenon called Simpson’s paradox [32].”
BackgroundMethodMeasuring forecast skill: is it real skill or is it the varying climatology?
Hamill, T., & Juras, J. (2006) · Quarterly Journal of the Royal Meteorological Society · doi:10.1256/qj.06.25 · Cited by 266 · see citing works
Tu, Y., Gunnell, D., & Gilthorpe, M. S. (2008) · Emerging Themes in Epidemiology · doi:10.1186/1742-7622-5-2 · Cited by 261 · see citing works
“Simpson's Paradox Simpson's paradox [13], or Yule's paradox [14], is a well known statistical phenomenon.”
BackgroundEfficacy and safety of tigecycline for the treatment of infectious diseases: a meta-analysis
Tasina, E., Haidich, A., Kokkali, S., & Arvanitidou, M. (2011) · The Lancet Infectious Diseases · doi:10.1016/s1473-3099(11)70177-3 · Cited by 258 · see citing works
Maximum Likelihood in Three-Way Contingency Tables
Birch, M. W. (1992) · Springer series in statistics · doi:10.1007/978-1-4612-4380-9_33 · Cited by 255 · see citing works
Species richness, species-area curves and Simpson's paradox
Scheiner, S. M., Cox, S. B., Willig, M. R., Mittelbach, G. G., Osenberg, C. W., & Kaspari, M. E. (2000) · Cited by 254 · see citing works
Tu, Y. (2004) · American Journal of Epidemiology · doi:10.1093/aje/kwi002 · Cited by 253 · see citing works
“Simpson on the way in which the relation between two variables changed after a third variable was factored into a two-by-two contingency table (17).”
BackgroundBrooks, J. S., Waylen, K. A., & Mulder, M. B. (2013) · Environmental Evidence · doi:10.1186/2047-2382-2-2 · Cited by 253 · see citing works
Computational characterization of B-cell epitopes
Rubinstein, N. D., Mayrose, I., Halperin, D., Yekutieli, D., Gershoni, J. M., & Pupko, T. (2007) · Molecular Immunology · doi:10.1016/j.molimm.2007.10.016 · Cited by 251 · see citing works
Metodolog�a de la Investigaci�n Social Cuantitativa (2015)
(2018) · Revista de educación y derecho · doi:10.1344/reyd2018.17.13 · Cited by 245 · see citing works
Estimation of the Neutrality Index
Stoletzki, N., & Eyre‐Walker, A. (2010) · Molecular Biology and Evolution · doi:10.1093/molbev/msq249 · Cited by 243 · see citing works
Papili Gao, N., Ud-Dean, S. M. M., Gandrillon, O., & Gunawan, R. (2018) · Bioinformatics · doi:10.1093/bioinformatics/btx575 · Cited by 235 · see citing works
“Simpson’s paradox (Simpson, 1951)], thereby presenting new means for understanding biology.”
BackgroundHaining, R. (2003) · Cambridge University Press eBooks · doi:10.1017/cbo9780511754944 · Cited by 241 · see citing works
Direct and Indirect Causal Effects via Potential Outcomes*
Rubin, D. B. (2004) · Scandinavian Journal of Statistics · doi:10.1111/j.1467-9469.2004.02-123.x · Cited by 324 · see citing works
“A third tenet is that we need to be explicit about assumptions because human beings, naturally, are very bad at dealing with uncertainty, and repeatedly fall prey to 'paradoxes' such as 'Simpson's Paradox' (Simpson, 1951).”
BackgroundActor-observer asymmetries in explanations of behavior: New answers to an old question.
Malle, B. F., Knobe, J., & Nelson, S. E. (2007) · Journal of Personality and Social Psychology · doi:10.1037/0022-3514.93.4.491 · Cited by 233 · see citing works
“The trait hypothesis was tested as an unweighted average of separate tests for unintentional behaviors and intentional behaviors, averting data patterns known as Simpson's paradox (Simpson, 1951).”
MethodIdentifying causal effects with proxy variables of an unmeasured confounder
Miao, W., Geng, Z., & Tchetgen, E. J. T. (2018) · Biometrika · doi:10.1093/biomet/asy038 · Cited by 233 · see citing works
“Simpson’s paradox (Simpson, 1951) is an elegant illustration of the type of bias that may arise in causal inference subject to unmeasured confounding.”
