On the Link Between Sufficient-cause Model and Potential-outcome Model

Suzuki, Etsuji; Yamamoto, Eiji; Tsuda, Toshihide

doi: 10.1097/EDE.0b013e3181febc5c
Author Information

Department of Epidemiology, Okayama University Graduate School of Medicine, Dentistry and Pharmaceutical Sciences, Okayama, Japan, etsuji-s@cc.okayama-u.ac.jp (Suzuki)

Department of Information Science, Faculty of Informatics, Okayama University of Science, Okayama, Japan (Yamamoto)

Department of Environmental Epidemiology, Okayama University Graduate School of Environmental Science, Okayama, Japan (Tsuda)

Supplemental digital content is available through direct URL citations in the HTML and PDF versions of this article (www.epidem.com).

Article Outline
Back to Top | Article Outline

To the Editors:

The sufficient-cause model and the potential-outcome model have now become cornerstones for causal thinking in epidemiology. The link between these 2 models has been discussed.1–3 We present details of one specific situation to elucidate these 2 different causal approaches. We demonstrate a link between these 2 models under 2 binary causes X1 and X2 and a binary outcome D.

We let Dx1x2(ω) denote the potential outcomes for individual ω if, possibly contrary to fact, there had been interventions to set X1 = x1 and X2 = x2. For each individual ω, there would thus be 4 possible potential outcomes D11 (ω), D10 (ω), D01 (ω), and D00 (ω), which results in 16 (= 24) possible response types.1,4

We could enumerate 9 different types of sufficient causes for D along with certain background causes Ak: A1, A2X1, A3X2, A4SYMBOL, A5SYMBOL, A6X1X2, A7X1SYMBOL, A8SYMBOLX2, and A9SYMBOL, where we let SYMBOL denote the complement of Xi.1 Here, the symbol Ak denotes a set of all components or factors other than the presence of X1, SYMBOL, X2, and SYMBOL, that may be required for a particular mechanism to operate. An individual is at risk for sufficient cause k if Ak is present. We can thus enumerate 512 (= 29) possible risk-status patterns for the sufficient causes.2

We show in the eTable (http://links.lww.com/EDE/A440) a complete enumeration of the 16 response types and the 512 risk-status patterns for the sufficient causes. Although a particular risk-status pattern in individual ω suffices to fix a response type, the converse is not true.2,4,5 Indeed, potential outcomes are quantities specific to individuals, whereas the sufficient-cause model refers to mechanisms. Nonetheless, response types 7, 8, 10, 12, 14, 15, and 16 correspond to a unique risk-status pattern.5

In some cases, the effects of Xi may be in the same direction for all individuals. We say that X1 and X2 have positive monotonic effects on D if Dx1x2(ω) is nondecreasing in x1 and x2 for all individuals ω, ie, Dx1x2(ω) ≥ Dx′1x′2(ω) for V6ω whenever x1 ≥ x′1 and x2x2.5 Under the assumption of positive monotonic effect of X1 and X2, the individuals of response types 3, 5, 7, and 9 through 15 are excluded; and individuals of response types 1, 2, 4, 6, 8, and 16 may remain. These individuals can have, at the maximum, 446 risk-status patterns for the sufficient causes, including A4SYMBOL, A5SYMBOL, A7X1SYMBOL, A8SYMBOLX2, and A9SYMBOL. By contrast, we will say that the assumption of no preventive action of X1 and X2 holds if sufficient causes A4SYMBOL, A5SYMBOL, A7X1SYMBOL, A8SYMBOLX2, and A9SYMBOL are all absent so that neither SYMBOL nor SYMBOL acts in a sufficient cause.4 These remaining individuals can have, at the maximum, 16 risk-status patterns for the sufficient causes. Therefore, the absence of SYMBOL and SYMBOL in any sufficient cause is stronger than the assumption of positive monotonic effects of both X1 and X2. Thus, the concepts of “positive monotonic effect” and “no preventive action” should be clearly distinguished; the former is defined in the potential-outcome framework, while the latter is defined in the sufficient-component cause framework. Recently, VanderWeele6 distinguished these concepts by using the terms “counterfactual monotonicity” and “sufficient-cause monotonicity,” respectively.

Consideration of the correspondence between the 2 models should allow greater insight to facilitate use of each model in the appropriate contexts, clarifying the strengths of each model. Our explication would also facilitate understanding of the recent findings on the identifiability of particular sufficient causes and response types.5,7,8 As the duality between the 2 models shows, the different approaches to causality provide complementary perspectives, and can be employed together to improve causal interpretations.

Back to Top | Article Outline


We thank Tyler J. VanderWeele for his helpful comments on the earlier version of this letter.

Etsuji Suzuki

Department of Epidemiology

Okayama University Graduate School of Medicine

Dentistry and Pharmaceutical Sciences

Okayama, Japan


Eiji Yamamoto

Department of Information Science

Faculty of Informatics

Okayama University of Science

Okayama, Japan

Toshihide Tsuda

Department of Environmental Epidemiology

Okayama University Graduate School of Environmental Science

Okayama, Japan

Back to Top | Article Outline


1. Greenland S, Poole C. Invariants and non-invariants in the concept of interdependent effects. Scand J Work Environ Health. 1988;14:125–129.
2. Flanders WD. On the relationship of sufficient component cause models with potential outcome (counterfactual) models. Eur J Epidemiol. 2006;21:847–853.
3. VanderWeele TJ, Hernán MA. From counterfactuals to sufficient component causes and vice versa. Eur J Epidemiol. 2006;21:855–858.
4. Greenland S, Lash TL, Rothman KJ. Concepts of interaction. In: Rothman KJ, Greenland S, Lash TL, eds. Modern Epidemiology. 3rd ed. Philadelphia: Wolters Kluwer Health/Lippincott Williams & Wilkins; 2008:71–83.
5. VanderWeele TJ, Robins JM. The identification of synergism in the sufficient-component-cause framework. Epidemiology. 2007;18:329–339.
6. VanderWeele TJ. Attributable fractions for sufficient cause interactions. Int J Biostat. 2010;6:5. doi:10.2202/1557-4679.1202
7. VanderWeele TJ, Robins JM. Empirical and counterfactual conditions for sufficient cause interactions. Biometrika. 2008;95:49–61.
8. VanderWeele TJ. Epistatic interactions. Stat Appl Genet Mol Biol. 2010;9:1. doi:10.2202/1544-6115.1517

Supplemental Digital Content

Back to Top | Article Outline
© 2011 Lippincott Williams & Wilkins, Inc.