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 X 1 and X 2 have positive monotonic effects on D if D x1x2(ω) is nondecreasing in x 1 and x 2 for all individuals ω, ie, D x1x2(ω) ≥ D x′1x′2(ω) for ω whenever x 1 ≥ x′1 and x 2 ≥ x′2.5 Under the assumption of positive monotonic effect of X 1 and X 2, 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 A 4 SYMBOL, A 5 SYMBOL, A 7 X 1 SYMBOL, A 8 SYMBOL X 2, and A 9 SYMBOL. By contrast, we will say that the assumption of no preventive action of X 1 and X 2 holds if sufficient causes A 4 SYMBOL, A 5 SYMBOL, A 7 X 1 SYMBOL, A 8 SYMBOL X 2, and A 9 SYMBOL 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 X 1 and X 2. 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.
We thank Tyler J. VanderWeele for his helpful comments on the earlier version of this letter.
Department of Epidemiology
Okayama University Graduate School of Medicine
Dentistry and Pharmaceutical Sciences
Department of Information Science
Faculty of Informatics
Okayama University of Science
Department of Environmental Epidemiology
Okayama University Graduate School of Environmental Science
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