BackgroundDevine, P. G., Forscher, P. S., Cox, W. T. L., Kaatz, A., Sheridan, J., & Carnes, M. L. (2017) · Journal of Experimental Social Psychology · doi:10.1016/j.jesp.2017.07.002 · Cited by 226 · see citing works
“…departments made 101 hires (47% women).2 To protect against potential spurious effects arising from collapsing across separate, independent hiring units (i.e., Simpson’s Paradox; Pearl, 2000; Simpson, 1951), the formal test of our hypothesis treats departmental clusters as the unit of analysis.”
BackgroundYaari, G., Heiden, J. A. V., Uduman, M., Gadala-Maria, D., Gupta, N., Stern, J. N. H., O’Connor, K. C., Hafler, D. A., Laserson, U., Vigneault, F., et al. (2013) · Frontiers in Immunology · doi:10.3389/fimmu.2013.00358 · Cited by 228 · see citing works
Sample Sizes for Clinical Trials
Julious, S. A. (2009) · Chapman & Hall/CRC eBooks · doi:10.1201/9781584887409 · Cited by 227 · see citing works
Improving standards in brain-behavior correlation analyses
Rousselet, G. A., & Pernet, C. R. (2012) · Frontiers in Human Neuroscience · doi:10.3389/fnhum.2012.00119 · Cited by 222 · see citing works
“One such scenario could lead to the Simpson’s paradox: a correlation present in different groups is reversed when the groups are combined (Simpson, 1951; Frontiers in Human Neuroscience www.frontiersin.org May 2012 | Volume 6 | Article 119 | 3 analyses; the third line contains Spearman correlation…”
BackgroundThe Simpson's paradox unraveled
Hernán, M. Á., Clayton, D. G., & Keiding, N. (2011) · International Journal of Epidemiology · doi:10.1093/ije/dyr041 · Cited by 222 · see citing works
The single-cell eQTLGen consortium
van der Wijst, M. G. P., de Vries, D., Groot, H. E., Trynka, G., Hon, C.-C., Bonder, M. J., Stegle, O., Nawijn, M. C., Idaghdour, Y., van der Harst, P., et al. (2020) · eLife · doi:10.7554/elife.52155 · Cited by 215 · see citing works
“However, caution is warranted when using this information, as their effect on GRN reconstruction depends on the quality of these data priors (Siahpirani and Roy, 2017; Simpson, 1951) and priors derived from bulk data may not hold true at the single-cell level (Buenrostro et al., 2015).”
BackgroundSpencer, C. C. A., Plagnol, V., Strange, A., Gardner, M. J., Paisán‐Ruiz, C., Band, G., Barker, R., Bellenguez, C., Bhatia, K. P., Blackburn, H., et al. (2010) · Human Molecular Genetics · doi:10.1093/hmg/ddq469 · Cited by 213 · see citing works
“This is an example of a general phenomenon known in statistics as Simpson’s paradox (8,9): association for one variable is seen when the analysis is stratified by another variable, whereas in a marginal analysis either no association is observed or it is in the opposite direction.”
BackgroundA Product-Quality View of the Linder Hypothesis
Hallak, J. C. (2010) · The Review of Economics and Statistics · doi:10.1162/rest_a_00001 · Cited by 216 · see citing works
“This result is a manifestation of Simpson’s paradox, a well-known case of association reversal due to aggregation (Simpson, 1951).”
Conditional and Marginal Models: Another View
Lee, Y., & Nelder, J. A. (2004) · Statistical Science · doi:10.1214/088342304000000305 · Cited by 213 · see citing works
Guénancia, C., Lefebvre, A., Cardinale, D. M., Yu, A. F., Ladoire, S., Ghiringhelli, F., Zeller, M., Rochette, L., Cottin, Y., & Vergely, C. (2016) · Journal of Clinical Oncology · doi:10.1200/jco.2016.67.4846 · Cited by 213 · see citing works
“Use of meta-analyses as an adjustment method avoids a bias called Simpson’s paradox, which can be encountered in simple additions of patients and populations of each individual study.(27) Network meta-analyses combine the information from direct evidence (pairwise comparisons) and indirect evidence (comparing B to C f…”
Method
- 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
- 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
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
