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. 2025 Jan 10;15:1605. doi: 10.1038/s41598-024-84702-8

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© The Author(s) 2025

Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, 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 you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. 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-nc-nd/4.0/.

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Fig. 6 — Analysis of the XGBoost model with TreeSHAP on the simulation test set for the SEL model with variables. (a) shows that the SEL covariate is considered, by far, to be the most important covariate in the model. The order of the other covariates remains the same when we train the model without our SEL variable (i.e. considered as not available). The Partial Dependence Plot in (b) depicts the relationship learnt by the model between the target variable Y and the SEL variable. We can see that it recovers the quadratic relationship between and Y as defined in Eq. (2).

Inline graphic — Analysis of the XGBoost model with TreeSHAP on the simulation test set for the SEL model with variables. (a) shows that the SEL covariate is considered, by far, to be the most important covariate in the model. The order of the other covariates remains the same when we train the model without our SEL variable (i.e. considered as not available). The Partial Dependence Plot in (b) depicts the relationship learnt by the model between the target variable Y and the SEL variable. We can see that it recovers the quadratic relationship between and Y as defined in Eq. (2).