The logic of Simpson’s paradox

Bandyoapdhyay, Prasanta S.; Nelson, Davin; Greenwood, Mark C.; Brittan, Gordon G.; Berwald, Jesse

doi:10.1007/s11229-010-9797-0

Cited by 25 publications

(21 citation statements)

References 17 publications

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“…Simpson's paradox (hereafter SP) has been formally analyzed by mathematicians and statisticians (e.g., Blyth, 1972; Dawid, 1979; Pearl, 1999, 2000; Schield, 1999; Tu et al, 2008; Greenland, 2010; Hernán et al, 2011), its relevance for human inferences studied by psychologists (e.g., Schaller, 1992; Spellman, 1996a,b; Fiedler, 2000, 2008; Curley and Browne, 2001) and conceptually explored by philosophers (e.g., Cartwright, 1979; Otte, 1985; Bandyoapdhyay et al, 2011). However, few works have discussed the practical aspects of SP for empirical science: How might researchers prevent the paradox, recognize it, and deal with it upon detection?…”

Section: Introductionmentioning

confidence: 99%

Simpson's paradox in psychological science: a practical guide

Kievit¹,

Frankenhuis²,

Waldorp³

et al. 2013

Front. Psychol.

414

299

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The direction of an association at the population-level may be reversed within the subgroups comprising that population—a striking observation called Simpson's paradox. When facing this pattern, psychologists often view it as anomalous. Here, we argue that Simpson's paradox is more common than conventionally thought, and typically results in incorrect interpretations—potentially with harmful consequences. We support this claim by reviewing results from cognitive neuroscience, behavior genetics, clinical psychology, personality psychology, educational psychology, intelligence research, and simulation studies. We show that Simpson's paradox is most likely to occur when inferences are drawn across different levels of explanation (e.g., from populations to subgroups, or subgroups to individuals). We propose a set of statistical markers indicative of the paradox, and offer psychometric solutions for dealing with the paradox when encountered—including a toolbox in R for detecting Simpson's paradox. We show that explicit modeling of situations in which the paradox might occur not only prevents incorrect interpretations of data, but also results in a deeper understanding of what data tell us about the world.

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Section: Introductionmentioning

confidence: 99%

Simpson's paradox in psychological science: a practical guide

Kievit¹,

Frankenhuis²,

Waldorp³

et al. 2013

Front. Psychol.

414

299

View full text Add to dashboard Cite

show abstract

“…This detailed insight could not be revealed through average analysis/analysis of fixed effects (ANOVA or regression analysis). 25 Furthermore, the average analysis is prone to measurement errors from Simpson's paradox [37][38][39][40][41] and that by averaging among the study clusters (in this context, high-swellers and low-swellers and their rate of deswelling), other relationships within the data may be masked or reversed. In addition, the mixed model analysis can concurrently investigate the impact of other factors or covariates that might FIGURE 4.…”

Section: Overall Discussionmentioning

confidence: 99%

Mixed Model Analysis of Between-Subject Variability in Overnight Corneal Swelling and Deswelling With Silicone Hydrogel Lenses

Moezzi

Hutchings

Fonn

et al. 2018

Invest. Ophthalmol. Vis. Sci.

