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. 2024 Mar 28;24:79. doi: 10.1186/s12874-024-02196-4

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© The Author(s) 2024

Open Access This 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/. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated in a credit line to the data.

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Fig. 1 — Balance of observable X_m and unobservables X_u1, X_u2 before and after adjustment. Note: Median, interquantile range box, and outliers of the standardized mean differences after 1,000 simulations. SMD standardized mean difference. IPTW Inverse probability treatment weighting. PSM Propensity score matching