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. 2023 Sep 28;13:16279. doi: 10.1038/s41598-023-43618-5

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© The Author(s) 2023, corrected publication 2023

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/.

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Simulated data curves for normal aging (red line) and MS (blue line). Description of simulated spline shape based on piecewise linear mixed models with age varying slopes. Red line represents normal aging and blue line represents MS. This illustrates the ‘ground truth’ spline shape that should be found by the mixed spline model. A High-resolution version of this Figure can be found in Supplementary Information 2.