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Versions of Lord’s paradox (Lord, 1967) are described in Glymour (2006); Hernández-Díaz, Schisterman, and Hernán (2006); Senn (2006); Wainer (1991). A comprehensive analysis can be found in Pearl (2016a).

Paradoxes invoking counterfactuals are not included in this chapter but are no less intriguing. For a sample, see Pearl (2013).

References

Appleton, D., French, J., and Vanderpump, M. (1996). Ignoring a covariate: An example of Simpson’s paradox. American Statistician 50: 340–341.

Crockett, Z. (2015). The time everyone “corrected” the world’s smartest woman. Priceonomics. Available at: http://priceonomics.com/the-time-everyone-corrected-the-worlds-smartest (posted: February 19, 2015).

Glymour, M. M. (2006). Using causal diagrams to understand common problems in social epidemiology. In Methods in Social Epidemiology. John Wiley and Sons, San Francisco, CA, 393–428.

Grinstead, C. M., and Snell, J. L. (1998). Introduction to Probability.

2nd rev. ed. American Mathematical Society, Providence, RI. Hernández-Díaz, S., Schisterman, E., and Hernán, M. (2006). The birth weight “paradox” uncovered? American Journal of Epidemiology 164: 1115–1120.

Julious, S., and Mullee, M. (1994). Confounding and Simpson’s paradox. British Medical Journal 309: 1480–1481.

Lindley, D. V. (2014). Understanding Uncertainty. Rev. ed. John Wiley and Sons, Hoboken, NJ.

Lord, F. M. (1967). A paradox in the interpretation of group comparisons. Psychological Bulletin 68: 304–305.

Pearl, J. (1988). Probabilistic Reasoning in Intelligent Systems. Morgan Kaufmann, San Mateo, CA.

Pearl, J. (2009). Causality: Models, Reasoning, and Inference. 2nd ed. Cambridge University Press, New York, NY.

Pearl, J. (2013). The curse of free-will and paradox of inevitable regret. Journal of Causal Inference 1: 255–257.

Pearl, J. (2014). Understanding Simpson’s paradox. American Statistician 88: 8–13.

Pearl, J. (2016a). Lord’s paradox revisited — (Oh Lord! Kumbaya!). Journal of Causal Inference 4. doi:10.1515/jci-2016-0021.

Pearl, J. (2016b). The sure-thing principle. Journal of Causal Inference 4: 81–86.

Savage, L. (1954). The Foundations of Statistics. John Wiley and Sons, New York, NY.

Savage, S. (2009). The Flaw of Averages: Why We Underestimate Risk in the Face of Uncertainty. John Wiley and Sons, Hoboken, NJ.

Senn, S. (2006). Change from baseline and analysis of covariance revisited. Statistics in Medicine 25: 4334–4344.

Simon, H. (1954). Spurious correlation: A causal interpretation. Journal of the American Statistical Association 49: 467–479.

Tierney, J. (July 21, 1991). Behind Monty Hall’s doors: Puzzle, debate and answer? New York Times.

Wainer, H. (1991). Adjusting for differential base rates: Lord’s paradox again. Psychological Bulletin 109: 147–151.

Annotated Bibliography

Extensions of the back-door and front-door adjustments were first reported in Tian and Pearl (2002) based on Tian’s c-component factorization. These were followed by Shpitser’s algorithmization of the do-calculus (Shpitser and Pearl, 2006a) and then the completeness results of Shpitser and Pearl (2006b) and Huang and Valtorta (2006).

The economists among our readers should note that the cultural resistance of some economists to graphical tools of analysis (Heckman and Pinto, 2015; Imbens and Rubin, 2015) is not shared by all economists. White and Chalak (2009), for example, have generalized and applied the do-calculus to economic systems involving equilibrium and learning. Recent textbooks in the social and behavioral sciences, Morgan and Winship (2007) and Kline (2016), further signal to young researchers that cultural orthodoxy, like the fear of telescopes in the seventeenth century, is not long lasting in the sciences.

John Snow’s investigation of cholera was very little appreciated during his lifetime, and his one-paragraph obituary in Lancet did not even mention it. Remarkably, the premier British medical journal “corrected” its obituary 155 years later (Hempel, 2013). For more biographical material on Snow, see Hill (1955) and Cameron and Jones (1983). Glynn and Kashin (2018) is one of the first papers to demonstrate empirically that front-door adjustment is superior to back-door adjustment when there are unobserved confounders. Freedman’s critique of the smoking — tar — lung cancer example can be found in a chapter of Freedman (2010) titled “On Specifying Graphical Models for Causation.”

Introductions to instrumental variables can be found in Greenland (2000) and in many textbooks of econometrics (e.g., Bowden and Turkington, 1984; Wooldridge, 2013).

Generalized instrumental variables, extending the classical definition given in our text, were introduced in Brito and Pearl (2002).

The program DAGitty (available online at http://www.dagitty.net/dags.html) permits users to search the diagram for generalized instrumental variables and reports the resulting estimands (Textor, Hardt, and Knüppel, 2011). Another diagram-based software package for decision making is BayesiaLab (www.bayesia.com).

Bounds on instrumental variable estimates are studied at length in Chapter 8 of Pearl (2009) and are applied to the problem of noncompliance. The LATE approximation is advocated and debated in Imbens (2010).

References

Bareinboim, E., and Pearl, J. (2012). Causal inference by surrogate experiments: z-identifiability. In Proceedings of the Twenty-Eighth Conference on Uncertainty in Artificial Intelligence (N. de Freitas and K. Murphy, eds.). AUAI Press, Corvallis, OR.

Bowden, R., and Turkington, D. (1984). Instrumental Variables. Cambridge University Press, Cambridge, UK.

Brito, C., and Pearl, J. (2002). Generalized instrumental variables. In Uncertainty in Artificial Intelligence, Proceedings of the Eighteenth Conference (A. Darwiche and N. Friedman, eds.). Morgan Kaufmann, San Francisco, CA, 85–93.

Cameron, D., and Jones, I. (1983). John Snow, the Broad Street pump, and modern epidemiology. International Journal of Epidemiology 12: 393–396.

Cox, D., and Wermuth, N. (2015). Design and interpretation of studies: Relevant concepts from the past and some extensions. Observational Studies 1. Available at: https://arxiv.org/pdf/1505.02452.pdf.

Freedman, D. (2010). Statistical Models and Causal Inference: A Dialogue with the Social Sciences. Cambridge University Press, New York, NY.

Glynn, A., and Kashin, K. (2018). Front-door versus back-door adjustment with unmeasured confounding: Bias formulas for front-door and hybrid adjustments. Journal of the American Statistical Association. To appear.