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. 2021 Jun 9;9:133. doi: 10.1186/s40168-021-01034-9

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© The Author(s) 2021

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. 5 — Simulation results for the unbalanced data with a set-level covariate of scenario (5). “no ID” means not adjusting for the set ID (second strategy in scenario (5) of Table 1). The power and sensitivity of methods using the strategy not adjusting for ID but adjusting for X^set (third strategy) are very similar to the power and sensitivity of their counterparts using the “no ID” strategy; thus, only the latter is shown