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PURPOSE.To model between subject variability of corneal swelling (CS) and deswelling after overnight wear of silicone hydrogel (SiHy) contact lenses. METHODS.A total of 29 neophyte subjects wore 12 SiHy lenses with central transmissibility range of 31 to 211 Dk/t units on separate nights, in random order, and on one eye only. The contralateral eye served as the control. Central corneal thickness was measured using digital optical pachymetry before lens insertion, immediately after lens removal on waking, then 20, 40 minutes, 1, 2, and 3 hours later. Mixed modelling was conducted for simultaneous analysis of group and between-subject effects of CS and deswelling.RESULTS. The best model for overnight CS versus Dk/t was linear with a random intercept showing constant between-subject differences in CS for different Dk/t values. The best fit for corneal deswelling versus time was a curvilinear random intercept and random slope model. About 90% of the total between-subject deswelling variance in either lens or control eyes was due to the intercept variability with much less (~10%) being due to the variability of the individual deswelling rate (slope). Subject age, sex, and ametropia were not predictors of individual corneal swelling in the swelling versus Dk/t analysis. Age, however, was a significant (inverse) predictor of the rate of corneal deswelling, only in lens-wearing eyes. CONCLUSIONS.A large proportion of variability in corneal swelling is because of subject-specific differences in corneal response to hypoxia. This shows that ''low swellers'' and ''high swellers'' actually do exist.Keywords: corneal swelling, corneal deswelling, between-subject variability, pachymetry, mixed modeling C orneal swelling is regarded as one of the main indices of corneal physiologic change as a result of corneal oxygen deficiency produced by contact lens (CL) wear.1,2 The postlens tear oxygen tension in soft lenses is dependent on the oxygen diffusion through the lens material 3,4 and the effect of the tear pump on tear mixing to equilibrate the oxygen tension under a soft lens is insignificant. 5,6 Therefore, oxygen diffusion through contact lenses plays a vital role in maintaining corneal health and normal physiology in soft lens wear. 7,8 Corneal oxygen deprivation may lead to corneal swelling (thickening) from water absorption by the corneal stroma. This is believed to primarily be from the increased stromal osmotic gradient resulting from the accumulation of lactic acid from anaerobic metabolism in the cornea.9 It would appear self-evident, therefore, that the level of lens induced corneal swelling is inversely related to the oxygen transmissibility of the contact lens. 10The average overnight corneal swelling of 3%-4% occurs in response to eye closure 11 in nonlens wearers 12-15 and sleeping with a contact lens on the eye, further deprives the cornea of the oxygen supply from the palpebral vasculature, maximizing the hypoxic stress and potentially leading to increased corneal edema. 16 Even silicone hydrogel (SiHy) lenses ...

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“…There have been several investigations examining the nature of the Simpson's paradox including the relationship with causal inferences[29, 30]. Evaluations of these causality interpretations have also been conducted by other investigators [31, 32]. Recommendations to reduce or avoid Simpson's paradox have been suggested, including a detailed checklist for each research process stage[33] and the use of cross-sum ratio rather than the cross-product ratio (odds ratio) based on a 2 × 2 table[34].…”

Section: Discussionmentioning

confidence: 99%

The occurrence of Simpson's paradox if site-level effect was ignored in the TREAT Asia HIV Observational Database

Kerr

Chandrasekaran

Huelgas

et al. 2016

Journal of Clinical Epidemiology

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Background In multi-site HIV observational cohorts, clustering of observations often occur within sites. Ignoring clustering may lead to “Simpson's paradox” (SP) where the trend observed in the aggregated data is reversed when the groups are separated. This study aimed to investigate the SP in an Asian HIV cohort and the effects of site-level adjustment through various Cox-regression models. Methods Survival time from combination antiretroviral therapy (cART) initiation was analysed using four Cox models: (i) no site adjustment; (ii) site as a fixed effect; (iii) stratification through site; and (iv) shared frailty on site. Results A total of 6454 patients were included from 23 sites in Asia. SP was evident in the year of cART initiation variable. Model (i) shows the hazard ratio (HR) for years 2010-2014 was higher than the HR for 2006-2009, compared to 2003-2005 (HR = 0.68 vs 0.61). Models (ii)-(iv) consistently implied greater improvement in survival for those who initiated in 2010-2014 than 2006-2009 contrasting findings from Model (i). The effects of other significant covariates on survival were similar across four models. Conclusions Ignoring site can lead to SP causing reversal of treatment effects. Greater emphasis should be made to include site in survival models when possible.

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The logic of Simpson’s paradox

Cited by 25 publications

References 17 publications

Simpson's paradox in psychological science: a practical guide

Simpson's paradox in psychological science: a practical guide

Mixed Model Analysis of Between-Subject Variability in Overnight Corneal Swelling and Deswelling With Silicone Hydrogel Lenses

The occurrence of Simpson's paradox if site-level effect was ignored in the TREAT Asia HIV Observational Database

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