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. 2021 Sep 1;27(4):588–631. doi: 10.1007/s10985-021-09530-8

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© The Author(s) 2021

Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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Fig. 1 — The directed acyclic graph (DAG) in a represents a restrictive data generating assumption on the observed data structure such that there are no common causes of the event of interest and the competing event at any time. The extended DAG in b is an augmented version of the graph in a representing a treatment decomposition satisfying the generalized decomposition assumption. The bold arrows encode deterministic relationships