The Annals of Statistics

General theory for interactions in sufficient cause models with dichotomous exposures

Tyler J. VanderWeele and Thomas S. Richardson

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The sufficient-component cause framework assumes the existence of sets of sufficient causes that bring about an event. For a binary outcome and an arbitrary number of binary causes any set of potential outcomes can be replicated by positing a set of sufficient causes; typically this representation is not unique. A sufficient cause interaction is said to be present if within all representations there exists a sufficient cause in which two or more particular causes are all present. A singular interaction is said to be present if for some subset of individuals there is a unique minimal sufficient cause. Empirical and counterfactual conditions are given for sufficient cause interactions and singular interactions between an arbitrary number of causes. Conditions are given for cases in which none, some or all of a given set of causes affect the outcome monotonically. The relations between these results, interactions in linear statistical models and Pearl’s probability of causation are discussed.

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Ann. Statist., Volume 40, Number 4 (2012), 2128-2161.

First available in Project Euclid: 23 January 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62A01: Foundations and philosophical topics
Secondary: 68T30: Knowledge representation 62J99: None of the above, but in this section

Causal inference counterfactual epistasis interaction potential outcomes synergism


VanderWeele, Tyler J.; Richardson, Thomas S. General theory for interactions in sufficient cause models with dichotomous exposures. Ann. Statist. 40 (2012), no. 4, 2128--2161. doi:10.1214/12-AOS1019.

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  • [1] Aickin, M. (2002). Causal Analysis in Biomedicine and Epidemiology Based on Minimal Sufficient Causation. Dekker, New York.
  • [2] Bateson, W. (1909). Mendel’s Principles of Heredity. Cambridge Univ. Press, London.
  • [3] Bliss, C. I. (1939). The toxicity of poisons applied jointly. Annals of Applied Biology 26 585–615.
  • [4] Cayley, A. (1853). Note on a question in the theory of probabilities. London, Edinburgh and Dublin Philosophical Magazine VI 259.
  • [5] Cayley, A. (1889). A theorem on trees. Quart. J. Math. 23 376–378.
  • [6] Cordell, H. J. (2002). Epistasis: What it means, what it doesn’t mean, and statistical methods to detect it in humans. Hum. Mol. Genet. 11 2463–2468.
  • [7] Cox, D. R. (1958). Planning of Experiments. Wiley, New York.
  • [8] Dedekind, R. (1897). Über Zerlegungen von Zahlen durch ihre größten gemeinsamen Teiler. Gesammelte Werke 2 103–148.
  • [9] Flanders, D. (2006). Sufficient-component cause and potential outcome models. Eur. J. Epidemiol. 21 847–853.
  • [10] Fukuda, K. (2005). cddlib reference manual. Technical report, EPFL Lausanne and ETH Zürich. Available at
  • [11] Greenland, S. and Brumback, B. (2002). An overview of relations among causal modelling methods. Int. J. Epidemiol. 31 1030–1037.
  • [12] Greenland, S. and Poole, C. (1988). Invariants and noninvariants in the concept of interdependent effects. Scand. J. Work Environ. Health 14 125–129.
  • [13] Koopman, J. S. (1981). Interaction between discrete causes. Am. J. Epidemiol. 113 716–724.
  • [14] Mackie, J. L. (1965). Causes and conditions. American Philosophical Quarterly 2 245–255.
  • [15] Marcovitz, A. B. (2001). Introduction to Logic Design. McGraw-Hill, New York.
  • [16] McCluskey, E. J. Jr. (1956). Minimization of Boolean functions. Bell System Tech. J. 35 1417–1444.
  • [17] Novick, L. R. and Cheng, P. W. (2004). Assessing interactive causal influence. Psychol. Rev. 111 455–485.
  • [18] Pearl, J. (2000). Causality: Models, Reasoning, and Inference. Cambridge Univ. Press, Cambridge.
  • [19] Phillips, P. C. (2008). Epistasis—the essential role of gene interactions in the structure and evolution of genetic systems. Nat. Rev. Genet. 9 855–867.
  • [20] Quine, W. V. (1952). The problem of simplifying truth functions. Amer. Math. Monthly 59 521–531.
  • [21] Quine, W. V. (1955). A way to simplify truth functions. Amer. Math. Monthly 62 627–631.
  • [22] Robins, J. (1986). A new approach to causal inference in mortality studies with a sustained exposure period—application to control of the healthy worker survivor effect. Math. Modelling 7 1393–1512.
  • [23] Robins, J. M. (2000). Marginal structural models versus structural nested models as tools for causal inference. In Statistical Models in Epidemiology, the Environment, and Clinical Trials (Minneapolis, MN, 1997). IMA Vol. Math. Appl. 116 95–133. Springer, New York.
  • [24] Rosenbaum, P. R. and Rubin, D. B. (1983). The central role of the propensity score in observational studies for causal effects. Biometrika 70 41–55.
  • [25] Rothman, K. J. (1976). Causes. Am. J. Epidemiol. 104 587–592.
  • [26] Rothman, K. J. and Greenland, S. (1998). Modern Epidemiology. Lippincott-Raven, Philadelphia.
  • [27] Rubin, D. B. (1974). Estimating causal effects of treatments in randomized and nonrandomized studies. J. Educ. Psychol. 66 688–701.
  • [28] Rubin, D. B. (1978). Bayesian inference for causal effects: The role of randomization. Ann. Statist. 6 34–58.
  • [29] Rubin, D. B. (1990). Comment on J. Neyman and causal inference in experiments and observational studies: “On the application of probability theory to agricultural experiments. Essay on principles. Section 9” [Ann. Agric. Sci. 10 (1923) 1–51]. Statist. Sci. 5 472–480.
  • [30] Splawa-Neyman, J. (1990). On the application of probability theory to agricultural experiments. Essay on principles. Section 9 [Ann. Agric. Sci. 10 (1923) 1–51]. Statist. Sci. 5 465–472. Translated from the Polish and edited by D. M. Da̧browska and T. P. Speed.
  • [31] Taylor, J. A., Umbach, D. M., Stephens, E., Castranio, T., Paulson, D., Robertson, G., Mohler, J. L. and Bell, D. A. (1998). The role of N-acetylation polymorphisms in smoking-associated bladder cancer: Evidence of a gene-gene-exposure three-way interaction. Cancer Research 58 3603–3610.
  • [32] VanderWeele, T. J. (2010). Empirical tests for compositional epistasis. Nat. Rev. Genet. 11 166.
  • [33] VanderWeele, T. J. (2010). Epistatic interactions. Stat. Appl. Genet. Mol. Biol. 9 24.
  • [34] VanderWeele, T. J. (2010). Sufficient cause interactions for categorical and ordinal exposures with three levels. Biometrika 97 647–659.
  • [35] VanderWeele, T. J. and Hernán, M. A. (2006). From counterfactuals to sufficient component causes and vice versa. Eur. J. Epidemiol. 21 855–858.
  • [36] VanderWeele, T. J. and Knol, M. J. (2011). Remarks on antagonism. Am. J. Epidemiol. 173 1140–1147.
  • [37] VanderWeele, T. J. and Robins, J. M. (2007). The identification of synergism in the sufficient-component cause framework. Epidemiol. 18 329–339.
  • [38] VanderWeele, T. J. and Robins, J. M. (2007). Directed acyclic graphs, sufficient causes, and the properties of conditioning on a common effect. Am. J. Epidemiol. 166 1096–1104.
  • [39] VanderWeele, T. J. and Robins, J. M. (2008). Empirical and counterfactual conditions for sufficient cause interactions. Biometrika 95 49–61.
  • [40] VanderWeele, T. J. and Robins, J. M. (2010). Signed directed acyclic graphs for causal inference. J. R. Stat. Soc. Ser. B Stat. Methodol. 72 111–127.
  • [41] VanderWeele, T. J., Vansteelandt, S. and Robins, J. M. (2010). Marginal structural models for sufficient cause interactions. Am. J. Epidemiol. 171 506–514.
  • [42] Vansteelandt, S. and Goetghebeur, E. (2003). Causal inference with generalized structural mean models. J. R. Stat. Soc. Ser. B Stat. Methodol. 65 817–835.
  • [43] Vansteelandt, S., VanderWeele, T. J. and Robins, J. M. (2012). Semiparametric tests for sufficient cause interaction. J. R. Stat. Soc. Ser. B Stat. Methodol. 74 223–244.
  • [44] Wiedemann, D. (1991). A computation of the eighth Dedekind number. Order 8 5–6.